Strings and Monopoles in Strongly Interacting
~ARCHWE
Gauge Theories
by
OF TECHNOLOGY
Ethan Stanley Dyer
JUL 0 1 2014
Submitted to the Department of Physics
I LIBRARIES
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
at the
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June 2014
@ Massachusetts Institute of Technology 2014. All rights reserved.
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Department of Physics
May 8, 2014
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Associate Professor
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2
Strings and Monopoles in Strongly Interacting Gauge
Theories
by
Ethan Stanley Dyer
Submitted to the Department of Physics
on May 8, 2014, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
Abstract
In this thesis we discuss aspects of strongly coupled gauge theories in two and three
dimensions. In three dimensions, we present results for the scaling dimension and
transformation properties of monopole operators in gauge theories with large numbers
of fermions. In two dimensions, we study (0,2) gauge theories as a tool for constructing
string backgrounds with non trivial H-flux. We demonstrate how chiral matter content
in the gauge theory allows the construction of infrared fixed points outside of the usual
Calabi-Yau framework, and further derive consistency relations for a special class of
torsional models.
Thesis Supervisor: Allan Wilfred Adams III
Title: Associate Professor
3
4
Acknowledgments
There are many people without whom this thesis would not be possible.
I would
like to thank my family, especially my parents, Barbara and Sam Dyer; grandparents, Betty and Ira Dyer; and girlfriend Gabrielle Lurie for their continual support of
my interest in physics, and tolerance of long work hours. I would also like to thank
my advisor, collaborator, and thesis committee member, Allan Wilfred Adams III,
for his encouragement, critiques, and undying enthusiasm for all things physics, as
well as for his help in navigating the world of academia. I am deeply indebted to
my collaborators, Jaehoon Lee, Mark Mezei, Silviu Pufu, and Sho Yaida, who have
been instrumental in shaping my graduate experience and research, and a pleasure
to interact with. I am also grateful to my fellow center for theoretical physics(CTP)
classmates who have made the past few years a joy both academically and socially,
and the CTP faculty who have provided answers to countless questions, especially
John McGreevy and Jesse Thaler who helped guide my research on numerous occasions. I would like to express my appreciation to the CTP administrative staff, Joyce
Berggren, Scott Morley, and Charles Suggs who have helped me in many ways, and
without whom I would most likely still be locked out of my office. I would like to give
a special thanks to my committee members, Allan Adams, Hong Liu, and Michael
Williams for their willingness to read this thesis. Lastly, I would like to acknowledge
the United States taxpayers and private donors, without whose support physics could
not go on.
Thank you
5
6
Contents
1
19
Introduction
1.1
1.2
Monopoles and Confinement in Three Dimensions . . . . . . . . . . .
21
1.1.1
Monopoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
1.1.2
Fate of the IR . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
Strongly Coupled Gauge Theory for Chiral Strings . . . . . . . . . . .
27
1.2.1
Consistancy conditions from world-sheet and space-time
. . .
31
1.2.2
Gauge Linear Sigma Models: The basic idea . . . . . . . . . .
33
1.2.3
Lorentz Symmetry, Supersymmetry, and Anomalies in Two Dim ensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Monopole Operators in Strongly Coupled Gauge Theories
43
. . . . . .
47
Introduction . . . . .
2.2
Monopole operators via the state-operator correspondence
2.4
43
. . . . . . . . . . . . . .
2.1
2.3
38
2.2.1
Classical Monopole Backgrounds
. . . . . . . . . . . . . . . .
48
2.2.2
Three Dimensional Gauge Theories with Fermions . . . . . . .
50
2.2.3
Quantum Monopole Operators . . . . . . . . . . . . . . . . . .
52
. . . . . . . . . . . . . . . . . . . . . . . . . .
54
Free energy on S 2 x R
2.3.1
Setup
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
2.3.2
Gauge Field Effective Action . . . . . . . . . . . . . . . . . . .
58
. . . . . . . . . . . . . . . . . . . . . . . . .
63
2.4.1
The fermion determinant . . . . . . . . . . . . . . . . . . . . .
63
2.4.2
The Faddeev-Popov determinant
. . . . . . . . . . . . . . . .
67
2.4.3
The gauge fluctuations determinant . . . . . . . . . . . . . . .
69
Functional determinants
7
2.5
2.6
2.7
2.8
2.9
3
2.4.4
Combining the subleading terms in the free energy
. . . . . .
81
2.4.5
Summary and an example . . . . . . . . . . . . . . . . . . . .
84
Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
86
2.5.1
A systematic study of monopole stability in QCD 3 . . . . . . .
88
Monopole operator dimensions . . . . . . . . . . . . . . . . . . . . . .
92
2.6.1
Monopole operator dimensions in QED . . . . . . . . . . . . .
93
2.6.2
Monopole operator dimensions in U(Nc) QCD . . . . . . . . .
93
Other quantum numbers of monopole operators
. . . . . . . . . . . .
2.7.1
Quantum numbers of monopole operators in QED
2.7.2
Quantum numbers of monopole operators in U(Nc) QCD ...
. . . . . .
95
96
106
Monopoles in general gauge theories . . . . . . . . . . . . . . . . . . .
109
2.8.1
Anomalous dimensions for general groups . . . . . . . . . . . .
110
2.8.2
Exam ples . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
116
D iscussion . . . . . . . . . . . . . . . . . . . . . . . . . . .
131
2.9.1
Summ ary
. . . . . . . . . . . . . . . . . . . . . . .
131
2.9.2
Confinement and chiral symmetry breaking . . . . .
133
2.9.3
QED and and algebraic spin liquids . . . . . . . . .
136
Chiral Gauge Theory for Stringy Backgrounds
139
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
139
3.2
Generating dH in a (0,2) GLSM
. . . . . . . . . . . . . . . . . . . .
142
3.2.1
Torsion in (0, 2) NLSMs . . . . . . . . . . . . . . . . . . . . .
142
3.2.2
Adding dH to a (0,2) GLSM by hand: the Green Schwarz
3.3
mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
144
3.2.3
On the geometry of GS GLSMs . . . . . . . . . . . . . . . . .
147
3.2.4
Generating dH in a garden-variety (0, 2) GLSM . . . . . . . .
149
Verifying Quantum Consistency in a Special Class of Models . . . . .
154
3.3.1
The M odels . . . . . . . . . . . . . . . . . . . . . . . . . . . .
154
3.3.2
Methodology
. . . . . . . . . . . . . . . . . . . . . . . . . . .
158
3.3.3
Gauge Invariant Model . . . . . . . . . . . . . . . . . . . . . .
160
8
3.4
3.3.4
Anomalous Model with Green-Schwarz Mechanism
. . . . . .
171
3.3.5
Multiple U(1)s . . . . . . . . . . . . . . . . . . . . . . . . . .
175
.
179
.
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
183
A Monopole Harmonics
A.1 Definition and Properties of Monopole Harmonics . . . . . . . . . . .
183
A.1.1
Scalar Harmonics . . . . . . . . . . . . . . . . . . . . . . . . .
183
A.1.2
Spin s Harmonics . . . . . . . . . . . . . . . . . . . . . . . . .
185
A.1.3
Spin 1/2 Harmonics . . . . . . . . . . . . . . . . . . . . . . . .
186
A.1.4
Spin 1 Harmonics . . . . . . . . . . . . . . . . . . . . . . . . .
187
189
B (0,2) Details
B.1 Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
189
B.1.1
Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . .
189
B.1.2
Superfields . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
190
B .2 A ction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
192
B.3 OPEs
. . . . . . . . . . . . . . . . . . . . . . . . ...
..
194
. . . .
. . 194
B.3.1
Operator Product Expansion with single anomalous U(1)
B.3.2
Operator Product Expansion with multiple U(1)s . . . . . . . 196
B.4 Quantum Chirality . . . . . . . . . . . . . . . . . . . ..
9
..
. . . .
197
10
List of Figures
1-1
Anomalous diagrams in four and two dimensions . . . . . . . . . . . .
2-1
We plot the terms in the infinite sum over j (2.74) that give the matrix
40
q' = 1/2, Q = 1, and J = 35/2.
element [Kq ,(a)] u for q = -1,
We show the stage of the calculation where all the finite sums (over
6q, ', 1, and j') in (2.74) have been done and only the infinite sum
over j remains. The dots represent the actual terms in the sum, while
the solid line is the asymptotic expansion of the summand to 9(1/jl 8 )
that we determined analytically. We perform the infinite sum by zetafunction regularization of the asymptotic form for
j
> jc, where
j,
is
the value below which we use the numerical values of the terms in the
sum. We check the numerical precision by changing jc and we reach our
goal of 10-1' precision by choosing j,
to get the free energy with 102-2
3
~ 40. This precision is needed
precision.
. . . . . . . . . . . . . . .
77
The eigenvalues of K ,(Q) for some example q, q' and low J values as
a function of Q. Zero eigenvalues corresponding to pure gauge modes
are omitted.
Note that the eigenvalues are monotonic in J and Q,
hence it suffices to examine the Q = 0 behavior of the lowest J mode
for stability. Also note that in both examples
IQI
;> 1 and the two
lowest lying J modes have one non-zero eigenvalue, while higher J
modes come with two eigenvalues. (The smaller number of eigenvalues
corresponds to the reduced size of the matrix K ,(Q).)
11
. . . . . . . .
81
2-3
We plot the ratio of the non-zero eigenvalues Ajgauge(Q) of the gauge
kernel divided by their asymptotic behavior Aaymp(Q). We chose q
-1,
q' = 1/2 for this example. Because
IQI
= 3/2 the J = 1/2, 3/2
modes contribute one eigenvalue, while for higher J eigenvalues come in
pairs. We used the same colors to plot the pair of eigenvalues for these
higher J modes. Because the ghosts give a contribution proportional
to Aaymp(Q) this plot shows that the low energy modes are the most
important in determining the free energy. . . . . . . . . . . . . . . . .
2-4
83
We plot the subleading term in the free energy, 6F(q, q') for q =
-1,
q' = 1/2 as a function of the cutoff A. We extrapolate to 1/A -+ 0
by fitting the data points by a second order polynomial. Our results
are reliable to 10-
2-5
3
precision.
The lowest eigenvalue A
=
. . . . . . . . . . . . . . . . . . . . . .
86
Kqaq 1 (0) of the aAb component of the
gauge field fluctuations around the GNO monopole background (2.18).
We have marked explicitly the plane z = 0. The region where this
eigenvalue dips below zero corresponds to an instability of aAb. If this
eigenvalue is positive, then the action for a
2-6
b
is positive-definite.
.
.
90
A summary plot of the stability of GNO monopoles. A GNO monopole
with charges {qi,.
.
. qNpe
is stable provided that all pairs (qa,
qb)
cor-
respond to (open or filled) black circles, and it is unstable otherwise.
We denote Qab = qa - q, as in the main text. The orange dots correspond to values of (qa, qg)
for which Kq$a-
1
(0) < 0, i.e. the effective
action for aab has a negative mode with J = jQabI - 1. The open and
filled black circles correspond to values of (qa, qg)
for which there is no
such negative mode. The difference between the open and filled black
circles is that for the filled ones the lowest angular momentum mode
has J = lQab| - 1, while for the open ones the lowest value of J is IQabI.
12
91
2-7
The weight lattice of SU(2) (blue dots) as well as the lattice of all
possible monopole charges (dotted circles). The monopole charges are
defined modulo the action of the Weyl group, which in this case is Z2
and acts as reflections about the origin. . . . . . . . . . . . . . . . . .
2-8
118
The SU(2) monopoles appearing as black dotted circles in Figure 27. In the presence of N 1 fundamental fermions these backgrounds are
all stable, and we list the scaling dimensions A of the corresponding
monopole operators.
2-9
. . . . . . . . . . . . . . . . . . . . . . . . . . .
119
The SO(3) weight lattice (blue dots) and its dual lattice (dashed circles). The weight lattice is a sublattice of the SU(2) weight lattice in
Figure 2-7. The dual lattice contains more monopole charges q than
the dual lattice of SU(2).
As in the SU(2) case, the Weyl acts by
reflections about the origin, so it provides the identification q
-
-q on
the set of monopole charges. . . . . . . . . . . . . . . . . . . . . . . .
119
2-10 The SO(3) monopoles appearing as black dotted circles in Figure 2-9.
Here, we consider these backgrounds in the presence of N1 fermions
transforming in the three-dimensional fundamental representation of
SO(3).
The orange circles correspond to unstable backgrounds. For
the stable backgrounds (represented in black by various shapes), we
list the scaling dimensions A of the corresponding monopole operators. 120
2-11 The weight lattice of SU(3) (blue dots) as well as the lattice of all
possible monopole charges (dotted circles). The monopole charges are
defined modulo the action of the Weyl group, which in this case is S 3
and is generated by 120 degree rotations as well as reflections about
the q2 axis.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
121
2-12 The SU(3) monopoles appearing as black dotted circles in Figure 2-11.
Here, we consider these backgrounds in the presence of Nf fermions
transforming in the three-dimensional fundamental representation of
SU(3).
The orange circles correspond to unstable backgrounds. For
the stable backgrounds (represented in black by various shapes), we
list the scaling dimensions A of the corresponding monopole operators. 122
2-13 The weight lattice of Sp( 4 ) (blue dots) as well as the lattice of all
possible monopole charges (dotted circles). The monopole charges are
defined modulo the action of the Weyl group, which in this case is (Z 2 ) 3
and is generated by reflections about the q1 axis, q2 axis, and the line
that makes a 45 degree angle with the qi axis. . . . . . . . . . . . . .
123
2-14 The Sp( 4 ) monopoles appearing as black dotted circles in Figure 2-13.
Here, we consider these backgrounds in the presence of N
fermions
transforming in the four-dimensional fundamental representation of
Sp( 4 ). The orange circles correspond to unstable backgrounds.
For
the stable backgrounds (represented in black by various shapes), we
list the scaling dimensions A of the corresponding monopole operators. 124
2-15 The weight lattice of SO(5) (blue dots) as well as the lattice of all
possible monopole charges (dotted circles). The monopole charges are
defined modulo the action of the Weyl group, which, as in the Sp( 4 )
case, can be identified with the (Z 2 )3 generated by reflections about
the qi axis, q2 axis, and the line that makes a 45 degree angle with the
qi axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
125
2-16 The SO(5) monopoles appearing as black dotted circles in Figure 215. Here, we consider these backgrounds in the presence of N fermions
transforming in the five-dimensional fundamental representation of SO(5).
The orange circles correspond to unstable backgrounds. For the stable backgrounds (represented in black by various shapes), we list the
scaling dimensions A of the corresponding monopole operators.
14
. . .
126
2-17 The weight lattice of G2 (blue dots) as well as the lattice of all possible
monopole charges (dotted circles). The monopole charges are defined
modulo the action of the Weyl group, which in this case is D6 (the
dihedral group of order 12) and is generated by 60 degree rotations as
well as reflections about the line that makes a 45 degree angle with the
qi axis. . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . ..
. . . 127
2-18 The G 2 monopoles appearing as black dotted circles in Figure 2-17.
Here, we consider these backgrounds in the presence of Nf fermions
transforming in the seven-dimensional fundamental representation of
G 2 . The orange circles correspond to unstable backgrounds. For the
stable backgrounds (represented in black by various shapes), we list
the scaling dimensions A of the corresponding monopole operators.
. 128
2-19 In the top right corner we show the number of (inequivalent) stable
monopoles for SU(3) gauge theory with nrfundNf fundamental fermions
and nadjNf adjoint fermions as a function of the ratio nrfuna/nadj. The
solid line is divided into five regions that correspond to the diagrams
on the left and bottom, where we show explicitly which monopoles are
stable in each region. The dashed line is a continuation of the solid
line for larger values of nfund/naj, but in this region we do not show
explicitly which monopoles are stable. . . . . . . . . . . . . . . . . . .
15
130
16
List of Tables
1.1
Monopole operator dimension Aq for monopole charge q in QED 3
1.2
Estimates of the smallest number of fermions,
QCD 3
Nd*nf
.
26
for which the IR
is in a deconfined quantum critical point . . . . . . . . . . .
27
1.3
Standard model gauge charges for a fermion generation . . . . . . . .
41
2.1
Monopole operator dimension Aq for monopole charge q in U(1) gauge
of
theory.
2.2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6F(q, q') for various values of q and q'. The orange dots mean that the
corresponding W boson is unstable.
2.3
93
. . . . . . . . . . . . . . . . . .
95
The transformation properties of the first few (bare) monopole operators under the flavor SU(N)
global symmetry of QED 3 with Nf
flavors. The dimensions of the irreps were calculated using (2.131). All
these monopole operators are singlets under spatial rotations.
2.4
. ...
105
Estimates of the smallest number of fermions, Nd*nf for which the IR
of QCD 3 with gauge group G is in a deconfined quantum critical point.
Results are listed for various rank one and two gauge groups.
17
. . . .
134
18
Chapter 1
Introduction
Strongly interacting gauge theories are ubiquitous in physics, perhaps the most familiar example of a strongly interacting gauge theory is quantum chromo dynamics in
1
four space-time dimensions, QCD 4 . At high energies, this theory describes how the
six flavors of quarks interact with gluons. At low energies, it describes the spectrum
and interaction of hadrons, including the proton and neutron, which make up the
nuclei of atoms. The theory can be described by the Lagrangian:
1
LQCD 4
-
F
6
F" +
(iq.,
(1.1)
- maqaqa).
Q=1
At first blush, it is not clear why this theory describes strong interactions in the
infrared (IR). The coupling that dictates the strength of interactions, g, is classically
dimensionless, and so one might expect that its strength is insensitive to energy scale.
Furhtermore, very similar theories, such as QED 4 become weekly interacting. This
puzzle was resolved many years ago [1-3]. Though classically dimensionless, quantum
effects conspire to drive g to large values in the IR. In particular, the
#
function which
describes how the coupling g changes with energy is given at one loop by:
'We will use the notation QEDD and QCDD throughout the introduction to denote abelian and
non-abelian gauge theories with matter in D space-time dimensions
19
,3QCD4,
W1d
dlog(A)
- _ 11 - - Nf
3
167r 2
(1.2)
,
indicating that g becomes large at low energies. This strong coupling leads to a
very rich structure for the IR physics, the perturbation theory in g breaks down, and
we must appeal to other techniques to describe the zoo of interacting baryons and
mesons that result from the confinement of quarks.
Viewing this story from the other direction, it is an amazing simplification that the
messy low energy physics of hadronic resonances can be described by the relatively
simple theory of quarks at high energies. This ultraviolet simplification allows for
relatively easy computation of some aspects of the strong interactions.
The focus of this thesis is not with the strong interactions in four dimensions, but
rather with strongly interacting gauge theories in two and three space-time dimensions. As we will see, these strongly interacting gauge theories play important roles
in string theory, condensed matter physics, and mathematics, as well as serving as
toy models to help understand features of gauge theories in four dimensions, such as
confinement.
A particularly important feature of gauge theories in lower space-time dimensions,
is that strong coupling at low-energies is almost unavoidable.
In particular, the
classical scaling dimension of the gauge coupling is:
(1.3)
[g 2] = 4 - D,
from which we can create the dimensionless coupling, Ag = g/E
2
D/ 2 , where E is an
appropriate energy scale. As E becomes small, A9 becomes large, indicating that the
theory runs strong in the infrared and interesting low energy phenomena are possible.
This thesis addresses two such classes of strongly coupled gauge theories. The first
such class consists of three dimensional gauge theories with fermionic matter. These
gauge theories, denoted by QED 3 in the abelian case and QCD 3 in the non-abelian,
20
display interesting strong-coupling dynamics, such as confinement or chiral symmetry
breaking for certain choices of matter content and gauge group. The second class of
theories are two dimensional gauge theories with a particular amount of supersymmetry, suited to describing heterotic string theory. In these, an asymptotically free
gauge theory provides a UV completion of a 2d strongly interacting fixed point.
In the remainder of the introduction, we will provide background on the two
classes of theories described, beginning with gauge theories in three dimensions and
then moving down to two, setting the stage for the results in chapterss 2 and 3 which
make up the body of the thesis.
1.1
2
Monopoles and Confinement in Three Dimensions
A major focus of this thesis will be on three dimensional gauge theories with fermionic
matter. As mentioned above, these models are often strongly interacting at low energies, and perturbative analysis in the gauge coupling is typically not feasible. Nevertheless, three dimensional gauge theories provide a useful setting in which to try
and understand field theoretic phenomena such as confinement [6], chiral symmetry
breaking [7,8], and duality [9]. Furthermore, three dimensional gauge theories have
recently emerged as candidate descriptions of quantum phase transitions in two spatial dimensions [10].
The aim of chapter 2 is to examine the possible low energy
descriptions of gauge theories with large amounts of fermionic matter.
There is a natural guess for this low energy physics based on classical dimensional
analysis. To understand this guess, let's discuss QED 3. We will introduce the tools
to address general gauge groups when appropriate. QED 3 with NJ fermions can be
described by a Lagrangian,
2
The work appearing below has appeared previously in [4] and [5].
21
I
LQED 3 = -
Nf
FvFM" + Z P@
ba.
(1.4)
a=1
Here, F,, = iAv-avA, is the usual electro-magnetic field strength, and a= ,...,Nf
1
are a collection of complex two component fermions, with the covariant derivative
given by:
D
=
=
-
i
(1.5)
We would like to understand the behavior of such a theory at low energies.
As
mentioned above, e2 has dimensions of mass in three dimensions, and so we might
guess that the low energy limit is the same as the e -+ oo limit.
way, the kinetic term for the gauge field, FF""
Said another
has scaling dimension four, and so
is a classically irrelevant operator. We therefore might expect that the low energy
description of the theory can be understood by dropping the gauge kinetic term from
(1.4).
Nf
LIR = i (
00c'.
(1.6)
a=1
We will refer to this guess for the IR theory as a deconfined critical theory. In
addition to their interest as toy models for four dimensional physics, such deconfined
theories have been suggested as describing critical points in quantum phase transitions
of spin systems [10,11]. For large amounts of matter, our work in chapter 2, as well
as previous work by [12-15], indicate that the deconfined critical theory is the correct
description of the low energy physics. For small numbers of fermions, however, the
fate of the infrared is far less clear.
As mentioned above, three dimensional gauge theories can exhibit diverse IR phenomena not described by the deconfined fixed point, (1.6).
The seminal work of
Polyakov [6] showed that the classical argument leading to (1.6) can miss crucial
22
quantum effects. In particular, even though the operator F,,Fi" is irrelevant there
are additional operators, called monopole operators, which can become relevant, causing the theory to confine. More generally, understanding whether these additional
monopole operators are relevant plays a crucial role in understanding the low energy
physics of three dimensional gauge theories, giving insight into confinement in quantum field theory, as well as constraining whether the deconfined critical theories are
viable descriptions of phase transitions. The bulk of chapter 2 is devoted to calculating the dimension of monopole operators, as well as their other quantum numbers,
as a way to understand the low energy physics of the gauge theories in which they
occur.
In the next few subsections we introduce monopole operators in more detail and
explain the techniques used in chapter 2 to compute the scaling dimension and transformation properties of these operators. In the last subsection, we give a preview of
the results of chapter 2 and comment on the implications for the IR of the deconfined
critical theories.
1.1.1
Monopoles
As alluded to above, monopole operators play in important role in understanding the
structure of many gauge theories. We will be interested in them because they can
become relevant, in the RG sense, and destabilize the deconfined fixed point described
by (1.6), as such, they provide a valuable probe of low energy physics.
In four space-time dimensions, monopoles are a localized sources of magnetic flux.
A particularly nice choice for the gauge potential describing a localized source of flux
was given by Wu and Yang [16].
Au(l)
JA(N) q(1 - cosO)do
A(s) =q(-1 - cos )dO
23
if6$ir,
if 0 ,0,
(1.7)
which gives the field strength,
FU(1) = q sin Od9 A do.
As was originally realized by Dirac, [17], the charge of a monopole, 47rq =
(1.8)
fS2 T,
is not arbitrary, but is quantized. With the conventions we have chosen, q E Z/2.
We wrote down, (1.7), thinking of it as a static source of magnetic flux in four
dimensions, but as it is time independent, it can also be thought of as a field configuration in three Euclidean dimensions. In this thesis, the term monopole will always
refer to an object localized in three Euclidean dimensions.
We will not be so concerned with this classical gauge configuration, but rather with
the quantum monopole operator associated to this background. More explicitly, given
a classical field configuration such as (1.7) we can construct a local gauge invariant
operator Mq, often referred to as a disorder operator, [18], by performing the path
integral subject to the boundary conditions that near the operator, fields approach a
given classical configuration. In chapter 2 we give a more careful explanation of this
construction.
For non-abelian gauge theories we can consider more general monopoles known as
Goddard-Nuyts-Olive (GNO) monopoles [19]. For gauge group G, these monopoles
are given by:
(1.9)
AGNO = HAu(l),
where H is an element of the Lie-Algebra of G. We can use a gauge transformation
to rotate H into a diagonal form. For example, for G = U(Nc) we can rotate H into
the form:
H = diag(q
conditin
, q2,
)- e.
In this case, the Dirac quantization condition becomes, qi E Z/2, more generally,
24
we can write H = qah', where {h } form a basis for the Cartan subalgebra of the
Lie algebra of G. The charges, qa must be chosen such that e4 1riqaha = 1.
This is
the non-abelian generalization of the Dirac quantization condition. The collection
{qi, q2 , ...
, qr}
is gauge invariant and characterizes the GNO monopole.
There is a much coarser characterization of monopoles that is often important.
The collection {qi, q2 ,...
,
qr} are known as GNO charges. Monopoles may also carry a
topological charge, Qtp. For abelian monopoles this is precisely the charge, QtP
=
q,
and the conserved current is given by:
JA =
1
47r
"PF.(1.11)
For U(N) monopoles, the charge is the sum of the GNO charges. QtiO = E
qj, more
generally the topological charge is an element of -r,(G).
For a nice introduction to non-abelian monopoles, see [20].
1.1.2
Fate of the IR
Given a particular gauge theory, we would like to understand what effects monopole
operators have on the renormalization group (RG) trajectory. The strategy that we
will take in chapter 2 is to assume that the low energy physics is described by the
deconfined critical theory, (1.6), and check the self consistency of this assumption.
As explained below, we are able to identify and compute the dimensions of monopole
operators at the deconfined fixed points. For some choices of gauge group and matter
content we find that these fixed points are self consistent, that is the dimension of all
monopole operators is greater than 3 and so they are all irrelevant, however for small
amounts of matter, and small gauge groups, we find that the monopole operators are
relevant and represent an instability of the IR fixed point.
The technique for calculating the dimension of monopole operators to leading
order was pioneered in [14] and extended to calculate the dimension of the lowest
charge monopole in QED 3 in [21]. The idea, as we explain in chapter 2, is to map
the calculation of the scaling dimension, Am on R' to the computation of an energy
25
on S 2 x R, which can in turn be evaluated by performing the Euclidean path integral
on S 2 x R.
1
AM = lim -- log(ZE)-
(1.12)
Here:
ZE =
D[A]D[Vt]D[V]e-sE[AVb,'t] =
D[A]e-NSEff[A],
(113)
is the Euclidean partition function on S 2 x R.
N 1 appears in this expression multiplying the whole action, we can thus perform
a saddle point expansion in 1/N where the saddles are just the classical monopole
configurations described above.
One distinguishing feature of non-abelian gauge theory is that not all classical
saddles are minima. For some classical saddles, the quadratic fluctuations of gauge
fields are unstable, and so these saddles do not correspond to monopole operators in
the low energy theory.
For QED 3 the monopole dimensions are given in Table 1.1.
|Iq|l
Aq
0
0
1/2
1
0.265 Nf - 0.0383 + 0(1/N)
0.673 N1 - 0.194 + 0(1/N)
3/2
1.186 N1 - 0.422 + 0(1/N)
1.786 N - 0.706 + 0(1/N)
2.462 N - 1.04 + 0(1/N)
2
5/2
Table 1.1: Monopole operator dimension Aq for monopole charge q in QED
3
The most relevant (smallest dimension) monopole is the charge q = 1/2 monopole.
With:
A 1 / 2 = 0.265 N1 - 0.0383 + 0(1/N 1 ).
26
(1.14)
The assumption that monopoles are irrelevant is only valid if, A1/ 2
3, which
requires Nf > 12 and we predict an instability for smaller N 1 . We can run similar
arguments for other gauge groups.
The predictions are given in Table 1.2, where
Nd"""f is the number of fermions for which the assumption of the deconfined theory
becomes self consistent. We present the complete results in chapter 2.
Gauge group
N
onf
U(1)
U(2)
12
14
SU(2)
SO(3)
SU(3)
8
8
10
Sp(4)
10
SO(5)
10
G2
6
for which the IR of
Table 1.2: Estimates of the smallest number of fermions, N d,*f
f
QCD 3 is in a deconfined quantum critical point.
In the next subsection we move away from gauge theories in three dimensions and
begin discussing the two dimensional gauge theories which make up the second half
of this thesis. We return to three dimensions in chapter 2.
1.2
Strongly Coupled Gauge Theory for Chiral Strings
In this section we will lay the groundwork for the models introduced in chapter 3.
In the previous section we used an asymptotically free gauge theory to understand
interacting conformal field theories in three-dimensions, here we will use a class of
gauge theories in two dimensions to study novel two-dimensional CFTs. The conformal field theories we will discuss are particularly relevant for string theory, where
they describe potential backgrounds for the heterotic string.
Perturbative string theory describes the propagation of one-dimensional strings
in a (typically) higher dimensional target space in a similar fashion to how nonrelativistic quantum mechanics describes the motion of point particles in space.
27
Non-relativistic point particles moving in d spatial dimensions can be described
by the action:
SNR
Here i = 1, ...
, d,
Jdt
(1.15)
( m4ii - V(qi)).
and V describes some potential for the point particles. This
action can be thought of as defining a 0 + 1 dimensional field theory, where the
coordinates of the particle, qg(t) are the fields. This is a one dimensional quantum
field theory describing maps from the one dimensional world line of the particle into
the d dimensional target space.
Similarly, the gauge fixed form of the Polyakov action describing the motion of
strings in target space takes the form:
Sp = - 1
(1.16)
d2x,#kia#qi.
Here, the two dimensional coordinates, (xO, x 1 ) parameterize the two dimensional
world sheet traced out by a string moving in time. The fields,
#i(xO,
x 1 ), i = 1, ...
give the coordinates in the D dimensional space-time, of the point (x 0,
,D
1).3
The action, (1.16), describes strings propagating in D flat space-time dimensions.
Consistency of the theory puts constraints on the allowed values of D. The simplest
string backgrounds describe the propagation of strings in either 26 flat dimensions,
for the bosonic string, or 10 for the superstring.
As our universe appears to be
four dimensional this has lead to the search for more general backgrounds of string
theory.
A particularly geometric approach to solving this problem is to consider
strings propagating in four flat non-compact directions, with all remaining directions
3
We have been slightly careless here, for the sake of a simple analogy. For the non-relativistic
point particle the indices i only run over the spatial coordinates, and are thus contracted with the
Euclidean metric, while in the case of the string, i runs over all of space-time, and is contracted
with the Minkowski metric. We have also brushed all of the gauge fixing required to arrive at (1.16)
under the rug. There is a much closer parallel between the relativistic string and the relativistic
point particle
28
compactified. Such a theory can be encoded in a non-linear sigma model describing
propagation of strings in this geometry,
SNLSM
~-
1
d 2 x (Gjj(0)am#'a1 + Bij(O)E M±
+ . . .) .
(1.17)
The background field, Gj(#) is the target space metric, while Bij(#) is an antisymmetric two form. The dots represent other possible background fields coupled
to world sheet fermions. One coupling of particular significance for the models con-
sidered below, is the possibility of a background gauge field, Ai(#).
The detailed
expansion for the models we will consider can be found in chapter 3. As the noncompact directions describe Minkowski space, we often use (1.17) to refer to purely
the compact directions.
The theory as written in (1.17) does not describe a consistent quantum theory
of strings for every choice of background fields. Rather, quantum consistency of the
string theory imposes constraints on G, B, A, and any other interactions included.
The consistency conditions are equivalent to demanding that (1.17) describe a modular invariant conformal field theory with a particular central charge.
This is an example, of a much more general statement, that any two dimensional
conformal field theory with the right central charge (a parameter in the conformal
algebra) defines a consistent background of string theory. In this way, the study of
perturbative string theory is intimately tied to the study of two dimensional conformal
field theories.
One rich application of string theory has been the interplay of two dimensional
CFT, on the one hand, with the target space geometry on the other. The additional
structure provided by two dimensional super-conformal field theories, has provided
tools for understanding and classifying manifolds.
the case of conformal field theories with
P2
This has been most fruitful in
= (2, 2)4 supersymmetry, where confor-
mal field theory, and string theory more generally have lead to many new tools for
4
The notation K2 = (M, N) is described in subsection 1.2.3.
29
understanding a class of geometries known as Calabi-Yau manifolds. However these
Calabi-Yau compactifications, with (2, 2) supersymmetry only represent a small patch
of the string landscape. We will be interested in exploring two dimensional theories
with N2 = (0, 2) supersymmetry. These models are appropriate for compactifications of the heterotic string to four dimensions which preserve at least A4 = 1 target
space supersymmetry [22]. One hope for these models is that they will shed light on
geometric structures beyond the Calabi-Yau framework.
Though we can write down non-linear sigma models for these (0, 2) theories, in
general, an NLSM described by (1.17) can be difficult to study, as the metric and
other background fields my induce complicated interactions. The main tool we will
use to study these non-linear sigma models is the gauge-linear sigma model (GLSM).
In the context of string theory, the GLSM was introduced by Witten in [23], however
the basic idea behind the GLSM is much older. As we illustrate in subsection 1.2.2
the purpose of the GLSM is to realize the non-linear sigma model as an effective low
energy description of a simpler, linear gauge theory. In this way, we will again be
concerned with gauge theories, in this case as a convenient UV description of NLSMs
appropriate for the heterotic string. 5
The main focus of chapter 3 will not be with phenomenology of the heterotic
string, or even with a detailed analysis of specific (0, 2) geometries, but rather with
understanding the implications and constraints of chirality in two dimensional, supersymmetric, abelian gauge theories. As we explain, just as fermions can have a
handedness in four dimensions, fermions in two dimensions can be labeled by their
chirality.
One of the interesting, and under-explored aspects of (0, 2) compactifi-
cations is taking full advantage of this chirality. It is this feature, and especially
anomalies resulting from chiral fermions, that allows for the construction of models
that have the potential to expand the space of geometries string theory is able to help
understand.
In the remainder of this section we will lay out some of the basic tools necessary to
5
These same gauge theories have also featured prominently in recent developments attempting
to use two dimensional field theory to understand four-manifolds [24].
30
understand these models. We begin by reviewing the geometric constraints imposed
by target space and world sheet supersymmetry on the compact internal geometry.
We will then give a simplistic introduction to the GLSM, demonstrating how a nonlinear sigma model can arise as the low energy description of an asymptotically free
gauge theory. We then set up the language of two dimensional supersymmetry as
well as discussing gauge anomalies in two dimensions. We close by discussing how
the interplay of gauge anomalies with classical gauge non-invariance allows the models
we introduce in chapter 3 to go beyond the typical non-chiral geometric framework
and exhibit properties generic in chiral theories that have only begun to be explored.
1.2.1
Consistancy conditions from world-sheet and spacetime
Before launching into the discussion of the gauge theories we will use to explore
conformal field theory, let's discuss what constraints conformal invariance puts on
the non-linear sigma model itself. The most well known example of this is when we
look for consistent, supersymmetric theories of the form (1.17) with H = dB = 0. In
this case, the leading order consistency conditions for the six dimensional manifold, is
that it be a compact Kiihler manifold 6 that satisfies the vacuum Einstein equations.
R%3 = 0.
(1.18)
7
Such manifolds are known as Calabi-Yau manifolds.
More generally, the lowest order consistency conditions on (1.17) will reduce to
Einstein's equation for the metric, with the other background fields sourcing the
energy momentum tensor, as well as additional constraints on the background fields
themselves.
6
A Kahler manifold is a special class of complex manifold, where a particular two form (the
Kshler form) is closed.
7
The bars on the indices indicate the fact that supersymmetry requires the compact manifold
to be complex, and thus we have a splitting into holomorphic indices and anti-holomorphic indices.
This is analogous to the coordinates z = x + iy and = x - iy on the plane.
31
There are typically two approaches to finding these conditions. One approach is
to study the two dimensional field theory described by (1.17) directly. Here, both
the cancellation of sigma model anomalies, and the condition that the theory be
conformal impose conditions on the background fields [25-31].
There is a complimentary approach to understanding the NLSM consistency conditions, which relies on studying the ten dimensional target space description of the
string. Requiring that a given geometry solve the ten dimensional supergravity equations of motion also serves to constrain the form of the geometry [32].
Unfortunately, even if a solution to the perturbative equations is found to some
order, it may not satisfy the higher order equations.
Both the sigma model beta
function computations and the supergravity computations are perturbative in a',
and receive modifications up to arbitrarily high order in perturbation theory [33,33].
As such, it is useful to identify topological properties of the low energy manifolds
which will be unaffected by perturbative corrections.
For the case of Calabi-Yau manifolds, (1.18), implies the apparently weaker topological condition, 8
c1(TM) = 0.
(1.19)
It turns out that allowing for more general backgrounds does not alter this condition, but the background fields must satisfy additional constraints.
c1(V) = 0 mod 2,
8
(1.20)
1n writing the topological conditions that compact geometries must satisfy, we use the notation
of characteristic classes. These classes represent equivalence classes of closed forms. We have already
run into one such class in the discussion of monopoles c1 (Au(1)) = - [F]. The brackets represent the
fact that c1 (V) is only defined up to the addition of a closed form, but the integral, fs2 c1 (AU(l)) =
fs2 FU(1) = 2q is well defined, and is just twice the monopole charge. ch 2 also has an expresion
in terms of the field strength. These equivalence classes are invariants of the manifold, in the case
of TM, and the gauge configuration in the case of V.
A
32
and,
ch 2(TM) = ch 2 (V) ,
(1.21)
which follow from more restrictive geometric equations. Here, V represents the vector
bundle (gauge configuration) specified by the background gauge field A'(0).
Whether or not there is guaranteed to be a solution to the geometric equations
given a manifold that satisfies the topological restrictions is a difficult question. For
the special case of H = dB = 0, and thus a compact Kihler manifold with vanishing
first Chern class (c 1 (TM) = 0), the answer was given by Yau [34] as a proof of the
Calabi conjecture [35]. For this case there is guaranteed to be a solution to (1.18).9
Outside of this class of models it is difficult to proove whether a manifold which
satisfies all of the topological requirements, (1.19 - 1.21), admits field configurations
that satisfy the string consistency conditions. As such, it is important to develop other
techniques to study the flow of two dimensional sigma models, ultimately identifying
and describing candidate IR fixed points.
The gauge linear sigma model, which we introduce in the next section, provides
just such a technique, by realizing the conformal theory as the low energy fixed point
of a gauge theory, we are often able to address the existence of a conformal theory
in the infrared by an analysis of consistency in the gauge theory. The next section
is an introduction to the GLSM, which we use in chapter 3 to understand candidate
conformal theories which are not described by Calabi-Yau geometry.
1.2.2
Gauge Linear Sigma Models: The basic idea
As emphasized above, in string theory, we are often concerned with the physics of a
string propagating in some potentially complicated geometry. This can be described
in terms of a two dimensional non-linear sigma model (NLSM).
9
Yau, together with Uhlenbeck, also showed that given some assumptions about the vector bundle V, any vector bundle satisfying [J] A [J] Ac 1 (V) = 0 admits a connection, that satisfies the
supergravity equation of motion, Fj = F3 = giF3 = 0 [36].
33
1
LNLSM =
(1.22)
+...
2
Here 0', i = 1, . . . , D can be thought of as two dimensional scalar fields and as
maps from the two dimensional world-sheet into the D dimensional target space. Gij
is the target space metric, while the derivatives,
contracted with the flat metric,
the
#b to
0
, are two dimensional derivatives
. The dots represent other potential couplings of
background fields, as well as additional two dimensional fields.
The idea behind the gauge linear sigma model is to realize this potentially complicated non-linear sigma model as an effective low energy description of a simpler,
linear gauge theory at high energies.
This basic idea is quite old, however in the
context of string geometry it was introduced by Witten in [23]. Before introducing
the full machinery of the supersymmetric GLSM, it is nice to consider a few simple
examples which illustrate the basic idea.
O(N) Model:
The O(N) model is a theory of N scalar fields,
010I
= r
2
#,
constrained to be on a sphere,
.
This can be described by the Lagrangian:
LO(N)
-1
-
A (101
_
,
(1.23)
in the limit A -+ oo.
Alternatively, as A has dimension 2, we can think of the O(N) model as being
given by the low energy theory of the above Lagrangian.
Note that we can also solve the constraint
010I = r 2
consider the case of N = 3, and take the fields q1
explicitly. For simplicity lets
= X, 02 =
y, q3 = z. The constraint
is just:
x2 + y2 + z2 = r2,
34
(1.24)
describing a two sphere, S 2 in R'.
Taking spherical coordinates to solve this
constraint,
x = r sin ()
cos(6)
(1.25)
y = r sin(#) sin(9)
z = r cos(#) ,
and plugging into 1.23, yields:
LNLSM
Here, 60 = (0,
#)
=
1
-- Gj (0,0),a5 1 90
3
(1.26)
.
and G = diag(r 2 , r 2 sin 2 (0)) is the usual round metric on
S2.
In this way, we see that (1.23) reduces to the non-linear sigma model (1.26) at low
energies. In contrast, at high energies, we can neglect the interaction term in (1.23)
as [A] = 2, so we are left with just a free, linear theory.
2
As a final point on the O(N) model, note that Gij has an overall factor of r ,
2
similarly, if we rescale red we can pull out a pre-factor of r in (1.23).
In the path
integral, Z = f D[<]ew[1], and so r2 appears in front of the action, just as .. This
means that r controls the saddle point approximation, and perturbation theory is
only valid when r is large.
The above example didn't incorporate gauge symmetry, in order to get a feel for
the types of models that will be discussed in chapter 3 it is good to look at a relatively
simple example of a geometry that can be realized through a gauge theory at high
energies.
pN Model:
As apposed to the two dimensional O(N) model, which described maps from two
dimensional space to the sphere, SN1, the two dimensional PN model describes maps
from two dimensions to complex projective space, pN (also referred to as CpN).
The space, pN is the space of lines in CN+1-{O}.
35
More explicitly, if (Z', . . .
, ZN+1)
are coordinates for CN+1, and we introduce an equivalence relation,
(Z,... , ZN+1 )
(AZ 1 ... AZN+ 1 ) V A E C*,
(1.27)
then pNis given by:
pN
_
= (oN+1
{o})
(1.28)
Here, {0} denotes the origin in CN+1, and C* = C - 0.
We can use the N + 1 numbers, (Z 1 : Z 2 :
...
: ZN+1) to represent a point in pN,
keeping in mind that the overall scale has no meaning (thus the ":"
notation). These
are known as homogeneous coordinates. Alternatively we can work locally, defining
inhomogenous coordinates in say the ZN+1 5 0 region by z
=
Zi/ZN+1. This defines
a set of N coordinates on a patch of pN.
We can write down an NLSM describing maps into this geometry by using the
natural metric on 1pN called the Fubini-Study metric.
kk(1
Izk 2)J
G.(FS)
-.
=
'1
+
LNLSM
- 213
G
(1
-
ZZ j
+ E Izk 12)2
(1.29)
--
The NLSM is just,
~,2D~
43
(1.30)
Here, the NLSM is written in terms of N complex fields z'.
There is another useful way to view the space pN. Given the homogenous coordinates, (Z 1 :
...
: ZN+1), we can use the equivalence, (1.27), to fix EN 1 jZ
1
2
-
1,
this condition defines an S2N+1 inside of CN+1. This is not enough to totally fix the
equivalence, as multiplying by a complex phase doesn't effect E
means that pN can be thought of as
S2N+1
the coordinates by a phase.
36
1
2 = 1. This
subject to the equivalence of multiplying
PN
=
S 2N+1/U(1).
(1.31)
This perspective on pN allows us to write down a relatively simple theory, which
flows to (1.30) at low energies. Using the example of the O(N) model, we know how
to write down a theory which describes fields living on a 2N
LO(2N+2) = ~0pZI
2O - A
(
+ 1 dimensional sphere,
I ZI12 - r2).
(1.32)
Here we have written the 2N + 2 real fields of the 0(2N + 2) model in terms of
N + 1 complex fields ZI.
This theory has a global symmetry, ZI -+ e"ZI, but this is exactly the equivalence
we would like to impose to get pN, therefore gauging the symmetry, by introducing
a U(1) gauge field,
LpN = --
FF"
- DIIZIDA2 - A
7ZI|2
-
r2,
(1.33)
gives a gauge theory which flows to a sigma model on PN at low energies, and a
free theory at high energies.
These two examples illustrate the idea behind the GLSM, however the discussion
here has been purely classical, and there are subtleties in these models quantum mechanically that destroy this simple picture. The types of models we will be interested
in in chapter 3 are supersymmetric, and as a result, the quantum effects which are
crucial for understanding the O(N) and pN models are under better control. Before
introducing these models, let's briefly review fermionic physics in two dimensions.
37
1.2.3
Lorentz Symmetry, Supersymmetry, and Anomalies in
Two Dimensions
The focus of the second half of this thesis, will be on strongly interacting, supersymmetric gauge theories in two space-time dimensions. As such, it is useful to get a
little familiarity with Lorentz invariance and supersymmetry in two dimensions, we
record the detailed conventions in appendix B and here remind the reader of a few
basic facts about two dimensional physics.
Recall, that all massless relativistic fields, D satisfy the Klein-Gordan equation:
E14) = (-O +
a2)& = 0 .
(1.34)
Just as in higher dimensions this equation admits plain-wave solutions. An interesting feature of two dimensions, is that a general solution to (1.34) can be written
as a superposition of left moving and right moving waves. This can be seen by noting
that in the light cone coordinates,
X*
- I(XO i X1) .
2
(1.35)
The wave equation becomes:
0EI
= -4a+O-_
= 0,
(1.36)
and so admits solutions of the form 1(x) = D+(x+) + D-(x-).
This is a result of the fact that the defining representation of the Lorentz group
in two dimensions is reducible. Explicitly, Lorentz transformations act on the coordinates, x± as multiplication.
38
X-
e±o+X.
(1.37)
Similarly, fermions in two Minkowski dimensions can be written in terms of real
one component fermions,
I±(x), which transform as:
,0(x)= eg2±(A-lx).
(1.38)
For the models under consideration in chapter 3, we will be interested in supersymmetric theories in two dimensions. Supersymmetry is an extension of the Poincare
group that includes fermionic symmetry generators,
Q'
and
Q'
, in addition to the
usual bosonic generators, PA and M"'. As fermions come in two chiralities, t,
we
can consider two dimensional theories with some number, M, of negative chirality
supersymmetry generators,
are said to have
Q_',
and sum number N or positive,
QS.
These theories
2 = (M, N) supersymmetry.
We will be interested in the case
2 = (0, 2), relevant for describing the heterotic
string propagating in a supersymmetric target. In this case we can package the two
real supersymmetry generators into a single complex generator. At supersymmetric
conformal fixed points, this is enhanced to the (0, 2) superconformal algebra.
As (0,2) supersymmetry is chiral, the matter content of these theories is generally chiral as well. One aspect of theories with chiral fermions is the possibility of
anomalies.
An anomaly refers to a symmetry of the classical theory which is not
preserved quantum mechanically. Anomalies in global symmetries have important
implications for the content of the theory. For instance in four dimensions the decay
rate for the process
7ro
symmetry [37, 38].
In two dimensions, the U(1) axial anomaly results in a gauge
-+ 7 7 crucially depends on an anomalous global U(1) axial
invariant mass for the photon [39].
In four dimensions anomalies appear in correlators involving three currents, or
39
(a) Triangle anomaly in four dimensions
(b) Diangle anomaly in two dimensions
Figure 1-1: Anomalous diagrams in four and two dimensions
gauge fields, and show up diagrammatically in evaluation of triangle diagrams, such
as shown in Figure 1 - la. In two dimensions, they show up as diangle diagrams,
such as Figure 1 - 1b.
As apposed to global anomalies, which alter the phsyics of a theory, anomalies
in gauge symmetries are typically said to render a theory inconsistent.
As we will
see below, and more explicitly in chapter 3, care should be made in making this
statement.
It has been known since the work of Green and Schwarz that some theories with
apparently anomalous matter content are actually rendered non-anomalous by the
cancelation of quatum anomalies with a classical gauge non-invariance [40].We will
see that for chiral gauge theories in two dimensions, such cancellations are typical and
facilitate the construction of gauge theories for geometries outside of the Calabi-Yau
framework.
In chapter 3 we present the two dimensional story, but the basic pronciple can be
understood in the more familiar four dimensional context. To see this explicitly let's
consider the standard model. The standard model describes an SU(3) x SU(2) x U(1)y
gauge theory at high energies, which at low energies breaks to SU(3) x U(1)EM. The
fermionic content of this theory is three generations of quarks and leptons, where
each generation has the charges listed in Table 1.3.
It is not difficult to check that all of the gauge anomalies, which are proportional
to Tr(T1 {T 2 T3}), vanish. Here T is the generator of the ith gauge group. For example,
40
SU(3)
SU(2)
UL
3
2
dL
3
2
I
UR
U(1)y
U(1)EM
1
12
dR
33
VL
1
2
1
0
eL
1
2
1
-1
eR
1
1
1
1
Table 1.3: Standard model gauge charges for a fermion generation
the cancellation of the U(1)3 anomaly is given by:
0,j" oc Tr(Q3) = 3 (2(1/6)3
-
(2/3)3 + (1/3)3) + (-2(1/2)3 + (1)3) = 0.
(1.39)
leptons = 3/4
quarks = -3/4
One thing to note about the cancellation, is that it relies crucially on having
a full generation.
If one drops the contribution from the top quark, for instance,
the anomaly no longer cancels. This presents a puzzle. The top quark is about forty
times heaver than the bottom quark, and so we should be able to consider an effective
theory where we integrate out the top quark. Even worse, if we consider a theory
like the standard model, but where we are free to change the Higgs-quark Yukawa
couplings, we can take the mass of the top and bottom to infinity without changing
the mass of any other particles.
For energies above the scale of spontaneous symmetry breaking, but below the
mass of the arbitrarily heavy quarks, the theory is naively described by a gauge
theory with anomalous matter content which appears inconsistent.
On the other
hand, it should be consistent to integrate out the heavy quarks.
This puzzle was resolved by D'Hoker and Farhi [41,42], who showed that integrating out the massive fermions induces self interaction terms of the Higgs field which
conspire to cancel the gauge anomaly in the low energy theory.
41
This same phenomena is common in the two dimensional chiral gauge theories we
study, and as we explain in chapter 3 it is precisely this cancellation between quantum
anomaly and classical gauge non-invariance which allows these models to describe low
energy fixed points outside of the usual Calabi-Yau framework.
The rest of this thesis goes into more detail trying to understand the strongly
interacting gauge theories introduced above. In section two we explore monopoles
in three dimensional gauge theories with large amounts of matter. Using the 1/Nf
expansion we compute the scaling dimension and quantum numbers of monopoles for
a variety of gauge groups ultimately making predictions for the infrared behavior of
these gauge theories. In section three we move on to discuss chiral gauge theories in
two dimensions exploring the consistency conditions and geometric implications for
the low energy conformal field theories.
42
Chapter 2
Monopole Operators in Strongly
Coupled Gauge Theories
In this chapter, we cary out the investigation described above, identifying monopole
operators and studying their scaling dimension and quantum numbers in strongly
coupled gauge theories in three dimensions.
2.1
Introduction
In three-dimensional gauge theories, one can define local disorder operators by requiring the gauge field to have a certain singular profile close to the point where the
operator is inserted [13,14]. These operators are commonly referred to as monopole
operators, because in Euclidean signature the gauge field singularity looks like that
of a Dirac monopole [17] or a non-Abelian generalization thereof [19,43,44], as will
be the case in this chapter.
Monopole operators are of interest for many reasons.' As explained by Polyakov
in 3d Maxwell theory without matter, the proliferation of monopoles provides a mechanism for confinement [6,46]. If one adds enough fermionic or bosonic matter, how'We restrict ourselves to the study of monopole operators in three-dimensional gauge theory.
In four-dimensional gauge theories one can define line operators by requiring the gauge field to
asymptote to that of a monopole close to the line singularity [18]. These operators play an important
role in the geometric Langlands program; see, for instance, [45].
43
ever, the monopole operators become irrelevant in the renormalization group (RG)
sense [12-15], and in the deep infrared one finds a deconfined quantum critical theory.
As stressed in [47] (see also [48]), the existence of these deconfined quantum critical
theories relies on not having any monopole operators with scaling dimensions smaller
than three. Another reason why monopole operators are of interest comes from certain spin systems whose low energy physics is described by an emergent gauge theory,
such as the CPN model [10,11]. In these gauge theories, monopole operators can act
as order parameters [49,50] for second-order quantum phase transitions that cannot
be described within the Landau-Ginzburg-Wilson paradigm [10,11]. The scaling dimensions of these monopole operators constitute interesting critical exponents that
can also be computed directly from quantum Monte Carlo simulations of the spin
systems [51-53].
Monopole operators play a prominent role in supersymmetric theories as well.
For instance, in the model introduced by Aharony, Bergman, Jafferis, and Maldacena
(ABJM) [54] (see also the related model in [55]), it was shown that when the ChernSimons level is k = 1 or 2, there exist BPS monopole operators that are Lorentz
vectors and have scaling dimension precisely equal to two [55-57].
In other words,
these operators are conserved currents. The existence of these conserved currents is
what makes possible an enhancement in the amount of supersymmetry from K = 6,
which is the manifest supersymmetry of the ABJM Lagrangian, to K = 8, which
is the expected amount of supersymmetry that follows from M-theory. In the same
ABJM model, as well as in many other superconformal field theories with gravity duals
[58-67], monopole operators are also needed to match the spectrum of supergravity
fluctuations in the bulk, and indeed certain holographic RG flows are triggered by
operators with non-vanishing monopole charge [68]. Monopole operators also play
important roles in various supersymmetric dualities (see for example [69-71]) and
mirror symmetry (see for example [14, 72]), where the duality transformations map
them to more conventional operators. It is important to know the quantum numbers
of these monopole operators if one wishes to check these dualities.
The goal of this chapter is to study monopole operators in (non-supersymmetric)
44
three-dimensional QCD with gauge group G (which includes QED in the case G =
U(1)) and N1 flavors of fermions transforming in some representation of G.
We
study these operators perturbatively to next-to-leading order in 1/Nf. While in the
absence of matter fields, 3d gauge theory with any compact gauge group is believed to
confine [6,46,73-76], in the presence of a sufficiently large number of matter fields the
theory flows to an interacting conformal field theory (CFT) in the infrared (IR) [77].
We are interested in studying monopole operators at this interacting IR fixed point.
We want to answer the questions: How many independent monopole operators are
there, and what are their quantum numbers?
Of course, starting with any monopole operator, we can take its product with
various gauge-invariant local operators built out of the fermions, and construct new
monopole operators this way. Throughout this work, however, we will focus only
on the "bare" monopole operators, namely those that cannot be written as such
composites.
Like in any CFT, one can use the state-operator correspondence to identify the
space of local operators that can be inserted at a given point on R3 with the Hilbert
space of states on S2 x R. In general, if one defines a monopole operator by requiring
the gauge field to have a fixed behavior close to the insertion point, the resulting
operator will not have a well-defined scaling dimension. It is quite subtle, in general, to associate a certain monopole profile to an operator with well-defined scaling
dimension, or, equivalently, to a certain energy eigenstate on S2 x R. We will discuss this subtlety in Section 2.2. As we will explain, in the large N limit that we
study, the subtlety is ameliorated by the fact that the gauge field fluctuations are suppressed, and one can indeed say that certain energy eigenstates on S 2 x R correspond
to monopole operators. However, not all possible non-Abelian generalizations of the
Dirac monopole can be associated with linearly-independent monopole operators of
well-defined scaling dimension.
As we will review in Section 2.2.1, the non-Abelian generalization of a Dirac
monopole involves several discrete parameters referred to as Goddard-Nuyts-Olive
(GNO) charges [19]; the monopoles are also classified topologically by iri(G) [78], and
45
unless G = U(1), there are infinitely many GNO monopoles in the same topological
class. One of our main results is that, at least in the limit of large Nf, only certain
sets of GNO charges yield independent bare monopole operators. These sets are the
ones for which the corresponding monopole background is stable, in the sense that
it is a local minimum of the gauge effective action on S 2 x R. A surprising result is
that we find more than one independent monopole operator per topological class.
The monopole operators must transform as representations of the global symmetry
group, which includes the conformal group and the flavor group. The quantum numbers under the conformal group are the spin and the scaling dimension. We devote a
significant part of our work to computing the scaling dimensions of the monopole operators to second order in 1/Nf. For clarity, we first present our computations in the
case where the gauge group is G = U(Nc) and where the fermions are two-component
complex spinors transforming in the fundamental representation of U(Nc). We later
generalize our computations to other gauge groups and/or other representations of the
fermion flavors. Our work improves on existing results in the literature: in the QED
case, N, = 1, the leading large N 1 behavior of the monopole operator dimensions
was found in [13]; for the monopole with lowest charge, the first subleading correction was computed in [21]; lastly, in U(Nc) QCD with N1 fundamental fermions, the
dimensions of the monopole operators at leading order in N 1 were found in [72]. Related computations can be found in [12,15, 79] in non-supersymmetric theories, and
in [14, 55,56,65] in a supersymmetric context.
We also calculate the representations of the monopole operators under the flavor
symmetry group, but only in the case where the gauge group is G = U(Nc). In this
case, the flavor group is SU(Nf). We find that the monopole operators transform in
irreducible representations of SU(Nf) whose Young diagrams are rectangles with a
number of rows equal to N 1 /2 and a number of columns that depends on the GNO
charges. Our results apply, of course, in particular to the QED case, N, = 1, and agree
with the results of [13] for the monopole of smallest charge, but disagree with [13] on
the monopole with two units of charge. (Our computation for monopole operators
with greater than two units of charge is a new result.) We also find disagreement
46
with the results of [72] in the case of the simplest GNO monopole in U(Nc) QCD.
The rest of this chapter is organized as follows. Section 2.2 is a rather non-technical
and highly recommended read that includes a definition of monopole operators in 3d
gauge theory (in particular in QED3 and QCD 3), as well as a discussion of how these
operators can be studied via the state-operator correspondence. In Section 2.3 we set
up the computation of the scaling dimensions of the monopole operators in U(Nc)
QCD
with Nf fundamental fermion flavors as an expansion in 1/Nf.
To evaluate
these scaling dimensions through order O(NO), we need to compute three functional
determinants corresponding to the fluctuations of the fermions, of the ghosts, and of
the gauge field. We study the effective actions of these fields in Section 2.4. The gauge
field effective action is not positive-definite for all sets of GNO charges, thus making
certain GNO monopole backgrounds unstable and the corresponding monopole operators poorly defined. We discuss this stability issue in Section 2.5. For the monopoles
that are stable, we collect the results on their scaling dimensions in Section 2.6. Our
results include the QED case N, = 1 as a particular case. In Section 2.7, we find
how the monopole operators transform under the SU(N) flavor symmetry group. In
Section 2.8 we generalize the results of Sections 2.3-2.6 to other gauge groups and/or
representations of the fermions. Lastly, we end with a discussion of our results in
Section 2.9. The reader interested only in the results can skip Sections 2.3 and 2.4.
2.2
Monopole operators via the state-operator correspondence
We now start by addressing some of the preliminaries necessary for studying properties
of monopole operators. In Section 2.2.1, we introduce classical monopole backgrounds
in both Abelian and non-Abelian gauge theories. In Section 2.2.2, we then review
the gauge theories of interest for this chapter and highlight the role played by large
Nf in studying them. Lastly, in Section 2.2.3, we introduce carefully the monopole
operators that we will study in the rest of the chapter, and discuss two ways of
47
defining them that become equivalent in the limit of large Nf.
2.2.1
Classical Monopole Backgrounds
To begin discussing monopole operators more explicitly, it is convenient to first think
The simplest and perhaps most familiar such back-
about classical backgrounds.
grounds can be constructed in Abelian gauge theory as follows. A monopole of charge
q in a U(1) gauge theory in three dimensions is a rotationally-invariant background
A for the gauge field A, where the field strength F
=
dA integrates to 47rq over any
two-sphere surrounding the center of the monopole. For a monopole at the origin, we
can write the gauge field and its field strength in spherical coordinates as
A(N)
F = q sinOdO A do
A(S)
q(1
q(-1-cos)d
where the expressions A(N) and A(S) satisfy dA(N)
everywhere away from 0 =
7r
if 09# 7r,
cosO)do
= dA(S)
if 6O0,
-
F and are defined
(the North chart) and away from 0 = 0 (the South
chart), respectively. 2 In the overlap region, these two expressions differ by a gauge
transformation, A(N) - A(S) = dA, with gauge parameter A = 2q.
The condition that
this gauge transformation is well-defined, namely that the same U(1) group element
eiA is associated both with
4
and 0 + 27r (assuming that the U(1) gauge group is a
circle of circumference 21r), implies the quantization condition q E Z/2.
For a gauge theory with gauge group G, one can define similar monopole backgrounds by simply considering a U(1) subgroup of G for which one can construct a
monopole just like (2.1) [19]. For instance, if the gauge group is G = U(Nc), we can
write
A
=
H(±1 - cos 9)d#,
2
(2.2)
The expression for the monopole background is given in the dual coordinate basis {dr, dO, d}.
It is also common to present this background in the frame basis {r, 0, $}, where it takes the form
A(N) _ q
soS
and
qrsin
in the North and South charts, respectively.
48
where H is a constant N, x N, Hermitian matrix, and the two possible signs correspond
to the North and South charts, as in (2.1).
Requiring that on the overlap region
between the two charts, the expressions for the gauge field in (2.2) should differ by a
U(Nc) gauge transformation, implies e 4
= 1. Making use of the gauge symmetry,
riH
we can always rotate H to the diagonal form
H=diag{qi,q2, ... ,qNc,
with qi
q2
---
(2-3)
The condition e 4 riH = 1 implies qa E Z/2 for all a.
In general, for a gauge group G there exist monopoles of the form (2.2), with H an
element of the Lie algebra of G. Using the gauge symmetry, H can always be rotated
into the Cartan of the gauge group [19].
More explicitly, if ha (with a = 1,... ,r,
where r is the rank of the gauge group) is a basis for the Cartan subalgebra, then
H can be written as H = EIl
qaha for some set of numbers qa. This rotation does
not completely exhaust the gauge symmetry, as the Weyl group acts non-trivially on
the ha, and consequently on the qa as well. One should therefore regard as equivalent
any two sets of qa that are related by a Weyl group transformation. The collection
of numbers {qa} are called GNO charges after the authors of [19]. The GNO charges
must satisfy the quantization condition exp [47ri
>E
qaha] = 1, where "exp" is the
usual exponential map defined on the Lie algebra and valued in the gauge group. It
is equivalent to say that exp [47ri Er=1 qaha = 1 in any representation of the gauge
group, where the ha are now matrices, and "exp" is the matrix exponential.
It is worth noting that the GNO charges qa are not all conserved, or equivalently,
they do not all provide a topological characterization of the singular gauge configurations (2.2).
Indeed, there exists a much coarser classification of monopoles by
elements of the fundamental group 7ri(G) [78].
As we discussed, in order to make
sure that we have a well-defined monopole background (2.2), we need to specify a
gauge transformation (i.e. an element of the gauge group G) in the overlap region
between the North and South charts. Since the overlap region has the topology of a
circle, these gauge transformations are classified topologically by maps from a circle
49
into the gauge group, or in other words by elements of 7ri(G).
this topological charge is the same as the GNO charge q.
In the case of U(1)
When G = U(Nc), the
topological charge can be derived from the current
JItop =
(2.4)
, trF"P,
1
which is conserved provided that the field strength F,, satisfies the (non-Abelian)
Bianchi identity. It follows that in this case it is only the sum qtop =
.=,
q, that is
a conserved topological charge, as opposed to all the individual q,. In other words,
in a non-Abelian gauge theory there are several GNO monopoles (in fact, infinitely
many) belonging to the same topological class.
2.2.2
Three Dimensional Gauge Theories with Fermions
To make the discussion of monopoles more concrete, let us focus our attention on
a specific class of three-dimensional gauge theories.
The class of theories whose
monopole operators we want to study is QCD 3 with gauge group G and N fermion flavors transforming in some representation of G. These theories have a parity anomaly
if N 1 is odd [80-82], so we will restrict the following discussion to the case where N
is even. When the gauge group is G = U(Nc), the Lagrangian in Euclidean signature
is
Nc
L=42
Nf
(Fab)2 +
gYM a,b=1
N
N.
N
a,b=1
01
(#Q,(ioabO, + Aab)b,c]
(2.5)
Here, the indices a, b are fundamental color indices, and a is a flavor index, while the
two-component spinor indices on the fermions and gamma matrices are suppressed.
When N 1 is sufficiently large, this theory flows to a CFT in the infrared [77]. This
CFT can be studied by simply erasing the Yang-Mills term from the action, which by
dimensional analysis is an irrelevant operator. We can therefore write the Lagrangian
50
for this CFT as
Nf
Nc
a=1
a,b=1
+
,
CFT
A3
ab) pb,]
-
(2.6)
In the infrared theory, in the absence of the Yang-Mills term, the only role played by
the gauge field is that of a Lagrange multiplier that imposes the constraint that the
non-Abelian current vanishes, jab(X)
= 0.3
Being a conformal field theory, the fixed point (2.6) can be studied on any conformally flat space. The Lagrangian on such a space differs from (2.6) only in that the
partial derivative ,, should be replaced by the covariant derivative V,. The gauge
field components Aajb, considered as components of a one-form in the coordinate basis, remain invariant under the Weyl transformation used to map the theory on R'
to that on a different conformally flat space.
For large numbers of fermion flavors, this theory simplifies. Indeed, integrating
out the matter fields, the effective action for the gauge field takes the form
Seff[A = -Nf log det(i ab yV, + .yIAab)
(2.7)
The factor of Nf in front of the action means that we can perform a semiclassical
expansion about any saddle point of (2.7) that is also a local minimum of the effective
action, with Nf playing the role of 1/h. As in any such expansion, the typical size of
fluctuations about the saddle is of order vh, or 1/VNI in our case, as can be seen
from expanding (2.7) around the saddle point configuration and examining the term
quadratic in the fluctuations. 4
The monopole backgrounds introduced in Section 2.2.1 are rotationally symmetric
about their center and invariant under conformal inversions, as one can easily check.
These properties guarantee that they are saddle points of the effective action (2.7)
on any conformally flat space. It is not guaranteed, however, that they are all local
3
In the quantum theory, this constraint translates into the condition that (xIjablx) = 0 for all
physical states
4
IX).
Note that for a saddle point of the effective action that is not a local minimum, the fluctuations
would grow with time and eventually become large.
51
minima of the effective action, which is a fact that will become important in studying
monopoles in the quantum theory.
2.2.3
Quantum Monopole Operators
In the previous two subsections we established the existence of monopole saddles in
a class of three-dimensional gauge theories. We now explain how to define local operators with well-defined scaling dimensions that are associated with these saddles in
the infrared CFT. As we explain below, for small Nf there is a tension between defining an operator that corresponds to a classical monopole background, and defining
an operator with definite scaling dimension. In particular, the operator most easily
identified with a monopole background does not have definite scaling dimension, but
rather corresponds to a sum of such operators. At large Nf, however, this tension is
alleviated, and we can indeed associate an operator of fixed scaling dimension to a
monopole background.
Before delving into monopole operators, let us briefly review what we know about
local operators in the IR theories of interest.
The most familiar local operators
are those that can be written as gauge-invariant combinations of the fundamental
fields, such as 0 =
operators [18].
Z
Y
4
Pa.
These operators are sometimes referred to as order
In addition to order operators, one can also define local disorder
operators, which cannot be written simply in terms of the fundamental fields. Rather
than being defined as local products of fields, disorder operators can be thought of as
creating singularities for the fundamental fields. In the context of the path integral,
we can define a disorder operator inserted at a point p by integrating only over field
configurations that asymptotically approach a prescribed singular configuration in a
neighborhood of p.5
For the classical monopole backgrounds described in Section 2.2.1, the gauge field
has such a localized singularity. We can thus define a local disorder operator asso'Concretely, this procedure can be realized by cutting out a ball of radius e about the point
p, fixing the boundary conditions for fields on the surface of this ball, and only integrating over
fluctuations outside the radius E. Away from the insertion point, this regularized disorder operator
acts just like a local operator. A similar prescription in four dimensions was used in [18].
52
ciated to a monopole background by requiring that the gauge field asymptotically
approach that of the classical background near the insertion point of the operator. In
this way we can indeed associate a quantum operator with a classical monopole background. Unfortunately, the disorder operator so defined does not transform nicely
under the conformal symmetries present at the infrared fixed point. In a CFT it is
convenient to work in a basis of local operators with definite spin and scaling dimension. As we will see, the disorder operator defined above does not have a definite
scaling dimension, but rather can be written as a sum of operators with definite
scaling dimension.
To see that the disorder operator cannot in general have a well-defined scaling
dimension, it is convenient to change perspectives from operators on R 3 to states on
S 2 x R by using the state-operator correspondence.
In a CFT, the state-operator
correspondence maps operators inserted at the origin of R 3 to normalizable states on
S2 x R. The R coordinate r is interpreted as Euclidean time and is related to the
radial coordinate r on R 3 through r = er. The scaling dimension of an operator on R3
is identified with the energy of the corresponding state on S2. Restricting to disorder
operators, the correspondence identifies the disorder operator defined by boundary
conditions at a point in R3 to the state on S2 x R given by a wave-functional on field
space with delta-function support on the classical field configuration at r = -oo.6
This state is not an energy eigenstate, but rather a superposition of energy eigenstates.
The wave function, which is localized about the classical configuration at
early times, spreads out at late times. The corresponding operator on R 3 is therefore
a sum of operators with definite scaling dimension. In a generic theory, there is no
principle that singles out any one operator in this sum, and correspondingly there is
a significant distinction between a disorder operator defined by boundary conditions
and an operator of definite scaling dimension.
At large Nf, however, the situation is better. The monopole background is a
classical saddle, and for large Nf the gauge fluctuations are suppressed. If the sad6
If we regularize the disorder operator by smearing it over a sphere of radius E, as in footnote 5, the
wave-functional would have delta-function support on the classical field configuration at r = log e.
53
dle is stable, the state corresponding to the disorder operator is close to an energy
eigenstate. It is the operator of definite scaling dimension corresponding to this energy eigenstate that we refer to as the monopole operator for the remainder of this
chapter. 7
In the next section we explain how to use the path integral on S2 x R to calculate the energy of eigenstates associated with stable saddles, and in Section 2.5 we
determine which saddles are stable.
2.3
Free energy on S 2 x R
In the previous two sections we discussed at some length the precise definition of
monopole operators at the infrared conformal fixed point of QCD 3 with many flavors
of fermions. In summary, for the GNO backgrounds (2.2) on S2 x R that are local
minima of the gauge field effective action, and only for those backgrounds, there
exist several degenerate lowest-energy states whose wavefunctions are highly peaked
around the saddle (2.2); it is these states that, via the state-operator correspondence,
get mapped to the bare monopole operators on R3 whose properties we want to
study. We will refer to these states on S2 x R as "ground states" in the presence
of the monopole flux (2.2).
The use of the term "ground states" can be justified
only in large NJ perturbation theory, where one can define a Fock space of states for
every stable GNO configuration, and these Fock spaces mix only non-perturbatively
in 1/Nf.
One aspect of the state-operator correspondence is that the scaling dimensions of
operators on R 3 are equal to the energies of the corresponding states on S2 x R. In
particular, the scaling dimension A of a bare monopole operator equals the ground
7
1n the U(1) case each operator in the decomposition of the disorder operator must carry the
same topological charge. As such, even away from large Nf, there is a natural operator with definite
scaling dimension to associate with a monopole background, namely the operator corresponding
to the lowest energy state with the given topological charge. As discussed in Section 2.2.1, for
non-Abelian theories there are many classical backgrounds with the same topological charge, and
so one would be able to identify only one monopole operator per topological class this way. In
supersymmetric theories, however, it may be possible to identify BPS monopole operators with
certain GNO backgrounds after performing a Q-exact deformation of the theory to weak coupling
(see, for example, [55,65]).
54
state energy in the presence of some constant GNO flux through the two-sphere.
The goal of this section is to exploit this equality in order to calculate the scaling
dimensions A. In later sections, we will calculate the other quantum numbers of the
bare monopole operators.
The ground state energy in the presence of some constant GNO flux can in turn
be calculated by performing the path integral on S 2 x R. More explicitly, we have
A = - log Z[A]
= F[A],
(2.8)
where Z[A] is the Euclidean partition function on S2 x R in the presence of the
background A, and F[A] is the corresponding ground state energy (or free energy). 8
In principle, the quantity log Z[A] should be understood as the limit
log Z[A] = lim - log Z6[A],
(2.9)
where Z[A] is a similar partition function on S2 x S 1 calculated after first compactifying the R direction into a circle of circumference 3. In practice, as we will see, it is
easy to isolate the leading term in the large
3 expansion of log Z,3 [A] while working
directly on S2 x R.
Our procedure for calculating F[A] consists of three steps:
1. We perform the path integral over the matter fields. Integrating out the matter
fields generates a gauge-invariant effective action for the gauge fluctuations and
leads to a sensible 1/Nf expansion.
2. We fix the gauge by introducing Faddeev-Popov ghosts.
3. We evaluate the path integral over the gauge fluctuations and the ghosts at
next-to-leading order in 1/Nf.
We now provide an explanation of this procedure, while the next section is devoted
to the details of the calculation.
8
The expression (2.8) should be taken to include only perturbative contributions in the 1/N
expansion. All non-perturbative contributions in 1/Nf should not be taken into account.
55
2.3.1
Setup
As discussed previously, the IR conformally-invariant action for QCD with U(Nc)
gauge group and N 1 flavors of complex two-component fermions is
Nf
S
d
N.
VfH-
[OT',yp
abV + A al+ al,)4b,a]
,
(2.10)
a=1 a,b=1
where, in anticipation of having to study this theory in the presence of a background
monopole flux, we split the non-Abelian gauge field Aab into a sum between a background Aab and fluctuations aab. Here, the indices a, b are color indices, and a is a
flavor index. The spinor indices on the fermions and gamma matrices are suppressed.
The action (2.10) can be more compactly written as
S[A; a,44 , 0]
=
So[A; 0t,4] +
d3 X V
Z
ab
(211
a,b
where So[A; 4t,40] is the action (2.11) with gauge fluctuations set to zero, and
3a=
Z:
?4,c74,,a:
is the non-Abelian covariantly conserved current.
(2.12)
Double colons stand for normal
ordering. As in any gauge theory, we can write the partition function as
Z [A]
=
V 1
Vol (G)
Da Dt
DV) exp [-S [A; a, Vt, V],
(2.13)
with Vol(G) being the volume of the group of gauge transformations, which we need
to divide by because we do not want to count gauge-equivalent configurations multiple
times.
Let us set up our conventions for this calculation. We write the standard line
56
element on R' in spherical coordinates as
s3= d
e2 [d
2
+ d92 + sin 2
2
= e (dq
sin 9 cos # sin 0sin 4 cos 0)
]
(2.14)
We want to calculate the partition function on
x R. The metric on S2 x R is
S2
obtained by rescaling the R3 metric (2.14) by e- 2r:
dsS2xR - dO2 + sin2 6 dq2 + dr 2
(2.15)
.
Recall that the dynamics of a CFT is insensitive to such a rescaling. We will be
doing calculations with spinors on the curved space S2 x R, hence we need to specify
a frame ei. We obtain the frame by the conformal transformation of the standard
frame ei = dxi on R3
(2.16)
e = e-dx
iy= o-&, where the a are the Pauli matrices.
We choose the set of gamma matrices
All subsequent formulae for spinors are understood to follow these conventions. A
point on S 2 x R will be denoted by x = (r,9, #). Sometimes we will also use the
decomposition x = (r,ft), where fi is a unit vector pointing to a point on S 2 . The
covariant derivative on S2 x R will be denoted by V,. The gauge covariant derivative for a fundamental fermion 0' and current jab (which transforms in the adjoint
representation of U(Nc)) is
N.
[D,)]a =
D Ab=
(DA)] ab
where [A,,
j=]a - Zl
1
V (A/ Jcb
=
(v,
jab
Vjab
-
iab
i
1
-
i
)
b
O b 1(2
1 7)
[A,, j]ab
is the matrix commutator.
c AA)
As explained in the previous sections, for a U(Nc) gauge group, the most general
57
monopole background can be taken to be
A"a
=
diag{qi, q2 ,... qNJ}U(1)
AU(1)
with q
q2
2.3.2
Gauge Field Effective Action
--
(1 cos 0)do
(-I - cos 0)do
if 9
(2.18)
7r,
if 0 4 0,
qN, and qa E Z/2.
The first step in our general procedure for evaluating the ground state energy F[A] is
to integrate out the matter fields. Doing so yields a gauge-invariant effective action
for the gauge field fluctuations. The gauge effective action is defined in such a way
that the partition function is simply
Z[A] = Vol(G)
Da exp -Seff[a]].
(2.19)
Comparing with (2.13), using the decomposition of the action in (2.11), and expanding
in powers of a, one can write
Seff[a]
=
-log Zo[A] +
.
n=1
ni
K(Inz
d
(
/7VYagb(x) ()j(x)
,
COnnI
a
(2.20)
where the correlators on the right-hand side are evaluated using the action of free
fermions in the background A, namely the action So[A; V), Of] introduced above. In
other words, the effective action Seff [a] is the generating functional of connected correlators of the current operator jP in this theory of free fermions. The quantity Zo [A]
appearing in (2.20) is the partition function associated with So[A;
4P, Ot];
it is just a
Gaussian integral, which evaluates to
ZO[A] =
D'tDoexp[-So[A; , t]] = (det(iO")))N"
58
(2.21)
Here, the subscript (A) denotes a background gauge covariant derivative, as in (2.17)
with A replaced by A.
For us, the GNO monopole background (2.18) is static as well as invariant under
rotations and time reversal, and therefore the one-point function of the current operator must vanish, (j"(x)) = 0 (see also the last paragraph of Section 2.2.2). Therefore,
the term linear in a in (2.20) vanishes. In general, the term quadratic in the a does
not vanish, and its coefficient is given by the current-current correlator
Kp'cd(xx')
(2.22)
'
This current-current correlator should be thought of as an integration kernel that
defines an operator on the space of square-integrable one-forms on S 2 x R.
The kernel Kf"c(x, x') can be written more explicitly in terms of a quantity
Gq(x, x'), which can be identified with the Green's function of a single fermion in an
Abelian gauge theory in the presence of a monopole background (2.1), namely
G,(xx') = (*(x)pt(x')) =
x
U
.
(2.23)
(This theory would be described by the action (2.10) with Nc = Nf = 1 and a = 0.)
Indeed, substituting the normal-ordered expression (2.12) into (2.22), and noticing that the contractions between the fermions take the form
6
a,56 abGqa(X,
(/,a(X)>4,(X'))
=
x'), one obtains
K11c(x, X') = N 6 &c6 adKq(x, x),
(2.24)
with
s in
With t(x,e x')
wec(ax
tr (hand
'Gi. (x, x'))a.
(2.25)
With the expression (2.24) in hand, we can write the effective action for the gauge
59
field fluctuations as
Seff [a] = Nf tr log(i;
A)) +
Jdx
d d3 x' g(X) g(x')aabt(x)Kq
x,.)a.b(X/)
(2.26)
The ellipses denote terms with higher powers of a, which one can easily show are also
proportional to Nf. We are now in business. That the quadratic part of the action is
proportional to Nf means that the typical gauge field fluctuations are a oc 1/V/N-,
and we can calculate Z[A] (see (2.19)) approximately at large Nf using a saddle point
approximation.
We can now try to perform the integral over the gauge fluctuations by keeping
only the terms up to quadratic order in a in Seff[a], and write down the free energy
on S2 x R as:
1 " tr log K" + O(l1/N).
F[A] = -Nf tr log(i(A)
2
(2.27)
The quotation marks are meant to emphasize that the O(NO) term is only rough,
because we ignored the issue of gauge invariance when we performed the integral over
the gauge fluctuations. To obtain a more explicit answer, we now proceed to a more
careful analysis of gauge invariance.
Just like the original action (2.10), the effective action Seff[a] is invariant under
gauge transformations that in the gauge sector act as
(Al + a,) -+ i U&iIU + U(A + a,,)Ut.
(2.28)
Correspondingly, the integrand in (2.19) has flat directions corresponding to these
gauge transformations. Therefore, one cannot simply identify the functional integral
f Da exp[-aKa] with the determinant of the kernel K, as this kernel has many
eigenvalues that vanish.
60
It is most convenient to work in background field gauge by imposing the condition:
(2.29)
D(A)all = 0.
This condition distinguishes one gauge configuration in every gauge-equivalence class,
so if we restrict our integral over a to configurations that satisfy (2.29) then the
integrand e-sff[a] will no longer have any flat directions. The condition (2.29) does
not exhaust, however, the group of all possible gauge transformations, because there
are residual gauge transformations that leave a completely untouched. These residual
gauge transformations form the isotropy group H(A). The Faddeev-Popov trick is to
insert
1 = Det' (-D(A)D(A+a)p) X V
A
~
J
Vol (H(A)) jA
DU 6 [D(A)aU,1]
,
(2.30)
into the path integral, where Det' denotes the functional determinant with zero modes
omitted, and af = iUaUt + U(A, + a)Ut - A, is the gauge transformed a,.
Changing variables9 a -+ au in the partition function (2.19), inserting (2.30), and
then renaming au -4 a gives:
[A]
Vol (H(A))
Da e-sff [a]-sFPJI
[DA)a]
Det' (-D
)D(A)A
,
(2.31)
with
SFp[a] = -
Tr'log (-D
A+a)D(A+a),)
,
(2.32)
where Tr' is a trace over the non-zero modes. The factor in the parenthesis in (2.31)
multiplying the delta-function is precisely the inverse of the Jacobian factor that one
obtains when taking D (A) outside of the delta-function. We can therefore perform
the path integral (2.31) by integrating only over configurations that satisfy (2.29)
9
Note that both the measure and the action Seff[a] are invariant under this change of variables.
61
(and thus removing by hand all the flat directions), provided that we supplement
the effective action Seff[a] by a term SFp[a] exhibited in (2.32) that comes from the
Faddeev-Popov procedure. Just like Seff[a], SFP[a] can be expanded in powers of a:
SFp[a] = -
1
Tr' log (-D (A)D(A)p) + O(a2 ).
(2.33)
The term linear in a in this expansion vanishes by an argument based on the symmetries of the background (2.18) similar to the one that showed that the linear term in
a in (2.26) vanished.
Before presenting the answer for the free energy, we note that the factor of
1/Vol(H(A)) should be ignored. This factor would be relevant if we computed the
partition function on S 2 x
S1,
where the S' circle has circumference 3, because in this
case every generator of HWA) would contribute a factor proportional to 3 to Vol(HWA)).
Thus, log Za[A] would receive a contribution proportional to log 3 from every generator of HA). However, these contributions disappear when we consider the limit in
(2.9).
Evaluating (2.31) in the saddle point approximation, we have
F[A] = N1 Fo[A] + SF[A] + 0(1/N1 ),
(2.34)
JF[A] = FFP[A] + Fgauge[A] ,
where Fo[A] is the fermion determinant
Fo[A] =
-
T log(j$
)
(2.35)
.
6F[A] denotes the subleading term in the free energy.
It is a sum of two terms,
namely FFP[A], which is the Faddeev-Popov determinant
FFP[A] =
-
1
Tr'log (-D()D
2 Tr' lo
62
,
7(ADA
(2.36)
and Fauge[A], which is the gauge fluctuation determinant
Fgauge[A]
=
1
-Tr'log K
2
.
(2.37)
The fermion, Faddeev-Popov, and gauge field determinants will be calculated in
Sections 2.4.1, 2.4.2, and 2.4.3, respectively.
2.4
2.4.1
Functional determinants
The fermion determinant
We now start by calculating more explicitly the leading term in (2.34), the fermion
determinant Fo[A] defined in (2.35). This term arises from evaluating the partition
function Zo[A] = e-N
Fo[A]
of non-interacting fermions in the background (2.18). See
(2.21).
Examining the action So[A;
4, V)/] more closely, we see that because we have taken
the monopole background (2.18) to be diagonal in the color indices, it is not only the
fermions of different flavor that decouple from one another, but also those of different
color.
Each fermion
4',
is only coupled to an Abelian monopole background of
charge qa. Because the fermions are non-interacting, the ground state energy can be
written as
N.
Fo(qa),
FO[A =
(2.38)
a=1
where F(q) denotes the ground state energy of a single fermion in an Abelian monopole
background of charge q. The quantity F(q) = - log Z(q) can be computed from the
partition function corresponding to this free fermion,
Z(q) =
DOD4fes(V
),
S(q) =
Jds
Vg(x)Oty(iV, + qAN'))V.
(2.39)
This computation was performed in [13,21] as part of studying the scaling dimensions
63
of monopole operators in U(1) gauge theory with N fermion flavors. We now review
this computation briefly, partly because such a review will keep our presentation selfcontained, and partly because in doing so we will also introduce some notation that
will become useful in the following sections.
In order to evaluate the integral in (2.39), we should first decompose the fermion
field 0 into a suitable basis of spinor fields.
Since translations in the Euclidean
time direction are a symmetry of the action, it is convenient to consider modes with
harmonic time dependence,
4
oc e-w.
Finding a basis for the angular dependence
of 0 requires more thought. If 0 were instead a complex scalar experiencing the
same monopole flux qAU(l), one could use the basis of monopole harmonics Yq,em" (i),
which were defined in [16,83] (see also Appendix A.1) as eigenfunctions of the gaugecovariant Laplacian on S 2 . Here, f >
IqI
is the angular momentum, and mt ranges
from -f through f. To find a complete basis for a field of a different spin s, we can
work in the frame (2.16) obtained by conformal transformation from R3 and expand
every component of the spin s field in terms of the monopole harmonics Yq,tm(h).
More conveniently, we can use the usual angular momentum addition rules to work
in a basis where the quantum numbers are {j,m, e, s},
j being the total angular
momentum and m the eigenvalue of J 3 . For spinor fields where s = 1/2, we have
j =f
- 1/2 or j = f + 1/2, and we can define
=
Yqj-
-1
Tq,jm
,
+
2|+
S2
1
,
2)
(2.40)
j+1-m
Sq,jm =
-q3j 2(j+1)
.
q,j+I,m-.
,
+1+m
2(j+1)
y
jq9
=2--2
1
q,j+!,m+
where Yq,em are the scalar monopole harmonics. Note that for total angular momentum j =
jqj -
1/2 we have only the Sq,jm harmonics, which in this case have orbital
angular momentum f =
Iqi, while
for larger
64
j
we have both Sq,jm, with orbital angular
j+1/2, and Tq,jm,
the Sq,m start at j = 1/2.
momentum f =
q = 0,
with orbital angular momentum f = j - 1/2. For
Expanding the fermion O(x) in the basis (2.40),
t
Z
O(x) =
+
(qj'52(w)Sjm()
T'-(w)Tqjm(f)) e-iwT ,
(2.41)
j 2Jqj-1/2 M=-j
M (w), one finds that the action S(q)
with anti-commuting coefficients 'IV (w) and
defined above can be written in almost diagonal form, because the gauge-covariant
Dirac operator only mixes the modes Sqjm and T,jm with the same
j
and m [13,21]:
C3m
t
(
''
>1
(2.42)
)JM
Here, Gqj(w) is a matrix that can be identified with the inverse propagator. When
j
> Iql - 1/2, dj
(w) is a 2 x 2 matrix given by [13,21]
qw_
- +
Gq,j(w)
=
-q2
2
-z i+
\
bj)Fj+
TI
\
ly;
2
+2
(2.43)
when
j
= |q| - 1/2, Tq,jm does not exist, and Gq,j(w) should be thought of as a 1 x 1
matrix equal to the bottom-right entry of (2.43), namely
q5j(w) = qw/
qi.
The path integral (2.39) becomes a Gaussian integral over the Grassmannian
coefficients V
Fo(q)=-
and ' T). Performing this integral yields
dw
2-7
(2j+1) log
W+
+
-q2
-2
2ql(log4,
j=lq\+i
(2.44)
65
where the first term corresponds to
the contribution from j =
tqI
j
> jqj - 1/2 while the second term represents
- 1/2. As one can easily check, the arguments of the
logarithms are nothing but det Gq,j (w) , with Gq,j(w) defined in (2.43). The prefactors of the logarithms in (2.44) come from summing over the allowed values of
m.
The integrals over w are divergent, but they can be regularized by analytic continuation,
J
27r
J
log (w2 + b2)
&
27rds
(W2 + b2 ) 8 S=o
=
IbI.
(2.45)
Using this identity to define the regularized expression, the free energy reduces to
/
00
Fo(q) = - E
\
(2j + 1) (j +
-q2.
(2.46)
i=lql-'
This sum is still divergent and can be regularized by various methods, such as by the
Abel-Plana summation formula as in [13] or by zeta-function regularization as in [79].
The result can be written in terms of an absolutely convergent sum and the Hurwitz
_'(n + a)-
zeta function ((s, a) =
Fo(q)=-
(2j + 1)
as
(j
2
- q2 -
(2j + 1)2 + q2
j=lql-'
(2.47)
q(2q - 1)(q + 2)
6
In Section 2.6 we will tabulate this sum for a few values of q; see Table 2.1. Knowing
F(q), one can easily calculate the leading term in the large Nf expansion of the ground
state energy in the presence of our GNO background using (2.38).
Now that we have a handle on the leading order computation let us move on to
the next-to-leading order contribution.
66
2.4.2
The Faddeev-Popov determinant
The next-to-leading order computation of the free energy has two contributions given
by the second and third terms in (2.34). The second term represents the contribution
from the Faddeev-Popov ghosts, while the third term comes from the determinant of
the gauge field fluctuations. Of these two contributions, the Faddeev-Popov one is
considerably simpler, because it involves the determinant of a local operator, and we
will discuss it first.
The Faddeev-Popov contribution to the ground state energy,
1
FFP
can be written as -log
ZFP [A],
log Det' (-D (A)D(A),p) ,
(2.48)
where ZFP [A] is the partition function for an anti-
commuting scalar ghost field c valued in the Lie algebra of the gauge group:
-Sghost
ZFP[A] =
Dc
~
IsghostSghost
1
=
2
/d
l1
3XC
ab
[
01A ab
g(x)3
2
a,b=1
(2.49)
Evaluated in the GNO monopole background (2.18), the ghost action becomes
N,
Sghost
d3 X
[
g(X)
-Z(qa
-
qb)Auj)] Cab
2
(2.50)
a,b=1
The interpretation of this formula is that the diagonal components caa are free real
scalar fields, while the off-diagonal components cab, a :
b, whose complex conju-
gates are Cba = Cab*, are free complex scalar fields experiencing an Abelian monopole
background (qa - qb)AU(l).
To diagonalize the action (2.50) we should expand the ghost fields ca in terms of
the monopole harmonics YQ,jm introduced in the previous section, with
67
Q
= qa - qb.
Explicitly, writing
C '()Yqa -q,JM(h)e
cab(X) =
W
(2.51)
2J,M
and using the fact that the monopole spherical harmonics YQ,JM have eigenvalue
J(J + 1)
-
Q2 under the gauge-covariant Laplacian [16,83] on S 2 , we can put the
ghost action in the form
d
Sghost1
a,b=1
I C (Q)12
[Q 2
J(J+ 1) -(qa
-qb)
2
(2.52)
]
J,M
Note that here the sum over J runs only from j9a - qbI to infinity, and the sum over
M runs from -J to J, as appropriate for the spin-J representation of SU(2).
The contribution to the free energy can now be computed by integrating over the
Grassmannian coefficients CjM in (2.49). Because the ghost fields Cab do not mix,
the result will be a sum
N.
FFP [A] = E
FFp (qa,
qb)
a,b=1
(2J + 1) log
FFp(q, q
(J + 1) - Q2
+2
Q=-q-q'
J=IQI
(2.53)
We will postpone evaluating this expression until after combining it with the contribution coming from the gauge field fluctuations in Subsection 2.4.4.
68
2.4.3
The gauge fluctuations determinant
We now turn our attention to the third term in (2.34), which is also the hardest to
compute. 10 Combining (2.37) and (2.24), we can write this term as
Nc
Fgauge[A] =
E Fgauge(qa, qb)
(2.54)
a,b=1
1
Fgauge(q, q') = Tr' log ]Cqqf
2qq
where we recall that the quantity Kqq can be written in terms of the Green's function
Gq(x, x') of a single fermion in an Abelian monopole background with charge q as
K",(x, x') = -r(y"Gq(x,
')y'Gt,(x, x')).
(2.55)
Therefore, in order to evaluate (2.54), we should first write down an explicit expression
for Gq(x, x'), and then describe how to use it to construct Cqq, and find its eigenvalues.
In evaluating (2.54), we will not be able to find simple analytical formulae such
as (2.47) or (2.53), and instead we will have to resort to numerics. In the rest of this
section we aim to provide enough details on the steps one has to take in implementing
these numerics, and the cross-checks that can be performed. We will postpone the
numerical results until Section 2.6.
Green's Functions
The expression for the Green's function Gq(x, y) of a single fermion in a charge q
Abelian monopole background can be read off from Fourier transforming back to
position space the inverse Gq,j(w)
=
Gq,j(w)-' of the expression in (2.43). It is not
OWe develop a method slightly different from the calculation for N, = 1, IqI = 1/2 done in [211.
That approach, although less straightforward, has the advantage of producing simpler formulae than
the approach of this chapter. However, it seems hard to generalize the method of [21] to the present
case.
1
69
hard to check that this inverse is
Gq')
Gqj (w)
w 2 + (j + 1/2)2-
q2 '
with Gq,j(w) as in (2.43). The quantity Gq,j(w) is a 2 x 2 matrix if j > jqj - 1/2.
When
j
=
q-
1/2, it is a 1 x 1 matrix equal to q/(Iq Iw). The Green's function is
then
Gq(x, X')
0 (X')
=(X)
Tt
(W')
(2.57)
(T,jm(h) Sqjm(fl)) Gqj(w)e-i-'T')
= -
where it should be understood that when
j
= qj - 1/2 and Gq,j(w) is a 1 x 1 matrix,
we should only consider the Sq,jm modes.
The Green's function (2.57) is a 2 x 2 matrix whose indices are the spinor indices
that we have been consistently suppressing. By combining (2.57) with the explicit
expressions for the monopole harmonics (2.40), one can see that each entry in Gq(x, X')
can be written as a sum over products of two monopole spherical harmonics:
'~ (x, x')
=
~
Z
gM;,11(T
-
T1
(2.58)~0Yt/
j,M~efEj-j'
with coefficients g him-,e(T
-
') that can be easily worked out by integrating with
respect to w in (2.57). (In (2.58), mi = 1/2 or -1/2 if i = 1 or i = 2, and similarly
for Mk.)
The details of expression (2.58) are crucial for calculating the numerical value
of the monopole scaling dimensions, but not so essential if one is only concerned
with understanding the general structure of the calculation. For understanding the
70
structure of the calculation, we can write down (2.58) schematically as
00
G,(x, x') =
(2 x 2 matrix oc Y... (h)Y(*...(f'))
,
(2.59)
j=JqJ-1/2
where we emphasized that Gq is a 2 x 2 matrix (because it has spin indices). Each
entry can be written as a single infinite sum of products between a monopole harmonic
with charge q at A and the conjugate of a monopole harmonic with charge q at h'.
There are additional finite sums in (2.59) that have not been indicated.
The careful reader may notice that the position space Green's function Gq (x, X') is
not unique, because one needs to specify a pole-passing prescription in performing the
w integral in (2.57). While usually in Euclidean signature the poles of the propagator
are off the real axis and the Fourier transform provides a well-defined position-space
Green's function, in our case we have zero-energy modes that generate a pole for the
propagator on the real axis. We choose the prescription for passing around this pole
given by principal value integration, whereby
J
d
-
27r
-
sgn(r - r') .
--
W
2
(2.60)
This prescription respects CP invariance of the monopole vacuum, in which the
Green's function is an expectation value.
The kernel IC
Now that we have an expression for the Green's function, it is a straightforward
matter to write down the kernel K
(x, y) = -Tr(-yG(x, y)'"G ,(x, y)) and compute
its eigenvalues. Since the Green's function is a sum over products of two monopole
harmonics, the kernel
00
Cqq, (x,
x')
/Cqq,
is a sum over products of four spherical harmonics
00
(3 x 3 matrix oc Y,... (h)Yq,,... (ft')Y *...(f')Y* (i)),
=
j=JqJ--! j'=\q'J-6
(2.61)
71
where we emphasize that this kernel can be written as a 3 x 3 matrix (since the tangent
indices p and v run over three values), and that each entry of this matrix contains two
infinite sums of products involving four spherical harmonics. It is straight-forward,
but tedious, to work out the precise form of
ACqq,
(x, x') given the precise form of the
Gq(x, X').
The object K",(x, x') should be thought of as an integration kernel that acts on
a space of vector fields aO(x) on S 2 x R by
[Kqqa]"(x) =
x')a,(x').
d x'v -g(x,)t(x,
(2.62)
Actually, the expansion (2.61) reveals that the a"(x) must not be regular vector
fields on
S2
x R, but rather sections of a more complicated vector bundle. Indeed, if
we pass from the North chart where AU(1) = (1 - cos 9)d# to the South chart where
A(s
= (-1-cos O)d#, a scalar monopole harmonic Yqfm picks up a phase, Y()
=
Y,(Nh)e-21qO, as appropriate for how a field with charge q should transform under a
gauge transformation A
= A
+ dA, with A = -24.
KYv(S)
qqtql (X, X,) = KYV(N)
Xe(.3
Consequently, we have
(2.63)
i
Imposing the condition that both aP(x) and [Cqqa]1'(x) transform in the same way
when passing from the North to South chart (because otherwise Kqq, would not be a
well-defined operator), we see that (2.62) implies
ap(s)(x) = a (N)
In other words, a"(x) carries charge
Q
2
- iQ,
Q
/q-
(2.64)
under AU).
That Kqq, (x, x') acts on vector fields carrying charge
Q
could have been anticipated
from the form of the effective action (2.26). Indeed, in (2.26) we see that Kqq, (with
q = q. and q' = q.) is multiplied on the right by the a' b components of the gauge
field fluctuations. In the background AUM), these components carry precisely charge
72
Q
= q.- q=
q
-
q, as can be seen by an argument similar to the one in Section 2.4.2
that showed that the ghost fields cab carry charge
Q
as well. We will call the off-
diagonal components a ab W bosons in the following.
We are interested in finding the eigenvalues of the kernel Aqq,. To do so, we should
make use of translational and rotational symmetry. Translational symmetry in the
Euclidean time direction means that if we expand aP(x) in Fourier modes, the kernel
Kqq,
will not mix modes with different frequencies. Similarly, rotational symmetry
along the S 2 directions implies that
ICqq,
only mixes modes that transform in the
same representation of SU( 2 ),ot. As per (2.64) above, a good basis for the angular
dependence of a"(x) is given by the vector monopole harmonics with charge
Q.
We
saw in the spinor case that we can construct spinor monopole harmonics from scalar
harmonics with any given charge Q. A similar construction can be performed for
the vector monopole harmonics [84] (see also Appendix A.1).
While in the spinor
case we had two independent sets of spinor harmonics, Tq,jm and Sq,jm with orbital
angular momentum t = j + 1/2 and
j-
1/2, respectively, we now have three sets of
vector harmonics with total angular momentum quantum numbers (J,M) and orbital
angular momentum L:
UQJM(f),
L = J+l,
J >ifQ
JJQJ ,
VQ',M (,) ,
L = J,
J>
IQ,
1,
L=J-1,
WQJM(ft),
if JQJ = 0 or 1/2
if Q >0,
if QI=0,
(2.65)
J IQI+-1.
That the orbital angular momentum is L means that in a frame basis one can write
down the components of the harmonics (2.65) in terms of scalar monopole harmonics
YQ,LML, where ML E {M - 1, M, M + 1}. Since we must always have L > IQ1, we
obtain the allowed ranges in (2.65). Note that the harmonics with J =
defined only when
IQ|
IQI
- 1 are
> 1; these harmonics will play an important role shortly.
73
We can thus expand a in terms of vector monopole harmonics (2.65) and Fourier
modes in r:
>
a(x) =
aM(Q)UQJM(h) + ajm(Q)VQ,jm(h) + ajm(Q)WQ,JM(h)] e-QT
Jm
(2.66)
with coefficients aum(Q), ajm(Q)) and ajm(Q). Then the operator
ACqq,
is almost
diagonal and mixes together only modes with the same J, M, and Q:
dx d x'
g(x)Vg(x')a,(x)*C"(x, x')av(x')
t
aim(Q)
dQ
E ajm (Q)
aim(Q)
Kj(2.67)
Kq A,(i)
M(Q)
J,M
Gm(p)
aw(tiw
When J ;>
IQ
.
JMy(Q)j
+ 1 there are three such modes for each M, ICqq, acts in this 3-
dimensional space as the 3 x 3 matrix K ,(Q). When J =
IQ,
the modes correspond-
ing to WQ,JM are absent, and K ,(Q) is a 2 x 2 matrix. Lastly, when J =
IQI -1,
both
VQ,JM and WQ,JM are absent, and K ,(Q) is a 1 x 1 matrix. Because of rotational
invariance, K ,(Q) is independent of M.
From (2.67) it is not hard to extract an inversion formula for the components of
the matrix K ,(Q):
[K q(Q)]xy2IicrJ(Q - Q')
=
d x 3d x''g(x))
(x')XQ",JM(f)*K ,4v(xx')Ym (')eirnr',
(2.68)
where X, Y E {U, V, W} denote the indices of K 1 ,(Q).
This expression is rather
unwieldy, especially after plugging in the explicit formula (2.61) for the kernel KIqq,
74
which yields, schematically,
Kq ,(Q)
=
J
Cdd...(Q)Y ,....()Y* ...(O)Y4,....(")Y*...(')Yq,,....(f)YQ,...(
.i
3,
(2.69)
This formula for K ,(Q) involves two angular integrals over a product of six monopole
spherical harmonics, two infinite sums exhibited explicitly in (2.69), as well as several
finite sums that were omitted.
We are not discouraged and still determined to evaluate (2.69) as efficiently as
we can. We can simplify (2.69) by using rotational invariance, which, as mentioned
above, implies that K ,(Q) is independent of M. So we might as well compute
K q,(Q) after averaging over M. Writing (2.68) as
[(K ,(Q)]xy27r6(l
=
-2J
J 1
+1
(
-
Q')
d3 x d3 x' /g7Y
JV..4
M=-Jf
g(x'Xj(f)*K,,,,x,
1I
IY,y(ftei,
1IIIX'IQ4JV\
\I4,i
(2.70)
and plugging in the explicit form of Cqq, we again obtain an expression of the schematic
form in (2.69). This time, however, because we averaged over M, this expression is
rotationally-invariant and the integrand depends only on the relative angle between
ft and ft'. The integral with respect to ft is therefore independent of ft', so we can
choose f' to point in the
direction and replace the integral with respect to ft' by a
factor of 47r. Using
Yq,im(s)
=
3
q,-m
+
4ir
we get rid of three of those pesky monopole harmonics in (2.69).
(2.71)
The remaining
angular integral over the product of three harmonics can be evaluated using some
75
properties of monopole harmonics [83].
(2.72)
Yq,,m(h)* = (-1)q+mY-,,,-m(h)
and
fdn Y,tm(h)Y,,,/M1(h)Yq/f'PM//(h)
21 + 1)(2' + 1)(21" + 1)
4ir
V
q' q ")
m
(2.73)
where
is the Wigner 3j symbol. After using these identities, K,, (Q)
can be put in the form:
+
K ,()
=ZZ
3'+i
2
1(Q)
S 2,...(Q)
2'
L
-q'
-Q
(2
E'
L
-q + 5q q' - Sq Q
(2.74)
where L = J - 1, J, or J + 1 (depending on which component of K ,(Q) we are
computing), and where the coefficients C 33... (Q) have fairly complicated expressions
that will not be reproduced here. The lesson to be learned is that one can write
K 1 ,(Q) explicitly in terms of two potentially infinite sums (over j and
finite sums over products of 3j symbols. In fact, for fixed
j
j') and several
and J the sum over
j'
is
finite, because the 3j symbol vanishes if its arguments are not triangular. There is
therefore only a single infinite sum over j in the expression for Kq ,(Q), which can be
evaluated using zeta-function regularization. The terms in the infinite sum are shown
in an example in Figure 2-1 for one matrix element.
76
0.00
%
0
0.01
*
q
1, q'
S1, J
1 2
352
0.02
0.03
10
20
30
40
Figure 2-1: We plot the terms in the infinite sum over j (2.74) that give the matrix
element [K4J,(Q)]G for q = -1, q' = 1/2, Q = 1, and J = 35/2. We show the
stage of the calculation where all the finite sums (over Jq, 1', 1, and j') in (2.74) have
been done and only the infinite sum over j remains. The dots represent the actual
terms in the sum, while the solid line is the asymptotic expansion of the summand
to Q(1/j 8 ) that we determined analytically. We perform the infinite sum by zetafunction regularization of the asymptotic form for j > jc, where jc is the value below
which we use the numerical values of the terms in the sum. We check the numerical
precision by changing j, and we reach our goal of 10-1 precision by choosing jc ~ 40.
This precision is needed to get the free energy with 10-3 precision.
Properties of Kj, (Q)
On general grounds, the matrix K ,(Q) should satisfy certain properties that can be
used as checks on the explicit formulae (2.74). For instance, K 1 ,(0) is Hermitian,
K't,(Q) = K,,(Q), and due to invariance of the monopole background under CP, one
can show that
[K ,q(Q)]xy = [K'q,,q_,(Q)]xy = (-1)--[K!,(Q)]xy.
(2.75)
Here, the range of X, Y E {U, V, W} depends on whether K q,(Q) is a 3 x 3, 2 x 2, or
1 x 1 matrix.
It follows from gauge invariance that K ,(Q) has a zero eigenvalue."
To leading
"When J = IQI - 1 KJ,(Q) is a 1 x 1 matrix that does not vanish. For J = IQ it is a 2 x 2 and
for J > IQI +1 it is a 3 x 3 matrix with one zero eigenvalue. In the following we assume J > IQI +1.
77
order in the large Nf expansion the current conservation equation takes the form:
0 = D A)ji
alX - i [A,
=
i(qb - qa) AU()] ja,
= [-
(2.76)
where we dropped terms proportional to the gauge fluctuation a and used that the
monopole background is diagonal in the gauge indices.
defined in (2.22) is a current two point function.
current in K,"
The gauge kernel, K"',
Applying (2.76) to the second
we get
o=
+ i(q
-
q')A 71)]
",(x,x')
(2.77)
where we used (2.24). Note that in this Ward identity the delta function is absent, as
follows from Lorentz covariance and dimensional analysis. From (2.77) we determine
the eigenvector with zero eigenvalue of K , (Q). Indeed, inverting (2.68) we obtain:
K
(x,x') =
(
d2
Q
XiJM(h) [K ,, ()]xy
Y 5JMfih)
JM X,YE{U,V,W}
(2.78)
and then acting with the derivative in (2.77), we obtain:
o=
(
[K ,(Q)]xy
YE{U,V,W}
Y"JM(h')* ei'
+ iQA
.
(2.79)
-
We compute the divergence of vector spherical harmonics in Appendix A.1.4. Thus,
the eigenvector with zero eigenvalue of K,, (Q) is:
(+ iQ)
K ,(Q)
(
(J+1)(2J+1)
Q(1
- iQ)
J(J+ 1)
J2
=
Q2
J(2J+1)
78
-
0 .
(2.80)
This property provides an essential check of our numerical results.
The same result can be understood in a different way. The gauge field effective
action should be gauge invariant, hence pure gauge modes should be zero eigenvectors
of the real space kernel. We set the gauge fluctuation to be pure gauge by taking
a= D(A)YQ,je-in in (2.62) to get
0
= [AKqq,
(D(A)YQJe-iT')' (x)
=
dox'
g(x')AC,(xx')
(2)e-
(A)y
(2.81)
Plugging in for C" the formula (2.78) we obtain for the zero eigenvector of Kj , (Q)
the following expression:
UQjM(h) ei'r
I
dx
WQyM (i) ein'r
(DbA)YQ,JM (ft)e
n'tr
W",m (h) * e
-(L+
= 27r(Q - Q')
2
(L+1) -Q
2
(L + 1)2L)
1)(2L+1)
Q(1 -if)
V/L(L +1)
(L +1M)
(2
_Q2
(2L +1)
(2.82)
This expression agrees with (2.80). What this formula says is that the zero eigenvector
is a pure gauge mode D (A)YQ,jme-6Q
in the vector monopole harmonic basis. The
calculation above is by no means an independent derivation of (2.80), as current
conservation follows from gauge invariance.
79
Eigenvalues and determinant of gauge field fluctuations
Having computed K ,,(Q), it is now easy to find the eigenvalues of the kernel K",(x, x'):
they are simply the eigenvalues of Kq, (9) for every J and Q, and they come with
multiplicity 2J + 1. To have the terms in (2.54) we need to compute:
Tr'log Kqq, =
J>
(2J+ 1) log det'K,()
1
,
(2.83)
J=11-1
where det' indicates that we should only take the product of the non-zero eigenvalues.
As shown in (2.80) the presence of pure gauge modes result in one zero eigenvalue for
Kq, (Q), and we have to drop the zero eigenvalue as our prescription is to integrate
only over gauge inequivalent configurations.
The matrix Kq ,(Q) is not necessarily positive definite. If it has a negative eigenvalue that signals an instability, the corresponding gauge fluctuation gives a wrong
sign Gaussian integral in the partition function and makes the free energy complex.
In Section 2.5 we discuss the instances when this happens. For illustration, we plot
some of the low J eigenvalues in two examples. Figure 2-2a shows a stable monopole
background, while Figure 2-2b an unstable one.
The expression (2.83) is not yet ready to be put on a computer due to various
divergences. We find it convenient to combine it with the Faddeev-Popov determinant
and introduce a UV cutoff first.
Note that (2.75) implies that Tr'log Kqq, = Tr' log
?Cqq,
Fgauge(q, q') = Fgauge(q', q)
.
which further implies
(2.84)
This property is the consequence of CP invariance. It is also not hard to show that
Fgauge(q, q') = Fgauge(-q, -q').
80
ALgauge
0.25J = Q + 1, Larger Eigenvalue
J=Q - I
0.05
0.5
.0
1.5
1.0
(a) A stable example with q
2.0
=
-1,
q' = 1/2.
0.20ue
-
J= Q + 1, Larger Eigenvalue
J = Q + 1. Smaller Eigenvalue
0.15-
J=Q
J=Q - I
0.10
0.05-
0.00
- - - - ' -
- - ' - - - - ' - - - - '
1.5
1.0
2. '
-0.05(b) An unstable example with q = 1/2, q' = 3/2.
indicated in orange.
The instability is
Figure 2-2: The eigenvalues of K ,(Q) for some example q, q' and low J values as
a function of Q. Zero eigenvalues corresponding to pure gauge modes are omitted.
Note that the eigenvalues are monotonic in J and Q, hence it suffices to examine the
Q = 0 behavior of the lowest J mode for stability. Also note that in both examples
IQI > 1 and the two lowest lying J modes have one non-zero eigenvalue, while higher
J modes come with two eigenvalues. (The smaller number of eigenvalues corresponds
to the reduced size of the matrix K I(Q).)
2.4.4
Combining the subleading terms in the free energy
In the 1/N expansion of the free energy (2.34) there are two terms at O(N?) order,
the ghost and the gauge fluctuation contribution. Both contributions involve a sum
81
of N terms; see (2.53) and (2.54). Each term takes the form in (2.53) and (2.83):
FFp(q, q')
:(J+
2,7
2
1) lgfI+1
24g
(2.85)
J=IQI
Fgauge(q, q')
Tr'log Kqqi
where we used the notation
2
Q=
=
2
d-
J~~j-1qq
27r
(2J+ 1)log det'K ,(Q)
q-q'. These expressions only determine a meaningful
free energy if KJ(Q) only has positive eigenvalues apart from the zero eigenvalue
corresponding to pure gauge modes for all J and Q. If there is a negative eigenvalue,
there is an instability that will be discussed in Section 2.5.
Firstly, let us consider the large J, Q behavior of the eigenvalues A auge(Q) of
K ,(Q), the product of which gives det'K, (Q). For J and Q large we get
Agauge(Q)
asymp(
J(J+ 1) - Q2 + Q 2
16
-
(2.86)
which gives a divergence when integrated over Q and summed over J. We notice
however, the appearance of the Faddeev-Popov determinant (2.91). Note that the
ghost determinant comes with a negative sign and we end up with the ratio inside
the logarithm
SF(q, q')
FFp(q, q') + Fgauge(q, q')
1
2
d
27
(2±)0g
J=IQI+
(2 J +1)log
det'K ,(Q)
(2.87)
q+...2
J(J + 1) - Q + Q2
where we introduced the notation 6F(q, q') for the sum of the gauge and ghost contributions and the dots stand for the low J modes that do not pair up nicely with the
1
ghosts.1 2 The two beautifully combine to give a well-behaved result for large J, Q.
In Figure 2-3 we show that the eigenvalues of the gauge kernel, Ajgauge(Q) asymptote
-1 and J = IQI cases has to be treated separately with zeta function regularization.
12 The J =
13 Note that according to (2.86) the integrand goes to the constant - ZJ-IQI+1 (2J + 1) log 256.
In zeta-function regularization, the Q integral of a constant vanishes, hence constants do not give
any contribution.
82
Agauge / Aasymp
1.04'
-
%%J=1/2
--- J=3/2
1.02
J=5/2
-
J=7/2
1.00
-
J=9/2
0.98--
J=11/2
0.96
J=13/2
0
5
10
15
Figure 2-3: We plot the ratio of the non-zero eigenvalues Agauge (Q) of the gauge kernel
divided by their asymptotic behavior A'ymp(2). We chose q = -1, q' = 1/2 for this
example. Because IQI = 3/2 the J = 1/2, 3/2 modes contribute one eigenvalue,
while for higher J eigenvalues come in pairs. We used the same colors to plot the
pair of eigenvalues for these higher J modes. Because the ghosts give a contribution
proportional to Aymp(Q) this plot shows that the low energy modes are the most
important in determining the free energy.
to (2.86). Because [AJymp(q)]
2
asymptotes to that of the ghost contribution we see
that the main contribution to the free energy is from the low energy modes. At high
energies the ghosts cancel the contribution coming from the gauge fluctuations.
To complete the evaluation of the subleading terms we have to introduce a cutoff
that makes the integral definite and the sum finite. Because for large J and Q we are
probing the UV of the field theory where it should not matter what manifold we are
working on, we use a relativistic cutoff
J(J+ 1) - Q2 + 02 < A(A + 1) .
(2.88)
With this cutoff the sum and the integral in (2.87) are convergent. Evaluating (2.87)
for different A and extrapolating to A -+ oo we obtain our final result for the subleading term in the free energy. An example is given in Figure 2-4. We give a systematic
collection of results in Section 2.6.
83
2.4.5
Summary and an example
In this subsection we summarize the key formulae in the evaluation of the S 2 x R free
energy. We repeat the 1/Nf expansion of the free energy (2.34)
F[A] = Nf Fo[A] + SF[A] +
(2.89)
(1/Nf).
Fo[A] is the fermion determinant in the monopole background given by (2.38) and (2.47):
Nr
Fo [A]
=
(Fo(qa)
a=1
FO(q) = -
- q2 -
(2j + 1) (
E
2
(2j + 1) 2 + q2 )
(2.90)
q(2q - 1)(q+ 2)
6
6F[A] is the sum of the gauge and ghost contributions. The Faddeev-Popov determinant is given by (2.53), while the determinant of gauge fluctuation is obtained
by (2.54) and (2.83):
Ne
6F[A] =
66F(qa,qb)
a,b=1
6F(q, q')
FFp(q,q')
FFp(q, q') + Fgauge(q, q')
-1
(2.91)
dQ
Q2J+1)
log [J(J+i1)-2 +Q
2]
J=IQI
Fgauge(q, q')
2
STr' log
Kqq =
(2J+ 1)logdet'Kjq,(Q)
1
2J2r
J=IQI-1
We combine the subleading terms before evaluating (2.87) numerically.
In this subsection we examine an example in more detail to illustrate some of the
steps sketched in the previous subsections. We make the simple choice G = U(2) and
84
qi = 1/2, q2 = -1.
The leading contribution is (2.90):
Fo[A] = Fo(qi) + F(q2 ) = 0.265 + 0.673 = 0.938 ,
(2.92)
where we numerically evaluated (2.90). The list of Fo(q)'s will be given in Table 2.1.
The subleading term is a sum of four terms
5F[A] = 6F(q1 , q1) + &F(q
1 , q2 ) + F(q2 , q1) + 6F(q2 ,q 2 )
.
(2.93)
We pick 6F(q2 , q1) from this sum to illustrate the calculation. The ghost contribution
is known analytically (2.91). To calculate the gauge contribution we need to determine
the gauge kernel Kj,qq()
numerically. For every J and Q we need to construct this
matrix. This construction involves an infinite sum, and the procedure is explained
around Figure 2-1. In that figure we display the matrix element [Ki, 1 (Q)] u for a
representative choice of J and Q. We need to know this kernel to
10-12
precision.
Gauge invariance determines the eigenvector with zero eigenvalue of the matrix
Kjq2 '(Q)
analytically (2.80). This eigenvector provides a powerful check of the result
and whether we indeed achieved the precision advertised. It turns out that matrices
Kq,(Q) can be reused in the computation for general gauge groups discussed in
Section 2.8.1.
We calculate the eigenvalues of the matrices KjqI (Q) numerically.
Figure 2-3
shows a few eigenvalues for our choice of qi, q2 . We drop the zero eigenvalue, and
combine the ghost and gauge contributions as explained in Subsection 2.4.4. Finally,
we calculate the sum over J and the integral over Q in (2.91) for different UV cutoffs A
defined in (2.88). All the divergences have been regularized in previous steps in zetafunction regularization and the free energy is finite as we take 1/A -+ 0. 6F(q2 , q1 )
as a function of 1/A is plotted in Figure 2-4.
The terms that can appear in (2.91) for
85
jqj,
jq'j < 2 will be presented in Table 2.2.
SF(-1,1/2)
-0575
-0580
-0585
-05901
-0595
-0.6"0 0A6.0
0.1
02
0.3
0.4
0.5
1/A
Figure 2-4: We plot the subleading term in the free energy, SF(q, q') for q = -1, q' =
1/2 as a function of the cutoff A. We extrapolate to 1/A -+ 0 by fitting the data
points by a second order polynomial. Our results are reliable to 10- precision.
We can find the terms needed in (2.93) from that table:
JF[A] = -0.0383 - 0.574 - 0.574 - 0.194 = -1.38 .
(2.94)
We conclude that in U(2) gauge theory the dimension of the GNO monopole operator
with charges qi = 1/2, q2
=
-1 is:
A = 0.938 Nf - 1.38 + 0 (1/Nf) .
(2.95)
We discuss the results for monopole operator dimensions more systematically in Section 2.6.
2.5
Stability
In the previous section we studied the effective action for the gauge field fluctuations
in the presence of a GNO monopole background (2.18). We noticed that the effective
action for the W bosons (off-diagonal components of the gauge field) is not always
positive-definite (see Figure 2-2), which is to say that certain classical monopole
86
backgrounds are unstable. In this section, we discuss this instability in more detail
and characterize which sets of GNO charges yield an unstable background.
The instability of certain GNO backgrounds should come as no surprise, as similar instabilities have been studied in related examples. Indeed, it is well-known that
GNO monopoles in Yang-Mills theory in flat space are generically unstable [20,85].
To characterize the unstable configurations, recall that the GNO monopoles organize
themselves into classes of topologically-equivalent backgrounds, where each class corresponds to an element of the first fundamental group of the gauge group,
7ri(G).
In
the case G = U(Nc), we have wr,(G) = Z, and there is a discrete topological charge
that can be identified with the sum of the GNO charges,
Nc
qtop =
qa.
(2.96)
a=1
The monopoles that were shown in [20,85] to be unstable in Yang-Mills theory in flat
space were those with Iqa - gbJ > 1 for at least one pair of GNO charges (qa, qb). It
is not hard to convince oneself that each topological class with charge qtop contains
precisely one stable monopole background.14 All the rest are unstable.
It is important to note that the flat-space instability of monopoles in Yang-Mills
theory discussed in [20,85], as well as the instability we noticed in the previous section,
occurs only at low frequencies and for W bosons with total angular momentum J =
Iqa -
qbl
-
that Iqa
-
1.
qab
(This is the lowest value of the total angular momentum provided
>
1-see (2.65).)
That the instability is at low frequency and low
angular momentum means that it is a property of the infrared dynamics. Different
non-Abelian gauge theories with different IR dynamics can therefore have different
sets of stable/unstable GNO configurations. It just so happened that in the case of
Yang-Mills theory in flat space it was all the W bosons with J = Iqa -
1 that
qbJ -
were unstable. In a different theory, on the other hand, it could be that not all these
lowest J modes are unstable. To assess stability, one has to examine the effective
14The stable background has qa
the other a > (q mod N,).
=
= [qt 0 p/Nc]
87
+ 1/2 for a
; (q mod Nc) and a =
-
1/2 for
action for the gauge field fluctuations, as well as the fluctuations of other fields, and
see whether there are any negative modes.
In the case studied in this chapter, namely the IR fixed point of QCD 3 at large Nf,
the question of stability is much richer than in pure Yang-Mills theory in flat space. 15
We find that in contrast to the pure Yang-Mills case, in QCD 3 there are multiple
stable monopoles per topological class. As we will discuss later, even topologically
trivial gauge groups such as SU(2) admit stable monopole backgrounds.
2.5.1
A systematic study of monopole stability in QCD 3
In performing a more systematic study of the instability of monopole backgrounds
in large Nf QCD with gauge group U(Nc), let us first note that at leading order in
Nf, where one can treat the gauge field as a background and ignore its fluctuations,
there are no instabilities, as this is just a theory of non-interacting fermions. To
decide whether or not a given GNO background is stable, it is important to consider the subleading 1/Nf effects described by the effective action for the gauge field
fluctuations.
In the previous section we have developed a whole machinery needed to study
the eigenvalues and eigenfunctions of the quadratic action for the fluctuations of the
gauge field around a GNO monopole background (2.18). In brief, each component
a,,(x) of the gauge field fluctuation can be expanded in terms of Fourier modes in
the Euclidean time direction as well as the monopole vector harmonics UTT'
Vg'J,
' 5 One could wonder how many stable monopoles there are in pure Yang-Mills theory on S 2 x R.
Since Yang-Mills theory in three dimensions is not conformal, one cannot simply borrow the flat
space result, so a separate analysis is needed. We find that if the gauge group G = U(Nc), the
quadratic action for the a b component of the gauge field fluctuations around the GNO monopole
(2.18) has eigenvalues oc
+ J(J +1) - (q, - qb) 2 (for physical modes) or 0 (for pure gauge modes),
where Q is the frequency and J is the total angular momentum. There is an instability at low Q for
J = 1q. - qb - 1, so the situation is identical to that of Yang-Mills theory in flat space.
88
and W',JM defined in (2.65):
aI'ab)
=
J[,JM(Q)USab,JM()
+ ay'JM(Q)Voab,JM()
+ ab,JM()W'IJM
e-T.
(2.97)
For the fluctuation a b(X), we should take Qab =qa - qb. The quadratic action for
the coefficients ab,JM takes the form
abJ
ab,Jm
aab,JM(Q)
a,b=1
Kj
.
b'y
(2.98)
JM
aab J(Q),aW
()
The matrix Kj, (Q) can be computed tediously by following all the steps presented
in the previous section. As demonstrated in the example presented in Figure 2-2, the
eigenvalues of Kjq (Q) increase with both Q and J. To check whether the action for
a"b is positive-definite, it is therefore sufficient to calculate Kj.'q (Q) for Q = 0 and
the lowest attainable value of J = Jab. If IQabI < 1, this lowest value is Jab = Pad;
if IQabI
1, it is Jab = IQabI - 1.
We have computed numerically KJab (0) for all possible values of qa and qg in
the range -10
< qa, qb < 10. From our numerics, we find that it is only the modes
with Jab = IQabI - 1 and IQadi > 1 that are sometimes unstable. We have plotted
Kaqb
(0) in Figure 2-5 as a function of qa and qb. In Figure 2-6 we have indicated
the stable region in black and the unstable region in orange.
A GNO monopole
labeled by charges {qi, q2, - - - , qNc} is stable if every pair of charges lies in the stable
region displayed on the plot.
As can be seen from Figure 2-6, we find two stable regions in the qa-qb plane. The
89
SAgauge
01
qa
-10-5
Figure 2-5: The lowest eigenvalue A = Kf' 1 (O) of the a component of the gauge
field fluctuations around the GNO monopole background (2.18). We have marked
explicitly the plane z = 0. The region where this eigenvalue dips below zero corresponds to an instability of aab. If this eigenvalue is positive, then the action for aab is
positive-definite.
first such region is where
Iqa
- qbI <1,
(2.99)
which, as mentioned above, is the same stability condition as in Yang-Mills theory
in flat space.
For these values of the charges there is no instability because the
problematic mode with angular momentum J = Iqa - qa| - 1 is simply absent. The
second stable region is new and unexpected. It occurs where qa and q6 are comparable
in magnitude and of opposite sign. Asymptotically, at large values of qa and qb we
can estimate that the second stable region is where
-tan73 0 <$
< - tan170 .
(2.100)
qa
The existence of the second stable region implies that there are several stable
GNO monopoles per topological class. Indeed, the first stable region alone implies
that each topological class contains at least one stable GNO monopole, and hence
90
qb
:0*
.* **
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=
thFistne of6 Asmar econd th stableit reio mean thtmhrecnpbes mor
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uand
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wth chNge Imonopo.es are indexedby arvddal pairofGOcags (q, q) andrtespn toogca
chr
isd black cicls and +tr is usa inintehuerwe stablne GNO monopoles hathae
itooeogicytivifo
because
a onopoiveoes
J = 0b li in the regio GNt
(210)and
siity
rgo with
GN.O moops.tiAll
monopole
Figur2-6:a
tAt summar
peel ofte
factlerylak topologc clssn thi thlesoy conain an infinhichthe ube nof stablnegNOiv
wistha
lo sin
t45e inted
monoes,
bhecausferecee
the onoole
wihnixd
file We onlac
clesQab
teunalse
otwisea.
dote
discus~,
orafid)blc
irces, and sbity
ones phlnes anwila necessaril lie e in ~ -1 hl o the regio (210osmttialntlres
Havingle finis the setabiloityaalyclss. woeaowmplet U2)ue thesoof
tion2.2
Iln thet follong, essaalyzeiheMinkosk
91
Shec-
time00 evoluto.c Int quantumq
theory all field modes fluctuate about the monopole background. If the background
is stable, at large Nf the wave functional is supported on gauge configurations close
to the background, as the typical size of the fluctuations is 0(1/
Nf). Conversely, if
the background is unstable, the unstable mode grows exponentially. The wave functional that started out as having delta function support on the background spreads,
and ends up with broad support. In this case, it is more useful to decompose the wave
functional in terms of energy eigenstates. The spread in energy will be wide, compa16
rable to the potential energy difference to a nearby local minimum. The Euclidean
path integral will be dominated by the lowest energy eigenstate at the bottom of a
nearby local minimum, as a result we get the free energy of another monopole. Note
that there is no topological obstruction to this scenario, as every topological sector
has at least one stable monopole background in it.
One could still consider the disorder operators corresponding to unstable monopole
backgrounds. The decomposition of the corresponding state into energy eigenstates
translates into this disorder operator being a sum of operators that have a big range
of scaling dimensions.
In correlation functions, at long distances such a disorder
operator would behave as the operator with the lowest scaling dimension from this
sum, i.e. another monopole operator in the same topological sector (or the identity).
2.6
Monopole operator dimensions
In this section we collect the results for monopole operator dimensions in U(Nc)
gauge theories. We first exhibit the QED case N, = 1 explicitly in Subsection 2.6.1,
and then we present the results for N, > 2 in Subsection 2.6.2. For details on how
to obtain the results collected in this section we refer the reader to the example in
section 2.4.5.
16
For very late times tunneling has to be taken into account. We neglect tunneling effects in this
discussion.
92
2.6.1
Monopole operator dimensions in QED
In U(1) gauge theory all monopole backgrounds are stable because the monopole
charge is a topological quantum number. To obtain the scaling dimensions of the
corresponding monopole operators, one should simply set N, = 1 in the formulae
summarized in Section 2.4.5. See Table 2.1 for the scaling dimensions Aq of the
monopole operators with |qj < 5/2.
|qj
Aq
0
0
1/2
0.265 Nf - 0.0383 + O(1/Nf)
1
0.673 N1 - 0.194 + O(1/Nf)
3/2
1.186 Nf - 0.422 + O(1/Nf)
2
1.786 Nf - 0.706 + O(1/N)
5/2
2.462 Nf - 1.04 + 0(1/Nf)
Table 2.1: Monopole operator dimension Aq for monopole charge q in U(1) gauge
theory.
Part of these results are not new: the O(Nf) contributions to the scaling dimensions given in Table 2.1 were first obtained in [13], while the subleading correction to
the dimension of the monopole operators with
2.6.2
jqj =
1/2 was also obtained in [21].
Monopole operator dimensions in U(Nc) QCD
As mentioned before, in U(Nc) gauge theory not all GNO backgrounds A specified
by the charges {qi, q2 , ... , qNcj are stable. Stability is a dynamical question, and we
presented the criterion for stability in Section 2.5-see Figure 2-6. For the stable
backgrounds, we can compute the scaling dimension A = F[A] of the corresponding
operators using the formulae (2.89), (2.90), and (2.91), which we repeat here for the
reader's convenience:
F[A] = Nf Fo[A] + 6F[A] + 0(1/N),
93
(2.101)
where
Ne
Fo [A]
=
S Fo(qa),
Nr
6F[A] =
a=1
5 6F(qa, qb)-
(2.102)
a,b=1
The numerical values of Fo(q) are the same as the coefficients of Nf in the expressions
for Aq given in Table 2.1. For 6F(q, q'), see Table 2.2. Note that not all the entries
in Table 2.2 are numerical; some of them are instead orange dots, which indicate
an instability. According to the recipe of Section 2.5, if such a term features in the
second sum in (2.102), the corresponding monopole background is unstable and does
not correspond to a monopole operator with well-defined scaling dimension.
Using the values listed in Tables 2.1 and 2.2 one can determine the dimension of
any monopole operator with GNO charges obeying Iqal 5 2. For higher GNO charges
one has to construct larger tables.
Note that the subleading terms in Table 2.1 are equal to the diagonal entries in
Table 2.2. Note also that Table 2.2 has a reflection symmetry about the diagonal
F(q, q') = F(q', q) ,
(2.103)
as a consequence of CP symmetry (2.84), as well as a reflection symmetry about
origin, 6F(q, q') = 6F(-q, -q').
Let us consider a few examples:
" If we take {qi, q2,
, qN
{- /2, 0,... ,
0,
we have
Fo[A] = FO(1/2) = 0.265 ,
6F[A] = 6F(1/2, 1/2) + 2(Nc - 1) 6F(1/2, 0) ,
(2.104)
A = 0.265 Nf - 0.0383 - (N, - 1) 0.516 + O(1/Nf).
This monopole operator has the smallest dimension among all.
" If we instead took {qi, q2 , ...
, 0,... , 0}, we would find that there is no
, qNc
monopole operator with this GNO charge and well-defined scaling dimension,
94
-2
-3/2
-1
-1/2
2
-1.90
-1.63
-1.52
-2.16
3/2
-1.63
-1.26
-1.04
-1.05
*
1
-1.52
-1.04
-0.730
-0.574
1/2
-2.16
-1.05
-0.574
0
0
0
I
3/2
2
*
-0.857
-0.706
0
-0.592
-0.422
-0.857
0
-0.386
-0.194
-0.592
0
-0.338
-0.258
-0.0383
-0.386
*
0
-0.258
0
-0.258
*
*
*
-0.386
-0.0383
-0.258
-0.338
-0.574
-1.05
-2.16
-0.386
*
-0.574
-0.730
-1.04
-1.52
0
*
-1.05
-1.04
-1.26
-1.63
-2.16
-1.52
-1.63
-1.90
1/2
q
-1/2
-1
9
-0.592
-0.194
-3/2
-0.857
-0.422
-0.592
-2
-0.706
-0.857
Table 2.2: 6F(q, q') for various values of q and q'. The orange dots mean that the
corresponding W boson is unstable.
F(1, 0) =
because
..
Finally, if we consider {qi, q2 ,...
,
qN} = {1/2, 0,
...
,0,
-1/2}, the correspond-
ing dimension is
Fo[A] = 2Fo(1/2) = 0.530 ,
JF[A] = 23F(1/2, 1/2) + 26F(1/2, -1/2) + 4(Nc - 2) 6F(1/2, 0) ,
(2.105)
A = 0.530Nf - 0.753 - (Nc - 2) 1.06 + O(1/Nf).
2.7
Other quantum numbers of monopole operators
In the previous sections we calculated the energy of the ground state 17 on S 2 x
R localized around the GNO saddle (2.2). This computation used the equivalence
17As explained before, the use of the term "ground state" is not necessarily appropriate. We are
talking about the lowest energy states whose wavefunction at large N is highly peaked around the
saddle (2.2).
95
between the ground state energy and the thermal free energy at zero temperature, and
as such does not tell us much about the properties of the ground state, or equivalently,
about the quantum numbers of the operator corresponding to it. In this section we
fill this gap. Of course, the results presented here will only be valid for the GNO
saddles that do not have any unstable directions.
The states on S 2 must transform in representations of the conformal group and of
the flavor symmetry group. The flavor symmetry group of a theory of Nf fermions and
gauge group U(Nc) is SU(Nf). The conformal group on S2 x R is SO(4, 1), regardless
of whether the R coordinate is Lorentzian or Euclidean time. We choose to work in
Euclidean signature, even though time evolution is a unitary transformation on the
Hilbert space of states only in Lorentzian signature. We expect the bare monopole
operators that we studied in the previous sections to be conformal primaries. We will
now determine their spin and SU(Nf) quantum numbers.
2.7.1
Quantum numbers of monopole operators in QED
Before studying the quantum numbers of the GNO monopoles in QCD, it is instructive
to study the same question in the QED case, N, = 1, where the monopole operators
are labeled by the charge q E Z/2 and heuristically create the background (2.1).
The quantum numbers of the monopole operators in QED were calculated in [13] for
IqI
= 1/2. In this section we present the quantum numbers for arbitrary q. 18 The
result we will find is that the monopole operator of charge q transforms as a Lorentz
scalar, and as an irreducible representation of SU(Nf) given by the rectangular Young
diagram with Nf/2 rows and 2jqI columns:
Nf/2
{
.
(2.106)
21qj
We now explain the derivation of this result.
Because at large Nf the fluctuations of the gauge field around the background
18
Our work corrects a slight error in the analysis of [13] for general q.
96
(2.1) are suppressed, we should start by canonically quantizing the theory of free
fermions in this background, and worry later about the effects of having a dynamical
gauge field. The fermionic modes can be found by solving the Dirac equation
(i
+ A)0 = 0 .
(2.107)
To solve this equation, one can begin by expanding V) in terms of the spinor harmonics
Sq,jm(ft) and T,jm(ft), and Fourier modes in time, as in Section 2.4.1:
( t)=
S S
j=ql-.
"CjSq,jm(
) +c
"TTq,m(i) e~W
(2.108)
m=-j
with arbitrary w to be determined by solving the Dirac equation. When
j
;
I
+ 1/2,
the Dirac equation has two solutions (for every j, m and flavor a) with energy iwj
=
±Eq,j, where
Eq,
=
(j +
We can denote by c±)athe linear combinations of (c
energy ±Eq,j. In the quantum theory, the c
(2.109)
1/2)2 - q2 .
and cjm
corresponding to
a become anti-commuting annihilation
operators for the corresponding modes. Generically, there are 2j + 1 such operators
for both choices of sign and every a and j.
The case j = jq| - 1/2 is special because the spinor monopole harmonics Tq,jm are
absent, and the Dirac equation implies that the Sq,jm modes have energy iwj = 0.
In the quantum theory, the coefficients c(s)a which in this case we denote by cjm
for brevity, become annihilation operators for these zero-energy modes.
For each
flavor, there are 2j + 1 = 2 IqI zero-energy modes transforming in the spin
IqI
- 1/2
representation of the SU(2)rot rotation group. There are a total of 2 IqI Nf zero-energy
modes when we consider all of the flavors.
If there had been no zero-energy modes, the situation would have been quite
simple.
The theory would have had a unique rotationally-invariant vacuum1Q),
97
corresponding to a Dirac sea filled with particles with negative energy and containing
no particles with positive energy. In other words, this vacuum should be annihilated
by all annihilation operators for positive energy modes and by all creation operators
for all negative energy modes:
cMa|)
i> |ql
= cj;~QtIG) = 0,
-
.
(2.110)
One could build up the states in the Hilbert space by adding particles with positive
energy or removing particles with negative energy.
The existence of 2
IqI Nf
zero-energy modes, however, means that the theory of
free fermions in the monopole background (2.1) has, in fact, not just one, but
2 2|qlNf
degenerate ground states that satisfy the condition (2.110). Let us call the Hilbert
space spanned by these ground states g. One of the states in g is the Fock vacuum
I),
defined by (2.110) together with the requirement that it should be annihilated
by the annihilation operators cjm with j = |q. -
CimIQ)
=
j
0,
=
:
q
-
1
2
- .
(2.111)
The other linearly independent states in g can be obtained by acting with any number
of creation operators ct (with j
=
I- ), on the rotationally-invariant Fock vacuum
IQ). The full Hilbert space of the theory R is obtained by acting on the states of g with
any number of creation operators cm
for positive-energy modes and annihilation
operators c m)c for negative energy modes (j > IqI - 1).
This description of the Hilbert space W is correct assuming the gauge field is a
background field. For us, however, the gauge field is dynamical and its effect is to
remove some of the states in ?- (and consequently some of the ground states in !).
In the path integral language the gauge field appears in the action only as a Lagrange
multiplier that imposes the constraint j(x)
98
= 0. What we mean by this constraint
is that all correlation functions of the current should vanish in the full theory
0
= (j1(X1) j12(X2)
...
(2.112)
j1n(Xn)) full theory -
This equation looks perplexing at first sight, as we spent most of the chapter determining the gauge kernel K""(x, y) = -(j(X)j"(y))conn.
The resolution of this puzzle
is that K"" is determined by the current-current correlator in the free fermion theory,
where the gauge field is treated as a background.
In canonical quantization language we have a constrained system; the canonical
momenta conjugate to a"(x) vanish identically, and we should not define any oscillator
modes in the gauge sector. Instead, in analogy with the Gupta-Bleuler prescription,
we should require that the positive and zero energy part, j,
j,(x) =: V51t(x)-y,,04(x) : annihilates all physical states
of the current operator
Ix):
(2.113)
i(+) (X)Ix) = 0.
This requirement reduces the Hilbert space 71 introduced above to a smaller one
'Wphys, and the 22lqlNf-dimensional space of ground states g to
9
phys-
Understaning
the quantum numbers of the monopole operators means understanding what gphys
is, and how the SU(Nf) flavor symmetry and the rotation group SU(2)rot acts on it.
The (2.113) condition guarantees that (2.112) is satisfied. Of course, the expectation
value has to be taken between states in 'Wphys- 1 9
Using (2.108) and the definition of j"(x), one can obtain an explicit expression
for the current operator in terms of oscillators:
+()
cc,
-
CJmm) S,(,jm(i)'ySq,jm(n) + (non-zero modes),
(2.114)
where we wrote down explicitly only the contributions from the oscillators corresponding to the zero-energy modes with j =
19
IqI
- 1.
(From here on, it should be
1t is easy to check that imposing only the strictly positive energy part of j1 to annihilate physical
states is not sufficient to ensure (2.112). We have to require the stronger condition (2.113).
99
understood that j =
IqI -
1/2 and that m runs over -j through j unless otherwise
specified.) The quantity C appearing in (2.114) is a c-number corresponding to a possible normal-ordering ambiguity when taking the product of cg9
normal ordering ambiguity is present only when
j
=
j', m =
with c,'m,.
Such a
m', and a = a', because
any given cjm anti-commutes with all the other fermionic creation and annihilation
operators except for cjm.
The normal ordering constant is determined by CP-invariance to be C = Nf/2.
Indeed, creating a zero-energy mode is related by CP to destroying a zero-energy
mode, and if we want to quantize the theory in a CP-invariant way, we better treat
the creation and annihilation operators for zero-energy modes on equal footing. Doing
so means that instead of the expression in the parenthesis in (2.114) we should have
written
c-ic.c
-J 3m
,,-c Cmc
jm' im)
(2.115)
Anti-commuting the two factors in the second term, summing over a, and comparing
with (2.114) yields C = Nf/2.
Using the explicit expressions for the spinor monopole harmonics, it is straightforward to find the explicit position dependence of the zero-mode contribution to the
current operator (2.114). For instance, when q = 1/2, the zero modes have spin j = 0.
In the North chart, the expression for the only spinor harmonic with j = 0 is
1Cos 0
S1
(()
=o-2
(2.116)
e=1 sin
After plugging this expression into (2.114), a little algebra shows that the zero modes
do not contribute to jo(x) and jO(x), while the charge density p(x) = j T (x) is
q=
:
p(x)
=
7(
-
Nf/2) + (non-zero modes),
100
(2.117)
where K
c'ce is the operator that counts the total number of excited zero-energy
modes. More generally, the operator that counts the total number of fermions in the
zero-mode sector is
m
S
(2.118)
cm m
That the charge density p(x) annihilates the states means that !phys consists only of
the states in 9 for which k = Nf/2.
Similarly, when q = 1, the zero modes have spin j = 1/2. In the North chart, the
spinor harmonics are
S1 11(i)
=
1
'22
(
e
sin 0
S1
,
l/8e2 (1 - cos0)
1
'2
1
(L)
1 + cos 9219
I.
=
e' sin
2
)
(2.119)
Again, using these expressions one can show that the zero modes do not give any
contributions to jo(x) and jO(x), while the charge density is
q=1:
p(x)=
1 Y()*(
- N) -
Y-m (h)* Sm + (non-zero modes).
1
(2.120)
Here, K is the fermion number operator in the zero-mode sector defined in (2.118)
and Sm is the total spin of the zero-energy modes organized as states in the spin-1
angular momentum basis:
S, = -ct
S=
S_
1
C1
il
c'
c
72
2
= cot icia
2'
2
2 2
1
2 22
=
=
SX - iSy ,
c
X-
2
is,
= .1
2'
-v
2
2Sz,
(2.121)
.
The requirement that the charge density should vanish for any physical states implies
that
9
phys
consists only of the states in 9 that satisfy K = Nf and S= 0.
101
The expressions (2.117)-(2.120), as well as the characterization of !phys as a subspace of g, generalize to arbitrary q in the following way. From the creation and
annihilation operators in the zero-energy mode sector one can construct SU(Nf)singlet operators of the form
o=
"mM't3
(2.122)
'
m,m'
where
mm'
is a 21 q x 2 1qI Hermitian matrix. There are (2 |qI) 2 linearly independent
such matrices, and hence (2 |qj) 2 linearly independent operators 0, which organize
themselves according to irreducible representations of the rotation group SU( 2 )rt.
The representations that appear are precisely those in the product of two spin-j
irreps of SU(2), namely all the ones whose spin is between f = 0 and f = 2j. If we
Ojm,,
denote the spin-f operators by
then
OoO
is proportional to the total fermion
number K, 61m is proportional to the total spin Sm, and so on.
The expression for the charge density operator in (2.117) and (2.120) then generalizes to
21q|-1
p(x)
=
(
-
Iqj Nf)
+
S 5 Yjm- ()(O'm,
+ (non-zero modes),
(2.123)
9 m,
O
come with
where Ym (h) are the usual spherical harmonics, and the operators
specific normalizations. The precise normalization of
Oem,
is not essential for the
argument we are about to make. That the charge density annihilates all the states
means that out of the
2 2JqlNf
degenerate ground states in g we should only consider
the ones where
91X) =
IqI Nf IX)
and
Ofm, IX) = 0,
for t > 1 and all me.
(2.124)
The first requirement in (2.124) means that gphys contains only states of the form
IqjNf
4
iC|).
3c
i=1
102
(2.125)
where we act with precisely
Iqi Nf
fermion creation operators (out of the total of
2 1qI Nyf) on the Fock vacuum I2) defined in (2.110)-(2.111). The second requirement
in (2.124) requires more thought. It can be understood most simply by enlarging the
SU(2)rot symmetry to an SU(2 qi) symmetry, where the (21ql)
2
-
1 operators 61m,
generate SU(2 jql) and the states ct IQ) transform in the fundamental representation.
The second condition in (2.124) means that all operators (2.122) where Omm', is
a traceless Hermitian matrix should annihilate the physical states
IX)-20
This is
just the requirement that, infinitesimally, Ix) should be invariant under SU(2jql)
transformations. In other words,
consists of the states of g that are of the form
9phys
(2.125) and, in addition, are also SU(2
IqI)
singlets.
Each fermionic creation operator transforms as a fundamental of SU(N), so we
are looking for singlets under SU(2
IqJ)
in the product of
IqI
Nf fundamentals of
SU(Nf). There is a further wrinkle, however. As we are considering anti-commuting
creation operators, the states must be totally antisymmetric.
To count how many such states there are and see how they transform under
SU(Nf), it is convenient to introduce a bigger group that contains both SU(Nf)
and SU(2 qJ): if we make a list of all the zero-energy mode creation operators ciM,
we can consider SU(2
IqI Nf)
transformations under which cgi form a fundamental
vector. Similarly, the annihilation operators cym transform in the anti-fundamental
representation of the same SU(2
Iq Nf)
group.
The benefit of considering this larger group is that constructing totally antisymmetric states is simple. The states of Gphys are constructed by decomposing the antisymmetric products of
jqI N1 fundamentals
of SU(2
IqI Nf)
and selecting those which are singlets under the SU(2
to identify all the SU(2
20
IqI)
IqI)
under SU(2
IqI)
x SU(Nf)
factor. We therefore need
singlets in the decomposition of the rank-JqI N1 totally
Note that the non-traceless part of j is included in the operator 9.
103
antisymmetric representation of SU(2
IqI Nf),
(2.126)
IIq| Nf
under
SU(2 |q| Nf) D SU(2
Iq|)
(2.127)
x SU(Nf).
Such a group theory exercise is common in atomic physics where one needs to construct a totally anti-symmetric wavefunction for several identical particles with given
angular momentum and spin. In general, the rank r anti-symmetric representation
of SU(NM) decomposes under SU(N) x SU(M) as the sum (see, for example, [86])
ED (,/)
(2.128)
V
over all possible irreps with Young diagrams v with a total of r boxes (whose conjugates are denoted by I), such that v has at most N rows and F has at most M
rows. Each ordered pair (v, F) appears precisely once in this decomposition. For our
problem, we have
Jq+ Nf
-- -2,
2
Nf /2
(2.129)
21ql
where on the RHS the first Young diagram of any given pair corresponds to SU(2
jqJ)
and the second to SU(Nf). Of this infinite sum, we want to pick out the terms for
which the first factor is an SU(2
JqJ)
singlet. Only the diagram explicitly exhibited in
(2.129) has this property. Consequently, the states of gphys transform as the SU(Nf)
104
q = 1/2
q= 1
q = 3/2
Table
under
of the
under
D-(2)
ZD|
Nf = 6
N1 = 4
Nf = 2
|(3)
zp p(4)
B
(6)
(20)
-..
(20)
(175)
...
-(50)
P7(980)
...
2.3: The transformation properties of the first few (bare) monopole operators
the flavor SU(Nf) global symmetry of QED 3 with Nf flavors. The dimensions
irreps were calculated using (2.131). All these monopole operators are singlets
spatial rotations.
irrep whose Young diagram is a rectangle with Nf/2 rows and 2
Nf/2{IH
IqI
columns:
(2.130)
(Recall that we should only consider an even number of flavors in order to avoid a
parity anomaly.) The dimension of this irrep is
((ii + N)!)2
j= ((i + Nf /2)!)
2
(2.131)
See Table 2.3 for a few examples.
This discussion also shows that the physical ground states gphys are singlets under
the rotation group SU(2)rot. Indeed, SU(2)rot can be embedded as a subgroup of
SU(2
Iqi),
and the states that are SU(2
IqI)
singlets must also be SU(2)rot singlets.
We have thus found the quantum numbers of the (bare) monopole operators of
charge q in QED with Nf flavors. Their topological charge is q, and their conformal
dimensions were computed in the previous section at large N1 (see Table 2.1). In
this section we determined that these operators transform as the irrep (2.130) (see
105
also Table 2.3) under the flavor SU(Nf) symmetry and as singlets under the SU(2)rot
group of spatial rotations.
Quantum numbers of monopole operators in U(Nc) QCD
2.7.2
The careful analysis of the previous section can be generalized to the more complicated GNO monopole operators in U(Nc) QCD with Nf flavors in the fundamental
representation. As in the QED case, when Nf is large we can start by quantizing
the theory of free fermions in the GNO background (2.2), and then we can take into
account the effects of having a dynamical gauge field. The result is that the monopole
operators now transform in an irrep of the SU(Nf) flavor symmetry corresponding
to a Young diagram with Nf/2 rows and 2 EI
q,Iqaboxes in each row,
Nf /2{7§j
1(2.132)
2 EZaqaI
where {qa} is the set of GNO charges. In addition, the monopole operators are singlets
under the SU(2)rot rotation group. We obtained the same result in the Abelian case,
but now
Iqj
is replaced with Ea
Iqa.
The rest of this section provides the derivation
of these quantum numbers in the non-Abelian case.
carry a color index a = 1,..., N. in
In the non-Abelian case, the fermions #a,a
4
addition to the flavor index a. In the GNO background (2.2), the action for
ba'"
is the same as that of a QED fermion in an Abelian monopole background (2.1)
with charge q = qa. We therefore have 2 1qal Nf zero energy modes for each value
of a, with some corresponding creation operators c
c a
(here, ja =
Iqal
and annihilation operators
- 1/2 and ma ranges from -ja through ja). In addition to
the zero energy modes, we also have positive and negative energy modes. As in the
Abelian case, we can define the vacuum in the non-zero mode sector by requiring that
all positive-energy annihilation operators and all negative-energy creation operators
annihilate this vacuum. These conditions leave
(as appropriate for having 2Nf E
2 2Nf ZalqaI
degenerate ground states
qaI fermionic oscillators with zero energy) that
106
span a Hilbert space g. This Hilbert space has a Fock vacuum IQ), which by definition is annihilated by all C'.
The other linearly independent states in g can be
constructed by acting with any number of fermionic creation operators c',,
on
The analysis so far did not take into account the dynamical gauge field, which, as
in the QED case, acts as a Lagrange multiplier that imposes the constraint that the
positive and zero energy part of the current, jba+)(x), should annihilate all physical
states. This constraint reduces g to a smaller Hilbert space
9
phys,
whose transforma-
tion properties under the flavor group SU(Nf) and the rotation group SU(2)rt we
need to understand, as each state in gphys corresponds to a monopole operator.
The creation operators transform as fundamentals under SU(Nf). In the Abelian
case, we saw that it was useful to consider SU(2
IqI)
acting on the creation operators.
The condition of vanishing current in that case translated into two constraints that
select the states of
9
phys
operators be equal to
under this SU(2
Iql).
from g. The first constraint required the number of creation
JqI Nf, and the second required
that the states in 9phys be singlets
Similarly, in the case of U(Nc) we can consider the action of
SU(2 Ea IqaI) on the set of all creation operators of a fixed flavor. Note that this
group mixes fermions of different color and spin. We will argue that the condition of
vanishing current in this case translates into the following two constraints: each state
in
9
phys
is created by acting with Ea qaI Nf creation operators on the Fock vacuum,
and it should transform as a singlet under SU(2 Ea lqa ). The problem of finding
physical states is then just the same group theory problem we solved in the Abelian
case with IqI replaced by Ea Iq,I.
As in the Abelian case, each diagonal component
ja1t
imposes the constraint
that the number of creation operators of color a equal JqaI Nf and that the physical
states IX) are invariant under SU(2
lqal).
It will be more useful, however, to consider
the overall constraint coming from Eajjli, which says that the total number of
generators of all colors is Ea IqaI Nf. The other constraints coming from jia(
1
that the physical states are invariant under U(1)Nc- X Ha SU(2 Iqa 1).21
21
imply
We will
The factors of U(1) come from the separate particle number constraints for each color, with one
removed corresponding to the total particle number.
107
now argue that invariance of the physical states under this latter group enhances
to invariance under a full SU(2
aqaI)
when one also examines the off-diagonal
generators j(+) with b = a.
For simplicity we start by considering N, = 2, where the discussion above implies that the conditions coming from j(+) and j
SU(2
Iqil) x SU(2 jq2j).
require invariance under U(1) x
Recall that, in general, U(1) x SU(M) x SU(N) is a maximal
subgroup of SU(M + N). Therefore, if a state is a singlet under U(1) x SU(M) x
SU(N) and is annihilated by any other generator of SU(M + N), it is automatically
a singlet under the whole SU(M + N). For the N, = 2 case, the off diagonal current
jl,
provides at least one additional condition independent from the ones coming
from j
and j
, which required that the physical states be annihilated by the
generators of U(1) x SU(2 1q 1) x SU(2 1q
As such, the physical states must be
2 1).
singlets under the full SU(2
Iqil
+ 2 1q2 I).
For the general case, U(1)Nc-1 X
Ha SU(2 IqaI)
is not quite a maximal subgroup
of SU(2 Ea Iqa|). Instead, for each pair of indices (a, b) with a =A b, the subgroup
IqaI)
IqbI) C SU(2 qaI + 2 Iqb) is maximal.2 2
Repeating the
argument above from the N, = 2 case, the off-diagonal current j>,
b / a, is non-
U(1) x SU(2
x
SU(2
vanishing, so it provides an additional constraint on the physical states beyond the
invariance under U(1) x SU(2 qa|) x SU(2 IqbI) required by (,/ and '(.
The physical
states Ix) are therefore singlets under SU(2 qaI + 2 1qb1) for every pair (a, b). Iterating
this procedure for all pairs of color indices leads to the singlet condition under the
full SU(2 Ea IqaI).
Putting everything together, the states in
9
phys
are the SU(2 E.Iqa I) singlets
in the decomposition of the totally anti-symmetric tensor of SU(2N Za IaI) with
Nf Ea IqaI indices under SU(Nf) x SU(2
Za
IqaI) It follows that the states of 9phys
transform under SU(N) as the irrep (2.132). The corresponding monopole operators
are singlets of SU(2)rot because SU(2)rot is embedded in SU(2 Za IqaI) and we selected
only the SU(2 Ea IqaI) singlets.
22
We only need to consider pairs of indices, (a, b), where qa
the SU(2 jqi) factor is not present.
108
# 0 and
qb = 0. For vanishing charge
A generalization of these results to more complicated groups and/or representations of the fermion flavors is left for future work.
2.8
Monopoles in general gauge theories
In this section we generalize the computation of the dimension of GNO monopole
operators, as well as the stability analysis included in Section 2.5, to arbitrary gauge
groups. We will see that the computation proceeds analogously to the U(Nc) case,
and, moreover, no new ingredients are needed. In particular, to complete the study
of gauge field fluctuations around a monopole background in QCD 3 with gauge group
G, all that is needed are the properties of the kernel Cqq, (x, x') analyzed in the U(Nc)
case.
In gauge theory with gauge group G, the most general monopole background
centered at the origin is
A = H(±1 - cosO)dp,
(2.133)
where H is an element of the Lie algebra of G, and the two signs correspond to
the North and South charts. As explained in [19], each such configuration is gauge
equivalent to one where H points along the Cartan, namely
r
i=1
where r is the rank of the gauge group and hi are the Cartan generators. There is still
some remaining gauge redundancy in (2.134), as the Weyl group acts non-trivially
on the qj, and we should hence consider configurations of the form (2.134) only as
equivalence classes under the action of the Weyl group.
The Dirac quantization condition is [19]
IW)R
=
e4 7ri H
R
109
e
R
(2.135)
for any state Iw)R in any representation R of G. Here, w is the weight vector corresponding to Iw)R (such that hjlw)R
=
wilw)R), and so it belongs to the weight lattice
of G. The quantization condition (2.135) implies that q - w E Z/2 for any w. The
set of all q with this property form themselves a lattice that can be identified with a
rescaled version of the weight lattice of a dual group d. The group d is referred to
as the GNO dual (or Langlands dual) of G.
2.8.1
Anomalous dimensions for general groups
In general, the fermions transform in some representation R of G. Let us denote the
states of this representation by 1w), suppressing from now on the index R that we
introduced above. In terms of these states, the fermions can be decomposed as
(x) = ZbW)(x) jW,
(2.136)
wER
with 4w(x) being anti-commuting spinor coefficients. To avoid clutter, the flavor and
spinor indices are suppressed. Like in the U(Nc) case, having Nf flavors of fermions
has the only effect of multiplying the gauge field effective action by a factor of Nf.
Similarly, the gauge field background and fluctuations can be decomposed in terms
of the states in the adjoint representation of G. Some of the components point along
the Cartan generators hi, and some along the root directions E.:
T
A = q - hAU(l,
a=Z
i=1
a hi + E
a. E..
(2.137)
aEroots
The hi and EQ are defined such that they satisfy the standard commutation relations
[hj, hy] = 0
[hi, Ea] = ai Ea.
(2.138)
As in the U(Nc) case, the large Nf expansion is equivalent to an expansion in the
gauge field fluctuations a,. To leading order in Nf we can thus treat the gauge field
110
as a background and write the action for the fermions in the background A as
So [A; V), I] =
d3X~fg- E 7/4 iV + q w qU(1)>)bw.
W
(2.139)
Since this action does not mix fermions with different weights w, the Green's function
takes the form
(Vipw(x) ),(x'))
=
6wwGq.w(x, x') ,
(2.140)
where Gq.w(x, x') is as defined in (2.23).
The corrections to (2.139) come from the coupling between the fermions and the
gauge fluctuations, which is
Sint [a, Of, 0] = jd3x V9 E (w1w',Iw)
jww,,, ,
(2.141)
In complete analogy with the discussion of Section 2.3.2, we can obtain the effective
action for the gauge field fluctuations by integrating out the fermions. As in (2.22),
it will be useful to define the kernel
Ku"vv,., (X, Y)
V
UVIU Wjsw(Y))conn I
(2.142)
and rewrite this kernel in terms of the single fermion Green's function using (2.140).
We have
Ku",w,(x, y) = Nf 6v.6u, C",L,q.u(x, y) ,
with K
(2.143)
.qb (x, y) defined in (2.25). Finally, using (2.141), we can write the effective
111
action for the gauge fluctuations as
Seff[a]
=
d3xd3 y Vfg(x) /g (y) 1:(w'l a,(x) Iw) KM",.(x, y) (w Iav (y) Iw').
=
S([2]
) +S([a] + - --
Nf tr log(i
w,1j!
(2.144)
This expression is the analog of (2.26) from the U(Nc) case. We can be more explicit
and decompose the gauge fluctuations in terms of internal directions as in (2.137).
Using the fact that the states 1w) are orthonormal, we can write
)
a' (w'|Ec w),
(w'|a"(x)Iw) = all -w Jw + E
(2.145)
aEroots
where the short-hand notation a w
ai wi involves only a sum over the Cartan
=
components. Combining (2.145) with (2.144), we see that the cross terms between the
fluctuations along the Cartan and root directions are proportional to (wIEIw), which
vanishes, so these two sets of fluctuations decouple from each other. Furthermore,
the gauge field fluctuations in different root directions do not mix either, because
#
if
(w'|EaIw)(wIEa
3 Iw')
a or &'
w - a,
=
I
if 3 = -a and w' = w - a.
I(w - aE.Iw)12
(2.146)
We finally obtain:
Sf [a] =
+d
+ 1
aEroots
dy yx g(x)
2 1
aa,p(x)
g(y)
|(w-a|Ea lw)1
"wq.w(x, y)
(wwim
aj,g(x)
2
(
X
aj,,(y)
-
(
-~~i,j=1
y) a.,g(y).
Y
qw~.wc)(7Y
\
(2.147)
Note that this expression can be computed using only the kernel KI"A" (x, y) analyzed
112
in Section 2.4.
Having found the effective action for the gauge field, we can now compute the free
energy on S 2 x R by evaluating the path integral on this space in the saddle point
approximation. Let us first examine the fermion determinant term in (2.144), as this
term gives the leading contribution to the free energy. As can be seen from (2.139),
So[A; V), 44] decomposes into a sum where each fermion 0,/-
is only coupled to an
Abelian monopole background of charge q -w. In analogy with (2.38), we obtain
Fo[A] = E Fo(q -w)
(2.148)
,
where Fo(q) is the same quantity as defined in (2.47) that is equal to the ground
state energy of a single fermion in an Abelian monopole background of charge q. To
leading order in Nf, the ground state energy on S2 x R, or equivalently the scaling
dimension of the corresponding GNO monopole operator, equals
(2.149)
A = NfFo[A] + O(Ny).
As in the U(Nc) case, the O(NO) contribution to the scaling dimension (2.149)
receives contributions both from the Faddeev-Popov ghosts and from the gauge field
fluctuations. Let us start by examining the Faddeev-Popov contribution, which can
be computed from a generalization of the path integral in (2.49):
ZFP[A] =
Dc e-Sghost
Sghost
=
!
3X
lg
K&c -
i[Ac,
C]
IAC
-
i[A., c],
(2.150)
where c is an anti-commuting scalar ghost valued in the Lie algebra. In (2.150), (-1.) is
the standard inner product on the Lie algebra, defined such that (see for example [87])
(hiIhj) = J6j ,
(hilEa) = 0 ,
(EaIEa) = Jp6 .
(2.151)
Transforming in the adjoint representation of the gauge group, the ghosts c can be
113
decomposed just like the gauge field fluctuations in (2.137) into components along
the Cartan and components along the root directions:
r
c hi + 1
c=
c Ec.
(2.152)
aEroots
i=1
Using the commutators (2.138) and the normalization conditions (2.151), we obtain
the analog of (2.50):
Sghost
=
1
I
3
[
I
r
X
g(x)
l
U(1) C1 2
2
aEroots
i=1
(2.153)
Following the same steps as in Section 2.4.2, we find that the ghost contribution to
F is:
FFP[A] = -
[
r
(2J + 1) log [j(J + 1) +Q 2]
J=O
+
E
(2J + 1) log [J(J+ 1) - (q. a)2 + Q2]
aEroots J=q.a
(2.154)
As can be seen from the first line of this expression, the r ghosts in the Cartan
directions give equal contributions.
To obtain the full O(N2) correction to (2.149), we should also include the contribution coming from the gauge field fluctuations. We can split this contribution
as
Fgauge [Al
= FCartan +
Froot(a) ,
(2.155)
aEroots
where
Feartan is
obtained by integrating out the Cartan components of the gauge field,
and each Froot(a) comes from the component along the root a.
In general, all the Cartan contributions mix with one another. Performing the
114
same decomposition in vector spherical harmonics as in Section 2.4.3, we obtain
Fcatan
1T log
WiW
,Cq.w,q.w
/
=
J(2J+1)ogdet' (Z
W(2.156)
wiwK .w,q.w(Q)
J=1
where K is the same object as the one appearing in the U(Nc) result (2.85). For each J
and Q we now have to calculate the determinant of the matrix E
WjWy
Kq.,q.w ().
The dimension of this matrix is 3r x 3r because there are 3 spatial directions (or
equivalently, there are 3 vector harmonics UQ,JM, VQ,JM, and WQ,JM), and r Cartan
elements. As a sanity check, we note that all fluctuations in the Cartan directions are
regular vector fields on S2 x R, and therefore they can be decomposed in terms of the
Q = 0 vector spherical harmonics. The total angular momentum J hence takes integer
values. We also note that E wiw, K' .,q.w(Q) has r vanishing eigenvalues, as required
by gauge invariance, and that the contribution of the r uncharged ghosts exhibited
explicitly in the first line of (2.154) cancels the integrand in (2.156) asymptotically
at large J and Q, just as in the U(Nc) case.
As mentioned before, the fluctuations in the root directions decouple and can be
examined individually. Each such fluctuation gives a contribution equal to
Froot(a)
I Tr' log
(z
E
|(w - aIEaIw)| 2
(2J+1)logdet' (
I(w-
aEaW)2 K.w,.(w_)(Q) )
(2.157)
Unlike in the Cartan case, here we need to add the matrices K instead of taking their
tensor product, so in evaluating (2.157) we only need to calculate the eigenvalues of
a 3 x 3 matrix. A sanity check in this case is that the vector spherical harmonics that
appear in the decomposition of
Kq.w,q.(w-a)
have the same Q = q- w - q- (w - a) = q- a
for all the terms of the sum over w in (2.157). For every root there is a ghost that
115
cancels the integrand in (2.157) asymptotically at large J and Q.
We should emphasize that in preparing the results presented in Section 2.6 in the
U(Nc) case, we calculated the matrices Ki,(Q) for all
jqj
2 and Iq'I
2. It is
not hard to use the same matrices combined with the needed group theory data in
(2.156)-(2.157) in order to calculate the scaling dimension of any monopole operator
that has
Iq-wl
< 2 for all weights w of the matter representation.
We can also see easily which GNO backgrounds are unstable. The instability only
arises for the gauge fluctuations in the root directions for which
lowest J = Iq - al - 1 mode, K j7(q)a
(
Iq- al
> 1. For the
is simply a number, as the VQ,JM and
WQ,jm modes in (2.65) are absent. Hence the condition of stability is
0 <
Here, K
4
>
(w - aIEaIw)12 K"*(
-'w)(0) -
(2.158)
_a) is evaluated at Q = 0 because, as one can check, it is a monotonically
increasing function of Q; it is thus sufficient to check its sign for Q = 0. The expression
on the right-hand side of this equation can be easily evaluated in particular cases.
We now provide a few examples.
2.8.2
Examples
In this subsection we use the formulae derived in the previous subsection to obtain
some of the monopole dimensions for various gauge groups. We start with G = U(Nc)
and demonstrate the equivalence of the results obtained in the previous subsection
with those in the previous parts of the chapter. We then move on to discuss several
gauge groups with rank r = 1 and 2.
Another perspective on U(Nc) QCD with Nf fundamental fermions
As a first example, let us see how the G = U(Nc) results presented in Sections 2.5 and
2.6 fit within the general group framework of this section. The Nf fermions transform
116
in the fundamental representation of U(Nc), whose weight vectors w are
(2.159)
Wa = ea,
where a = 1, . . . , Nc, and the ea form the standard unit frame in RNc. In components,
eq = 6j. The adjoint representation has N, Cartan elements hi, as well as Nroots
a a= =e -
a:34b.
e ,
(We could identify the Cartan elements with aa
(2.160)
0.) Since the set of all possible
=
weights is the lattice ZNc, the Dirac quantization condition implies that q E (Z/2)Nc
In other words, the GNO monopoles are indexed by N, charges q, that are halfintegers.
Using the weights (2.159) and roots (2.160) in (2.148), (2.154), (2.156), and
(2.156), one can straightforwardly reproduce the U(Nc) formulae in (2.90) and (2.91).
In doing so, it is helpful to note that in (2.157) we have I(wC - aablEabIWc)
= 6 bc
SU(2) QCD with fundamental fermions
Our second example is where G = SU(2) with Nf fermions transforming in the
fundamental representation of SU(2). The group SU(2) has rank r = 1, so its roots
and weights are simply numbers. The weights of the fundamental representation are
w1 =-,
2
2
=--
1
2
.
(2.161)
The adjoint consists of a Cartan element and two generators with roots
a1 = w1 - w 2 = 1,
a2l
2 _ w = -1.
(2.162)
The weight lattice is generated by the fundamental weights (2.161) and is therefore
Z/2. The monopole charges that satisfy the condition (2.135) are all q E Z, modulo
117
the Weyl group-see Figure 2-7. The Weyl group consists of Z2 reflections about the
origin, so the monopoles with charge q and -q should be identified. Concretely, we
can think of a monopole with charge q as the background where
q/2
0
(±1 - cos9)do.
A=
0
(2.163)
-q/2
Figure 2-7: The weight lattice of SU(2) (blue dots) as well as the lattice of all possible
monopole charges (dotted circles). The monopole charges are defined modulo the
action of the Weyl group, which in this case is Z 2 and acts as reflections about the
origin.
All these monopole backgrounds are topologically trivial because 7r 1 (SU(2)) is also
trivial. However, they are all stable because the stability condition (2.158) reduces
to Kq/2,-q/2(O) > 0, which can be seen to be true from Figure 2-6.
The scaling
dimensions are
A = 2Fo(q/2)Nf + [6F(q/2, q/2) + 26F(q/2, -q/2)]
+ 0(1/N 1 ).
(2.164)
The numerical values of Fo(q/2) as well as 6F(q/2, tq/2) can be read out from Tables 2.1 and 2.2. See Figure 2-8 for specific examples.
SO(3) QCD with fundamental fermions
Our third example is where the gauge group is G = SO(3). The Lie algebra of SO(3)
is identical to that of its double covering, which is SU(2), but SO(3) has fewer allowed
representations than SU(2). In particular, the spinor representation considered in the
previous subsection is not a representation of SO(3). The smallest irrep of SO(3) is
118
&
A&%
IV~.
MW
Symbol
A
*
0.530Nf - 0.713 + O(1/Nf)
L
1.35Nf - 1.65 + O(1/Nf)
*
2.37Nf - 2.95 + O(1/Nf)
A
3.57Nf - 4.51 + 9(1/Nf)
*
60-
Figure 2-8: The SU(2) monopoles appearing as black dotted circles in Figure 2-7. In
the presence of Nf fundamental fermions these backgrounds are all stable, and we
list the scaling dimensions A of the corresponding monopole operators.
the fundamental, which in this case is the same as the adjoint. The weights are
w 1 = 1,
w 2 = 0,
w 3 = -1,
(2.165)
where w 2 corresponds to the Cartan element, and wi and w 3 to the roots.
The weight lattice of SO(3) is generated by the fundamental weights (2.165), so it
can be identified with Z. It is a subset of the weight lattice of SU(2). The quantization
condition (2.135) implies that in SO(3) the allowed values of q are q E Z/2, and not
just q E Z as was the case for SU(2). See Figure 2-9. As in the case of SU(2), the
qiw
Figure 2-9: The SO(3) weight lattice (blue dots) and its dual lattice (dashed circles).
The weight lattice is a sublattice of the SU(2) weight lattice in Figure 2-7. The
dual lattice contains more monopole charges q than the dual lattice of SU(2). As
in the SU(2) case, the Weyl acts by reflections about the origin, so it provides the
identification q
-
-q on the set of monopole charges.
Weyl group acts by reflections about the origin, so we should identify the monopoles
with +q. Unlike the case of SU(2), however, we now have a non-trivial fundamental
group, as 7r 1 (SO(3))
=
Z 2 . The topological charge is (2q) mod 2, so the extra values
of q that are allowed in SO(3) but not allowed in SU(2) correspond to topologically
non-trivial monopole backgrounds.
119
A
Symbol
10.530Nf - 0.554 + 0(1/Nf)
0
Unstable
Figure 2-10: The SO(3) monopoles appearing as black dotted circles in Figure 2-9.
Here, we consider these backgrounds in the presence of N1 fermions transforming
in the three-dimensional fundamental representation of SO(3). The orange circles
correspond to unstable backgrounds. For the stable backgrounds (represented in black
by various shapes), we list the scaling dimensions A of the corresponding monopole
operators.
The stability condition (2.158) reduces to KO,0 (0) > 0 in this case. As can be seen
from Figure 2-6, the only stable monopole background is that with Iql = 1/2. Note
that this stable monopole background is also topologically non-trivial. The scaling
dimension of the corresponding monopole operator is
A = 2Fo(1/2)Nf +
[SF(1/2,
1/2) + 26F(I
/2, 0)] + 0(1/Nf).
(2.166)
See Figure 2-10.
SU(3) QCD with fundamental fermions
Our next example is QCD with gauge group SU(3) and N1 fermions in the fundamental representation. The rank of SU(3) is r = 2, so the roots and weights are
points in R2
The weights of the fundamental representation are
=
1
(2
,
w2
=
-2
3 = (0,
6
3)
(2.167)
The adjoint consists of two Cartan generators, as well as six roots given by
± (1,7 0) ,
E
(
1 -4
2' 2
120
J
1
(2.168)
The weight lattice is generated by the fundamental weights (2.167). Dirac quantization implies that the monopole charges belong to a lattice generated by
1
2
(1,0) ,
f3
2
(2.169)
This lattice is the weight lattice of the GNO dual group SU(3)/Z 3. See Figure 2-11.
q 2 /W 2
Wi
0
*
0
0
0
We
3
0
0
S
S
We
0
2
*1
S
0
WI
.
0
WI
et
S
0
*
S
*
0
WI
0
0
0
0
~av~
0
S
0
0
WI
0
0
0
0
WI
S
0
*
0
S
0
0
0
0
0
.0
*
S
0
S
0
*1
0
S
S
*
0
6
WI
S
0
0
0
0
0
0
0
0
*
S
0
-34
.4
0
WI
.- 2
0
0
0
0
0
Wi
0
0
0
0
0
*
WI
0
We
WI
*
*
*
Figure 2-11: The weight lattice of SU(3) (blue dots) as well as the lattice of all
possible monopole charges (dotted circles). The monopole charges are defined modulo
the action of the Weyl group, which in this case is S3 and is generated by 120 degree
rotations as well as reflections about the q2 axis.
In this case, the Weyl group is S3 and is generated by rotations of 27r/3 and
reflections about the q2 axis.
The independent monopoles are thus given by the
points (qi, q2) in the monopole charge lattice modded out by the action of S3 . See
Figure 2-12 for which of these monopole configurations are stable and for the scaling dimensions of the monopole operators corresponding to the stable backgrounds
pictured. Since 7r,(SU(3)) is trivial, none of these monopoles carry any non-trivial
topological quantum numbers.
121
q2
o
o
*A
*
0
*
o
0
0
0
0
3
0
* 0
0 0 0
A
*0
... .... ....
+
0
*
A
o
0
* -3
0
0
0
+
0
_f
O
+
0
0
*-1
-
A
1.75 + O(l/Nf)
*
0.530Nf
l
1.20Nf
-
2.55 + O(1/Nf)
*
2.12Nf
-
5.38 + 0 (1l/Nf)
A
3.13Nf - 7.20 + O(1/Nf)
0
Unstable
-
0
0
0
A
Symbol
0
0
Figure 2-12: The SU(3) monopoles appearing as black dotted circles in Figure 2-11.
Here, we consider these backgrounds in the presence of Nf fermions transforming
in the three-dimensional fundamental representation of SU(3). The orange circles
correspond to unstable backgrounds. For the stable backgrounds (represented in black
by various shapes), we list the scaling dimensions A of the corresponding monopole
operators.
Sp(4) QCD with fundamental fermions
We can also consider the gauge group G = Sp(4) and Nf fermions transforming in
the four-dimensional fundamental representation of this gauge group. The rank of
2
Sp(4) is r = 2, so again the roots and weights are points in R . The weights of the
fundamental representation are
1 0
1 )
t V2- .
(2.170)
The adjoint consists of two Cartan elements as well as eight roots:
t v2 V2
{ 1
Q
i (V2, 0) ,1
122
± 0, ,2)
(2.171)
The charge lattice in this case is the same as the weight lattice, and is generated by:
1 0
,2 ) 1,
oj1
v, ). .
(2.172)
After scaling, the charge basis vectors also generate the weight lattice of SO(5), which
indeed is the GNO dual of Sp(4). See Figure 2-13.
NJ
1
'I
F
2
V.
0t
WI
147
W'
W
TV
&
WI
W
qllwl
-1
WIWI*~NJ
p
I
WI
Figure 2-13: The weight lattice of Sp(4) (blue dots) as well as the lattice of all possible
monopole charges (dotted circles). The monopole charges are defined modulo the
action of the Weyl group, which in this case is (Z 2 ) 3 and is generated by reflections
about the qi axis, q2 axis, and the line that makes a 45 degree angle with the qi axis.
Sp(4) is simply connected, so all GNO monopoles have trivial topological charge.
The stability of various monopoles along with their scaling dimensions are included
in Figure 2-14.
123
q2
0
0
N
0
V
0
0
V
A
+1
0
0
*
+
'~2
-~~'
0
W -
*
V
O O _
0.530N, - 1.75 + O(l/Nf)
l
1.06Nf - 2.18 + O(1/Nf)
*
1.88Nf - 4.29 + O(1/Nf)
A
2.69Nf - 5.16 + O(1/Nf)
V
3.72Nf - 7.87 + O(1/Nf)
3
.o
M
0
.
.
. . . . .
O VA+
0
0o
0
I'W
A
0
V o
a
0+
* o
*
E
Symbol
13
+
0
0
o
4.75Nf - 9.27 + O(1/Nf)
*
A
V 0
*
5.95Nf - 12.4 + O(1/Nf)
o
V
0
_
7.15Nf - 14.2 + O(1/Nf)
0o 0
N
N
o
Unstable
Figure 2-14: The Sp( 4 ) monopoles appearing as black dotted circles in Figure 2-13.
Here, we consider these backgrounds in the presence of Nf fermions transforming in
4
the four-dimensional fundamental representation of Sp( ). The orange circles correspond to unstable backgrounds. For the stable backgrounds (represented in black
by various shapes), we list the scaling dimensions A of the corresponding monopole
operators.
SO(5) QCD with fundamental fermions
Moving onto G = SO(5) with Nf fundamental fermions, the weights of the fundamental representation are:
t (1,0)
(2.173)
(0,0).
t(0,1),
,
The adjoint consists of two generators in the Cartan, as well as generators with roots
t (1, 1) ,
+ (1, 0)
i (1, -1)),
,
+ (0, 1) .
(2.174)
The charge lattice is generated by:
,0
0,
,
124
) .
(2.175)
See Figure 2-15.
q2/W2
0
0)
0.
0
0 .
0
Ce
0
0
~..
*...'
C'
~~1
~
Ci
~~1
~*i
Ci
".
-
q J.
...
I
0
o c~
~.i
o
'.a
C'
*
Ci
1;
-It
Figure 2-15: The weight lattice of SO(5) (blue dots) as well as the lattice of all
possible monopole charges (dotted circles). The monopole charges are defined modulo
the action of the Weyl group, which, as in the Sp( 4 ) case, can be identified with the
(Z 2 )3 generated by reflections about the qi axis, q2 axis, and the line that makes a 45
degree angle with the qi axis.
In this case the fundament group is non-trivial,
7r1 (SO(5))
= Z 2 . The topological
charge of a monopole with GNO charges qi, q2 is (2q, + 2q 2 ) mod 2. For SO(5), there
are only two stable monopoles. Monopoles of various charges are plotted in Figure
2-16.
G2 QCD with fundamental fermions
Lastly, we consider G = G 2 and Nf fermions transforming in the seven-dimensional
fundamental representation of G 2 . The weights of the fundamental representation are
+ (0, 1)
12
t 2 ' 2)
2v
2 '
125
12
2,'
(0,0) .
(2.176)
q2
0
0
0
0
0
0
0
0
0
0
0
0
M .
.1!!I.
'11
00
a,.
1
0
0
0
0
0
0
0
0
0
0
0
0
x
S0 0
0
0
04 0
0
1W
0-s
0
0
a.
0
0
0
0
0
0
0
0
-
1
4
-
Af. M
MW Y
-
.
-
0
30
0
0
0
A
Symbol
*
0.530Nf
L
1.06Nf
-
1.59 + O(l/Nf)
M7
0
0
0
0
0
0
0
0
0
0
0
0
0
-
1.86 + O(1/Nf)
Unstable
0
00-1
Figure 2-16: The SO(5) monopoles appearing as black dotted circles in Figure 2-15.
Here, we consider these backgrounds in the presence of Nf fermions transforming
in the five-dimensional fundamental representation of SO(5). The orange circles
correspond to unstable backgrounds. For the stable backgrounds (represented in black
by various shapes), we list the scaling dimensions A of the corresponding monopole
operators.
The adjoint representation is fourteen-dimensional and consists of two Cartan elements as well as the roots
t(0, 1) '
1
23 ' 2) 'I
t
,3 3
2)
2
12
2)
2
3
2)
(2.177)
± (v, 0)
,
The set of all possible monopole charges is generated by the vectors
( 1\)
,30)
vf3 1
'2
' 2J.
(2.178)
After scaling and rotating, the charge lattice is identical to the weight lattice, reflecting the fact that G2 is its own GNO dual. See Figure 2-17. Here, the Weyl group
is D6 (the dihedral group with 12 elements), which is generated by rotations by 60
126
degrees as well as reflections about the line that makes 45 degrees with the qi axis.
q2/w2
0*000
0
o c *
WI
0
WI
o o
0
~-'
-2'
0
o o
*
00 o
*t-2
'~1
oS
o1
0.
O
00
S
-*
O
0
1
0
10
0
WI
~,-I
[ 0
-*
~.I
0
?.~I
*
*.~l
0
Wi
0
q 1/w
0
*
Figure 2-17: The weight lattice of G2 (blue dots) as well as the lattice of all possible
monopole charges (dotted circles). The monopole charges are defined modulo the
action of the Weyl group, which in this case is D6 (the dihedral group of order 12)
and is generated by 60 degree rotations as well as reflections about the line that makes
a 45 degree angle with the qi axis.
G2 has a trivial fundamental group, and so there is no topological charge. The
stability of monopoles with different GNO charges as well as the dimensions of the
operators corresponding to the stable backgrounds are given in Figure 2-18.
127
q2
0
0
0
0
0
0
0
0
-
0
0
0 -0
0
0
0
0
0
0
o
0
0
0
*
*
0
0
0
I 1.06N -280+0(1/N)
1W
I' qW
A
Symbol
0
0
0
0
0
0
0
0
00
00
0
00
0
0
X)00
00
0
Unstable
0
0
0
Figure 2-18: The G2 monopoles appearing as black dotted circles in Figure 2-17. Here,
we consider these backgrounds in the presence of Nf fermions transforming in the
seven-dimensional fundamental representation of G2 . The orange circles correspond
to unstable backgrounds. For the stable backgrounds (represented in black by various
shapes), we list the scaling dimensions A of the corresponding monopole operators.
SU(3) QCD with adjoint and fundamental fermions
While in all of our previous examples, the matter fields were in the fundamental representation of the gauge group, we can also consider matter in other representations.
In this example we consider fermions that transform in the adjoint representation
of SU(3). The weights of the adjoint are just the root vectors (2.168). The set of
possible monopoles is independent of the matter representations, and so the charge
lattice is still generated by (2.169) and is shown in Figure 2-11.
23
Unfortunately, for Nf copies of the adjoint, there are no stable monopoles. The
absence of stable monopoles does not mean that the adjoint representation is unin-
teresting, however. There is no reason to restrict to matter in an irreducible representation, and we can consider theories with nadjNf adjoint fermions, and nfundNf
fundamentals. For nadj < nfund this theory should have many stable monopoles, as
The absence of stable monopoles is a common feature of larger representations of any gauge
group.
23
128
is the case for SU(3) with only fundamental matter. 2 4 For nadj > nfund the theory
should behave more like the theory with only adjoint matter, and have no stable
monopoles. Below we plot the number of stable monopoles as a function of the ratio
nfund/fnadj.
For small values of this ratio, the specific monopoles which become stable
are shown.
24 SU(3)
with only fundamental matter has infinitely many stable monopole backgrounds.
129
q2
Number of stable monopoles
1o1
_
2_0_00
0
0
0
0
01'0
- '
-!3 Z 2
o
01
0
o
0
o
o
0
0
0
0
0
0
0
0 2(
0
o
0
.......
.... ...
0
-3
0o
0
8
0
0
10
q1
0
0
0
0
q2
0
0
0
0
0S
0
*0
0
T0 2
-3
0
5
0
0
0
q2
0
0
0
0
0
0
0
oa
0
0
a
q
0
0
1
0
0
0
_I
0
0
0V
A
0
*
0
A
0
0
0
0
01-
1
0
02_
0
0
0
T
1
0
0
-3
0
0
0
-
-'
0
A
0
O
*|
0
0
0 0
+
0
-2
-1
0
* 0
*
Sa *
A
0I-
0 22 ~'1K~r~r
0
0
0
4
0
12
4
0
0
A
0
/~.
35
30
25
20
15
10
q2
0
-l --
I
0
0
0
II
3
0
0
0
0
0
Al
0-
0
0
0
-- V q,
-71
-2
3
61 .0
0
0
-
IV
0
0
0
0
0
0
0
0
0
IV
4
3
0
-3
0
0
0
4
0
0
A
0
0
0
0
0
0
-3
*
,
0
A
0
* 0
*
0
0
10
0
+
A
q,
0i
+
0
0
Figure 2-19: In the top right corner we show the number of (inequivalent) stable
monopoles for SU(3) gauge theory with nfndNf fundamental fermions and najNf
adjoint fermions as a function of the ratio nfund/nadj. The solid line is divided into
five regions that correspond to the diagrams on the left and bottom, where we show
explicitly which monopoles are stable in each region. The dashed line is a continuation
of the solid line for larger values of nfund/naj, but in this region we do not show
explicitly which monopoles are stable.
130
2.9
Discussion
2.9.1
Summary
In this chapter we studied properties of monopole operators in non-supersymmetric
QCD 3 and QED 3 with Nf fermion flavors. We worked in the limit of large Nf, where
gauge field fluctuations are suppressed and where the theory flows in the infrared
to an interacting CFT. At this infrared fixed point, we used the state-operator
correspondence to first define the monopole operators in terms of energy eigenstates
on S 2 x R and then to study their transformation properties under the conformal
and flavor symmetry groups. As we emphasized in Section 2.2, associating energy
eigenstates with certain GNO monopole backgrounds can. be done cleanly in the limit
of large Nf, provided that these GNO backgrounds are stable saddle points of the
effective action for the gauge field fluctuations.
We obtain three main results. Our first result is that only certain GNO monopole
backgrounds are stable saddle points in the CFT. In general, stability is a dynamical
issue that can only be decided by studying the effective potential for the gauge field
fluctuations. We provided the criterion for stability in Section 2.5 in the case where
the gauge group is U(Nc) (see Figure 2-6 for a summary plot). We later generalized
this criterion to theories with other gauge groups in Section 2.8. In all these theories,
we were thus able to identify precisely for which sets of GNO charges one can define
independent monopole operators, at least at large N1 . We found that many, but
not all, GNO backgrounds in each topological sector are stable. For every stable
background there is a Fock space of energy eigenstates on S2 x R whose wavefunctions
are localized around that background. Each such energy eigenstate corresponds to an
operator on R 3 . We further focused on the lowest energy eigenstate within every Fock
space and studied its quantum numbers. We referred to the operator that corresponds
to this state as a bare monopole operator.
Our second result is that we computed the scaling dimensions of the bare GNO
monopole operators in the 1/Nf expansion. The scaling dimension is a quantity
determined by the dynamics. It equals the ground state energy on S2 x R in the
131
GNO monopole background. We obtained the ground state energy by evaluating the
path integral on S2 x R to subleading order in the 1/N expansion. For large N the
monopole operators have O(Nf) dimension
A = Nf Fo + 5F +O(1/Nf)
(2.179)
Explicit results for A for various GNO charges can be found in Section 2.6 for U(Nc)
theories, and in Section 2.8 for general gauge groups. We expect the results we
obtained from the large N1 expansion to be reliable down to fairly small values N1 >
JF/Fo = 0(1).
This expectation is supported by the high accuracy of large Nf
computations for supersymmetric theories [88], where the answer can be compared
to exact results.
Our third result is that for the case where the gauge group is G = U(Nc), we
calculated all the other quantum numbers of the bare monopole operators. We found
that these operators are all spin singlets and that they transform in the irreducible
representations of the SU(Nf) flavor symmetry group given by the Young diagrams
N/2{IE
(2.180)
2 E.|qal
where the GNO charges are {qi, q2,
...
,
qN}. We can therefore completely charac-
terize the quantum numbers of the bare monopole operators in U(Nc) QCD with
Nf fundamental flavors: We know their topological charge, scaling dimension, spin,
and representation of the flavor symmetry group. It would be very interesting to
generalize this analysis to other gauge groups and matter representations.
These results are interesting in their own right as they teach us about the operator
content of QCD 3 . Using the knowledge that we gained, it is desirable to understand
the role that the monopole operators play in the dynamics of the theory.
132
2.9.2
Confinement and chiral symmetry breaking
From our results, we can learn about the following three theories, one of which describes confinement:
I. We can consider Yang-Mills theory coupled to Nf flavors of massless fermions.
This theory is super-renormalizable and asymptotically free, so it is well-defined
up to arbitrarily high energies. If we wish, we could think of it as an effective
field theory at large distances that arises from a lattice Hamiltonian that does
not allow mass terms for the fermions.
II. We can also consider a non-trivial interacting CFT. At large Nf, we can define
this CFT by erasing the Yang-Mills term from the action of (I), as we did
throughout this chapter. This description of the CFT should make sense as long
as this CFT can be achieved from (I) without any fine tuning, which should
happen for all Nf greater than or equal to some number N d*"f that we will
estimate shortly. Below Nj~onf, a non-trivial CFT may still exist, but a good
description for it may not be readily available.
III. A confined or partially confined theory, potentially with some number of Goldstone bosons coming from spontaneous flavor symmetry breaking. The description of this theory is intentionally vague, as it should be viewed just as an
alternative to (II) for describing the IR physics of (I).
Recall that we restrict our discussion to the case where the number of fermions is
even, because otherwise we would necessarily be breaking parity [81,82].
The first question we can ask is: When is the infrared physics of generic RG flows
starting from (I) described by the deconfined CFT (II), and when is it not? In other
words, we should estimate Nd*nf from the fact that for Nf ;> N deconf, all monopole
operators should be irrelevant, i.e. their scaling dimensions should be greater than
three. From (2.179) we find:
Nf
-O F
133
.
(2.181)
Here, the values of FO and 6F correspond to the monopole operator with the lowest
scaling dimension for a given gauge group and matter content. See Table 2.4 for a
Gauge group
N
"o
U(1)
12
U(2)
14
SU(2)
8
SO(3)
8
SU(3)
10
Sp(4)
10
SO(5)
10
G2
6
Table 2.4: Estimates of the smallest number of fermions, Nfeconf for which the IR
of QCD 3 with gauge group G is in a deconfined quantum critical point. Results are
listed for various rank one and two gauge groups.
few particular cases. As can be seen from this table, N econf is smaller for groups with
fewer monopoles.
We stress that the estimate (2.181) as well as the numbers given in Table 2.4 are
not relying on any assumptions about the physics at Nf < N
econf.
All we can tell
for sure is that in this case, Yang-Mills theory with Nf fermions does not generically
flow to the deconfined CFT (II).
It is possible to obtain an independent estimate of N d,,"f
sumptions about what happens for Nf < N
econf.
if we make some as-
We will do so only in the cases
where the gauge group is G = U(1) and U(2), and leave a more extensive analysis for
future work. As reviewed in [89], one expects that below Nof"", the SU(N) global
flavor symmetry should be broken to SU(Nf/2) x SU(Nf/2) x U(1). 2 5
25
A simple
1n [90,91] it is shown that if the number of fermions is N1 > 6 there must be massless particles
in the infrared. It is likely that these particles are Goldstone bosons corresponding to the symmetry
breaking pattern mentioned in the main text. This symmetry breaking pattern is usually referred
to as chiral symmetry breaking, even though there is no chiral symmetry for fermions in three
dimensions. The name "chiral symmetry" comes from the fact that if the same theory were realized
134
computation shows that the number of Goldstone bosons is
(2.182)
2
NG
2
As pointed out in [89], such a symmetry breaking pattern is constrained by the Ftheorem [92-95] ,26 which states that any three-dimensional Lorentz-invariant RG flow
from a UV CFT to an IR CFT should satisfy
(2.183)
FUV ;> FIR,
where Fuv (FIR) is the S3 free energy of the UV (IR) CFT.
To use the F-theorem, we can consider starting with Yang-Mills theory with
N econf - 2 fermions, which by assumption is the largest value of N for which
Nf
the IR theory consists of N2/2 Goldstone bosons. It is likely that the same IR theory
of Goldstone bosons can be obtained by starting with the CFT in (II) with N'eonf
fermions and giving masses to two of them. These masses should be of opposite sign
in order to preserve parity. The latter flow is the one for which we will use the Ftheorem. The F-theorem should of course hold for the flow from Yang-Mills theory
with Nf fermions to the theory of N'/2 Goldstone bosons as well, but in the UV
Yang-Mills theory is not conformal and should be assigned Fuv = oo.
For the flow between the deconfined CFT with N
econf
fermions and the IR theory
of N2/2 Goldstone bosons, Fuv can be read off from the results of [96]:
Fuv
(
+ 87r2
Nc Neconf
grN+deconf
N2
Vol(U(Nc))
Vol (U(1))NC + O(1/Nd"onf),
(2.184)
in 4d by pairing up the N1 Weyl spinors into N 1 /2 Dirac spinors, then the broken symmetry would
be chiral.
26
Note that this F stands for the S 3 free energy, and should not be confused with the S2 x R
partition function that was discussed in this chapter.
135
where for N, = 1, 2 we should use Vol (U(1)) = 27r and Vol (U(2)) = 87r3. Because
the IR theory is a CFT of Nf/2 free scalar fields, we have
N2
FIR =
N
2
(2.185)
Fscalar ,
with Fscaiar ~~0.0638 being the S3 free energy of a single real scalar field [93].
Using (2.184) and (2.185), we see that the F-theorem inequality (2.183) holds
for N donf
<
12 in the U(1) case and NeConf < 20 in the U(2) case. 2 7 This result
is consistent with, but less precise than, the values N
eonf
-
12 and N
econf
-
14
for G = U(1) and U(2), respectively, that we obtained from studying the scaling
dimensions of the monopole operators.
It should be stressed that the SU(Nf) global symmetry cannot be broken by the
same monopoles which cause confinement. If an operator spontaneously breaks a
symmetry, it necessarily is not a singlet under this symmetry, and so will not be
generated under RG flow. The connection, therefore, between the prediction from
chiral symmetry-breaking and that from monopole relevance is not clear.
2.9.3
QED and and algebraic spin liquids
The analysis of the previous subsection on the minimal value of Nf for which YangMills theory with Nf fermions flows to a deconfined phase considered only "generic"
such RG trajectories. This analysis can be refined in the case where ri(G) is nontrivial-and so certain monopole operators carry topological charges-by restricting
our attention to RG trajectories that are invariant under a subgroup of the corresponding topological symmetry. Under this extra assumption, Yang-Mills theory
with Nf fermions flows generically to a deconfined CFT provided that the monopole
operators that are invariant under the above subgroup are irrelevant; it does not matter whether the other ones are relevant or not. Consequently, the values of Nyeconf in
27
1n the U(1)
case, [89] obtained Nyeconf < 14 from considering a flow between supersymmetric
QED 3 and the symmetry broken phase of N'/2 Goldstone bosons. The bound Nfeco"f < 12 that we
obtain is more constraining than that of [89f, because we start from a UV theory with fewer degrees
of freedom.
136
this case would be smaller than the values obtained in the previous section.
The QED case N, = 1 provides a nice example relevant to algebraic spin liquids
[47,97]. In this case the topological symmetry is U(1)t 0 p, and the topological charge
is qtop = q E Z/2. It was suggested in [98] that if U(1) QED with Nf = 4 fermions
can be obtained as an effective theory of a spin system on the Kagome lattice, the
lattice symmetries are embedded into a Z 3 subgroup of U(1)t 0 p. So let us restrict our
attention to RG trajectories that preserve this Z3 subgroup as a symmetry. Under
the generator of this Z3 symmetry, a monopole of charge q is multiplied by a phase
equal to e4'7riq/3, so only monopole operators with q E 3Z/2 are invariant. If all
fermion mass terms are also forbidden by the lattice regularization, it then follows
that the IR theory is a deconfined CFT provided that all the monopole operators
with q E 3Z/2 are irrelevant. According to Table 2.1, these monopole operators have
scaling dimension greater than 3 for Nf > 4. This bound is less restrictive than
Nf > 12, which is what we obtained in the previous section by requiring that all
monopole operators should be irrelevant.
137
138
Chapter 3
Chiral Gauge Theory for Stringy
Backgrounds
In this chapter, we move down to two dimensions, using strongly coupled, chiral,
two-dimensional gauge theories, we construct candidate low-energy conformal field
theories that describe string backgrounds outside of the traditional Calabi-Yau framework.
3.1
Introduction
String theory has taught us a great deal about the quantum geometry of Calabi-Yau
(CY) manifolds. Central to this progress is the gauged linear sigma model (GLSM),
a formalism which translates quantum computations in Kdhler geometries into freefield-theory calculations in an auxiliary gauge theory [23].
Of course, CYs form a
set of measure zero in the full space of string compactifications, so it is natural to
wonder what we can say about the quantum geometry of more general non-CY manifolds. This question is particularly natural in the heterotic string, where a worldsheet
analysis should suffice. This motivates us to search for GLSMs for non-Ksihler' manifolds, with the goal of using them to study the quantum geometry of more general
'The term "balanced" is probably more appropriate [99], since all such 4d K = 1 compactifications come from balanced manifolds which may or may not be Kahler. The term "non-Kifhler" has
become standard, however, emphasizing that these manifolds need not be Kahler.
139
geometries.
At first glance, this seems quite challenging. Mathematically, the basic structure
of a GLSM is a Kdhler quotient of flat space, which naively should not be much
help in getting a non-Kihler manifold. Meanwhile, if the geometry is not Kdhler,
target space SUSY requires non-trivial 3-form torsion, H, which must satisfy the
Green-Schwarz (GS) Bianchi identity,
dH = a' (trR A R -TrF A F).
(3.1)
Correspondingly, the B-field itself must transform non-trivially under the full set of
spacetime gauge transformations. Furthermore, dH -/ 0 implies a non-trivial dilaton
profile [32], so that the worldsheet conformal symmetry is only realized non linearly.
Finally, the fact that tree- and one-loop effects compete in (3.1) means that some
cycles may be frozen near string scale, making a large-radius limit problematic. It
is difficult to see how all these effects could be incorporated into a GLSM. Indeed,
considerable effort has been devoted to adding bells and whistles to the GLSM to
mock these effects up [100-105].
In this chapter, we demonstrate a general mechanism for generating the modified Bianchi identity and related quantum effects in a garden-variety (0, 2) GLSM. 2
Surprisingly, this mechanism does not require any new ingredients. Rather, simple
quantum effects in every (0, 2) GLSM generate all the necessary features dynamically.
The basic mechanism first appeared in the study of 4d chiral gauge theories [41,42]. In
a chiral gauge theory, integrating out heavy fermions in chiral representations of the
gauge group will generically generate anomalous Green-Schwarz terms in the action.
These terms are essential for canceling the anomaly of the surviving light fermions.
In our (0, 2) GLSMs, such GS terms precisely generate the corrections to the B-field
transformation law which are required to satisfy the Bianchi identity, and which lead
to non-trivial H-flux and non-Kifhler hermitian metric.
The central claim of this chapter is that all of the features of a generic non2
For useful introductions and reviews of (0,2) sigma models see [106,107].
140
Kdhler model with H-flux can in fact be found within a standard (0, 2) GLSM. As
we shall see, non-trivial H-flux and a modified transformation law for the B-field
are automatically generated as needed by the mechanism sketched above. Indeed,
we will find that all of the various previously-preposed quasi-linear mechanisms for
generating H-flux in special cases emerge naturally when studying the low energy
effective physics of standard (0,2) GLSMs.
The rest of this chapter is organized as follows. In Section 2, we demonstrate how
the GS mechanism in a GLSM generates precisely the anomalous transformation law
for the B-field needed to ensure cancellation of the sigma model anomalies of the IR
NLSM. Along the way, we demonstrate that the GS models previously constructed
can thus be UV completed into completely pedestrian (0, 2) GLSMs; conversely, at
low energies and at generic points in the (0, 2) moduli space, a general (0, 2) GLSM
reduces to such an anomalous GS effective GLSM. In Section 3, we will study a
particularly simple class of such effective GLSMs in which the axial couplings are
entirely linear, and use them to explore the quantum consistency and semi-classical
geometry of such GLSMs. Furthermore, we find unanticipated topological constraints
on the existence of such models. The central ingredient in these models is a set of
Green-Schwarz axions, Y, playing the role of Stiickelberg fields for the anomalous
gauge symmetries. This allows us to avoid the subtleties associated with logarithms
and address both classical and quantum properties of the models. As we shall see,
these theories show every sign of running to good IR CFTs - more precisely, these
theories enjoy a (0, 2) superconformal algebra which closes in Q+-cohomology, as was
previously shown for the T 2 models in [101]. While the models on which we focus
are not generic, the lessons we learn can be readily applied to more general GLSMs
with non-trivial H-flux. We close in Section 4 with a summary and list of future
directions.
141
Generating dH in a (0, 2) GLSM
3.2
The goal of the present section is to demonstrate that all the ingredients necessary
to study models with non-trivial dH are already present in a garden-variety (0, 2)
GLSM. We begin by recalling how dH =4 0 and the modified Bianchi identity arise
in a general (0, 2) NLSM. We then review how these effect can be incorporated into
quasi-linear (0, 2) GLSMs by hand via Green-Schwarz anomaly cancellation, as first
demonstrated in [100,103]. We then argue that such quasi-linear models arise as effective descriptions of totally standard (and non-anomalous) (0, 2) GLSMs at generic
points in their moduli space. Concretely, moving along the (0,2) moduli space modifies masses for chirally-gauged fermions in the GLSM; integrating out the heaviest
fermions then generates non-linear Green-Schwarz terms in the action which realize
the non-trivial dH. This explains for example how dH $ 0 is generated in (0, 2)
deformations of (2,2) GLSMs, such as deformations of the tangent bundle. We also
comment briefly on the effective geometry of such models.
3.2.1
Torsion in (0, 2) NLSMs
Before we address the GLSM, let's recall how the modified Bianchi identity and the
c2 constraints arise in a non-linear sigma model (NLSM) with (0, 2) supersymmetry.
The action for a (0, 2) NLSM is given by [108]:
L
11
=
-
2
(G11 + B15) o9+'&0
j(flAP(&+7
y
where
(
2
/4 +
+ Fc±{IL&'9
2~ (+Gi(i
+
q$0'+ - (G1 - B11 ) a_#1a35
+ AI A CaO/I7)
+ _Y nAB(P+ Y-
+
WA-
JK
+i41 1 Op ++
+ GIL HLJK
=
GIL (KGJL+
+ A? 0 +s~
. ..
OJBKL ~ OKBJL)
(3.2)
and the ... refer
to four-fermi terms which we will not need for the moment.
By construction, this action is invariant under target space local Lorentz transfor142
mations, 3 target space gauge transformations, and target space Kalb-Ramond transformations:
()
-+
y_
-+U()B-
B -
B+ dwl.
(3.3)
Note that plugging these transformations into the Lagrangian does not leave the
action invariant - rather, we must supplement these transformations on the worldsheet
fields with a corresponding change in the background fields, G, B, n, A, and W. Only
under this combined transformation is the Lagrangian, classically, invariant.
Quantum mechanically, however, this is not in general a symmetry of the theory
due to an anomaly in the fermionic measure [109, 110]. Nonetheless, these theories
can be improved, order by order in a' [25-27], so as to respect a modified symmetry.
More precisely, while the theory is not invariant under (3.3), it is invariant under a
slightly different symmetry whose transformation rule for B is modified at order a',
JBj
where
6 ab,
M
~ a' (a[I
GMAM
- 0
w0)
(3.4)
parameterize infinitesimal local Lorentz transformations and gauge
transformations respectively. With this transformation law, Ho = dB is no longer
gauge invariant. Correspondingly, we must also modify the definition of H at order
a,
H = Ho + a' (Q 3 (W) - Q
3 (A)) ,
(3.5)
where Q3 are the Chern-Simons three forms for the spin and gauge connections. The
3
V
For the local Lorentz transformations it is convenient to introduce a vielbein, ey and define
a)
143
redefined H gives the modified Bianchi identity,
dH = a'(trR A R - TrF A F).
This is the worldsheet manifestation of the Green-Schwarz mechanism [40].
(3.6)
Note
that if we wish to construct a theory with a non-anomalous symmetry that reduces
to the classical transformations as a' -+ 0, these modifications are unavoidable. As
emphasized in [27] care must be used to see that supersymmetry is preserved.
In principle, then, there is no obstruction to studying non-Kdhler geometries with
intrinsic torsion via a worldsheet NLSM - the NLSM itself is perfectly well-posed,
supersymmetric, and non-anomalous under all the symmetries of interest. In practice,
however, this construction is not very useful for many computational purposes. First,
as usual with a NLSM, most things we would like to compute end up depending on
the physical metric; however, finding a solution to the Einstein equation and modified
Bianchi is even more difficult than finding a compact Calabi-Yau metric (and indeed
we still do not have a general proof of the existence of solutions to these equations
except in a few very special cases [111]). Second, since the Bianchi identity mixes
orders in a', it is not clear when, or if, perturbation theory around a classical solution
even makes sense. Computationally, then, the NLSM is just not enough.
3.2.2
Adding dH to a (0, 2) GLSM by hand: the Green Schwarz
mechanism
The obvious question, then, is how to implement H-flux and the modified transformation law for the B-field in a computationally effective GLSM. Ideally, this would
produce a GLSM which manifestly reduces to an NLSM of the above form at low
energies.
To begin, consider a (0, 2) GLSM whose classical action reduces, at lower energies,
to an effective NLSM for a complex manifold, X, with left-moving fermions valued
in sections of a holomorphic vector bundle, Vx. Importantly, since all the geometry
and topology of the IR CFT is generated by the gauge action, all of the potential
144
anomalies are similarly embedded within the gauge group [112]. This allows us to lift
the problem of tracking anomalies in the NLSM to the easier problem of identifying
gauge anomalies in the UV-free GLSM. This suggests that the anomalous Lorentz
transformation law for the B-field in the NLSM should lift to an anomalous gauge
transformation law for the B-field in the GLSM.
Our first job, then, is to locate the B-field in the GLSM. Happily, this is wellunderstood physics: the B-field in such a GLSM is controlled [23,113] by the axial part
of the FI coupling, Lo = 0a F+_. The resulting spacetime B-field is then B = 0"Wa,
where the w' are the (1, 1)-forms on the target space corresponding to the gauge
a. More generally, the hermitian (1, 1) form (which becomes the
field strengths, F+
complexified Kdhler class on the (2,2) locus) is J = J+iB = taWa, where ta
=
ra+i6a
are the FI parameters in the superpotential. Since Wa is closed and 0 is constant,
H=dB=0.
To introduce non-trivial H-flux, then, we can simply promote the Oa to dynamical
fields such that H = dB = d6a A Wa does not vanish identically [100,114,115]. Each
such dynamical Oa then represents a coordinate on an S1 on the target space of the
GLSM. 4 Notably, this also generates the dilaton gradients we expect from supergravity, as shown e.g. in [114]. Unfortunately, gauge invariance and single-valuedness of
the action require that d (d6a) = 0, and thus dH = 0.
This suggests a simple way [100-103] to build a GLSM with dH # 0: let Oa be
shift-charged under the worldsheet gauge symmetry,
Oa
- +a
+
Qab ab.
The resulting
B field now transforms non-trivially under the gauge symmetry,
B -c-+ B + QababWb
(3.7)
The worldsheet gauge-invariant H-flux is thus of the form
H = (d~a + QabAb) A a.
4
(3.8)
1n the UV 0 provides a new S' in the field space, in the IR this becomes an S' on the target
space of the NLSM to which the theory flows, with the gluing in of the S' specified by the gauge
action.
145
Taking a further exterior derivative then gives
dH = Qabwa
where wb
-= dAb
A b,
(3.9)
is the 2-form representing the gauge field strength under which 0 is
charged.
By making 9 charged, however, we have rendered the classical action (in particular,
the axial term LO) non-gauge-invariant. Fortunately, the variation of the axial term is
precisely of the form of a 2d anomaly: under a gauge transformation with parameter
a,
J64
ocl Qab aa F
.
(3.10)
It is thus possible to cancel the gauge-variation of the classical action against a quantum anomaly in the measure a la Green and Schwarz [100]', with L providing the
classical Green-Schwarz term.
This raises an obvious question: what is the gauge anomaly in a classical (0, 2)
GLSM measuring? It has long been understood that the gauge anomaly in a standard
(0, 2) GLSM is in fact a probe of the sigma model anomaly in the target space,
as follows. The anomaly in 2d comes from a di-angle diagram and thus defines a
quadratic form on gauge fields: A = Aab FT, F_. Since the gauge fields represent
the pullback to the worldsheet of 2-forms Wa in the target space, FTL_ =
q*wa,
the
anomaly thus defines a 4-form on the target, A = AabWaWb. A short computation (see
e.g. [100,103]) then verifies that the corresponding 4-form is the RHS of the heterotic
Bianchi identity, A = [trR A R - TrF A F].6
Choosing the gauge-transformation
of our dynamical theta angle to cancel the worldsheet anomaly then ensures that
5
Note that adding a GS term is not possible in a (2,2) model. Corespondingly, the fermionic
spectrum is necessarily non-chiral, which forbids any gauge anomaly. Such models necessarily have
dH =0.
'Technically, this result uses the natural Kdhler structure on the toric variety. In the presence
of torsion, the physical metric will in general not be this Kdhler metric. They will differ, however,
only by terms proportional to a, the loop counting parameter in the worldsheet; for the Bianchi
identity above, we need only the leading order result. Note that this argument is not reliable away
from large radius - however, away from large radius, the geometric picture is itself not reliable so
we should focus instead on the quantum consistency of the gauge theory.
146
we satisfy the target space Bianchi identity, with the vanishing of the net anomaly
four-form in cohomology corresponding to an integrability condition for a smooth H.
Rather poetically, then, the Green-Schwarz mechanism on the target space pulls back
to the GLSM as a Green-Schwarz mechanism on the worldsheet. This mechanism
was first used in [100] to build GLSMs for non-Ksihler T 2 -fibrations over K3 (which
geometries were first studied via supergravity in [111,116-118]) and was subsequently
exploited to study LG-orbifold points in the moduli space [103], to compute the
stringy spectrum [101], and generalized to non-abelian GLSMs [102].
3.2.3
On the geometry of GS GLSMs
We would like to argue that the Green-Schwarz mechanism for the worldsheet gauge
theory, together with the anomalous transformation law for the B-field under worldsheet gauge variations, reduces precisely to the anomalous transformation law for the
B-field and the corresponding Green-Schwarz mechanism of the NLSM. To see that
this indeed works out, we need to think more carefully about the effective geometry
of the GS GLSM.
Consider an arbitrary, potentially anomalous, (0, 2) GLSM. At the classical level,
this defines a classical NLSM by solving the equations of motion for all massive
fields and evaluating the action on-shell. By construction, the target space X of the
resulting NLSM is simply the quotient of the flat target space of the UV free fields
(call it Z) by the (complexified) gauge group, G, of the GLSM, i.e. X = Z'/G, where
Z' is Z with the fixed locus of G removed. If there is a superpotential turned on, we
further restrict to the vanishing locus of the superpotential in X, X w=o. The result
is a completely garden-variety classical (0, 2) NLSM.
Quantum mechanically, of course, this NLSM may be anomalous via the standard
sigma model anomaly. The key observation [112] is that, due to linearity of the UV
theory, the sigma model anomaly of the NLSM is embedded in the gauge anomaly
of the GLSM. Moreover, from general properties of the classical geometry associated
to a GLSM, we have a direct map between forms in the classical geometry, X, and
gauge field strengths in the GLSM. This allows us to translate the anomalous gauge147
transformation law of the axial GS term in the GLSM directly to an anomalous
transformation law of the B-field in the resulting classical NLSM. It also allows us
to relate the gauge-anomaly of the fermion measure in the GLSM to the sigmamodel anomaly of the fermion measure in the NLSM. The conditions for cancellation
of the GLSM anomaly in the UV then directly map to the conditions required for
cancellation of the sigma model anomaly in the IR NLSM as discussed above.
Of course, the above analysis only works at the classical level - matching the
two theories precisely would require showing that the full one-loop effective actions,
including un-protected kinetic terms, precisely match up to finite renormalizations.
The corresponding calculation is straightforward in principle, but quite involved in
practice, so we will not attempt it here. Instead, we make the following observation.
Consider a classical NLSM constructed from the gauge-invariant on-shell action of
a classical GLSM. In general, both the NLSM and GLSM will have quantum anomalies. As described above, we now have two ways of building non-anomalous models:
we can cancel the gauge anomaly in the GLSM via a GS term for the worldsheet
gauge symmetry; or we can cancel the sigma model anomaly of the NLSM directly
using standard NLSM techniques as described above. The question is whether these
two quantum modifications are equivalent or not - more precisely, whether they lie
in the same universality class in the deep IR. This can be represented in the following
diagram:
Classical GLSM
aImprove
GLSM
(3.11)
On-Shell
Classical NLSM
0"""
(0,2) NLSM
While it remains technically possible for these models to be inequivalent, being different would mean there is a new way to deal with sigma model anomalies which is
distinct from anything done before. We consider this highly unlikely, and thus happily conjecture that the non-anomalous GLSM and NLSM thus defined in fact flow
to the same CFT in the IR.
Finally, it is important to note that these GLSMs with non-trivial GS anomaly
148
terms are not in fact linear. When the mass of the heavy fermions is taken to infinity,
however, corresponding to running to a boundary of the vector bundle moduli space,
the only remaining non-linearity appears in the GS term itself, which will generically
include globally ill-defined logs. As we shall see in the next section, this need not cause
us to panic. However, various of the standard moves used in studying GLSMs must
be considered with some care in this quasi-linear setting. Happily, as we shall now
explore in detail, we can in fact embed these quasi-linear models within completely
standard (0, 2) GLSMs, with the quasi-linear models arising as low-energy effective
descriptions of part of the moduli space. The question thus reduces to asking whether
the corresponding limits of the vector bundle moduli are well-behaved.
3.2.4
Generating dH in a garden-variety (0,2) GLSM
Surprisingly, the GS couplings studied above, and the non-trivial dH they represent,
have been hiding in plain sight in almost all (0, 2) GLSMs studied to date.
For
example, consider a (0, 2) GLSM for a Calabi-Yau, X, at standard embedding, Vx =
Tx. Thanks to (3.1), standard embedding ensures that dH = 0. Crucially, such
models generically contain vector bundle moduli which deform the vector bundle away
from standard embedding, Vx = Tx. Away from standard embedding, however, we
no longer have trR A R = TrF A F. Consistency of the IR CFT, as reviewed above,
then requires that dH # 0. But the CY GLSM we started with has no anomaly, and
no GS terms, from which to derive a non-trivial dH. Where is the H-flux hiding in
this simple GLSM?
The key observation is that deforming the vector bundle boils down, in the GLSM,
to tuning a set of Yukawa couplings which control the masses of a host of charged
chiral fermions. When we study the system at energies beneath the mass of some
particular pair of fermions, it is appropriate to integrate them out. However, if these
heavy fermions transform in chiral representations of the gauge group (as for example
does the top quark in the standard model), then integrating them out leaves us with
an apparently anomalous spectrum of light fermions. However as observed in [42]
in the context of the standard model below the top mass, this would-be anomaly
149
is precisely cancelled by a GS term generated when integrating out the heavy chiral
fermions. We are thus left with an effective GLSM in a Higgs phase with an anomalous
spectrum of light fermions plus a GS term which ensures cancellation of the anomaly.
This is precisely the form of the models constructed by hand above.
It is useful to see this happen in detail in a simple example. Consider a gardenvariety (0, 2) GLSM for a Calabi-Yau, X. Since the specific model will not be important for what follows, we will be relatively schematic for the moment, focusing
only on the details we will need. For clarity of presentation, we restrict attention to
a model with a single U(1). The field content includes a set of chiral superfields, 4D,
a set of fermi fields, FA, and a U(1) vector with fermi field strength T. Let the
charges of the chiral and fermi fields be Q, and
qA
respectively (some of which may
vanish), and let the fermi multiplets satisfy the chiral constraint, D+L-A
= EA
(DI),
as usual. The corresponding Lagrangian has kinetic terms
Lkin =
-
qAV1(ieQIV~
1 2f
D- I4e+ e^ IF-AJ-A +
d2 0 y
e
2 j
'
-T,
2
,
(3.12)
together with a chiral superpotential of the form,
Lw =-
d+
A ]LAJA(4)
+ 1tT)
+ h.c.
(3.13)
Here, the JA(D) are holomorphic functions of the oD, with net gauge charge -qA
such that the superpotential is gauge-invariant. Chirality of the superpotential then
requires that EA JA = 0. The conventions and component expansions are listed in
Appendix (B.1).7
Since the fermions in such a (0, 2) gauge theory live in chiral representations of
the gauge group, there will in general be a gauge anomaly. Under a general gauge
transformation with gauge parameter A, the effect of the anomaly is to shift the action
by,
Oc
f
QiQI - EqAqA,
A=
dO+j AT + h.c. ,
I
7
(3.14)
A
A useful tool for performing (0, 2) component expansions in mathematica with examples may
be found at http: //www. mit . edu/-edyer/code. html.
150
as can be verified by a standard one-loop calculation. As discussed above, we are
interested in UV GLSMs which are completely free of anomalies, so we hereby demand
that A = 0, as is the case, for example, at standard embedding.
Among the terms that appear in the action are a host of Yukawa couplings,
I
d9+
(DHgqFQ-q
(3-15)
+ h.c. ,
where 4Dq is a chiral field of charge q, FQq is a fermi field of charge
Q-
q,
'1)H
is a
chiral field of charge -Q which gets a non-zero vev (01qH 0) on the chosen patch of
the target space, and p is a tunable modulus. Such terms can arise, for example, from
IPAJ^A(4)
terms in the action.'
Below the scale set by p(Oq$Hj0), we can integrate
out '1 q and FQ-q to get an effective action for the remaining light modes.
For example, consider a (0, 2) model for a Calabi-Yau at standard embedding,
corresponding to a (2,2) point in the moduli space. At such points, the mass terms
for the fermions are non-chiral,
Lyuk
= v
J d+ (a1 G FLP + Hi-G) + h.c. oc
91 G (7r-04 + ?PUr+) + .(3.16)
where the 7r± have charge -Q, the V)+ have charge 1, and G = 0 defines a hypersurface
in X. Since the scalar coefficient is the same for both pairs of fermions, the masses are
the same, so while integrating out either pair of unequally charged fermions would
generate an anomaly, the two pairs are degenerate so there is no regime in which it
makes sense to integrate out one but leave the other.
On the other hand, if we turn on a deformation which breaks this accidental (2,2)
supersymmetry, these two chiral mass terms will no longer be degenerate. A simple
such (0, 2) deformation, for example, replaces &1G in the FI term with a general
function, (01 G + p JI). The coupling p represents a modulus of the vector bundle,
with the precise geometry of this deformation being encoded in the functional form
8
Similar (non-superpotential) terms arise from or EAr^ terms deriving from the kinetic terms for
the fermi multiplets; the basic effect is directly analogous (though details can be phenomenologically
different), so we focus for simplicity on the superpotential case for the moment.
151
of J1 . Adding this deformation into the action then gives,
Lyuk
oc
&1 G (7_0i) + (O1G + p JI) (0,+)
+ ...
(3.17)
When IL = 0, we recover (2,2) supersymmetry and a non-chiral mass spectrum. When
p is non-zero, on the other hand, the masses of the two pairs of chiral fermions will be
different. When
[
is large we will generally find that one pair is heavy and should be
integrated out while the other remains light and fluctuates. At energies well below the
mass of the heavy fermions, the spectrum of surviving light fermions is thus explicitly
anomalous.
As the heavy fermions are gauge-charged, integrating them out requires some care.
One method is to use the non-vanishing Higgs field to change variables to uncharged
fields,
4DIQq
FD
=((DH)'QF
(3.18)
which we can then integrate out without concern. However, in doing this change of
variables we pick up a Jacobian factor of the form,
DqEqDrQ-q =
Dpff>
e 6f
1f('H)T
(3.19)
where C is such that the gauge variation of the resulting GS term precisely cancels
the effective action of the remaining light fermions. (See [41,42] for a lovely example
of this effect from integrating out the heavy top quark in the standard model.) These
anomaly-canceling GS terms then descend, in the deep IR, to non-trivial dH satisfying
the Bianchi identity, as discussed in Sec. 2 and in [100,103].
Crucially, for the special case of standard embedding (or indeed at any (2, 2) point
in the (0, 2) moduli space), the Yukawa couplings are tightly constrained such that
the physical fermion masses and interactions are non-chiral. There is thus never any
regime of parameter space or energy in which the set of fermions above, or below,
a given mass is chiral. When integrating out the massive fermions, then, all such
152
H-flux generating terms must cancel.
Note that the resulting effective model has an interesting limit in which we send
our bundle parameter to infinity, p -+ oo. In this limit, the massive fermions entirely
decouple and we are left with precisely the theory we started with, minus the fields
<bq and IQq, but plus the GS term I2 GS = C log(C1H)T. But this is precisely the form
of the theories studied in [104,105,119]: the anomaly arising from the measure for
a chiral family of fermions is cancelled by a logarithmic Green-Schwarz term, where
the argument of the log is a function of the charged scalar fields in the theory. It
should now be clear that virtually all such models can be orchestrated by Higgsing
and integrating out a chiral set of fermions, as above. It is also clear what physics
lies behind the singularities at points where the logarithms diverge: such singularities
signal the return of the again-massless fermions we had previously integrated out:
death from the UV.
Note, too, that the resulting models should be treated with some caution. In particular, the (0, 2) GLSM with which we began is as well-behaved as one could hope.
Fore example, it enjoys a non-singular topological chiral ring which varies smoothly
with the moduli, and the unbroken (0,2) supersymmetry together with the linear
model structure ensre that world sheet instantons do not generate a spacetime superpotential [120,121]. However, once we take the limit p -+ oo, the assumptions going
into those arguments no longer trivially hold, so these results must be re-evaluated.
Doing so, however, is hard: since the GS terms are non-linear, and indeed logarithmic, computing the quantum OPEs does not trivialize in the UV. This motivates us
to search for more tractable variants of these models which have more gentle, and
computable, physics in the UV.
153
3.3
Verifying Quantum Consistency in a Special
Class of Models
As we have seen, moving to a generic point in the (0, 2) moduli space leaves us
with a non-Kihler manifold supporting non-trivial H-flux. Around such points, it is
sometimes convenient to integrate out the heavy fermions, generating a quasi-linear
Green-Schwarz model in which the structure of the H-flux is more manifest. The
price of this simplicity is losing the manifest good-behavior of the original GLSM. It
is thus illuminating to verify that these effective GS models are in fact good quantumconsistent GLSMs which flow to candidate CFTs in the IR.
In this section we marshall evidence that these GS effective theories do, in fact,
flow to good quantum-consistent CFTs in the IR. For simplicity, we focus on a special
class of such GS models which are fully linear (avoiding the logarithms which generically appear in such effective models). Borrowing a technique from Silverstein and
Witten [122], and following a similar analysis to that in [103,114], we identify a chiral
left-moving conformal algebra in the UV that has all the properties needed to flow to
a left-moving conformal algebra in the IR. The existence of this algebra is equivalent
to the vanishing of anomalies for specific left moving and right moving symmetries.
We briefly review the justification of this technique. We then present the details of
the calculation in the case of a single U(1) gauge group, then generalize the results
to the case of multiple U(1)s. We also compute and record the left and right central
charges, and vector bundle rank, of the IR fixed point theory as a function of the
charges in the original quasi-linear model. We begin by identifying the linear models
of interest.
3.3.1
The Models
Consider the following models, which will be the focus of the rest of this section.
As above, we begin with chiral superfields, 4D,
fermi fields, I'_A, and a set of U(1)
vectors with fermi field strengths, Ta, under which the matter fields have charges Qa
154
and qA. The fermi multiplets again satisfy the chiral constraint,
D+F-A
= EA ('1 ).
The Lagrangian includes kinetic terms,
fd 0 (
2
1
Lkin =-
"
+
Aj-A
1-
(3.20)
where the ea are the gauge couplings, together with a chiral superpotential of the
form,
Lw = -J
Here, the
JA(JD)
d+
LA
jA (D) + 1taT a
+ h.c.
(3.21)
are holomorphic functions of the 4b, with net gauge charge -qA
such that the superpotential is gauge-invariant. Chirality of the superpotential then
requires that
EAJA
= 0. The conventions and component expansions are listed in
Appendix (B.1).
Since the fermions live in chiral representations, the gauge symmetry will again be
anomalous with anomaly matrix Aal =
QaQb -
q"aql. Instead of setting the anomaly
to zero, however, we now add to the model a set of Green-Schwarz axions, Y, with
shift-charges
Qa,
6
AY
D-Y
=
-iQaAa
=
oYi+
(3.22)
2
(o- Va +iAa),
(3.23)
standard gauge-invariant kinetic terms,
Ly =
J
d2 9+k 2 (Y +Y + Q
aVa
)D-(Yi
-
Y1)
(3.24)
and a set of non-gauge-invariant Green-Schwarz terms,
LGS
The
d,
±aC2 b]lVaAb
-
al
a
+
(3.25)
specify the axial couplings of the axions to the gauge fields. Note that the
anomaly itself is strictly symmetric, while the axial term in LGS has a priori no
symmetry. The purpose of the VA term in LGS is to cancel the antisymmetric part
155
of the axial term, leaving the symmetric part to cancel the quantum anomaly.
The Y fields may be thought of as Stiickelberg fields for the anomalous gauge
multiplets. In particular, for any anomalous U(1) we can introduce a shift charged
field, Y, together with suitable GS couplings, so that the anomaly cancels the variation
of the classical action. Note that setting the gauge Y = 0 we return to the original
anomalous theory, though now with an explicit mass term arising as the legacy of Y's
kinetic term. For more details about quantizing anomalous theories and applying the
Stiickelberg method to an anomalous action in the non-supersymmetric case see [123].
Example: A Single U(1)
To gain a little familiarity with these shift fields, let's take a brief look at the classical
geometry in a simple gauge invariant case, a single U(1) and shift field of charge,
Q = 1, but no GS coupling, and then highlight the subtlety when including the
anomaly. The action and component expansions are given in Appendix (B.2). For
this simple model, the bosonic potential is given by:
U
(Z
QI,0I12 + k2(y
+ 2)-r
+
E
JAI2 +
A
EA12.
(3.26)
A
The classical moduli space is obtained by restricting the field configuration to minimize the bosonic potential: D = 0, JA = 0, and EA = 0. In focusing on the D-term
constraint, we see that there are no compact models with a single U(1) and a shift
field. There always exists a runaway direction. The vanishing of JA and EA cannot
help with compactness, as the holomorphic hyper-surface of a non-compact complex
manifold is either a discrete set of points or non-compact.
Now consider the same model as above, but with an anomalous fermion content
cancelled by a GS term for Y,
LGS
_
dO+TY + h.c.
(3.27)
One might naively be tempted to plug in the Wess-Zumino (WZ) component expan156
sion for T and Y.
--
LGS
2
(D(y + g) - iF+-(y - y) +iA-X++iiA\+)
(3.28)
Yielding,
e2
D
2 (1]Q 012 + (k 2 - d)(y + #
=
)
I
This doesn't make sense classically, however. The Lagrangian is not gauge invariant,
and so fixing WZ gauge in the classical action is not possible.
It is of course possible to do away with gauge symmetry completely and write down
the full component expansion. After appropriate field redefinitions, the component
action is:
=
IGS
2
(D(y + ) - iF+-(y - 9) + iA-X+
i -+)
- (D(s + 9) - iF+-(s- 9) + iA-(+ + i-(+) .
2
+
(3.29)
Where s and (+ encode the unfixed parts of the gauge field. This is going too far,
however. This action, and the corresponding modified D term,
D
=
-
e2
(ZQI,12+(k2 - d)(y + 9) +C(s + g) - r),
(3.30)
should not be thought of as a good starting place for analyzing the topology and
geometry. In writing these classical expressions down we have neglected crucial one
loop effects, not least of which is the fact that the full theory is gauge invariant.
157
Y multiplets vs T 2 multiplets
In the models introduced in this section, the Green-Schwarz axion field is a (0, 2)
chiral boson with a shift gauge symmetry:
Y = y + V20+X+ - ij+O+y,
e" E C*.
(3.31)
This is a particular supersymmetric completion of the GS axion. Instead, we could
have chosen a different completion. For example, the torsion multiplet,
E
in [100,
101,103] of the T 2 models is another completion,
E
= d + VfO++
- i9+ +a+O,
9 = 01 +i9
2
E T 2.
(3.32)
Choosing a different supersymmetric completion of the GS axion has a natural interpretation in terms of quotient actions of the target space. fi = (0, 2) ensures that the
target space is a complex manifold; precisely what the complex structure is follows
from the action of SUSY on the real scalars; fixing this SUSY action then fixes the
action of the complexified gauge group, and thus determines the topology of the quotient. Inequivalent SUSY completions of the GS-axion thus correspond to different
quotient actions. The effective geometry of the T 2 multiplet is analyzed carefully
in [100].
3.3.2
Methodology
The algebra of a (0, 2) superconformal field theory consists of a left moving stress
tensor, 'TL, a right moving stress tensor, T, right moving supercurrents, g:, and a
right moving R current JR. We will rely on the existence of an additional left moving
current, JL, to have some control in the IR. These operators satisfy the following
OPE algebra:
158
JL(X-)JL(y-)
rl-$
JR(X')JR(Y')
-
JR(X')gR'(Y')
-±+Y
gR+(X+)g;i(y+)
2
2 (x
X+)± 2JRY+
+
TL~~x)TI,(y)~~
~2
TL7L(x-)(X
+
TL(!r)T(X)Ty)
I
(X--y-
RX)T(
)2
)
-
+)'+(+y
-
+2T(+
_R
(x4~W
)
T
+)'
(+y
2
+
O+JR(Y+)
+
+Y
+ 9TL(Y-)
+ -Y
TL@§-)JL(y-)
,.
JL()
L
(X--y-)
2
+ '9-JL(y-)
TR (X+)JR (Y+)
X--y-
3R(Y+)
(X+-y+)
In this section we identify theories which are believed to flow to such superconformal
field theories with nontrivial central charge and vector bundle rank at their IR fixed
point. We find these theories by constructing models that posses such an algebra
in cohomology even in the UV. Though this is neither necessary nor sufficient to
guarantee the existence of the IR algebra, it is usually taken as strong motivation
[101,102, 114, 122].
The (0, 2) supersymmetry algebra contains the anti-commutation relation:
{Q+,Q+}
9
=
2P+.
(3.34)
1t is conceivable that the UV (0, 2) algebra does not describe the IR fixed point; further operators
may appear near the fixed point, the theory may become trivial, etc....
159
(3.33)
As such, elements of Q+ cohomology are in one to one correspondence with left moving
10
ground states. Furthermore, correlators of cohomology elements are protected. This
motivates searching for the left moving components of the superconformal algebra
in cohomology.
It turns out that it is more convenient to identify states in D+
cohomology rather than
Q+ cohomology.
This is not a problem, as the two operators
are conjugate.
Even after finding candidate chiral currents, one might imagine that calculating
the OPEs would be difficult in the presence of a superpotential and gauge interactions.
As it turns out this is not an issue. Due to the magic of supersymmetry, there is a
dramatic simplification when considering left moving ground states. As explained
in [122], the superpotential comes with a dimension-full parameter, M.
By power
counting, any term in the operator algebra that contains a factor of A must also
contain a factor of x 2 and so vanishes in the x+
-+
0 limit, while the gauge interactions
flow away in the UV. This means that we can use the free field OPEs in calculating
the algebra.
So far this has just been a discussion of the left moving part of the algebra. As
we will see in the following sections, however, the existence of this left moving algebra relies heavily on having a non anomalous R-symmetry. Once this R-symmetry is
discovered, the rest of the right moving algebra is guaranteed, as long as supersymmetry is preserved. In the next few sections we walk through the construction of the
currents and the calculation of their algebra for the models of interest.
3.3.3
Gauge Invariant Model
To begin with, let's examine the case of a single U(1) and shift field coupled in a
gauge invariant fashion. We consider a model with fairly generic field content:
4,
F-A,
Y
(3.35)
Q+ commutator, L =
£o+t[Q+,0], then correlators of Q+ cohomology elements are independent of t. A specific realization
of this will allow us to compute the OPEs.
10Imagine that the action depends on some parameter, t, multiplying a
160
with the usual chirality constraints
D+Y =0,
+I
D+L-A =
(3.36)
V 2 EA.
For now, we will focus on a single U(1) gauge field. The action is
L
A
J
d2 0+
J
d9+
(3.37
+ h.c.
AF-Aj
2
Here EA EA JA
eQIVD-5 + i k(Y + Y + V)D-(Y - Y) + E e ^V)
i
A
=
0 in order to preserve supersymmetry.
This action is classically gauge invariant. In what follows we will write down the
conditions for the gauge symmetry to be anomaly free.
Equations of Motion
The equations of motion for these models are:
U+
(AVL-)
5+ (eQIVPA)
=
(3.38)
l jA
=
-41 E
+ iv/
L.AIjA
(3.39)
e^Vr-AaIEA
A
A
J +eiv
k 2 D+ (-Y)
EA ^
(3.40)
YEA
-A
A
+
-
D
Q
+EeAt-A
I
-A
+ kQ2+i
A
Global Symmetries
We are interested in models which posses global symmetries,
JDI
-+ e-ia'E4D,
in ad-
dition to the gauge symmetry. In particular we want candidate U(1)L and U(1)R
symmetries. We need further constraints on EA and
ant under these global symmetries.
charges; jA (Y - irE, e-i'f()
-
jA
for our theory to be invari-
First of all, IF-A and
eW/AEJA;
while EA and
rA
jA
must have opposite
must have the same
charge; EA (Y - iKe, e-iaEPI) = e-iAEEA. These relations imply the following quasi-
161
homogeneity conditions.
aS
&IaI
5
Where
LDI,
and
3
K
A
JA
+ KOYJ A + IAJA
O11&9EA + KOyEA - /AEA
=
0
(3.42)
=
0
(3.43)
is the charge of the shift field, Y, a 1 are charges for the chiral scalars,
are charges for the fermi multiplets, F-A.
One needs a little more care for the R-symmetry due to the fact that both 0+ and
D+ have R charge +1.
Thus the
jA
Preserving R-symmetry requires F-A JA to have R charge +1.
have R charges
+ 1, while the EA have R charges
-OAR
OAR
+ 1. The
quasi-homogeneity conditions corresponding to the R-symmetry are:
a
1
+ KRYJ
5&pIIo9IEA +
+ AR
A
I Ra9YEA
-
(OAR~
_
)jA
+ 1)EA
0
Now that we have our equations of motion and quasi-homogeneity conditions we
can begin to search for the chiral superconformal algebra. In particular we will identify
chiral superfield currents J+ and T++, whose lowest components are gauge invariant
conserved currents which satisfy the conformal OPE relations.
We begin by discussing the various U(1) currents in our theory,
U(1) Currents
In our class of models, there are three U(1) symmetries of particular interest: U(1)G
gauge, U(1)R and global U(1)L symmetry. Each of these plays a critical role in constructing the IR theory and the (0, 2) superconformal algebra. The gauge symmetry
effects a quotient of the target space, and ensuring gauge invariance at the quantum level is crucial. The U(1)L symmetry descends to the IR and can be used for
the GSO projection in string backgrounds. As we will see, having a non-anomalous
U(1)R current is essential for constructing the chiral, left moving stress tensor.
162
Let us start by exploring the gauge current.
=Q
=
+
1
1 D- 1 --
qA'-AI-A
- ik 2 D_9
(3.44)
A
=
jG
Qj1+O1
i
Qi +j
Q
-
+ ik 2 P+y
(3.45)
Note, the current is neutral, so it is conserved both partially and covariantly. The
superfield completion is
(-i eQIVQiD
JG=
i
\
Ar
_ A
+ k 2 EDY)
(3.46)
A
QiND+ (eQIv&i) + k 2 D+ (Y +
B+
J(
e
+ i
(3.47)
.
It is nice to see that J- is D+ trivial. Thus the interesting contribution of the
gauge current in cohomology comes purely from JG.
In order to check the chirality of JG, recall the EOM:
1 + (0-y)
=
5 eQIVQi4DjD_4
+i
22
S
^AP-AIP-A + k2 TY
=
i4q.,48)
A
JG is exact up to equations of motion. DU+JG = 0 follows from the fact that B2 = 0.
Another way to see this is to apply T+ directly to JG. Using the classical EOM and
the quasi-homogeneity conditions, one can explicitly check that J
is chiral.
For any other global symmetry, in particular for U(1)L, the procedure to find a
gauge invariant chiral current is similar and we get:
J
=+
e ^
eQIVa i D .. + i
+ k iLD _Y
.
(3.49)
A
Classical chirality, U+JA|EOMs
=
0, can be checked using the EOM and the quasi-
homogeneity conditions.
The R-current is a little trickier, as the component fields come with different R
163
charges. The lowest component of the R-current is:
#
ij+ =
5
3
-i E
-
AY-Y-Ak-A+k2
RD-
+
(3.50)
A
iij
=
-(
(aI)+01
-
i (a
-
1)
2 +X+ (3.51)
- k2RD+y - ik
+±i+±)
which is gauge invariant and conserved.
Further Modification of Chiral Currents
A classically chiral current for a given symmetry, Js+, can fail to be chiral in a quantum
theory. The supersymmetric extension of the chiral anomaly, known as the Konishi
anomaly [124], in 1+1 dimensions 1, tells us that
As T
(D+J+) = ( 7 )(3.52)
where As is the anomaly coefficient for the given symmetry.
Generically it is not possible to remove this anomaly by redefining the current in
a manner that preserves gauge invariance. As noted in [114], however, when there
is a shift charged field in the game we can remove the anomaly. This freedom to
redefine the current is easy to understand by looking at the bosonic components. In
two dimensions the anomaly is given by o9j" oc c""o,A,. It is possible to define a new
conserved current, j'" = j#- Ed"'A, but this is not gauge invariant. If we have a shift
field, 0, at our disposal we can do better and define, j" = j" -
"'(o9,0 + A,). This
is gauge invariant and conserved. This new current corresponds to adding a term of
the form O&"F,, to the quantum effective action and then improving the resulting
current. Note, that this addition to the action is just the form of a Green-Schwarz
term.
The superfield completion of this effect is:
J + s, i+ + i 47rApnDY.
1See, for example Appendix C of [114] and section 3 of [120].
164
(3.53)
The modified currents is no longer (classically) chiral:
B+
is++ i 47r DY
=
8,7r
(3.54)
T.
If we choose ANs = As, however, the classical non-chirality of this modified current
exactly cancels the one-loop contribution 1 2 .
Stress Tensor
Now that we have understood the U(1) currents, let's move on to constructing the
chiral stress tensor.
The unimproved stress tensor can be obtained from Ndether's procedure, however
this is not gauge invariant. Using integration by parts, the gauge invariant superfield
completion of the stress tensor is:
T +
-
2
Ta _ +
eQIVD_4DD_di + i
I4e
S e*^V,-AE-
_A + k2EYDY
A
This is equivalent to promoting the gravitational stress tensor to superspace.
Acting with T+ on To+ and using the EOM and quasi-homogeneity conditions
we get:
1
2
+T+
(-A).
(3.55)
Thus To+ is not quite chiral. We would like to improve it to a chiral stress tensor
without ruining the conservation.
To achieve this, recall that a conserved current JP can be improved to J'" by
adding the divergence of an antisymmetric operator.
1,
J't = J1 + &vKl"t
-
j't
= a, (Jt + 0,K EAv]) = 0
(3.56)
12For some theories, further modification of currents is possible. Adding iID_Y, for instance,
works for theories in which JA and EA are independent of Y as the EOMs give T5+ (D-Y) = 0.
One example of this kind of theory is the HK [114] model.
165
.
We want to find a K"] that renders T++ chiral. In our case this will be of the form,
CAW&LY.
As demonstrated by Silverstein and Witten [122], given an R-symmetry, finding
K"I" is a simple matter. Consider the following, non-conserved, current:
I0Z
I
e
~-Z
R
(
E
-
qAV
1)-AL-A
(OA
k2 R_Y
.
(3.57)
A
Where ai are the R-charges of the chiral scalars,
#3
are the R-charges of the left-
moving fermions in the fermi multiplets, and ,R denotes the R-charge of the shift
field.
We emphasize that this is NOT the R-current. This would be a regular chiral
current for a global symmetry under which all components of each superfield transform
with the same charge. However, as the quasi-homogeneity conditions corresponding
to the R-symmetry are different from those for a standard global symmetry, the given
current is not quite chiral.
+R+ EOMs = -2/
(3.58)
J-AJA,
In fact, J1+ is not chiral in exactly the correct way to compensate for the non-chirality
of TO+. Now, we are able to construct a chiral stress tensor:
T++
+
=
1
-2
+
0T +
(
(Te
-
rv-qAV
2 TOa-T + E
(4&
I
e
a.e
I
eQ ' DADA
-
-|+i
+ j
A
~
WA + 1)-A-A
2
7
F-A + k2D2Y)
~AD+
k2 R
hY
A
(3.59)
166
such that
15+T++ EOMs
=
(3-60)
0-
The lowest component is:
_
-++
, D-IID-I + i
IY-AD_--_A
+ k-y-Y
+ 2ED-\I
SA
(
+8 a((a
o'b#D-4I+ i
+ 1)A-A7--
+ k2
r
YD~) (3.61)
A
which we identify as flowing to the IR stress tensor TL.
To summarize, exploiting the existence of an R-symmetry and its quasi-homogeneity
conditions, we are able to identify a chiral, gauge invariant, and left moving stress
tensor.
Operator Product Expansion for Chiral Operators
Now that we have constructed candidate chiral operators, let's check the operator
algebra. We consider a larger algebra, containing the gauge currents as well as the
stress tensor and U(1)L current. The OPEs of the gauge current give the gauge and
chiral anomalies. When calculating the Operator Product Expansion of the currents
and stress tensor, we may use the free field OPEs for our component fields.
In the UV, where e -+ 0, we can rescale our gauge field strengthT -+ eT to go
to a free theory. The chiral operators in the free theory and the R-current are given
below. We have not included the potential modifications of section (3.3.3) because,
as mentioned, they correspond to using the Green-Schwarz mechanism, which we
address in the next section.
167
j+
O 1941--I+E q A-A,-A
=
1 ,1
a
JL+
-
ik
2a~
A
I
i01 +
IAY -A-Y-A - ik2
E
L
A
I
k 2ay&- + !
++
O -\
A
+-
+-
a
1
-+
=R
(a
Sa
-
+ k2RO
1A-AI-A - ik2 % O4 +
- +
-
~ + 1)!-AY-A
)
11
E
JR+
A
I
-
A
(a
-IV)+,r~~)
168
+ ik2 IKRO±g
-
k2 ,j+X±
(3.62)
The singular part of the operator product expansion for relevant operators are:
j+(x) j+(Y)
jG(x) j-(y)
_
- )2
(QIQI
(-
_
- )2
(Q L~
j(x)
jj+j(Y)
j++(y)
(
aic~
qA3A
-
E
A
1_
A
I
t++(W t++(Y
3
2(
y-)
2(x-
y-)3
(
(
t++
I
2
(X-
- y-)2
-_1)2_
X+
I
Qa R
Qi - I
A
2- (x
W)-
1) +
E(1
33(#
_R2
_
-
I
(3(af -
8(x--y-
(a
-
I
++(Y)
I
2
-)
Ra
j+ (x)
/
A
1X )
(x
qAqA)
A
(E
j 111(X)
-
(-
-
-
)2
_y-)
(x - y-)
)2)
a ++
(3.63)
(X- - y-)
Conditions for Conformality
As mentioned previously, classically chiral operators can be anomalous.
In order
to check the chirality at the quantum level one should investigate whether chirality
holds within correlation functions. It can be shown that checking this is equivalent
to the vanishing of the most singular terms in the gauge current OPE relations, see
appendix B.4. Requiring the existence of a chiral, (0, 2) superconformal algebra yields
169
-
the following anomaly cancellation conditions:
U(1)G U(1)G:
QiQi
qAqA
-
I
=
0
=
0
A
U(1)G U(1)R:
I
A
Qiac
U(1)G U(1)L:
-
I
5qA
A
Lf (aR - 1
U(1)L U(1)R:
# A-
I
= o.
(3.64)
A
The charge and rank of the vector bundle in the IR theory may also be gleaned from
the leading singularities of the OPEs.
rL
JL(x) JL (Y)
(3.65)
2
(x- - y-)
1
CL
T () TL(y)
2 (x- - y-) 4
1(
+
2
2T(y) +
(x-y)
(X- - y-)
1a T(y) (3.66)
R
(3.67)
JR(X) J(y)
The last equation is equivalent to calculating the leading coefficient of the
j+j+ OPE
and subtracting the leading coefficient of the j-j- OPE. This gives one third of the
right moving central charge UR
=
ICR.
It is conventional to normalize
L=
-2t++.
Then the UV OPEs tell us that:
CL
1)2-1
=
(
I
rL
(3.68)
A
AOR
a
=
CL - CR = EA
)
(3.69)
=
I
Note,
2
A
I
CR
13c
1
-
(3.70)
A
EZ
1, which is an RG invariant quantity as expected.
170
3.3.4
Anomalous Model with Green-Schwarz Mechanism
Now that we have warmed up with the classically gauge invariant case, lets consider
adding to the action the following non-gauge invariant piece, to incorporate the GreenSchwarz mechanism.
LGS =
dO+Y-T + h.c.
4
(3.71)
Here we take d to be real which represents a convention for the periodicity of the
imaginary part of Y. The motivation behind adding this term is that this classical,
gauge variant piece will cancel against the quantum anomaly. This is reasonable as
the classical variation of the added term has exactly the same form as an anomaly.
We follow the same procedure as before, but now with the GS term. The equations
of motion are slightly modified.
qVrA
U5+ (e^v
(3.72)
-jA
JA
U+ (eQIVD_4 j
e^A
vr
+ iVE
A
k2 T+ (D-Y)
-
j
i -A AC
iV E
A
+1-S
(3.73)
A
eZ^V
_A Ee
-
-
T
(3.74)
A
+ iE
I
I
eAVqp
-A
-A
+ k 2 2XY
A
It is interesting to note that acting with D+ on the last equation and using the
quasi-homogeneity conditions reveals:
dT = 0.
(3.76)
When the shift field is charged and the GS mechanism is in play, the field strength
multiplet vanishes on-shell.
171
Modification of Currents: Stress tensor and U(1) currents
The action is no longer gauge invariant. As a consequence, the canonical currents
transform under the gauge variation. All is not lost, however. Thanks to the shift
field we may improve the currents to be gauge invariant.
Let us consider the contribution from the GS term to the currents. An interesting feature of the GS term is that, though it destroys classical gauge invariance, it
preserves the global part of the symmetry. The action shifts by a total derivative.
Recall that when the action changes by a total derivative under a symmetry
transformation,
6L(x) = ajK",
the conservation of currents get shifted.
,,jm(x) = 6L(x)
6S
-
0a(x )
(x) = 4pK"(x)J#a(x)
6#a(x)
9,y (j"(x) - K"(x)) = 0.
The shift field transforms as y
J
=G
-+
(3.77)
y - ir, and the GS term leads to the variation:
1
rav
2
-
0-v+),
(3.78)
implying K+ = -!dv_. Thus the effect of the GS term on the currents is:
2
j+ _ j+ + Idrv_,
(3.79)
which is not gauge invariant. This is plausible as the GS term is not gauge invariant.
Luckily, as mentioned, the shift field allows us to improve the current to a gauge
172
invariant form.
j+ -+
(3.80)
j+ - idna-y
The supersymmetric completion of this is:
J+ 3 J+ - idtD_ Y.
(3.81)
Notice that this is exactly the form of the current modification in 3.3.3. Thus with
the GS contribution our supersymmetric currents can be written as,
QvQj4j)Aj
Q1V
J+
J+
(zI
QIVo
i'I-0hi
eqAV
qAr-A
i e^ o
E-A + k2E)Y
-A-A + k
2
'CY +
)y
(.2
Y (3.83)
A
These are gauge invariant and chiral, as before.
A little more care is required to construct the stress tensor.
Recall that the
canonical stress tensor is not always gauge invariant. In fact, it can in general only
be improved to be gauge invariant on shell. Varying with respect to the metric,
however, produces a gauge invariant, symmetric, conserved stress tensor. Indeed this
is one way to derive the expression for TO++, (3.55). This method of constructing the
stress tensor is particularly nice. The Green Schwarz term is metric-independent and
so does not contribute to this definition of the stress tensor. The improvement term
does change. Just as for the regular currents we have:
+ -+
J+-i
173
DY.
(3.84)
Putting this together yields an expression for T++
Tit
-
ia-r&
_2
\
+
QI
eQIVDibhi
-i +
-
+i
I
(-
-A + k2 _yE)
-~q
e*^Vp-AD
A
+ i
eQIc1Di
Z ev
RA + 1)-A-A +
2
R
A
(3.85)
The chirality of this expression is ensured by the vanishing of the field strength
multiplet.
Conditions for Conformality
With the improved gauge invariant currents, we can obtain constraints on the existence of an IR conformal algebra. As in the gauge invariant case (B.4), we check that
the chirality condition is not anomalous. The details of the calculation are relegated
to Appendix (B.3), but the results are presented below. For our model with GS term
to flow into a (0, 2) superconformal field theory, we require the following anomaly
cancellation conditions:
U(1)G U(1)G :
SIQI
-
I
U(1)G U(1)R :
qAqA
- 2C =
~
qAOA
A
QI
~-~
I
U(1)G U(1)L :
E
I
~ 2
R
A
Qa' -
5
qA/A -
2CrL
=
0
A
a L (aRf- 1)
U(1)L U(1)R :
0
I
#
-
-
2L
d
=0.
(3.86)
A
When there is a non trivial IR theory, the central charges and the rank of the vector
bundle can also be identified, including the modifications from the GS term, this
174
gives:
CL
(3(a
=
-
1)2
-
1) + E (1 - 3()2)
I
rL
-
I
(3.87)
2
.
(3.88)
_2
(3.89)
A
=2
A
I
3.3.5
6d
A
E0a
CR
-
A
Multiple U(1)s
Now that we have warmed up with a single gauge group and shift field, we are ready
to take on the more general case. The action is written in equations (B.17-B.22), but
we repeat it below.
1C
k +1
=
-+ 1W
+,CGS
+ LFI
(3-90)
Where:
Cgk
=
I
a
d20+
a Ta
(3.c
4a
20+
2
Y7 + QaVa) E)_ (y 1- Y) +
1
2 (y+
eQia
£21
5e
A-A
A
Lw
'GS
f=
2
=
+
I
ALJ
[a
b]I
+ h.c.
(3.c
A
~/
2
0+
ab,l
ta
LFI
_
al
-ab
f
dTaYl
+ h.c.
(3.
a,l
do+Ta + h.c.
(3.E
4aI
175
Equations of Motion
The equations of motion differ only by the addition of extra indices.
D+ (e
D+
A
(eQV-aD
JP
-,
=
aF-A)
)
(3U
iIJ
--
+ i-
A
A
k2D+ (D-I)
-A~lJ
+ iv
r
A
1
2e2 (D+Ta
-
A -
_QE
(3.
dalx
a
A
2(k
Q - _da)(y +y + QbV) +
bVbQI12 +
D+Ta)
(3.
_ A ":-AalEA
A
S
1
I
a blv
b,1
(3.
2e (D+-a
S
+ D+a-Ta)
?-Q I> +
I
(kQ +
2i5e
q
(aQbIl
~Th~) 5
(Y
I
I
-
dal )
b,1
(3.1(
_Ar _A
A
Combining the last two of these yields:
1
2a
+a-Ta
=
S
+
eQ0bQ(ID-
- daly)
(kQiI
5:eA
2
S
b A
a
-
(3.101)
±i+i eil~gi-AIF-A
A
Notice that acting with D+ on the above equation, we find:
0 =
(3.102)
Qf )bT
Modification of U(1) currents
Now let us look at the various currents. Constructing the gauge invariant conserved
currents is a little more involved. The results are:
5
j~a±T
QVbQ D~I
+I i
iaVb)
b,I
eq~bqp
A
176
+S
a
+Ual&
+
b1 a
iJ=
QaegVaLo
(b1D
1
+ iJ~
-A
q
+
-A
E (i2 (
KLl
Yi
+
1
b
A
Here the Us are chosen to make the various currents gauge invariant:
ual Qb = EdblQa1
S~
1
t
UQ
dbl K L
1
= E
(3.103)
(3.104)
1
It is not true that every model admits Us satisfying these conditions. Consider,
for example, a model consisting of two U(1)s, with a single shift charged field Y,
charged under a single factor, Qa = (0,2), with a particular GS term specified by
Oa = (1, 0). This gives the symmetric anomaly matrix and anti-symmetric matrix:
0 1
d(a Qb)=
C
0
d[aQb
=
-1
1
(3.105)
0
Bosonic component fields of the GS Lagrangian can be written as:
CGS =
OE""Fu - 2cI"Aj,,A 2,.
(3.106)
The variation of this term is symmetric and can be cancelled off of otherwise anomalous fermion content, rendering the total anomaly zero. Despite this apparent good
behavior, the term contributes to the U(1) currents in a gauge variant fashion.
oc
9e"Ai +...
(3.107)
Since Y doesn't shift under the first U(1), this cannot be improved to a gauge invariant
expression. There exists no U for this theory.
177
(Ab ~
0-1
The existence of gauge invariant currents imposes non-trivial constraints on the
matrix Cal. Of course, if Q is invertible, i.e. if there is a shift charged field for every
U(1) factor, then Ua and UL exist. These constrains should imply non-trivial restrictions on the types of geometries that can be realized via (0, 2) Green-Schwarz GLSMs.
It would be interesting to understand the larger implications of these equations and
whether they have a natural mathematical interpretation.
The stress tensor is defined in a similar fashion as before, however now the existence of a gauge invariant R current requires a UR. The stress tensor is given by:
T++
1
( Z
T
'Da
( -Ya +RAir.
+
(9+
L4
t-Is
+
\
2
Ta
+L
ai 4DID -1AXY
DI +
rl
(Ab
V a
-I
-
+i
Se
a-
-ADV-r-A +
k DYD-Y
A
IEe
C
eQ
-
A
iE e
A
(WA
A
+ 1)L-A1LA
+5 k
(3.108
iDa/b)).
With:
EURlQb =5bl.
1
(3.109)
1
T++ is manifestly gauge invariant. Chirality follows from the equations of motion
and the quasi-homogeneity conditions.
Conditions on Quantum Chirality: Multiple U(1)s and Shift Fields
When there exist improved gauge invariant currents, we can use them to obtain
constraints on the existence of an IR conformal algebra. The details of the calculation
are in Appendix (B.3) and only the results are presented below.
178
The anomaly cancellation conditions for multiple U(1)s and shift fields are:
QaQb -
U(1)a U(1)b :
q'4qi - 2
u(a Qb)
-0
A
I
U() U(1)R :
Qi- q
-
I
Qia
I
2
IR
d~xf=
0
A
U() U(1)L :
I
A
aI a
U(1)L U(1)R :
-
I
SL I
A
A
L3~~-
~,~
(1O
1
One can also calculate parameters of the CFT and central charges get contribution
from multiple gauge generators and shift fields
V2 11)
CL
- 1)
a
rL
ll
Rr
2
a
(3.112)
=
I
A
cR5
Again CL
3.4
-6
#OR2
1)
A
-
CR
is
A
a-
A
a
(3.113)
manifestly invariant.
Conclusions
In this chapter, we have argued that non-Kiihler geometries with H-flux satisfying
the modified Bianchi identity can be described by standard (0, 2) GLSMs without any
additional structure. At low energies, the non-trivial H-flux is realized through a set
of effective Green-Schwarz terms canceling an anomaly in the fermion measure. These
Green-Schwarz terms arise by integrating out charged fermions which are massive in
the local patch of the moduli space; the Green-Schwarz terms ensure the cancellation of the total anomaly after truncating to the (would-be anomalous) spectrum of
surviving light fermions. This clarifies, for example, how dH -, 0 arises in (0, 2) deformations of (2,2) models corresponding to deformations of the vector bundle away
from standard embedding. This improved understanding of the familiar (0, 2) GLSM
179
then allows us to realize various previously-constructed quasi-linear models for nonKdhler manifolds with torsion [100, 102, 104, 105] as effective descriptions of certain
patches of the moduli space of elementary (0, 2) GLSMs.
As an independent check of the consistency of these effective descriptions, we
studied the quantum consistency of a simple class of such quasi-linear models involving
an anomalous gauge group together with anomaly canceling Green-Schwarz axions.
Effectively, this linearizes the GS sector. This allowed us to show, following [101,122],
that GS anomaly cancellation in these models ensures the existence of an off-shell
.N = 2 superconformal multiplet whose OPE algebra closes correctly within
Q±-
cohomology. We then used these OPEs to compute the central charge of the (0, 2)
SCFT to which the theory is expected to flow in the IR.
Many questions remain about the geometry and moduli space of such (0, 2) GLSMs
and their quasi-linear effective descriptions. At a technical level, it would be reassuring to explicitly derive the equivalence between the NLSM and GLSM completions of
models with trR A R -A TrF A F by computing the full one-loop effective action of the
GLSM and verifying that it is in the same universality class as the standard NLSM
construction of [25, 27]. It would be surprising if these construction do not agree,
since that would give us a new way to complete naively anomalous string theories to
good compactifications.
Much more interesting are questions about the global moduli space of generic
(0, 2) GLSMs. What, in the GLSM, distinguishes models which topologically admit
a Kdhler structure (such as small deformations of (2,2) models) from models which
do not (such as the non-Ksihler T 2 -fibration GLSMs of [100])? Relatedly, when can
two such models (one admitting a Kdhler structure, one not) be realized as different
phases of a single underlying GLSM connected by a smooth quantum transition?
More generally, what is the global moduli space of a (0, 2) GLSM and does it contain
multiple inequivalent (2,2) sub-loci which are embedded smoothly? To this end, it
would also be of great interest to be able to compute topological invariants for specific
non-Ksihler manifolds using the structure of the (0, 2) GLSM.
Another interesting question is whether worldsheet duality along the lines of [113,
180
125-127] might be applicable to the GS axion models. On the surface it would seem
less than useful - under such an abelian duality, axial couplings of a scalar to a gauge
field are exchanged with canonical couplings, but in the present models our GS scalars
are both axially and canonically coupled. However, the role of the anomaly may alter
the naive dualization. Even if the anomaly just goes along for the ride, however, such
dualities may prove helpful in patching together a clear picture of the moduli space
of these theories.
There are clearly many interesting geometric questions waiting to be attacked
through more careful analysis of the torsional moduli space of (0, 2) GLSMs. We
hope to return to them soon.
181
182
Appendix A
Monopole Harmonics
A.1
Definition and Properties of Monopole Harmonics
A.1.1
Scalar Harmonics
Definition
In this Appendix we review some properties of the monopole harmonics. We start
with the scalar harmonics introduced in [16,83]. The monopole harmonics Yq,m (h)
are eigenfunctions of the angular momentum operator in the presence of a monopole
background of charge q, (2.1). In this background, the angular momentum operator
takes the form:
LZ =
-
q
2q
L=
_V2 +
- 2 (cos0 - 1)Lz.
(A.1)
The scalar monopole harmonics are defined to satisfy the relations:
=
£(
=
mY,em().
+ l)Yq,em(?),
(A.2)
LzYq,em(n^)
183
We can write the solutions to these equations explicitly in position space:1
Y m(ft) = 2m1
'V
(2( + 1)(f - m)!(f + m)!
r(t - )(t + )
(1 +
x)-m p(-q-m,q-m)
(cos )e(m+q)o.
(1-x)q+m f+m
(A.3)
It is sometimes convenient to write the monopole harmonics in bra-ket notation.
Yq,tm(^)
=
(0, #0, m)
.
(A.4)
Identities
The scalar monopole harmonics have a number of useful properties [83]. Under charge
conjugation the monopole harmonics transform as:
q~fm(ii)
=
(
Y-q,f,-m(t) .
)
)q+m
(A.5)
When evaluated at the north pole, the scalar harmonics satisfy.
Yq,i,m()
= 5 q,-m
' 4wr
+
(A.6)
Gauge invariant products of monopole harmonics also satisfy integral relations. The
monopole harmonics are normalized such that
Yq,tm(ii)J
2
= 1.
'Recall that ft is a unit vector parameterizing the two-sphere and just shorthand for 0, <.
184
(A.7)
The integral of a product of three monopole harmonics is given by
dftY,iM(f)Yq/ 7fm/ (h)Yq//j,/1/
(h)
1)(2/1"+ 1)
(2f=+ )(2f'/
---11 ) (+/'+
f
47r
M
A.1.2
q
mJC;
£/
M'
(A.8)
q
q' q"/
Spin s Harmonics
Now that we have the scalar monopole harmonics for arbitrary angular momentum,
Yqm (h).
It is easy to construct monopole harmonics of arbitrary spin using the
Clebsch-Gordon decomposition. Explicitly, we have:
S(
Isf; j,m)q
;mmi1s £;i,m)!S,m,)0
(A.9)
,me)q.
m 8 =-S mI=-f
Here, (s f; msmels t; j, m) is the usual Clebsch-Gordan coefficient, which can also be
written in terms of the Wigner 3j symbol.
Ji
(ji j2; m1m2 |ji j2; j, M)
=
(-j1)2-+m
The Clebsch-Gordon coefficient is zero unless ji,
-
vanishes unless
j21 5
;>
Iqi.
i5
i
. (A.10)
Mil
inequality, ji
j2
2j+1
j2,
M2
-M
and j satisfy the triangle
ji + j2l. In (A.9) the scalar monopole harmonic |l, me)q
Together these relations imply that the state Is
t; j, m)q
only
exists for,
j ;> qI - s.
(A.11)
In this paper, we often decompose fields of fixed spin, s, and total angular momentum,
j, in terms of sums over orbital angular momentum, t. The only terms that contribute
185
have
lij - s1i t
j+s,
and f > jqj.
(A.12)
For large j, (A. 12) gives 2s + 1 states. For smaller j there are fewer allowed values
for f.
A.1.3
Spin 1/2 Harmonics
It is useful to have explicit expressions for the spin 1/2 monopole harmonics. The
number of independent states depends on the value of j.
j =Iqj - 1/2
From (A.9) we see that there are two states with s = 1/2 for each
j
when j ;> qi,
11/2 j ± 1/2; j, m). In position space these take the explicit form:
2+±Tq j-1/2,m-1/2(f)
(6, #11/2 j - 1/2; j, M)
Tqjm(?)
==-1/2,m+1/2(ii))
(A.13)
(0, #11/2 j + 1/2; j,m)
Sq,jm(f)
-
2+2j
j = jqj - 1/2
For j =
IqI
only the single mode Sqjm(n) exists.
186
-qj1/2,m+1/2()
A.1.4
Spin 1 Harmonics
The spin 1 case is similar to the spin 1/2 case, except that now there are two special
values of j,
j
= q - 1 and j = q. The vector harmonics take the form:
j > qI
U+m-1)(j+m) Y
,
2j(2j-1)
(9,4#11 j - 1; j, m)
Wq,jm(n)
(i-m)(i~m)y
=
V
j(2j-1)
q,j-
/
\
_(i-m+1)U+m)y
q,j,M-1
2j(j+)
(0,1 j; j,m)
(-m)(+ml)y 1
1Vjm(n) =
V(-, +j)
/U-M)U+m+1)y.
_2j_(4+1)+
V'
q,
/j-m+1)(j-m+2) Y
_j-m+1)
++
U(+m+l)(i+m+
(2j+2)(2j+3)
,M+1(
g
j~lm
q,+,-
(2j+2)(23)
(0, #11 j + 1; j,m) = Uq, jm(h) =
(A.14)
iii
q,j m i
3)
2
+1)Y j
q,j+l,m
)y
4
q,j+1,m+1(n,,
j = jql
If j = jqj we only have the last two modes, Uqjm(i) and Ve jm(ft). If j = q = 0,
only the Uqjm(ft) mode is non vanishing.
j = jqj- 1
In the case
j
=
IqI -
1 only the mode Uq,jm(iz) is non-zero. This mode plays a key
role for the stability analysis of monopoles.
In order to check gauge invariance of the effective action, (2.20), it is useful to have
187
an expression for the divergence of the harmonics. The gauge covariant divergence of
the vector monopole harmonics with charge q is
(ec~j(9'
-WT
~))=(jiW)
j
j(+
0))
,(e-wUjm(O,
DA
(e-iwrVtm(O, #)) =
-
(j + 1)2 _ 2
1)(2j+ 1)(j++1)-q
iw
-
(j1 +
e
Yq,jm (0,)
j(2j+1)
188
Yq,jm(,
,
),
((A.O15)
Appendix B
(0, 2)
B.1
B.1.1
Details
Conventions
Coordinates
1
0
We work in 1+1 space-time dimensions with coordinates x , X and Lorentzian metric.
For most of the calculations we use light-cone coordinates. The relevant formulae are:
X*=1
(X0
± X ),
09± = 90 +
1,
g+
(1c)6+=
=
-
1
1
(B.1)
We have superspace coordinates, 0+, and 0+. Integrals are normalized as:
j
d 2 00+#+
I
dO++
= -
d6+6+ = 1.
(B.2)
We also define superspace operators:
D+
0 + i9+,
Q+
0+ - i
D+ =
a
Q+-
a+
=
+,
189
a
+ - i0+a+
+
i+a+.
(B.3)
These satisfy:
D
= 14
=0
(B.4)
Q2 =0
Q2=
{D+,D +} = 2ia+
{Q+, Q+} = -2ia+
{Q+, D+}
{Q+, D+} = 0.
=
When symmetrizing and anti-symmetrizing indices we take:
M(a,b)
M[a,b] = I (Mab - Mba) ,
B.1.2
=
(B.5)
(Mab + Mba)
Superfields
A chiral superfield, <D, satisfies b+<D = 0. A Fermi superfield, F, satisfies D+F
=
v'fE.
These have component Expansion:
D =0
F
=
+ /2+
+ - iO+#+a+o
--\20+G -
(B.6)
f2+E - iO+#+a+-.
(B.7)
We also use a shift charged chiral field, Y. We take it to have the component expansion:
Y = y + VO4+X+ - i9+6+a+y.
(B.8)
In addition to the chiral fields there are gauge fields. As apposed to the (2, 2) case,
which requires a single superfield, there are two (0, 2) vector fields, A, V. In WessZumino gauge these have the component expansion.
V
= o+U+v+
(B.9)
A = v_ + \0_++
20+#+D.
190
(B.10)
From this we can construct the fermionic, gauge invariant field strength, T.
T =
=
D+(A - iaV)
-iV-
(A
(B.11)
+ v0+(Fo1+ iD) - iO+6++A-)
Foi = ovi - 91vo is the gauge field strength.
We have both conventionally charged fields, 4, F, and shift charged fields, Y. The
gauge transformations with chiral gauge parameter A are summarized bellow.
<D
-+ e-'QA4D
F
-+
Y
-+
V
-+ V+i(A-A)
(B.12)
-iqAFp
Y-iQA
A A+&_(A+A)
A
(B.13)
The different qs,
Q,
q, and Q represent our naming conventions for the charges of
chiral, fermi, and shift fields respectively. To facilitate writing down a gauge invariant
action in superspace, it is convenient to define the following covariant derivatives:
D_<D
=
D-F =
DY
(9- + Q(V + iA)
(
+ q(oV + iA)) r
<D
(B.14)
(B.15)
Q
= aY + -(a-V +iA).
2
Lastly, for the FI parameter we use
t = r + i,
0 - 0 + 27r.
191
(B.16)
Action
B.2
Here, for completeness, we record the full action we use for the shift charged field,
and the complete component expansion for the gauge invariant case.
The action is given by:
L
where
12
9k
=
(B.17)
Lgk+Lm+w+LGS+LFI
contains the gauge kinnetic terms, Lm contains the kinetic terms for the
matter as well as the gauge interactions. Lw is the superpotential, LGS is the GreenSchwarz term , and the LFI is the conventional Fayet-Iliopoulos(FI) term.
Lgk
=-
d
TaTa
20+
(B.18.
a
Lm
=-1
d 20+
JE)1_(b +i
e
Y
Y +
aa
-(yl
YI ) + E e
A T
(B.19'
Lw
d+
-
IPA
+ h. c.-
A
LGS
L FI
5[aQblVa Ab
+E
abl
-
-
E5ta
d+Ta
dal
J+T
aYI + h.c.
(B.21
a,l
(B.22.
+ h.c.
a
Restricting to the case of d = 0, i.e. the classically gauge invariant case, the
component expansion in Wess-Zumino gauge can be organized as:
192
L
=
Lb
=
-- La +
- L
F
2F1a
=
a
+
_
21
a
a
Lf
Li
2e 2a
ik
--aa+A-a
2
aF01a
+
+ i
V)+D-/+l + i
(AaX+i +
D+5,D_#+
k 2DD-y
Sk
5
+i-X+i+ i E
-AD+7-A
A
-aX-t)
al
La L
I D1
2e2
=
=
+ 1|
Qa/k2 (Y1
+
(2|2
+ 9i)
-
ra )
+
IGA|2+
a
a
Li
a
(1:
E-A
+
" E(GAJA+
N2-A
A
1jAV+
, +
1
SOJAX+l + h.c.)
+ h.c.) -
Qa (A1 -a)+
I,a
-
'-
1
OIEA
-
|EA|2
A
The auxiliary Lagrangian can be put in a more standard form by solving for the
auxiliary fields.
La
=
-
a
GA
S
(B.27)
EA1 2
A
A
Da
2
EIGA12 -
I I-(?~
(B.28)
2
Q kQ
(y
-
(B.29)
JA
-
Thus the bosonic potential becomes
U =E
EQai#i
2
+
EQak2(y
+ 91)
193
-
+
ra
A A
S lEA IB.30)
A
x+jajEA+
B.3
OPEs
We consider a complex scalar with action
-capoapo = f(+09-0+a-4
Lb =
and a fermion with action, Lf = ic a/&- 3±. Also we consider the action of the form
S= !
d2
47r
(B.31)
L(x)
Then the two point functions are:
1
-log(x - y) 2
=
(B.32)
C
cX
(B.33)
y
Operator Product Expansion with single anomalous
B.3.1
U(1)
Let us list the currents and stress tensor that we want to calculate OPE's with
QIOI-b1 +
-G
I
+ 47 9GY
-
i6y
A
-i
L
ik 2
qAk-A7-A
IO
+-
A
-A
-A
-
2
-
7L
i Ld
A
+Ia
-
a
I
i
+
YAa-Y-A
+
k~ay~
-Aa-A
+ 1)j'-A-Y-A + k 2 KR
+Ei5(O3R
AR
IA-A-A
qoI
R=
5
A
4jR
i
&q1qi+
I
_
-
2rR&
+
-A-
4
-
0
y
)
R
9y +
a-'
RdY
A
(a RoI I
-
i (
1)
+IV)b±)
OPEs for the currents are:
194
+i(k 2K~g+y + ik
2
+X+) _
R
a )
j-(x)j-(y)
- )2
_
(-
2
-
(x-
QIQI
-
Qi
-
qAqA - 2)
qAA
(z
(z
j+ (X)jL+(Y)
L
2d
-
PLL
,sL
2(KR)2 )
R aRZ_,QR3R
a
-)2
_
(-
2C
A
a
i7 (x)i+ (Y)
-
A
y
(X
a
2
(a R- 1)2
Ra-
rl j
2(x-
(z
3
1
j+
G
2 (x
-
2
2(x-
y-)3
(x)t++(Y)
8(x-
S
-
-E
IAL3
-
2
KLIR)
A
4
y-)
1(
2
(x -y-)
a
1
i 2
2(x--y-)2
t++
A
-j+
G
-
(
ell j
2KR)
I
I
y-)
)
1+
I
2(x-- y-)
E (3(
t++
(X-
-
Y-)2
-1)2-
1) +E(1-3(0)2)
A
-t++ \
(- _ _ .)
195
6c(R
)2
+
y
A_
(B.31
B.3.2
Operator Product Expansion with multiple U(1)s
After rescaling Ta -4 eaTa in the deep UV, the free operators we calculate OPEs are
-AT-A
+ I qA
A
I
~
-
-iI
-
-k 4
-
al
I
A -A7-A
+
I
i
-1
-
-
-U &
A
2-
I
I
A
+- ka&yll + 1
O-A
)
'Aaa-ka
I
(I
4j+
JR
OA
I
-A1-A IyI-y
A!--Y-
A
(a
=R
A~->A +
i
k
gRlay)
A
R -E
=i
A + 1)A
1
IO+
-
i(
1 -a
2K~a~- l +
Ld A-
2~ 2
~~)
79I+I)
uRlay
a
1
(ki
+
1
I
A a
i
+i
± + -
Rigxi+
1
(B.36)
196
Then the free OPEs of the currents are:
1-1
-j
(x
aqb
2
A
Q)
-2
A
1
1+
ja+(X)jl+ Y
j+(X)j+(y)
11_
Qia
-2
(x
qvAL/f~
-
(ual M
-
A
I I
A
+ uLlQa)
I
A#
)
- 21
j+ (X)i+ (Y
2
I~a R
( aRa
\I
11_
j- (x)j- (y)
j+(x)t++(y)
(X+
2(x-
a RaR
2 (x2(x-
a
y-)3
j+
2 (x- - y-)2
8(x
-
2
(X-
(uaR
Pa
+ u1Qa))
A111
(u'L1M +UR
-5OAOA
1 aj+
2 (x- - y-)
+ X
-_y-)2
(1 - 3(
1) +
(I
t++
-1
B.4
)
-
A
1 a2ja±
2 (x--y-)
2
_-
aqo!
-a
\Ld
L
1
t++ (x)t++(Y)
1)
1)2
I
y-)3
-
/RS
a
A
-+)
2G
j+( x)t+(y)
I
A
)2) -6
'1 +
I
+-t++
)
Quantum Chirality
As mentioned above, classically chiral operators can be anomalous. In this section
we check that chirality holds within correlation functions. If we only focus on supersymmetric vacua then:
([I,J(x)] 0(y)) = -(J(x)
197
[I-+, 0(y) ])
(B.38)
I
2-:
In order to verify chirality we want an operator 0(y), where the action of D+ is
known, following [101], [122], we choose 0(y) =
OT(y).
Using the equations of motion D+O-T = 2ie2 JG+(y). A necessary condition for
quantum chirality is:
(D+J(x) a_.T(y))
2
-(J(x) D+O-f(y)|EOMs) = -2ie
=
B.39)
Thus quantum chirality requires the vanishing of the leading singular part of the
J JG+ OPE. Let us first investigate T++. The lowest component of T++ and JG give:
jG+(X
++ (Y
2(x
-
Q
( IZeR
y-)
-
E
A)A)
A
2 (x - y
2G
2
2 (x-
y-)
(B.40)
Therefore requiring U+T++ = 0, relies on having:
QI
Q'a-
A
I
I
(B.41)
i3 = 0.
-qA
This condition is equivalent to a particular R-symmetry being non-anomalous. Thus
quantum chirality of the left moving stress tensor requires a non-anomalous R-symmetry.
Next let's consider J+, which has an OPE with the gauge current:
jG+(X) jL+
(Y)
(X
-
y-)
2
A
Thus non-chirality of U(1)L boils down to the vanishing of an L-anomaly:
U+j+ oC)
I
QICL _ 1
A
198
qAA) T.
(B.42)
For the gauge current J+ to be a quantum chiral operator, the same analysis gives:
+ (ZQIQI
-
ZAA)
T.
(B.43)
A
(I
In short, the quantum chirality of a current requires the various anomaly coefficients vanish. The existence of a non-anomalous gauge current, L current and R
current is necessary for quantum chirality of J+and T++, and tells us that the lowest
components of these operators are elements of cohomology, which become the U(1)L
current and left moving stress tensor for the IR CFT.
199
200
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