Lecture Notes on
Automorphic Forms and Physics
Miranda C. N. Cheng
[
[,\
Department of Mathematics, Harvard University,
Cambridge, MA 02138, USA
\
Department of Physics, Harvard University,
Cambridge, MA 02138, USA
Abstract
This is a type-up of the lecture notes for a series of lectures on
the topic “Automorphic Forms and Physics”, given in the preparatory school for the conference “Conformal Field Theory, Automorphic
Forms, and Related Topics” in Heidelberg in 2011. The target audience are advanced students and postodoc researchers in mathematics with basic knowledge in modular forms and automorphic forms.
Please email me if you spot any typo/mistake. Thank you!
1
Contents
1 Conformal Field Theory: a physicist’s introduction
1.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Two-Dimensional CFT . . . . . . . . . . . . . . . . . . . . .
1.3 Virasoro Algebra and Primary Fields . . . . . . . . . . . . .
1.4 Quantum States, Partition Functions, and Modular Objects
1.5 An Example: The Free Boson . . . . . . . . . . . . . . . . .
1.6 Note and References . . . . . . . . . . . . . . . . . . . . . .
2
. 2
. 4
. 5
. 7
. 9
. 11
2 Superconformal Field Theory, Indices, and Jacobi Forms
2.1 N = 1 Superconformal Algebra and the Witten Index . . . .
2.2 N = 2 Superconformal Algebra, Elliptic Genus, and Jacobi
Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Elliptic Genus from a Geometric Point of View . . . . . . . .
2.4 Some Examples: K3 and T 4 . . . . . . . . . . . . . . . . . .
2.5 Notes and References . . . . . . . . . . . . . . . . . . . . . .
11
. 12
3 Twisting and Orbifolding
3.1 Twisting . . . . . . . . . .
3.2 Orbifolding . . . . . . . .
3.3 An Example: Z2 Orbifold
3.4 Notes and References . . .
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14
17
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4 Second-Quantised Elliptic Genus, Siegel Modular Forms and
Borcherds’ Lift
26
4.1 Notes and References . . . . . . . . . . . . . . . . . . . . . . . 30
1
1
1.1
Conformal Field Theory: a physicist’s introduction
Generalities
CFT= QFT w. Conformal Symmetry
What is a Quantum Field Theory?
We can describe it in the Lagrangian and Hamiltonian formalisms as
follows.
2
What are a conformal transformations?
They are transformations preserving angles but not distances.
More precise, they are generated by the following transformations, where
we have written out the form of the corresponding finite transformations.
translation
dilation
rotation
special conformal transformation
x µ → x µ + aµ
xµ → αxµ
xµ → Mνµ xν
xµ −bµ x2
µ
x → 1−2b·x+b
2 x2
What are CFT’s good for?
1. Conformal symmetries give us much better control over the theory.
2. To study physical systems at the critical points or with approximate
conformal symmetry. It’s also the basis of (perturbative) string theory.
3. Mathematically, 2d CFT’s are factories producing modular objects.
4. AdS/CFT correspondence conjecture makes CFT a great laboratory
for the complicated physics of quantum gravity.
3
1.2
Two-Dimensional CFT
Conformal Transformation in 2D
T (z), T (z̄) : Energy-Momentum Tensor.
Given a QFT with its Lagrangian we can compute the energy-momentum tensor
(also called the stress tensor) measuring how the system reacts to a change of world-sheet.
For instance a change of time gives the Hamiltonian. We will consider the mode expansion
P
P
m
m
T (z) = m∈Z zLm+2
, T (z̄) = m∈Z z̄L̃m+2
.
4
Weyl Anomaly
1.3
Virasoro Algebra and Primary Fields
With the Weyl anomaly taken into account, the mode expansion of the
energy-momentum tensor T (z) satisfies the modified version of the algebra
of `m , called the Virasoro algebra with central charge c:
[Lm , Ln ] = (m − n)Lm+n +
c
(m3 − m)δn+m,0 .
12
(1.1)
The mode expansions L̃m of T̃ (z̄) satisfies another copy of the Virasoro algebra, a priori with a different central charge c̃. And the two copies of Virasoro
do not talk to each other.
From this we conclude that the following is true by definition.
In a 2d CFT, the “quantum states” are assembled into representations of
(2 copies, left and right, of ) Virasoro algebras.
More precisely, attached to a QFT is a vector space V , which we shall call
the space of quantum states, or the Hilbert (or Fock) space of the physical
theory. For the case when the QFT is a 2d CFT, we say that V decomposes
into representations of the Virasoro algebra.
5
To discuss CFT it is hence crucial to know a few things about the representations of the Virasoro algebra. To simplify the discussion we will focus
on only one copy of the Virasoro. The same consideration applies to the
other copy.
P
†
m
= (1/z̄)4 T (1/z̄)
Finally, defining the adjoint operators as T (z)† = m z̄L2+m
†
and expand both sides as Laurent series in z̄, we obtain Lm = L−m . The motivation for defining the adjoint in the above way is that, in the Hamiltonian
formalism z 7→ 1/z̄ is exactly the time reversal t 7→ −t in the cylinder coordinate. In this lecture we will only be concerned with unitary representations
where hv|α† α|vi ≥ 0 for all |vi ∈ V and all operators α.
The Ground State
Moreover, for the energy of the theory to be bounded from below (in
other words, to avoid pathology), we require that there is a unique vector
|0i ∈ V, such that Ln |0i = 0 for all n ≥ −1. This is equivalent as requiring
T (z)|0i to be well-defined as z → 0.
The Highest Weight Vector and the Verma Module
A highest weight vector is a vector |hi satisfying
L0 |hi = h|hi, Ln |hi = 0 for all n > 0 .
From it we can built the corresponding highest weight module, or Verma
module, spanned by the states of the form
L−k1 . . . L−kn |hi ,
ki > 0 .
State-Field Correspondence
This is a 1-1 correspondence between a state |vi ∈ V and a field v(z, z̄)
(which might be “composite” in the sense that they are not the map ϕ
mentioned in our introduction to QFT), via the condition
|vi = lim v(z, z̄)|0i .
z,z̄→0
This formula gives us a way to go from fields to state. In the other
direction, we have to employ the time translation, or the radial quantisation,
to reconstruct the whole field with it’s full z-dependence.
6
Primary Fields
Using the above dictionary, we can see that the fields corresponding to
the highest weight vectors |h, h̃i have the “nicest” transformation properties
under a general coordinate transformation:
vh,h̃ (z, z̄) = vh,h̃ (w(z), w̄(z̄))(∂z w)h (∂z̄ w̄)h̃ .
The fields with these properties are called primary fields. We hence see that
there’s a 1-1 correspondence between highest weight vectors in the Hilbert
space and the primary fields in the theory.
1.4
Quantum States, Partition Functions, and Modular Objects
What are we quantising?
Hence, the Hilbert space V is obtained by quantising LM =the free loop
space of maps S 1 → M .
Partition Function
A partition function, or Zustandsumme, of the theory, is defined by
Z(τ, τ̄ ) = TrV q ĤL q̄ ĤR
,
q = e(τ ) .
One way to think about it is to consider the Ĥ operator as measuring
the energy of the state, and τ /i = β = 1/T provides Boltzmann factor as
in statistical mechanics, where T denotes the temperature. The other way
to think about it is the following: Hamiltonian dictates the time translation
of the theory, and the operator e2πiτ Ĥ evolves the system for a duration −iτ
7
of time, and we want to know the trace of such an operator acting on the
Hilbert space V . From this point of view, this quantity also has a very natural
interpretation in terms of the Lagrangian formalism. In the latter point
of view, taking the trace means we are also imposing a periodic boundary
condition in the “time” direction. As a result, we end up performing a path
integral with a doubly periodic boundary condition
ϕ(w) = ϕ(w + 1) = ϕ(w + τ ) .
The first identification comes from the fact that Σ̃ = S 1 , and the second
comes from taking the trace. In picture, this is:
But the formulas we have are now explicitly invariant under the torus
mapping class group P SL(2, Z)! Hence we conclude that the partition function of a 2d CFT should have the modular invariance
aτ + b aτ̄ + b
a b
Z(τ, τ̄ ) = Z(
,
) for
∈ P SL(2, Z) .
c d
cτ + d cτ̄ + d
The functional integral we are (pictorially) computing is called a path
integral by physicists. The name is due to the fact that, for the special case
when Σ̃ is a point and hence Σ a line (the world-line), the integral over all
possible maps with given initial and final boundary condition is literally an
8
integral over all possible “paths” from A to B (A and B is the same for the
computation of partition function) in the target manifold M .
1.5
An Example: The Free Boson
To illustrate the previous discussion we will now do a simple example in
which M = R and the Lagrangian is free (of interaction terms). In this part
I will use the “physicists’ convention” 2 = π = i = 1. But apart from this
which can be fixed by carefully going through the steps, all formulas should
be correct.
We are looking at the maps ϕ̃ : S 1 → R. The free loop space has the
following coordinates
X
e(ns)ϕn .
ϕ̃(s) = ϕ̃(s + 1) =
n∈Z
The free Lagrangian is given by
Z
X
dϕ ∧ ?dϕ =
ϕ̇n ϕ̇−n − n2 ϕn ϕ−n .
L=
Σ
n∈Z
The physicist’s black magic called (canonical) quantisation involves identifying a natural symplectic structure of the phase space of the theory and
replacing the classical Poisson bracket by a (anti-) commutator like this kind
[ϕn , πm ] = iδn,m ,
where here we read out from the Lagrangian (or the Hamiltonian) that πn ∼
ϕ̇−n . Using instead the alternative basis
ϕn =
1
(an − ã−n )
n
,
πn = an + ã−n
with the commutator relation
[an , am ] = nδn+m,0 = [ãn , ãm ], [an , ãm ] = 0 ,
the Hamiltonian reads
H = π02 +
X
(a−n an + ã−n ãn ) .
n6=0
9
(1.2)
Using the Hamiltonian to time-evolve the system and go to the Heisenberg
picture, we finally obtain
X
X
∂z ϕ(z, z̄) = π0 z −1 +
an z −n−1 , ∂z̄ ϕ(z, z̄) = π0 z̄ −1 +
ãn z̄ −n−1 .
n6=0
n6=0
From this we can compute and discuss some basic quantities of this theory.
Conformal Symmetry
From the Lagrangian we can read out the energy-momentum tensor T (z) ∼
∂ϕ∂ϕ. We can check that its mode expansion satisfies the Virasoro algebra
with c = 1 = c̃. Essentially, this can be understood by using the zeta func1
when dealing with the ordering
tion regularisation 1 + 2 + 3 + · · · = − 12
ambiguity of quantisation.
Ground State
The ground state satisfies
an |0i = 0 = ãn |0i ,
for n > 0 .
Primary Fields
Some examples are given by ∂ n ϕ, ∂¯m ϕ, eikϕ , . . . .
Partition Function
We want to compute the quantity Z(τ, τ̄ ) = TrV q ĤL q̄ ĤR . Notice that the
Hamiltonian (1.2) splits into two parts: the “zero modes” π0 and the “oscillators” an , ãn , n 6= 0. It is clear that the latter part is simply given by the
tensor product
copies of the Heisenberg algebras and its contribution
Q of two
1
2
is simply | ∞
|
. The first part, on the other hand, doesn’t factor
n=1 1−q n
into the left- and right-moving part and π0 has a continuous spectrum which
leads us to doing the following integral
Z
dk −πk2 Imτ
1
e
∼√
.
2π
Imτ
10
Combining ingredients including the zero-mode energy −c/24 = −1/24, we
get
∞
1 1 2
1 −1/24 Y 1 2
.
=√
Z(τ, τ̄ ) = √
q
n
1
−
q
η(τ
)
Imτ
Imτ
n=1
Now we can check that this is indeed modular invariant. This is to be
contrasted with the partition function for the Heisenberg algebra, namely
when we forget about the zero-modes and take the chiral half. In that case
the partition function η(τ1 ) is a modular form with non-zero weight.
1.6
Note and References
There are many important omissions and prejudices of this first lecture. I will
excuse myself by evoking the fact that it is next to impossible to introduce
CFT in one lecture. First, to compliment the VOA perspective introduced
in the lecture series of Prof. Mason, my language is deliberately geometrical
and also non-algebraic. The hope is to provide the school participants two
complimentary sets of intuition which will hopefully both be useful. Second,
one thing I find a big pity is that I don’t have time to discuss the operator
product expansion (OPE) properties of the 2d CFT. This is an essential
feature of CFT and corresponds to the “mutual locality” condition in the
lecture of Prof. Mason. I encourage the reader to read about this.
As for references, there are obviously many excellent reviews and lecture
notes on CFT from a physics’ point of view. The material covered here can
largely be found in, for instance, [1].
2
Superconformal Field Theory, Indices, and Jacobi
Forms
There are many interesting extensions of the Virasoro algebras, which are
realised in interesting conformal field theories. The W-algebras, and algebras
with affine Kac-Moody symmetries are some of the familiar examples.
In this lecture we will focus on Virasoro algebra extended by supersymmetries. The presence of supersymmetry means that there is now an extra Z2
grading of the algebra: A = A0 ⊕ A1 with Ai Aj ⊆ Ai+j . Therefore, there is a
natural involution which maps a0 → a0 and a1 → −a1 for a0 ∈ A0 , a1 ∈ A1 ,
and the corresponding operator satisfies [(−1)F , a0 ] = {(−1)F , a1 } = 0.
11
The motivations for studying such supersymmetric extensions abound.
Some examples are: to have better control over the theory, to study the
topological properties of a manifold of interests, and the hope that supersymmetries of some sort are realised in the physical nature. From the point
of view of automorphic forms, however, studying SCFT also leads us to an
interesting class of modular objects: the Jacobi forms.
2.1
N = 1 Superconformal Algebra and the Witten Index
The simplest algebra with supersymmetry and containing the Virasoro as
a subalgebra is the so-called N = 1 superconformal algebra (SCA). The
meaning of “N = 1” should become clear in the next subsection where we
introduce SCA with N > 1.
The algebra reads
[Lm , Ln ] = (m − n)Lm+n +
{Gr , Gs } = 2Lr+s
m − 2r
Gm+r .
[Lm , Gr ] =
2
c
(m3 − m)δn+m,0
12
(2.3)
There are two types of “N = 1” SCA, corresponding to have r, s ∈
Z (Ramond, or R sector), or r, s ∈ 21 + Z (Neveu-Schwarz, or NS sector)
in the above formula. In either case, the Z2 grading gives [Lm , (−1)F ] =
{Gr , (−1)F } = 0. Correspondingly, we say Lm ’s are the bosonic and Gr ’s the
fermionic generators of the above algebra.
Hence, in a theory with N = 1 superconformal symmetry
dimenP in two
Lm
sions, apart from the energy-momentum tensor T (z) =
m∈Z z m+2
P which
Gr
satisfies the Virasoro algebra, we have an extra current G(z) =
r z r+3/2
which satisfies the above algebra given above.
A familiar setup where such a SCFT arises is when apart from the bosonic
fields ϕi (z) which might be thought of as local coordinates as a certain “target
space”, a complex manifold M , we also have their “fermionic partners” ψ i (z),
and the currents which generate the SCA are schematically given by T (z) =
1
(∂ϕ · ∂ϕ + ψ · ∂ϕ) and G(z) = ψ · ∂ϕ, where ∂ = ∂z . In this language, the
2
difference of Ramond versus the Neveu-Schwarz sectors corresponds to the
two different boundary conditions one can choose for the fermions as we go
around the “world-sheet circle” s → s + 1.
12
The statement that a 2d QFT has N = 1 superconformal symmetry
implies that its quantum Hilbert space V can be decomposed into representations of the SCA. To study the structure of the theory we will now discuss
a few properties of its basic representations.
Ramond Ground States
For the purpose of illustration we will from now on focus on the R-sector
of the algebra. The NS sector can be treated analogously.
As before, we require a ground state |φi to be a vector in V annihilated
by all the positive modes
Ln |φi = Gr |φi = 0 for all n, r > 0 .
Moreover, we impose the condition that they are annihilated by the fermionic
zero-modes as well:
G0 |φi = 0 .
From the algebra (2.3) we observe that G20 = L0 , hence the Ramond ground
states are automatically L0 -eigenstate with zero eigenvalues.
Witten Index
In a SCFT, the Ramond ground state is no longer unique. Since the
ground state structure is such a basic feature of a theory, we would like to
have some information of it. To this end Witten has defined the so-called
Witten Index which counts (with Z2 -grading) the number of ground states:
WI : N = 1 SCFT → Z ,
defined as
WI(τ ) = TrV (−1)F q L0 .
The definition is very analogous to the partition function, with the main
difference being the insertion of the “fermion number operator” (−1)F . This
insertion makes the sum collapse to a sum over the Ramond ground states,
for if a state |vi is not annihilated by G0 , the other state G0 |vi will have
the same eigenvalue under L0 and an opposite eigenvalue under (−1)F . As
a result, the contributions from |vi and G0 |vi add up to zero unless |vi is a
Ramond ground state.
13
From this argument, we see that the Witten index counts the graded
number of Ramond ground states and is in particular independent of the
parameter τ . Moreover, since the index takes value in a discrete set, it is
immediate that it has to be invariant under infinitesimal change in the moduli
space of the theory, at least generically 1 .
2.2
N = 2 Superconformal Algebra, Elliptic Genus, and Jacobi
Forms
In the last subsection we introduced the “minimal” extension of Virasoro
algebra by supersymmetry. There are also the so-called extended supersymmetries with N > 1. Recall that for the minimal case, we have included a
fermionic current G(z) on top of the bosonic energy-momentum tensor T (z).
In general we will add N such fermionic currents. Furthermore, there’s now
an extra automorphism, which we call the R-symmetry, that rotates different fermionic currents onto each other. Together we have, schematically,
an superconformal algebra generated by currents {T (z); G1 (z), . . . , GN (z);
R-currents}.
For N = 2, we denote the two fermionic currents by G+ (z) and G− (z)
and the U (1)-current rotating the two by J(z). The algebra reads
c
[Lm , Ln ] = (m − n)Lm+n + m(m2 − 1) δm+n,0
12
c
[Jm , Jn ] =
m δm+n,0
3
[Ln , Jm ] = −m Jm+n
n
[Ln , G±
− r) G±
r ] = (
r+n
2
±
[Jn , G±
r ] = ±Gr+n
c 2 1
−
(r − ) δr+s,0 ,
{G+
r , Gs } = 2Lr+s + (r − s)Jr+s +
3
4
(2.4)
and all other (anti-)commutators are zero. As before we have two possible
1
Sometimes there can be the so-called wall-crossing phenomenon occurring, which
means the Witten index jumps discontinuously at these very special subspace in the moduli
space. We will not have time to discuss this very important subtlety here.
14
periodic conditions for the fermions


2r = 0 mod 2 for R sector


(2.5)
2r = 1 mod 2 for NS sector .
Two comments about this algebra are in order here. First, we have now
two generators of the Cartan subalgebra: [L0 , J0 ] = 0. As a result, the
representations will now be graded by two “quantum numbers” that are the
eigenvalues of the L0 and J0 of the highest weight vector. The second new
feature is that there is a non-trivial inner automorphism of the algebra, which
means that the algebra remains the same under the following redefinition
c
Ln → Ln + ηJn + η 2 δn,0
6
c
Jn → Jn + η δn,0
3
±
G±
→
G
r
r±η
(2.6)
with η ∈ Z. If instead we choose η ∈ Z + 1/2 we exchange the Ramond
and the Neveu-Schwarz algebra. Note that the only operator (up to scaling
and the addition of central terms) invariant under such a transformation is
2
cL0 − J02 .
3
Again we will focus on the Ramond algebra and define the Ramond
ground states of N = 2 SCFT:
Ramond Ground States and the Witten Index
As before we require the ground states to be annihilated by all the positive
modes:
Ln |φi = Jm |φi = G±
r |φi = 0 for all m, n, r > 0 .
Moreover they have to annihilated by the zero modes of the fermionic currents
G±
0 |φi = 0 .
Again this condition fixes their L0 -eigenvalue to be
1 + −
c
{G0 , G0 }|φi = L0 −
|φi = 0 .
2
24
15
Repeating the same argument, the Witten index WI : N = 2 SCFT → Z,
defined as
c̃ ˜
WI(τ ) = TrV (−1)J0 q L̃0 − 24 ,
is τ -independent and counts the graded number of the Ramond ground state
in V . Compared to the definition for N = 1 CFT, there are the following
two changes: From the fact that the operator (−1)J0 has the same (anti)commutation relations with all the generators of the algebra as the involution operator (−1)F , we can simply declare (−1)J0 as the involution operator
in this case. Second, there is a shift of the ground state energy by −c/24
from the G±
0 anit-commutation relations.
Notice moreover that the Witten index for N = 2 SCFT acquires an
interpretation as computing the graded dimension of the cohomology of the
+ 2
+
+ †
+
−
c
Q+
0 operator, satisfying (Q̃0 ) = 0. For {G̃0 , (G̃0 ) } = {G̃0 , G̃0 } = L0 − 24 ,
the Ramond ground states have the interpretation as the harmonic representative in the cohomology. This fact underlies the rigidity property of the
Witten index and the elliptic genus which we will define now.
The N = 2 Elliptic Genus
It is fine to be able to compute the graded dimension of a cohomology,
but we can go further and compute more interesting properties of this vector
space. For instance, we have learned that the representations of N = 2 SCA
are labeled by two quantum numbers corresponding to the Cartan generators L0 and J0 . It will hence be natural to consider the following quantity
which computes the dimension of Q̃+
0 cohomology graded by the left-moving
quantum numbers L0 , J0 .
Definition: Elliptic Genus (CFT)
The elliptic genus of a N = (2, 2) SCFT is the quantity
¯
Z(τ, z) = TrVRR (−1)J0 +J0 y J0 q L0 −c/24 q̄ L̄0 −c/24
, y = e2πiz ,
(2.7)
where the “RR” in VRR denotes the fact that we are considering the Ramond
sector of the N = 2 SCA both for the left- and right-moving copy of the
algebra.
Modular Properties
16
As we argued before, a path integral interpretation of the elliptic genus
suggests it has nice transformation property under the torus mapping class
group. Moreover, the inner automorphism of the algebra (the spectral flow
symmetry) implies that the graded dimension of a L0 -, J0 - eigenspace should
only depends on its eigenvalue under the combined operator 32 cL0 − J02 .
Hence, the Fourier expansion of the elliptic genus should take the form
X
Z(τ, z) =
q n y ` c( 23 cn − `2 ) .
n,`
From these facts one can deduce that the elliptic genus of an N = (2, 2)
SCFT is a weak Jacobi form.
Definition: Jacobi Form
If the function φ(τ, z) : H × C → C transformas in the following way
under the Jacobi group SL(2, Z) n Z2 :
cz 2
a
b
k 2πit cτ
z
aτ +b
+d Z(τ, z)
,
∈ SL(2, Z)
Z( cτ +d , cτ +d ) = (cτ + d) e
c d
Z(τ, z + λτ + µ) = e−2πit(λ
2 τ +2λz)
Z(τ, z) ,
and have furthermore the expansion
X
Z(τ, z) =
λ, µ ∈ Z ,
f (4tn − `2 )q n y ` .
(2.8)
(2.9)
n≥0,`+t∈Z
with f (4tn − `2 ) = 0 for all 4tn − `2 < 0 (n < 0) with some integral or
half-integral k and t, then the function is called a holomorphic (weak) Jacobi
form of weight k and index t.
From this definition, we have the following fact
Fact: The elliptic genus of an N = (2, 2) SCFT with central charge
c = 6t is a weight zero, index t weak Jacobi form.
2.3
Elliptic Genus from a Geometric Point of View
To understand the relation between the elliptic genus defined above and
certain geometries, let us focus on a specific type of quantum field theory
called the (non-linear) sigma model. A QFT is called a sigma model if the
fields it studies are maps ϕ : Σ → M from a world-sheet Σ to a Riemannian
17
manifold M with metric gij , i, j, = 1, . . . , dimR M , and the Lagrangian of the
theory includes a term
Z
L=
gij dϕi ∧ ?dϕj + . . . .
Σ
In particular, a 2d CFT with dimR Σ = 2 probe the geometry of M and
its free loop space LM .
The constrained structure of supersymmetry means that not all such
models for any arbitrary target manifold M can be extended into a theory with supersymmetry. For our purpose we are interested in theories
with N = (2, 2) supersymmetry. Supersymmetry is achieved when M is
a Kähler manifold and the U (1) R-symmetry is identified with the diagonal
U (1) ⊂ U (d), d =dimC M . Moreover, we would like to require conformal
symmetry as well. The requirement of N = (2, 2) superconformal symmetry
restricts the manifold to be Calabi-Yau, and the elliptic genus of the corresponding SCFT computes essential topological quantities of the Calabi-Yau
manifold.
In fact, this quantity can be defined for any complex, compact manifold
in the following way:
Definition: Elliptic Genus (Geometry)
For a compact complex manifold M with dimC M = d, the elliptic genus
is defined as the character-valued Euler characteristic of the formal vector
bundle
V
∗
Eq,y = (y d/2 −y−1 TM
)⊗
(2.10)
N
V
N
V
N
∗
∗
−yq n TM ⊗
−y −1 q n TM ⊗
n≥1
n≥1
n≥0 Sq n (TM ⊕ TM ) ,
∗
where TM and TM
are the holomorphic and anti-holomorphic tangent bundle,
and
V
V2
2
V + . . . , Sq V = 1 + qV + q 2 S k V . . .
q V = 1 + qV + q
with S k V denotes the k-th symmetric product of V .
In other words, we have
Z
Z(τ, z; M ) =
ch(Eq,y )T d(M ) .
M
18
(2.11)
This topological quantity reduces to the familiar ones: the Euler number,
the signature, the A-roof genus of M , and the Hirzebruch χy -genus, when
specialising at the special values z = 0, τ /2, (τ + 1)/2 and τ = i∞ , respectively.
When M is a Calabi-Yau manifold and the supersymmetric sigma model
with M as target space is a N = (2, 2) superconformal field theory, one
can show that the above geometric definition of the elliptic genus and the
CFT definition coincides. On a heuristic level, this can be understood in
the following way: because of the rigidity property of the CFT elliptic genus
we can go to the “large volume limit” where the manifold M is locally flat.
Writing ϕi , ϕ̄ , i,  = 1, . . . , d as the local flat coordinates of M and ψ i , ψ ̄ their
“fermionic partner” with ϕin , ϕ̄n , ψni , ψn̄ their respective mode expansions, we
can think about the second, third, and fourth factor of the formal vector
̄
i
and
bundle (2.10) as coming from acting on the ground states with ψ−n
, ψ−n
̄
i
ϕ−n , ϕ−n respectively. Finally, the first factor comes from quantising the
fermionic zero modes ψ0 which generate a Clifford algebra.
Combining all we have said before, it follows that the geometric elliptic
genus is a weak Jacobi form of weight zero and index dimC M/2 when M is a
Calabi-Yau manifold. Indeed this can be shown directly using the geometric
definition (2.11).
2.4
Some Examples: K3 and T 4
There are two topologically distinct Calabi-Yau two-folds: K3 and T 4 . We
expect their elliptic genus to be weight zero weak Jacobi forms with index 1.
Coincidentally, the dimension of the space of such a form is one: C ⊗ϕ0,1 (τ, z)
where
X θi (τ, z) 2
.
ϕ0,1 (τ, z) = 4
θ
(τ,
0)
i
i=2,3,4
Hence we only need one topological invariant of the Calabi-Yau two-folds to
fix the whole elliptic genus. From
Z(τ, z = 0; T 4 ) = χ(T 4 ) = 0 ,
Z(τ, z = 0; K3) = χ(K3) = 24
and
ϕ0,1 (τ, z = 0) = 12
we obtain
Z(τ, z; T 4 ) = 0 ,
Z(τ, z; K3) = 2ϕ0,1 (τ, z) .
19
This clearly demonstrates the power of modularity in gaining extremely
non-trivial information about the spectrum of N = (2, 2) SCFT.
2.5
Notes and References
Elliptic genus for/from mathematicians: [2, 3, 4, 5] and from physicist [6, 7].
For a VOA point of view, see [8].
3
Twisting and Orbifolding
Consider a special situation when the target manifold M has a non-trivial
automorphism group G that is a finite simple group. There are two interesting things we can do in such a situation: one is the so-called “twisting” (or
“twining”) where more refined information about the spectrum can be obtained. The other is the so-called “orbifolding”, which is a procedure which
constructs a new conformal field theory, with now the orbifold M/G as target
space, from the old one with target space M . Of course, in general the orbifold is not a smooth manifold. However, as we will see, it doesn’t obstruct
us to construct the corresponding conformal field theory. Hence we say that
this type of singularity can be “dealt with” by string theory.
For simplicity we will limit ourselves to Abelian groups. We will comment
briefly on the non-Abelian orbifold in the end of this lecture.
3.1
Twisting
To understand the procedure of twisting (or twining), let us note that the free
loop space LM also inherit the automorphism group G. Upon quantisation,
the quantum Hilbert space V hence also has a G-symmetry. Moreover, from
the above description we expect the G action on V to commute with the
grading of L0 . Now, when we have a G-module Vn , apart from its dimension
we can also compute its character TrVn g, for g ∈ G. Moreover, knowing the
character for all conjugacy classes [g] of the group allows us to pin down the
action of G on it. This procedure of “twisting by g” leads to the computation
of the so-called “twisted (or twining) partition function” of the theory.
Apart from this Hamiltonian description of the twisted partition function,
it is obvious that it also allows a natural interpretation in terms of path
integral. Namely, instead of performing a functional integral over maps from
a torus into the target space with doubly periodic boundary condition, we
20
are integrating over maps with boundary conditions that are modified by g
along the Euclidean “time” direction.
Here we illustrate what we said in terms of pictures:
21
3.2
Orbifolding
Twisted Sectors
As mentioned before, we can construct a new CFT by quantising the free
loop space L(M/G) of the orbifold instead. A very useful decomposition of
the loop space can be found by considering the larger space
IM = {ϕ̆ : [0, 1] → M } .
The loop space L(M/G) of the orbifold can be identified with a subspace of
IM in the following way:
L(M/G) = {ϕ̃ ∈ IM |ϕ̃(1) = h · ϕ̃(0) for some h ∈ G} .
Using this identification, we see that the loop space has a natural decomposition
L(M/G) = ⊕h Lh (M/G)
defined by ϕ̃ ∈ Lh (M/G) iff ϕ(1) = h · ϕ̃(0).
Related to this loop sub-space is the following quantity Z(1 ; τ ), which
h
is an important building block of the orbifold theory:
Upon quantisation we should obtain the decomposition of the physical
Hilbert space
V G = ⊕h∈G V G,h .
We call V G,h the quantum Hilbert space of the h-twisted sector of the orbifold
theory. We might expect V G,h to be simply the quantisation of the space
Lh (M/G) seen as a subspace of IM . There is a very crucial subtlety we would
be overlooking if we take this viewpoint, however. Namely, the quantum
Hilbert space of the orbifold theory on M/G has to be, by definition, Ginvariant. This invariance can be achieved by, in a path integral language,
averaging over the boundary condition along the Euclidean time circle. From
this prescription, and summing over all twisted sectors, we finally obtain an
expression for the orbifold theory in terms of path integrals over maps into
the original manifold M :
22
23
For non-Abelian finite simple group G, the projection onto G-invariant
state to obtain V G,h is achieved by summing over the boundary condition
twisted by elements of the centralising subgroup C(h), and as a result we get
the modified formula
Z G (τ, . . . ) =
X
Z G,h (τ, . . . ) =
h∈G
1 X X
Z(g ; τ ) .
h
|G| h∈G
g∈C(h)
Modular Transformation
One natural question to ask is how these twisted partition functions and
twisted sector partition functions transform under the torus mapping class
group. Recall that we argued for the invariance under SL(2, Z) by evoking
the invariance of the path integral under such a transformation of the torus,
a natural way to answer this question is to again examine how the (now
twisted) path integral transforms.
Using the so-called box notation as above to denote the boundary condition, we see that upon a redefinition of the A- and B-cycles of the elliptic
curve
a b
A → cB + dA ,B → aB + bA ,
∈ SL(2, Z) ,
c d
the path integral with different boundary conditions transform into each
other as
aτ + b
a b
a b
) , γ=
∈ SL(2, Z) ,
Z(g ; τ ) = ε(γ, g, h) Z(g h ;
c d
h
g c hd cτ + d
where ε(γ, g, h) is a phase.
As a result, typically these Z(g ; τ ) are only invariant (up to a phase)
h
under a subgroup Γg,h of the full modular group. On the other hand, the
total partition function Z G (τ, . . . ) is invariant under SL(2, Z).
3.3
An Example: Z2 Orbifold
First we consider a close cousin of the free boson example of the first lecture:
the compactified boson. Namely, we quantise the loop space LS 1 : {ϕ̃ : S 1 →
R/Z ' S 1 } of a circle of radius R. The oscillator part is identical as before
24
and given by two copies of the Heisenberg algebras. The only difference now
is that the spectrum of the so-called “zero-modes” are no longer continuous.
Instead we have
X
∂ϕ(z) = 12 ( Rn + mR) +
ak z −1−k .
k6=0
The integer n comes from the requirement that the vertex operator eipϕ
has to be invariant under ϕ → ϕ + 2πR and hence the eigenvalue of p is
quantised to be p = Rn . In general, when M is the torus M = Rn /Λ with
some n-dimensional lattice Λ, the momentum p has to be in the dual lattice
p ∈ Λ∗ . The second integer m comes from the fact that we can now consider
the map ϕ̆ : S 1 → R with ϕ̆(s + 1) = ϕ̆(s) + 2πRm.
With this modification in mind, repeat the calculation in 1.5 we obtain
the partition function
X 1 ( n +mR)2 1 ( n −mR)2
1
q̄ 2 R
.
q2 R
Z(τ ) =
|η(τ )|2 n,w∈Z
From this theory we would like to construct an orbifold theory by quantising ϕ̃ : S 1 → S 1 /Z2 where the Z2 acts as x → −x. According to the above
prescription, we need to compute
Z Z2 (τ ) = 41 Z(+ ; τ ) + Z(− ; τ ) + Z(+ ; τ )Z(− ; τ ) .
+
−
+
−
It is now not difficult to compute these quantities, first we have of course
Z(+ ; τ ) = Z(τ ) .
+
(3.12)
It is also straightforward to compute the partition function twisted by the
Z2 symmetry: first the only zero-modes which survives the twisting Q
is when
1
m = n = 0. This collapses the theta function to 1. Second we have n 1+q
n
Q 1
instead of n 1−qn because the oscillators are all odd under the Z2 -action.
Putting it together we get
∞
−1/24 Y
Z(− ; τ ) = q
1 2
.
n
1
+
q
n=1
+
(3.13)
And then we have to quantise the twisted
P sectorψnϕ̃(s+1) = −ϕ̃(s). The mode
expansion now takes the form ϕ̃(s) = n∈Z z1/2+n . The ground state energy
25
1
can be computed using zeta-function regularisation 1 + 2 + 3 + · · · = − 12
as in the case of free bosons and yields 1/48. Moreover, quantisation of the
zero-modes ψ0 leads to representations of the (in this case 1-dimensional)
Clifford algebra and gives an extra factor of two. Putting things together we
obtain
∞
1/48 Y
Z(+ ; τ ) = 2q
(3.14)
∞
1/48 Y
Z(− ; τ ) = 2q
(3.15)
2
1
n−1/2
1
−
q
n=1
1
2
1
.
n−1/2
1
+
q
n=1
+
Combining all the results, we can check explicitly that Z Z2 is invariant
under the modular group.
3.4
Notes and References
See [9, 10] for a systematic discussion on orbifolds CFT. Note that, although
for the purpose of illustration we use the geometric language in which the
finite simple group G is a geometric symmetry, the orbifold construction
discussed here can be straightforwardly applied as long as G is a symmetry
of the Hilbert space V , a condition that is more general that the geometric
statement.
4
Second-Quantised Elliptic Genus, Siegel Modular Forms
and Borcherds’ Lift
In this final lecture we would like to apply the orbifold technique to the special
case of symmetric products, and demonstrate how such a consideration leads
us naturally to Siegel modular forms in H2 .
Consider the CFT on a specific (singular) orbifold S M = M N /SN , with
M a d-(complex)dimensional Calabi-Yau manifold. As mentioned before we
should start by studying the decomposition of the loop space L[h] (S N M ),
where the conjugacy classes [h] is now described in terms of the so-called
cycle shapes of the permutation group:
`1
`2
`r
(i1 ) (i2 ) · · · (ir )
,
r
X
s=1
26
`s is = N .
Applying the procedure described in the last part of the lecture to construct orbifold CFT, we can derive the elliptic genus of the SCFT on the
orbifold S N M . In fact in turns out that it is more natural to write it in
terms of the generating function [11]
∞
X
N =1
N
N
p Z(τ, z; S M ) =
Y
n>0,
m≥0,
`∈Z
c(2dnm−`2 )
1
,
1 − pn q m y `
(4.16)
where we treat p as a formal expansion parameter at this point, and c(2dn −
`2 ) is the Fourier coefficient for the term q n y ` in the elliptic genus Z(τ, z; M )
of M .
A few comments on this rather amazing identity are in order. First, by
looking at the picture of the cycle shapes we can see that these configurations
of the loop space have an alternative interpretation as describing multiply
wound strings moving in the space S 1 ×M . Hence, if we don’t fix the number
N of copies of M , we get an alternative interpretation of the generating
function for the elliptic genus for the S N M orbifold SCFT as the partition
function of the “second-quantised” string on S 1 × M . The term “secondquantised” refers to the fact that we do not restrict the number of strings
allowed in the Hilbert space. For example,
27
Therefore, we have the identity
BPS Partition Function for the Second-Quantised Strings on S 1 × M
c(2dnm−`2 )
Y
1
=
.
nqmy`
1
−
p
n>0,
m≥0,
`∈Z
Indeed, notice that the product side of the formula can be rewritten in
the following form
∞
X
N
N
p Z(τ, z; S M ) = exp
∞
X
N =1
p TN (Z(τ, z; M )
N
(4.17)
N =1
where TN is the Hecke operator mapping a weight zero, index t to a weight
zero, index N t weak Jacobi form. Explicitly it reads
aτ + b
1 X
Z(
, az) .
TN Z(τ, z) =
N ad=N
d
b mod d
The right-hand side has the natural interpretation as a sum over degree N
maps from T 2 to T 2 × M , and summing over N gives the free energy of
the free string theory. Hence, this gives a path-integral derivation of our
expression for the second-quantised partition function.
Second, the formula (4.16) also holds if we replace the orbifold S N M by
the Hilbert scheme M [n] of n points on M [12]. Third, this formula can
be viewed as a generalisation of the formula by Göttsche for the generating
function for the symmetric product Euler characteristic. By specialising to
the value y = 1, we indeed get
∞
X
pN −
χ(M )
24
χ(S N M ) =
N =1
1
.
η(τ )χ(M )
Finally, this formula defines an “almost” automorphic form in the Siegel
upper-half plane. To explain this,first recall the definition of the Siegel
τ z
upper-half plane: we say that Ω =
is in the Siegel upper-half plane
z σ
H2,1 if det(ImΩ) > 0 and Imτ > 0. In other words, when ImΩ is in the
“future light cone” of the (2,1) Lorentzian space. It is sometimes denoted by
28
H2 for the fact that it is also the space of normalised B-period matrix for
genus two Riemann surfaces.
Definition: Siegel Modular Form
If the function Φk (Ω) : H2,1 → C satisfies
Φk ((AΩ + B)(CΩ + D)−1 ) = (CΩ + D)k Φk (Ω)
for all
A B
C D
∈ Γ(2) ⊂ Sp(2, R) ,
then we say Φk (Ω) is a Siegel modular form of weight k on group Γ(2) .
With this definition in mind, the almost automorphic property says that,
up to a factor H(τ, Z) which depends only on the Hodge numbers of M , the
second-quantised elliptic genus of a even dimC M = 2d0 Calabi-Yau manifold
M is a Siegel modular form. More precisely,
Φ(Ω) = H(τ, z) ×
∞
X
pN Z(τ, z; S N M ) ,
p = eσ
N =1
is a Siegel modular form of weight − 21 χ0d0 (M ), with
χ0n (M )
=
2d0
X
(−1)n+m hn,m (M ) .
m=1
See [15] for the complete description of the theorem.
For instance, when M = K3 we have the weight − 21 χ01 (K3) = −10.
In fact, this connection between a weak Jacobi form and a Siegel modular
form can be viewed as a slight generalisation/specialisation of the so-called
Borcherds lift:
29
Borcherds’ Lift
Theorem [13]
P
Let f (τ ) = n c̃(n)q n be a nearly holomorphic form of weight −s/2 for
SL(2, Z) with integral c̃(n) ∈ Z and 24|c̃(0) if s = 0. The there exists a
unique % ∈ L such that the function F (Ω) : Hs+1,1 → C defined by
Y
c̃(hα,αiL )
F (Ω) = e(−h%, ΩiL )
1 − e(−hα, ΩiL )
α∈L+
is a meromorphic automorphic form of weight c̃(0)/2 for OM (Z)+ .
The readers should consult Borcherds’ paper for the details of the theorem, including the second part of the theorem regarding the poles and zeros
of F (Ω).
4.1
Notes and References
A shorter and extremely nice version of this last part of the lecture can be
found in [14]. From Borcherds’ lift, the subject is then very closely related to
that of generalised Kac-Moody algebras which we unfortunately do not have
time to cover. The readers should consult, for instance, [15, 16, 17] and the
rest of the sequel.
30
References
[1] P. S. D. Di Francesco, Philippe; Mathieu, Conformal field theory.
Springer-Verlag, New York, 1997.
[2] E. Witten, “THE INDEX OF THE DIRAC OPERATOR IN LOOP
SPACE,”. To appear in Proc. of Conf. on Elliptic Curves and Modular
Forms in Algebraic Topology, Princeton, N.J., Sep 1986.
[3] S. Ochanine, “Sur les genres multiplicatifs dfinis par des intgrales
elliptiques. (french) [on multiplicative genera defined by elliptic
integrals],” Topology. An International Journal of Mathematics 26
(1987) 143.
[4] E. Witten, “ELLIPTIC GENERA AND QUANTUM FIELD
THEORY,” Commun. Math. Phys. 109 (1987) 525.
[5] P. S. Landweber, ed., Elliptic curves and modular forms in algebraic
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[6] E. Witten, “On the landau-ginzburg description of n = 2 minimal
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(Nucl.Phys.B414:191-212,1994) , hep-th/9306096.
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[9] L. J. Dixon, J. A. Harvey, C. Vafa, and E. Witten, “Strings on
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31
[12] L. Borisov and A. Libgober, “McKay correspondence for elliptic
genera,” Annals of Mathematics 161 (2005) 1521.
[13] R. E. Borcherds, “Automorphic forms with singularities on
Grassmannians,”.
[14] R. Dijkgraaf, “The mathematics of fivebranes,” hep-th/9810157.
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Siegel modular forms,” math/9906190.
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32