arXiv:2312.04756v3 [hep-th] 16 Jun 2024
Prepared for submission to JHEP
Anomalies of 4d SpinG Theories
T. Daniel Brennana Kenneth Intriligatora
a
Department of Physics, University of California, San Diego
E-mail: tbrennan@ucsd.edu, keni@ucsd.edu
Abstract: We consider ’t Hooft anomalies of four-dimensional gauge theories whose fermion
matter content admits SpinG (4) generalized spin structure, with G either gauged or a global
symmetry. We discuss methods to directly compute w2 ∪ w3 ’t Hooft anomalies involving
Stiefel-Whitney classes of gauge and flavor symmetry bundles that such theories can have on
non-spin manifolds, e.g. M4 = CP2 . Such anomalies have been discussed for SU (2) gauge
theory with adjoint fermions, where they were shown to give an effect that was originally found
in the Donaldson-Witten topological twist of N = 2 SYM theory. We directly compute these
anomalies for a variety of theories, including general G gauge theories with adjoint fermions,
SU (2) gauge theory with fermions in general representations, and Spin(N ) gauge theories
with fundamental matter. We discuss aspects of matching these and other ’t Hooft anomalies
in the IR phase where global symmetries are spontaneously broken, in particular for general
Ggauge theory with Nf adjoint Weyl fermions. For example, in the case of Nf = 2 we discuss
1
anomaly matching in the IR phase consisting of h∨
Ggauge copies of a CP non-linear sigma
model, including for the w2 w3 anomalies when formulated with SpinSU (2)global (4) structure.
Contents
1 Introduction
2
2 Computing Generalized SpinG Structure ’t Hooft Anomalies
2.1 w2 w3 ’t Hooft anomalies
2.2 The ϕ
b Mapping Torus and its Anomaly Indicator
2.3 Review of the WWW w2 w3 “New SU (2) Anomaly”
2.4 Application to SpinGglobal (4) Structure Theories
2.5 Determining Anomalies via Reduction to WWW “New SU (2)” Anomaly
2.6 Review of the w2 w3 Anomaly of SU (2) with 2 Adjoint Weyl Fermions
(1)
2.7 Gauging Γg subgroups, e.g. SO(3)gauge with 2 Adjoint Weyl Fermions
7
8
9
12
14
16
18
21
3 Determining the w2 w3 Anomalies of Other Theories
3.1 SU (2)gauge with 2nf Weyl Fermions in the (2j + 1)-Representation
3.2 SU (Nc )gauge theory with 2nf Weyl Fermions in the Adjoint Representation
3.3 Spin(2nc + 1)gauge with 2nf Weyl Fermions in the Vector Representation
3.4 Spin(2nc )gauge with 2nf Weyl Fermions in the Vector Representation
3.5 Spin(Nc )gauge with 2nf Weyl Fermions in the Adjoint Representation
3.6 Other Ggauge Theories with 2nf Weyl Fermions in the Adjoint Representation
22
22
23
24
26
28
29
4 C Symmetry, Higgsing, and Abelian Gauge Theory
30
4.1 Cases where ϕ
b Includes Charge Conjugation Symmetry C
30
4.2 Anomalies in Abelian Gauge Theory
32
4.2.1 Example: SU (2) with 2nf Weyl Fermions in the Adjoint Representation 33
4.2.2 Example: SU (2) with Weyl Fermion in 4-Representation
34
4.3 Anomalies and Higgsing with Adjoints Fermions
35
5 Symmetry Matching in the SSB phase: Nf Adjoint Weyl Fermion QCD
5.1 Ggauge Theory with Nf Massless, Weyl adjoints: UV symmetries
5.2 Ggauge Theory with Nf Massless, Weyl adjoints: Possible IR Phase with SSB
5.3 Symmetry Matching for Nf = 2 in the CP1 Sigma Model (on Spin Manifolds)
5.4 Symmetry Matching on M4 = CP2 with Twisted, SpinSU (2)R (4) Structure
36
37
39
42
45
6
46
Symmetric Mass Generation
A Fractionalization and the Example of Spin(4nc + 2) Vector-QCD
A.1 Anomaly for Spin(4nc + 2) Vector QCD
A.2 Anomaly for Spin(4nc + 2) Adjoint QCD
–1–
48
51
53
1
Introduction
Global symmetries, including approximate and spontaneously broken symmetries, yield powerful tools, constraints, and insights for progress in physics. See e.g. [1–6] for reviews and
references on the broadened notions and applications of generalized symmetries in quantum
field theory. In quantum theories, symmetries can be projectively realized on the Hilbert
space, as happens with anomalous symmetries. ’t Hooft anomalies for global symmetries
are especially useful: because they are invariant under continuous, symmetry-preserving deformations, including renormalization group (RG) flow, they are often exactly computable
and usefully constrain the dynamics and the IR theory. E.g. non-zero ’t Hooft anomalies
for continuous symmetries cannot be matched by a symmetry-preserving, gapped phase –
and that is also the case for some discrete symmetries (whereas other discrete symmetry ’t
Hooft anomalies can be matched by a symmetry-preserving, gapped TQFT) [7–11]. Anomalies of d-dimensional QFTs on spacetime Md are usefully described via inflow from bulk
(d + 1)-dimensional topological theories on spacetime Nd+1 with boundary Md = ∂Nd+1 , see
e.g. [12–16]. For anomalies involving fermions, this relates the anomaly to the η-invariant
of the (d + 1)-dimensonal Dirac operator via the APS index theorem [17–19] which, if the
perturbative anomalies vanish, is classified by cobordism groups Ωd+1 [16, 20, 21]. We will
only discuss d = 4 dimensional QFTs in this paper.
Theories that are bosonic, or more generally admit a generalized spin structure, can
be placed on non-spin manifolds, e.g. M4 = CP2 , which can lead to additional ’t Hooft
anomalies. In this paper we will especially focus on such ‘t Hooft anomalies of 4d QFTs,
which involve Stiefel-Whitney classes of the gauge bundle, the background fields for global
symmetries, and/or that of the spacetime geometry. We focus on “w2 w3 ” ’t Hooft anomalies,
R
i.e. where the associated 5d anomaly theory terms are ∼ w2 ∪ w3 . Such anomalies have
been discussed in e.g. [10, 20, 22–25] and we further explore and illustrate them here.
An example of such a w2 w3 anomaly is the 4d purely gravitational ’t Hooft anomaly,
which is Z2 -valued and given by the 5d anomaly theory term [20, 22]
Z
A ⊃ iπκT,T
w2 (T N ) ∪ w3 (T N )
κT,T ∼
= κT,T + 2
(1.1)
N5
where the boundary of N5 is the spacetime 4-manifold M4 = ∂N5 , and T N refers to the
tangent manifold of N ; in what follows we will often denote w3 (T N ) by w3 (T M ) as the
anomalous phase is only dependent on the choice of M4 . The anomaly coefficient κT,T is the
Z2 valued, theory-dependent purely gravitational ’t Hooft anomaly; the gravitational anomaly
is non-trivial if the theory has κT,T = 1 mod 2. The Stiefel-Whitney classes w2 (T N ) and
w3 (T N ) are obstructions to the existence of a spin- and spinc -structure respectively on N5 .
All 4-manifolds M4 have w3 (M4 ) = 0 and the anomaly (1.1) trivializes if there is a spin
structure on M4 , since then w2 (T N ) = w2 (T M ) = 0. The anomaly (1.1) is a cobordism
⊃ Z2 , where
invariant and is classified by the cobordism group for bosonic theories as ΩSO
5
the SO refers to the global structure of the Lorentz group. On the other hand, in fermionic
–2–
theories with spin structure the vanishing of the anomaly is reflected by the fact that the
relevant cobordism group vanishes: ΩSpin
= 0. Two choices for 5-manifolds N5 that generate
5
R
SO
the nontrivial Ω5 = Z2 , with N5 w2 (T N ) ∪ w3 (T N ) 6= 0 mod 2, are called the Wu and Dold
manifolds1 N5 = SU (3)/SO(3) and N5 = CP2 ⋊ϕ S 1 ; the latter arises in the mapping torus
construction of the anomaly in [24] and in what follows here. The ’t Hooft anomaly (1.1) is
non-zero e.g. in all fermion electrodynamics [22, 27] on CP2 .
More generally, theories that (classically) admit a generalized SpinG (4) structure can be
formulated on non-spin manifolds such as M4 = CP2 , with non-zero w2 (T M ).2 A generalized
SpinG (4) structure is possible only if all dynamical matter fields satisfy a generalized spin
charge relation, with (−1)F acting the same as a Z2 central element of the gauge or global
symmetry G, i.e. with all bosons having even charge under the Z2 and all fermions having
odd charge. The SpinG (4) structure requires the gauge fields to satisfy a flux constraint
correlated with the geometry, w2 (G) = w2 (T M ). For the case of gauge G = Ggauge , this is
a constraint on the gauge field functional integral and (−1)F is effectively gauged so there
are no fermionic gauge invariant operators in the spectrum; the theory is actually bosonic.
For the case of G = Gglobal , w2 (G) = w2 (T M ) is a constraint on the background gauge
fields and the interpretation is somewhat different. The theory could have gauge invariant
fermionic operators but, as with twisting of supersymmetric theories, the stress-tensor can be
consistently modified by an improvement term involving the Gglobal currents so that activating
the w2 (G) = w2 (T M ) background gauge field flux effectively twists the spins and converts
the fermionic operators to be bosonic.
The w2 (G) = w2 (T M ) condition of generalized SpinG (4) structures relates the gravitational anomaly (1.1) to a mixed w2 w3 type ’t Hooft anomaly between G and gravity
Z
A ⊃ iπκG,T w2 (G) ∪ w3 (T N ) .
(1.2)
Irrespective of whether G is gauged or global, the interpretation of (1.2) is that of an ’t Hooft
anomaly in the generalized SpinG (4) structure rather than an inconsistency of the theory.
It does not spoil the consistency of the theory with ordinary spin structure on spacetimes
with w2 (T M ) = 0, and even with the generalized SpinG (4) structure on a spacetime with
w2 (T M ) 6= 0 one can always choose local counterterms to represent a mixed anomaly as
preserving the gauged symmetry, with the global symmetry projectively realized.
The new SU (2) anomaly of [24] (hereafter “WWW”) is an example of a w2 w3 ’t Hooft
anomaly. It arises in SU (2) gauge theory with Nf = 1 Weyl fermion in the j = 32 representation. The Z2 center of the gauge group acts on all fields as (−1)F , so the theory is bosonic: all
gauge invariant operators are bosons. The theory thus classically admits a SpinSU (2)gauge (4)
structure. As shown in [24], the quantum theory with generalized SpinSU (2) (4) structure has
the gravitational anomaly (1.1), and consequently also the ’t Hooft anomaly (1.2).
1
As (foot)noted in [26], SU (3)/SO(3) is called the Wu manifold, but it was Calabi who found it as a
generator of ΩSO
= Z2 . Wu had CP2 ⋊ϕb S 1 , which is now called the Dold manifold. The space SU (3)/SO(3)
5
also arises as the pion target space in SU (3) → SO(3) spontaneous symmetry breaking.
2
We can and will take CP2 as the representative 4-manifold with w2 (T M ) 6= 0, to activate the new anomalies.
–3–
Spin
For general 4d SU (2) gauge theory, the cobordism classification gives Ω5 SU (2) = Z2 ⊕
Z2 [16], corresponding to the original SU (2) Witten anomaly [28] and the WWW new SU (2)
anomaly [24]. The original Witten anomaly is a gauge anomaly (or ’t Hooft anomaly if SU (2)
is a global rather than gauge symmetry) arises if there are an odd number of Weyl fermions
in the D = (2j + 1)-dimensional representation of SU (2) for D ∈ 2 + 4Z; it can be seen
from the fact that an SU (2) instanton then has an odd number of fermion zero modes so its
corresponding ’t Hooft operator violates SU (2) and Lorentz symmetry.
The WWW [24] anomaly is a ’t Hooft anomaly in the SpinSU (2) (4) structure, which is
non-trivial if the number of fermion zero modes of the 4d Dirac operator in a SpinSU (2) (4)
flux background is not a multiple of 4; this is the case if there are an odd number of Weyl
fermions in representations of dimension D ∈ 4 + 8Z [24]. In particular, the SpinG ’t Hooft
anomaly is non-zero for an odd number of chiral fermions in the 4 (i.e j = 3/2) of SU (2) when
placed on M4 = CP2 which has w2 (T M ) 6= 0. The theory with ordinary spin structure on
M4 with spin structure w2 (T M ) = 0 is perfectly healthy (e.g. the instanton ’t Hooft vertex
has an even number of fermion zero modes) 3 . Deforming the SU (2) theory by adding an
adjoint-valued scalar field (preserving the spin-charge relation) and Higgsing SU (2) → U (1)
leads to all fermion electrodynamics with the same gravitational anomaly (1.1) [24, 30].
The present paper is on a class of Z2 -valued w2 w3 -type ’t Hooft anomalies that can
arise in theories with SpinGglobal (4) structure on M4 = CP2 in the presence of a non-zero
(1)
background gauge field B2 for a Z2 subgroup 4 of a one-form global symmetry Γg . Such ’t
Hooft anomalies were first discussed in [23] in the context of SU (2)gauge theory with Nf = 2
Weyl fermions in the adjoint of SU (2)gauge , which has a generalized spin structure is associated
with a SU (2)global (rather than SU (2)gauge symmetry). The ’t Hooft anomaly is activated
(1)
(1)
when a non-zero background gauge field B2 is turned on for the Γg = Z2 one-form global
symmetry (the center symmetry of SU (2)gauge ) when the SpinSU (2)global structure is used to
put the theory on a spacetime with w2 (T M ) 6= 0, e.g. M4 = CP2 [23]. Then WWW [24]
discussed a relation to their anomaly. Such anomalies will be generalized and applied here.
Generalizing [23], the new Z2 -valued ’t Hooft anomaly has an anomaly theory of the form
Z
A ⊃ iπκΓg ,T
B2 ∪ w3 (T N ) ,
κΓg ,T ∼
= κΓg ,T + 2 .
(1.3)
N5
(1)
The background B2 activates a Z2 -valued subgroup of the Γg one-form global symmetry
associated with the center of Ggauge , modifying the gauge fields to have w2 (Ggauge ) = B2 ,
so (1.3) is indeed a type of w2 ∪ w3 anomaly.5 The interpretation of the anomaly, as discussed
in [23], is in terms of the Z-valued integer lifts, B̃2 and w̃3 (T N ), of the Z2 -valued quantities
3
This was also noted e.g. in [29], where the N = 1 version of this theory was considered.
(1)
(1)
As we will discuss, there is no such anomaly if e.g. Γg = ZNg with odd Ng : it can then be cancelled.
R
5
There are also possible ‘t Hooft anomaly terms on non-spin manifolds of the form A ⊃ πiκ
A1 ∪ B2 ∪
N
N5
w2 (T N ) where A1 is the background gauge field for a discrete 0-form symmetry. We will not discuss such
anomalies in this paper, see [31, 32] for examples and details. See also [33–35] on anomalies and [36, 37] for
analysis of w2 w3 anomalies in BSM-inspired theories.
4
–4–
B2 and w3 (T N ). As in (1.3), we will often suppress the tildes on the integer lifts to avoid
clutter, when their presence should be understood from the context. The integer lifts arise
from defining differentials of cyclic-valued elements such as B2 and w2 (T N ) via the Bockstein
map; the cyclic-valued quantities are recovered via modding out by B̃2 → B̃2 + 2x and
w̃2 (T N ) → w̃2 (T N ) + 2y, with x and y integral-valued. The mixed anomaly (1.3) implies
that the quantum partition Rfunction picks up a minus Rsign under either the x or the y shift
above (i.e. either Z 7→ (−1) x∪w2 (T M ) Z, or Z 7→ (−1) B2 ∪y Z); the two options for how the
anomaly shows up differ by a local counterterm in the backgrounds [23].
For the example of SU (2)gauge with Nf = 2 adjoint Weyl fermions, ref [23] argued that the
theory must have κΓg ,T 6= 0 via a RG flow from twisted N = 2 supersymmetric Yang-Mills on
CP2 . The ’t Hooft anomaly (1.3) of that theory was presented in [23] as a reinterpretation of an
effect that arises in the context of Donaldon theory [38, 39] and correspondingly in the twisted
N = 2 supersymmetric gauge theory that leads to the Donaldson-Witten partition function,
ZDW [40–42]. To compute ZDW , the Z2 valued quantities B2 and w2 (T M ) must be lifted to
e2 and w̃2 (T M ), and as above one recovers B2 and w2 (T M ) by
Z-valued integer quantities B
e
e2 → B
e2 + 2x or w̃2 (T M ) → w̃2 (T M ) + 2y
modding out B2 and w̃2 (T M ) by Z2 valued shifts, B
with x and Ry integral. The fact that
ZDW picks up a minus sign under one or the other shift,
R
B ∪y
x∪w2 (T M )
or (−1) M4 2 was found in [38, 39] as coming from an orientation
either (−1) M4
change of instanton moduli space, and it was shown to be reproduced in the twisted N = 2
SU (2) Yang-Mills or the U (1) (monopole) calculation of the partition function ZDW , from
the fermions in the functional integral in [40–43].
We will discuss and calculate SpinG (4) ’t Hooft anomalies (1.3) for wide classes of theories. We show how the anomaly coefficients κΓg ,T can often be simply directly computed
by an adaptation of the method that was used in WWW [24] to compute the κT,T ’t Hooft
anomaly of SU (2) with a Weyl fermion in the 4. This method computes the anomaly from
the index of the 5d mod-2 Dirac operator on the mapping torus CP2 ⋊ϕb S 1 , with appropriate
fluxes for the SpinG (4) background and also additional fluxes to activate the anomaly (1.3).
Here ϕ
b is a Z2 valued isometry of CP2 and the flux background. The 5d mod-2 Dirac index
then reduces to a 4d Dirac index calculation in the flux background, with the fermion zero
modes all in 2d representations of an O(2) group [24]. The upshot is a simple way to compute
κΓg ,T : it is half (mod 2) of the number of fermion zero modes I of the 4d Dirac operator in
a flux background that activates the anomaly:
1X 2 1
1
κΓg ,T = I =
qi −
,
mod2 ,
(1.4)
2
4
4
i
where the Atiyah-Singer index theorem yields the last equation. The qi are flux-weighted
charges of fermion species i, associated with an appropriately chosen Spinc connection.
This simple method is generally applicable when the matter content consists of Weyl
fermions in symplectic representations of the Ggauge × Gglobal group. This allows for non-zero
’t Hooft anomalies, as there is no symmetry-preserving mass term. E.g. for SU (2)gauge with
–5–
Nf = 2 adjoint Weyl fermions, the SU (2)gauge and Lorentz invariant fermion bilinear is in
the adjoint of SU (2)global .
We will consider and directly compute ’t Hooft anomalies of the general form (1.3) that
can arise in theories with a SpinGglobal (4) structure on M4 = CP2 , generalizing the original
example [23] to wide classes of theories. The examples include e.g. general Ggauge theory
with Nf = 2 Weyl fermions in the adjoint, formulated with SpinSU (2) (4) structure on CP2 ,
with background gauge fields B2 for the one-form global symmetry Γg (the center of the
gauge group Ggauge ). These same ’t Hooft anomalies would arise in N = 2 SUSY YangMills with gauge group Ggauge when formulated with SpinSU (2)R (4) structure on CP2 , as in
the Donaldson-Witten topological twist, where SU (2)R ≡ SU (2)global is the R-symmetry of
N = 2, which acts on the two adjoint fermions as in the non-SUSY theory.
The outline of this paper is as follows. In Section 2 we review aspects of w2 w3 -type
anomalies and their computation from [23] and [24], and generalize the methods to be applicable to more general SpinG (4) theories. In Section 3 we consider a variety of classes of
gauge theories that admit generalized SpinG (4) structures, and compute the w2 w3 anomaly
coefficients of the theory on CP2 . We find:
• Section 3.1: SU (2) gauge theory with Nf = 2nf flavors of matter fields in the general
isospin j ∈ Z representation has a w2 w3 anomaly coefficient κΓg ,T = 12 Nf for Nf ∈ 4Z+2
if j ∈ 4Z + 1 or j ∈ 4Z + 2; κΓg ,T = 0 for j ∈ 4Z and j ∈ 4Z + 3
• Section 3.2: SU (Nc ) with Nf chiral fermions in the adjoint representation has a w2 w3
anomaly coefficient κΓg ,T is non-zero iff both Nc = 2 mod 4 and Nf = 2 mod 4
• Section 3.3: Spin(2nc + 1) gauge theory with Nf = 2nf chiral fermions in the fundamental vector representation has anomaly coefficient is κΓg ,T = nf mod 2.
• Section 3.4: Spin(2nc ) gauge theory with Nf = 2nf chiral fermions in the vector
representation has anomaly coefficient κΓg ,T = nf mod 2.
• Section 3.5: Spin(Nc ) gauge theory with Nf = 2nf chiral fermions in the adjoint
representation has κΓg ,T = nf Nc mod 2.
• Section 3.6: Sp(Nc ) for Nc = 1, 2 mod4 (but not Nc = 0, 3 mod4 ) and E7 gauge
theories with Nf = 2nf adjoint Weyl fermions have κΓg ,T = nf mod 2.
b mapping torus method when it needs to include
In Section 4, we discuss how to modify the ϕ
charge conjugation, as in cases with U (1) factors and for anomaly matching when the gauge
group is Higgsed. In Section 5, we consider anomaly matching if the global symmetries are
spontaneously broken, e.g. in theories with a global SU (2)R ≡ SU (2)flavor symmetry that is
spontaneously broken to a U (1)R ⊂ SU (2)R , resulting in a CP1 ∼
= SU (2)R /U (1)R non-linear
sigma model. We discuss anomaly matching in the sigma model, generalizing the analysis
in [23] to other theories. In Section 6, we discuss a class of chiral gauge theory examples
that are motivated by examples considered in [44–46] for possibly exhibiting symmetric mass
–6–
generation in 4d; we double the spectrum to obtain an extra SU (2)R global symmetry that can
be used for SpinSU (2)R (4) twisting. We note that the resulting theory has a w2 w3 anomaly,
so it cannot be completely trivially gapped in the IR. In Appendix A we will recall the
fractionalization technique of [24, 30] and provide details of its application to compute discrete
anomalies for a few examples quoted in the main text.
Computing Generalized SpinG Structure ’t Hooft Anomalies
2
In this section we will review and generalize aspects of the new ’t Hooft anomalies that can
arise for theories with generalized SpinG (4) structure on CP2 . The review will be based on
discussion and methods from [23, 24]. For a generalized SpinG (4) structure, all dynamical
fields in the theory are in faithful representations of
Gtotal =
Ggauge
× Gglobal × Spin(4)
Γg
(2.1)
Γ
where Spin(4) is the Euclidean Lorentz group, Γg is a subgroup of the center of Ggauge , and Γ
(1)
is a subgroup of the centers of all of the groups. The Γg quotient implies a Γg one-form global
symmetry acting on the Ggauge Wilson lines. A SpinG (4) structure is possible if Γ identifies
(−1)F ∈ Spin(4) with a central element −1 in the center of G = Ggauge or G = Gglobal . In the
case of SpinGgauge , the path integral over the Ggauge fields effectively gauges (−1)F and the
gauge invariant spectrum of the theory is bosonic; there is a constraint w2 (Ggauge ) = w2 (T M )
in the path integral. In the case of SpinGglobal , the interpretation is instead that of twisting
and there are background fields constrained to satisfy w2 (Gglobal ) = w2 (T M ). We will here
be especially interested in the new ’t Hooft anomalies that arise for SpinGglobal theories on
(1)
M4 = CP2 in the presence of background gauge fields for the Γg one-form global symmetry.
Anomalies can be exhibited by considering background gauge fields and geometry, subject to
the Γ identifications, and arise from the variation of the path integral measure.
Anomalies arise in the path integral description from the transformation of the functional
integration measure, which often reduces to an anomalous variation of zero modes in an
/ of the 4d
appropriate background. These zero modes can be read off from the index Ind[D]
6
Dirac operator on M4 .
For example, the U (1)A × G2gauge ABJ anomaly can be read off from the fermion zero
/ in the background of a Ggauge instanton7 . The ’t Hooft vertex’s fermion zero
modes of D
modes have net U (1)A charge, explicitly breaking U (1)A to a discrete subgroup even in a
trivial background. The Atiyah Singer index theorem implies that a Weyl chiral fermion ψr
R
in representation r has Nr = I(r) fermion zero modes in the kinst = 8π1 2 TrF ∧ F = 1
instanton sector where Trr T a T b = 2I(r)δab is the quadratic index (normalized to be an
integer8 ). The U (1)A that assigns charge qi = 1 to all fermions is broken explicitly by the
6
The more general APS index theorem, including the η invariant contribution [17–19] is not needed here.
For U (1)gauge there is no instanton and the anomaly leads to a non-invertible symmetry category [47, 48].
∨
8
E.g. for the adjoint representation Nadj = 2h∨
Ggauge , with hGgauge the dual Coexter number.
7
–7–
P
instanton to a ZN0 subgroup where N0 = i I(ri ) is the total number of fermion zero modes
in the instanton background. For consistent theories without gauge anomalies, N0 is always
1
even and the generator g of ZN0 has g 2 N0 = (−1)F . See e.g. [49, 50] for discussion of the
’t Hooft anomalies of such a ZN0 symmetry. As another example, gauge anomalies from
fermions come from an index theorem for the 6d Dirac operator but can nevertheless often
be detected from the 4d index in an instanton background, if the product of the fermion zero
modes in the instanton’s Hooft vertex is not Lorentz and/or gauge invariant, e.g. because of
having an odd number of fermion zero modes.
2.1
w2 w3 ’t Hooft anomalies
(1)
(1)
(1)
If the one-form group Γg in (2.1) is Γg ⊇ ZNg , and (−1)F ∈ Γ so the theory admits a
SpinG (4) structure, we can consider a ZNg version of the w2 w3 ’t Hooft anomaly in (1.3)
Z
Z
2πi
2πi
BNg ∪ δw2 (T N )
δBNg ∪ w2 (T N ) . (2.2)
or
A ⊃ κΓg ,T
A ⊃ κΓg ,T
2Ng N5
2Ng N5
(1)
(1)
(1)
Here BNg ∈ H 2 (N5 , ZNg ) is the Γg background gauge field for the ZNg ⊂ Γg subgroup
and w3 (T N ) = 12 δw2 (T N ) and N1g δBNg = βNg (BNg ) is the ZNg Bockstein of BNg . The
anomaly encodes a dependence of the path integral on the choice of integer lift B̃Ng or
w̃2 (T N ) where different choices of lift are related by the shifts B̃Ng → B̃Ng + Ng x and
w̃2 (T N ) → w̃2 (T N )+ 2y with integral-valued x and y. This generalizes the Ng = 2 discussion
in [23]. The two expressions for the anomaly in (2.2) differ by an integration by parts,
corresponding to modifying the 4d action by a local counterterm
Z
iπ
B̃Ng ∪ w̃2 (T M ) ,
k
Sc.t. =
(2.3)
Ng M 4
with k = κΓg ,T . The two presentations of the anomaly in (2.2) lead to a variation of the
path integral under shifts of the integer lifts B̃Ng → B̃Ng + Ng x or w̃2 (T N ) → w̃2 (T N ) + 2y
respectively:
κΓg ,T
Z → Z · (−1)
R
M x∪w2 (T M )
or
Z →Z ·e
2πi
κ
Ng Γg ,T
R
M BNg ∪y
,
(2.4)
with integral-valued x and y. This variation can be cancelled if κΓg ,T is even, so we restrict
to the case where κΓg ,T is odd. Additionally, for Ng odd we can use the counterterm (2.3)
with k = Ng − κΓg ,T to completely trivialize the anomaly, so we take also Ng even. The first
presentation shows that the anomaly is Z2 valued for all even Ng . In the second presentation,
anomaly appears to be ZNg -valued – but it really is not: we can shift κΓg ,T → κΓg ,T + 2 by
the counterterm with k = 2. This is why the anomaly is trivializable when Ng is odd: 2 is a
divisor of -1 in Z2n+1 .
Because the anomaly is Z2 valued and vanishes for Ng odd, it suffices as in (1.3) to
N
activate a flux in a Z2 subgroup of ZNg as BNg = 2g B2 where B̃2 → B̃2 + 2x. The method
discussed in the next section only probes the κΓg ,T anomaly for flux in a Z2 ⊂ Γg . We will
–8–
see that this anomaly trivializes9 if Ng ∈ 4Z. The w2 w3 anomalies can more generally also
be computed via the symmetry fractionalization techniques of [24, 30], as illustrated in the
appendix.
2.2
The ϕ
b Mapping Torus and its Anomaly Indicator
As discussed in [24], the global anomalies associated with gravitational ’t Hooft anomaly
coefficient can be computed from a mod-2 index of the 5d Dirac operator on the mapping
torus N5 = CP2 ⋊ϕb S 1 associated with the theory on M4 = CP2 , where ϕ
b includes the operation
ϕc.c. that acts as complex conjugation on CP2 along with a Weyl transformation to preserve
R
the flux backgrounds. This is natural, since the Ω5 bordism class with N5 w2 w3 6= 0 can be
generated by the Dold manifold N5 = CP2 ⋊ϕc.c. S 1 . We will here generalize this method to
directly compute other w2 w3 type anomalies, and apply the method to a variety of examples.
As in (2.1), consider a general theory where the fermions form a faithful representation
of a total group that involves the gauge group, global group, and Lorentz group, modded
out by Γ. We divide such theories into two cases: (1) when (−1)F ∈ Spin(4) is not in
the identification, i.e. (−1)F ∈
/ Γ; and (2) when (−1)F ∈ Γ. If (−1)F ∈
/ Γ, the theory
is fermionic, and does not admit a generalized spin structure so it cannot be put onto a
non-spin manifold. For such cases, a basic M4 with spin structure that exhibits the possible
anomalies is M4 = CP1 × CP1 . Our interest here is in the case where (−1)F ∈ Γ, so the
theory classically admits a SpinG structure; we can then consider the theory on the non-spin
manifold M4 = CP2 with discrete flux in G such that w2 (T M ) = w2 (G). We generally refer
to the process of elevating a G-connection to a SpinG connection or reducing a G-connection
to a G/Z connection with obstruction class w2 (G) as “turning on a discrete flux” in the group
(1)
G. We will be interested in cases where there is a one-form global symmetry Γg , coming
from the center of the gauge group (or subgroup thereof), and turning on an associated 2-form
background gauge field B2 is also a type of discrete flux, which is unconstrained by the SpinG
structure but must be activated to see the associated ’t Hooft anomaly. For reasons we will
(1)
discuss shortly, we will only consider activating background fluxes for Z2 ⊂ Γg , so we will
restrict to B2 ∈ H 2 (BGgauge /Γg ; Z2 ).
(1)
To activate the needed background fluxes, both for the SpinG structure and for Γg
background gauge fields, it suffices to consider a group U (1)Q that is embedded in the gauge
group and global symmetries. More precisely, the fluxes are in a discrete subgroup of U (1)Q
and, if the discrete flux is associated with an obstruction class, then U (1)Q is actually a Spinc
(1)
(1)
connection. For example, if the gauge group is SU (Ng ) and the theory has a Γg = ZNg one(N )
form center symmetry, then turning on a generic ZNg -valued non-zero background B2 g
gauge field is achieved by choosing U (1)Γg ⊃ Z(SU (Ng )) (which itself embeds in U (1)Q ) and
9
The Bockstein β = 21 δ for a Z2 -valued cochain trivializes if the co-chain has a Z4 lift. Nevertheless, theories
can have other types of non-trivial, mixed Z4 , Z2 ’t Hooft anomalies, e.g. via the discrete gauge version of the
R (e)
i
B ∧ B (m) mixed ’t Hooft anomaly. E.g. if we start with a theory with a Z8 global symmetry and
A = 2π
gauge a Z2 subgroup, then the anomaly of [51] will give a mixed Z4 , Z2 ’t Hooft anomaly.
–9–
(N )
turning on a discrete flux B2 g along this U (1)Γg by elevating the U (1)Γg connection to a
Spinc connection. More generally, we can turn on generic fluxes for Γ = ZN and one-form
group Γg = ZNg in (2.1), and write fluxes associated with non-trivial w2 (Γ) and w2 (Γg )
by embedding Γ ⊂ U (1)Γ and Γg ⊂ U (1)Γg which are generated by some charges QΓ , QΓg
respectively. We can then write the flux backgrounds as an explicit Spinc connection using
U (1)cQ ≡ QΓ U (1)cΓ + QΓg U (1)cΓg :
I
fZn
1
AQ = aZN QΓ + aZNg QΓg ,
= ,
(2.5)
2π
n
where fZn is the field strength for the Spinc connection aZn .
The number of fermion zero modes of the Dirac operator in the U (1)Q flux background
R
b M ) ∧ Trψ exp(FQ /2π),
/ = M A(T
is then given by the Atiyah Singer index theorem as Ind[D]
4
b M ) = 1 − p1 (T M )/24 + . . . , where Trψ runs over the Weyl fermions. We decompose
with A(T
P
the fermions into representations of U (1)Q as ψ = i ψi vi , where vi is a basis of eigenvectors
of the AQ flux, FQ · vi = qi vi ; our notation will be to put the 1/n fractional fluxes in (2.5)
into the normalization of the charges qi . The index theorem gives the number of fermion zero
modes I to be10
(P
q2
M = CP1 × CP1
I(R) = Pi i
(2.6)
2
1 2
1
(q
−
)
M
=
CP
i 2 i
4
b evaluated on CP2 , and the overall
The − 18 comes from the −p1 (T M )/24 term in A
relative factor of 12 between the two cases is because the 2-cycle in CP2 has intersection
pairing 1 whereas in CP1 × CP1 the intersection pairings are 2.
As in [24], we exhibit w2 w3 anomalies by considering the Dirac operator on the 5d mapping torus associated with a classical symmetry ϕ
b that acts on CP2 as complex conjugation.
To be a symmetry of the flux background, ϕ
b generically needs to be augmented with a G Weyl
group operation. Although ϕ
b is then a symmetry of the bosonic backgrounds, the fermion
/
zero modes of D can transform non-trivially under ϕ
b which may give rise to an anomaly.
We will show, as in [24], that the fermion zero modes of the 4d Dirac operator all form 2d
representations of O(2) under U (1)Q and ϕ,
b and that this implies that the Z2 valued w2 w3
anomaly coefficient is given by half of the number of zero modes counted in (2.6).
We can apply these considerations to both the M4 = CP1 × CP1 and the M4 = CP2 cases.
The standard mapping torus construction of anomalies involves a 5th dimension that is an
interval where the ends are identified with a bosonic symmetry transformation. For w2 w3
anomalies, the transformation is given by
10
ϕ
b = ϕc.c. ◦ W ,
(2.7)
Equation (2.6) is the formula for the index of the Dirac operator from the Atiyah Singer index theorem.
As mentioned in [24], the index of the Dirac operator on CP2 is equivalent to the number of zero modes. This
can be shown by a Lichnerowicz-type argument using the fact that CP2 has positive curvature and that we
are coupling to only self-dual gauge fields.
– 10 –
where ϕc.c. : (z1 , z2 ) → (z1∗ , z2∗ ) which is a symmetry of M4 . The Weyl group element W
compensates for the fact that ϕc.c. acts as complex conjugation on the fluxes associated with
the Γ, Γg -fluxes, where Γ and Γg are as in (2.1), as ϕc.c. : w2 (Γ), w2 (Γg ) 7−→ −w2 (Γ), −w2 (Γg ).
The W group element restores the fluxes to obtain a classical symmetry of the background,
since W also acts as charge conjugation on U (1)Γ , U (1)Γg . For example, given an embedded
U (1) ⊂ SU (2), the Weyl generator of SU (2) acts as charge conjugation on the embedded
U (1). This construction from [24] generalizes immediately for theories with fermions in
irreducible real or pseudo-real representations of Ggauge × Gglobal .
Note that such W ∈ Ggauge × Gglobal only exist when w2 (Γ), w2 (Γg ) are Z2 -valued fluxes.
The reason is that G acts trivially on Z(G) so the only way that there can exist W that
undoes the action of ϕc.c. on the fluxes w2 (Γ), w2 (Γg ) is if ϕc.c. only acts to change their
representative – i.e. if they are Z2 -valued. For vector-like matter representations, there is
a similar construction with W an outer automorphism charge conjugation symmetry, as we
will discuss and illustrate in Section 4.1.
b satisfies ϕ
b2 = 1, although ϕ2c.c. = W 2 = (−1)F , as seen from
As discussed in [24], this ϕ
the fact that ϕc.c. acts as a π-rotation on M4 = CP2 [24] (and similarly for M4 = CP1 × CP1 ):
if we parameterize M4 in real coordinates (z1 , z2 ) = (x1 + ix2 , x3 + ix4 ), complex conjugation
takes (x1 , x2 , x3 , x4 ) 7→ (x1 , −x2 , x3 , −x4 ), which is a π-rotation in the x2 −x4 plane. Likewise,
W that acts a π-rotation in the Lie algebra ggauge × gglobal of the fermions. Since W acts as
b
complex conjugation on U (1)Q , the fermions transform in reps of O(2) = U (1)Q ⋊ Zϕ
2 . The
group O(2) has only one- and two-dimensional representations. The nontrivial action of the
generators Q of U (1)Q , and ϕ
b of O(2), on the fermions implies that the I fermion zero-modes
1
must reside in 2 I copies of the 2d representations of O(2), and the generators act as
!
!
01
qi 0
, ϕ
b=
.
(2.8)
Q=
10
0 −qi
Thus ϕ
b acts on the zero-modes, and thus on the fermion measure of the path integral, as:
Z
Z
Z
ϕ
b
I/2
(2.9)
[dψ] . . . −−−−−−−−−→ [dψ] × det[ϕ]
b . . . = (−1)
[dψ] . . . .
Here 12 I is the number of 2d O(2) representations, with I the total number of fermion zero
b anomaly if there is a non-zero (odd) value of the
modes as given by (2.6). There is a ϕ
Z2 -valued anomaly indicator
σI :=
I
mod2 .
2
(2.10)
Since the anomaly is Z2 valued, it is always trivialized upon doubling the matter content.
The anomaly exhibited by (2.9) can be described in terms of a 5d anomaly polynomial/anomaly SPT phase given by a w2 ∪ w3 anomaly theory
Z
w2 ∪ w3′ ,
A ⊂ πi σI
(2.11)
N5
– 11 –
where w2 , w2′ are discrete fluxes associated with Γ, Γg and w3′ is the Bockstein of w2′ : w3′ =
Bock[w2′ ]. The anomalous phase can be realized by considering the SPT/anomaly polynomial
on the mapping torus N5 = CP2 ⋊ϕb S 1 which is twisted by the action of ϕ.
b Then
(w3′ )N = (w3′ )CP2 ⊕ (w2′ )CP2 ∪ (w1′ )S 1 .
Since
H
(2.12)
2
S 1 w1 = 1 on the mapping torus, the anomaly SPT reduces to the phase on CP :
S5d = πiσI
Z
N
w2 ∪ w3′ = πiσI
Z
CP
2
w2 ∪ w2′ ,
(2.13)
which implies the anomalous phase (2.9) of the path integral under ϕ.
b
2.3
Review of the WWW w2 w3 “New SU (2) Anomaly”
We here illustrate the the discussion of the previous section by reviewing the original case
from [24]: an SU (2)gauge theory with Nf = 1 Weyl fermion ψα in the j = 3/2 representation
of SU (2)gauge . This is a chiral theory in that it is impossible to write a mass term for ψα as
there is no quadratic, gauge invariant, Lorentz scalar operator. The classical, ABJ-anomalous
U (1)A = U (1)ψ global symmetry is broken by instantons to Z10 : the Dirac operator in flat
R4 in an instanton background has N0 = I2 (j = 3/2) = 10 fermion zero modes11 ; the fact
that this is even shows that the theory does not have the original SU (2)gauge anomaly. The
generator g of the Z10 global symmetry has g5 = (−1)F and the Z10 ’t Hooft anomalies
obstruct a symmetry preserving gapped phase [9]; this symmetry will not play a role here.
All dynamical fields form faithful representations of the symmetry group
Gtotal =
SU (2)gauge × Spin(4) × Z10
.
Z2 × Z2
(2.14)
One Z2 identifies g5 = (−1)F and the other identifies (−1)F = −1SU (2)gauge . So this theory
(unlike the SUSY version in [29]) satisfies a spin-charge relation and thus can be classically
formulated with a SpinSU (2) (4) structure on M = CP2 via a twist of the SU (2)gauge -bundle
to have discrete flux w2 (SU (2)) ∈ H 2 (M ; Z2 ) equal to the second Stiefel-Whitney class of M :
w2 (SU (2)) = w2 (T M ) .
(2.15)
For CP2 , w2 (T M ) is non-trivial and thus the SU (2)cgauge bundle is an SO(3) bundle with fixed
w2 (SO(3)) = w2 (T M ). Here we write the gauge bundle as SU (2)cgauge rather than SO(3)gauge
because the gauge group is not actually SO(3), and is more analogous to a Spinc connection
– moreover we will often simply write the gauge bundle as SU (2)gauge despite the non-zero
w2 (SU (2)) obstruction from being an SU (2) bundle. Restricting to a U (1)Q ⊂ SU (2)gauge ,
for the theory on CP2 we turn on a Spinc connection along
!
Z b
1
fZ2
b
aZ2 0
= ,
(2.16)
ASU (2) =
,
2π
2
0 −b
aZ2
H
11
For a Weyl fermion in the 2j + 1 dimensional representation, N0 = I(j) = 32 j(j + 1)(2j + 1).
– 12 –
corresponding to Q = T 3 . This background is preserved by the ϕ
b map
!
0 −1
ϕ
b = ϕc.c. ◦ WSU (2) , WSU (2) =
.
1 0
(2.17)
WSU (2) acts as the generator of the Weyl group acting on the Cartan generated by Q.
To determine the fermion zero modes and ϕ
b action on them, we decompose the fermion
into U (1)Q representations, which results in four fermions with flux × charges given by
3 1 1 3
qi =
, ,− ,−
.
(2.18)
2 2 2 2
The U (1)Q charges of the fermions are odd integers, consistent with the spin-charge relation
of a Spinc connection. Our notation is that the charges qi = 12 · 2 · Tii3 includes a factor from
R F
R
the half-unit of flux, Σ2 2π
= 12 Σ2 w2 (T M ) = 12 , that cancels a factor of 2 from rescaling
the U (1) ⊂ SU (2) charges to be integers (with the T ± generators of charge ±2); the upshot
is that the qi are simply the T 3 eigenvalues with the usual normalization for T ± charges. The
number of zero modes of the Dirac operator in this flux background on CP2 is
I(Rj=3/2 ) =
X 4q 2 − 1
i
=⇒
8
ψi
I(4) = 2 .
(2.19)
From this we can read off the anomaly indicator for the chiral fermion ψ in the 4 rep:
σI(4) =
1
I(4) mod2 = 1 .
2
(2.20)
The theory thus has a ϕ
b anomaly, corresponding to the gravitational ’t Hooft anomaly (2.11)
′
with w2 , w2 = w2 (T M ):
Z
w2 (T N ) ∪ w3 (T N )
A ⊂ πiκT,T
(2.21)
N
with κT,T = σI(4) = 1. Since w2 (T M ) = w2 (SU (2)) and this anomaly can also be interpreted
as a mixed gauge-gravity anomaly
Z
w2 (SU (2)) ∪ w3 (T N ) .
A ⊂ πi
(2.22)
N
Generalizing to SU (2)gauge with Nf fermions in the 2j + 1-rep, the index with w2 (SU (2)) =
w2 (T M ) on M = CP2 is
I(R) = Nf
j
X
4f 2 − 1
f =−j
8
=
Nf
(4j 2 − 1)(2j + 3) .
24
(2.23)
The new SU (2) anomaly κT,T = σI (R) = 21 I is odd only if Nf is odd and j = 23 + 4r with
r ∈ Z [24]. The SpinSU (2) (4) structure implies that κT,T = κSU (2),T .
– 13 –
2.4
Application to SpinGglobal (4) Structure Theories
Our main interest here are the w2 w3 anomalies of theories with SpinGglobal (4) structure, i.e.
those where all matter couples to bundles Gtotal of the form:
Gtotal =
Gglobal × Spin(4)
Ggauge
×
,
Γg
Γ
(2.24)
where Γ = Z2 that identifies (−1)F with a Z2 in the center of Gglobal . The generalized
SpinGglobal (4) structure involves using background gauge fields for the Gglobal symmetry to
twist the theory, allowing it to be put on non-spin manifolds like M4 = CP2 . There is also the
(1)
(1)
option to introduce a background gauge field B2 for the Z2 ⊂ Γg one-form symmetry. The
possible w2 w3 ’t Hooft anomalies are the gravitational one, with coefficient κT,T = κGglobal ,T ,
(1)
and a mixed ’t Hooft anomaly between gravity and the global one-form symmetry Γg ,
with coefficient κΓg ,T .12 Both anomalies can be probed by turning on appropriate U (1)cQ
background fluxes as in (2.5); for κT,T the Γg flux B2 can be set to zero, and for κΓg ,T
the U (1)cQ should include both the Γg and Γ fluxes. The anomaly coefficients are then
determined from the 5d Dirac operator on the CP2 ⋊ϕb S 1 mapping torus with appropriate
flux backgrounds, with ϕ
b that preserves the bosonic backgrounds, leading to the result (1.4)
in terms of the charges qi .
In the original example of [23], i.e, SU (2) gauge theory with Nf = 2 Weyl fermions in
(1)
(1)
the adjoint representation, the one-form global center symmetry is Γg = Z2 and the SpinG
structure involved Gglobal ≡ SU (2)R , with −1SU (2)R ∼ (−1)F by Γ = Z2 . In that example, all
of the discrete symmetries are Z2 , and the κΓg ,T ’t Hooft anomaly is Z2 valued: the partition
function can pick up a minus sign Z 7→ (−1)x∪w2 (T M ) Z or Z 7→ (−1)B2 ∪y Z under Z2 trivial
e2 → B
e2 + 2x and w̃2 (T M ) → w̃2 (T M ) + 2y of the integer lifts of the Z2 -valued B2
shifts B
and w2 (T M ) [23]. We discuss some general aspects in this subsection, and will then illustrate
the general discussion with the original example of [23] in a later subsection.
(1)
(1)
Suppose that the one-form global symmetry is Γg = ZNg , with 2-form background gauge
(N )
e (Ng ) , which reduces to B (Ng )
field B2 that is ZNg valued. The integer lift of B g is then B
2
2
2
e (Ng ) → B
e (Ng ) + Ng x with integer-valued x. The κΓg ,T anomaly only
if we mod out by B
2
2
arises for Ng even and it suffices to turn on a background gauge field B2 for Z2 ⊂ Γg , setting
(N )
N
e2 also induces an integer lift of B (Ng ) : B
e (Ng ) = Ng B
e2 .
B2 g = 2g B2 . An integer lift of B2 , B
2
2
2
The κΓg ,T mixed anomaly also involves the Z2 valued w2 (T M ), which has integer lift w
e2 (T M )
that reduces to w2 (T M ) if we mod out by w
e2 (T M ) → w
e2 (T M ) + 2y. As in [23] a non-zero
12
The reason there is no independent κΓg ,Γg anomaly is that it can equivalently be written in terms of
κΓg ,T because w2 (Ggauge ) ∪ w3 (Ggauge ) ∼ w2 (Ggauge ) ∪ w3 (T N ) for Z2 valued B2 ≡ w2 (Ggauge ). This follows
by a Steenrod identity for any Z2 valued cohomology class x2 ∈ H 2 (N5 ; Z2 )
Z
Z
x2 ∪ β(x2 ) =
ν2 (T N ) ∪ β(x2 ) .
(2.25)
N5
N5
nd
where ν2 (T N ) is the 2 Wu class, and β is the Z2 Bockstein map. For orientable N , ν2 (T N ) = w2 (T N ). See
e.g. [26] for further details.
– 14 –
κΓg ,T mixed anomaly will imply an anomalous phase transformation of the partition function
under either of these shifts, and the two options differ by a choice of local counteterm.
The global symmetry G in the SpinG (4) structure must have a Z2 subgroup that can
be identified with (−1)F by Γ. The Z2 subgroup of G can be viewed as the center of an
SU (2)global ⊂ Gglobal and the U (1)Γ in (2.5) is the Cartan of this SU (2)global . There is some
freedom in the choice of this SU (2)global ⊂ Gglobal , but a non-trivial κΓg ,T anomaly can be
misleadingly trivialized if there is a factor of two in the index of embedding of SU (2)global in
Gglobal ; we will discuss this further in the next subsection.
We will generally assume that the fermions are in reps Rgauge ⊗ Rglobal of Ggauge ×
Gglobal , and that upon restricting Gglobal → SU (2)global the fermions transform as Rglobal →
2⊕nf , where the Cartan of SU (2)global can be identified with U (1)Γ and the w2 (Gglobal ) =
w2 (SU (2)global ) = w2 (T M ).13 The U (1)cΓ flux × charge pairing of each SU (2)global fundamental is then given by 2 → 12 ⊕ − 21 . In the B2 = 0 background, the index of the Dirac
P
operator is proportional to I = 21 i (qi2 − 41 ) with qi = ± 12 , so I = 0. Since this reduction
Gglobal → SU (2)global preserves the SpinG structure, our computation shows that κT,T = 0
in gauge theories with SpinGglobal structure (assuming Rglobal 7→ 2⊕nf ). This is to have been
expected given that the WWW new SU (2) anomaly only arises for SU (2) representations
j = 23 + 4r for r ∈ Z [24] as reviewed in the previous subsection.
To compute the κΓg ,T anomaly, we need to additionally turn on non-zero B2 background
gauge field for the Z2 ⊂ Γg center symmetry, leading to the Z2 -valued obstruction class
B2 = w2 (Ggauge ), and the SpinGglobal structure requires flux w2 (Gglobal ) = w2 (T M ). The total
flux can be embedded in a U (1)cQ = U (1)cΓg + U (1)cΓ where the U (1)cΓg ⊂ Gcgauge connection
activates the center flux B2 = w2 (Ggauge ) and the U (1)cΓ ⊂ Gcglobal connection activates the
w2 (Gglobal ) = w2 (T M ) flux on CP2 . The U (1)Γg has generator that we can write as
QΓg =
X
QI H I ,
(2.26)
I
for some flux charges QI associated with B2 = w2 (Gcgauge ), where HI are Ggauge simple
co-roots generating a Cartan torus containing U (1)Γg . The fermion representations Rgauge
decompose to have QΓg charges qig . The fermions are in the fundamental of SU (2)global and
the SpinGGlobal (4) structure on CP2 shifts the U (1)Γg charges qig as
1
1
g
g g
g
⊕ qi −
≡ qi .
(2.27)
(qi , qi ) 7−→ qi +
2
2
The index theorem for the number of zero modes in this flux background with B2 = w2 (Gcgauge )
13
For gauge theories with Γ = Z2 , we generally have Gglobal ⊂ SU (2nf ) where Z2 ⊂ Z(SU (2nf )) 7→ Z2 ⊂
Z(Gglobal ). We can then restrict SU (2nf ) 7→ SU (2) where 2nf 7→ 2⊕nf while preserving Z2 ⊂ Z(SU (2nf )) 7→
Z2 ⊂ Z(SU (2)global ), and we can also generally restrict Gglobal 7→ SU (2) so that Rglobal 7→ 2⊕nf factors
through the lift to SU (2nf ).
– 15 –
and w2 (Gcglobal ) = w2 (CP2 ) is then given by the index theorem to be
1X 2 1
n
I(Rgauge ⊗ Rglobal ) =
qi −
= Nf I2 (Rgauge ) TrRdef [Q2Γg ]
2
4
2
ψi
X
n
= Nf I2 (Rgauge )
QI CIJ QJ .
2
a
(2.28)
Here we write TrR [T a T b ] = n I2 (R) Trdef [T a T b ] with n = I2 (R1def ) (e.g. for SU (N ) groups the
factor n2 = 1). The anomaly can thus be characterized by U (1)Γg ⊃ Z(Ggauge ) and matching
the Z(Ggauge )-ality of fermion matter representations with the divisors of their Dynkin indices.
(1)
(1)
Here we see that if Γg ⊃ Z4N , then the index will be a multiple of 4 when coupled to
(1)
(1)
(1)
a Z2 ⊂ Z4N ⊂ Γg background gauge field B2 . The reason is that the index I must be an
(1)
(1)
(1)
integer for the minimal Z4N background as well as for the Z2 ⊂ Z4N background. These are
related by QIZ2 = 2N QIZ4N so that IZ2 = 4N 2 IZ4N . This implies that for the backgrounds
(1)
(1)
(1)
with Z2 ⊂ Z4N ⊂ Γg , there will be no anomalous phase of the partition function under the
action of ϕ.
b
(1)
Recall from our discussion in Section 2.1 that a w2 w3 anomaly betwen Γg and SpinG (4)
structure implies that the partition function depends on the choice of integer lift of B2 or
e2 7→ B
e2 + 2x or w̃2 (T M ) 7→ w̃2 (T M ) + 2y,
w2 (T M ), so that under a shift of integer lift B
partition function varies by
R
Z → Z · (−1)κΓg ,T M x∪w2 (T M )
or
R
Z → Z · (−1)κΓg ,T M B2 ∪y .
(2.29)
As we have discussed in Section 2.1, this anomaly only depends on the κΓg ,T mod 2 and
(1)
(1)
the phase can be generically activated for Z2 ⊂ Γg backgrounds. Therefore, since this
(1)
(1)
(1)
anomalous phase vanishes for any Z2 ⊂ Z4N ⊂ Γg , it must be that κΓg ,T = 0 mod 2 and
(1)
there is no w2 w3 anomaly between Z4N and SpinG (4) structures.
2.5
Determining Anomalies via Reduction to WWW “New SU (2)” Anomaly
As discussed in the previous subsections, the w2 w3 anomalies in theories with SpinG (4)
structure, where G can be a gauge or global symmetry, can be understood in terms of a
SpinU (1)Q (4) structure for some appropriate U (1)Q subgroup of the gauge and global symmetry group: U (1)Q = U (1)Γg + U (1)Γ , as in the previous subsection. Here the U (1)cΓg
connection activates a Z2 -flux in Ggauge that is determined by w2 (Ggauge ) = B2 , and the
U (1)cΓ connection activates a flux in Gglobal determined by w2 (Gglobal ) = w2 (T M ). The
U (1)Γg can often be embedded in an SU (2)gauge ⊂ Ggauge and the U (1)Γ can often be embedded in an SU (2)global ⊂ Gglobal . Then U (1)Q can be embedded in a diagonal SU (2)diag ⊂
Ggauge × Gglobal ⊂ Ggauge × Gglobal . In such cases, the computation of the κΓg ,T ’t Hooft
anomaly coefficient reduces WWW anomaly calculation [24] for this SU (2)diag subgroup.
This method was already mentioned and used in [24] to reproduce the w2 w3 anomaly of
SU (2) with Nf = 2 Weyl adjoints [23] from the anomaly in the SU (2)diag of a matter field
– 16 –
in the j = 3/2 representation. We will review and further discuss this method here, and also
a caveat is that this method can fail to exhibit a possibly non-trivial w2 w3 anomaly if the
index of the embedding of SU (2)diag ⊂ Ggauge × Gglobal is even.
As in the discussion above, we turn on background gauge field B2 for the one-form center
symmetry Z2 ⊂ Γg and consider a SpinGglobal (4) structure with spin-charge relation Γ that
includes −1Gglobal ∼ (−1)F . We consider the anomaly
Z
(2.30)
A ⊂ πiκΓg ,T B2 ∪ w3 (T M ) .
Since such anomalies are Z2 valued, they can only be non-trivial if there is a Z2 ⊂ Γg . If we
restrict to SU (2)gauge ⊂ Ggauge and SU (2)global ⊂ Gglobal such that Z2 ⊂ Z(Ggauge/global ) 7→
Z(SU (2)gauge/global ), then the anomaly will be matched by identifying the fluxes:
w2 SU 2 gauge = B2 , w2 SU 2 global = w2 Gglobal = w2 (T M ) .
(2.31)
One can now exhibit anomalies by reducing SU (2)gauge ×SU (2)global to the diagonal SU (2)diag
subgroup: the original theory has an anomaly (2.30) if the reduced theory has a WWW in
SU (2)diag with B2 6= 0 which vanishes when B2 = 0. This follows from the fact that when
we restrict to the SU (2)diag ⊂ SU (2)gauge × SU (2)global theory with B2 6= 0, we identify:
w2 SU 2 diag = w2 SU 2 gauge = w2 SU 2 global = w2 (T M ) ,
(2.32)
so that the mixed anomaly (2.30) reduces to the WWW SU (2) anomaly (1.1):
Z
Z
A ⊂ πiκΓg ,T B2 ∪ w3 (T M ) 7−→ A ⊂ πiκΓg ,T w2 (T M ) ∪ w3 (T M ) .
(2.33)
So if the original theory had no WWW anomaly for Gglobal , and the SU (2)diag ⊂ Ggauge ×
Gglobal theory does have a WWW anomaly, then the original theory has κΓg ,T 6= 0 (2.30).
Thus, rather than computing κΓg ,T from the anomaly indicator σI by the mod-2 index
I of the Dirac operator in a U (1)Q flux background, we can simply use the results of [24]
which classified which SU (2) representations yield w2 w3 anomalies for fermions coupled to an
SU (2) gauge field: when Nf fermions transform under a representation R which pulls back
L ⊕r
to the SU (2) representation R →
d d , the anomaly indicator is given by
X
σI (R) = Nf
r8n+4 mod2 .
(2.34)
n≥0
This provides an easy way to determine whether or not a theory has the anomaly (2.30).
A caveat is that this reduction method can fail to exhibit the anomaly if the embedding
L (i)
of SU (2)gauge/global ֒→ Ggauge/global where Rgauge/global 7→ i Rgauge/global has an even index
of imbedding14 , µGgauge/global :
P
(i)
R
I
2
i
gauge/global
(2.35)
,
µGgauge/global =
I2 Rgauge/global
14
The index of embedding also determines if there are instantons in partially broken groups [53, 54].
– 17 –
where I2 (R) is the Dynkin index. In this scenario, the index of embedding determines the
minimal Chern (i.e. instanton) number of the SU (2)gauge ⊂ Ggauge sub-bundle, with minimal kG related by kSU (2)gauge = µGgauge kGgauge . So the index of embedding determines the
proportionality constant in I as shown in (2.28):
I(SU (2)gauge × SU (2)global ) = µGgauge I(Ggauge × SU (2)gauge ) .
(2.36)
The anomaly indicator σI (R) in (2.6) will be a multiple of µGgauge and hence the restriction
Ggauge → SU (2)gauge fails to compute the anomaly if the index of embedding is even15
µGgauge ∈ 2Z :
anomaly trivialized.
(2.37)
The restriction Gglobal → SU (2)global where Rglobal → 2⊕nf , if Gglobal ⊂ SU (2nf ) where
again 2nf → Rglobal , would not trivialize the anomaly. Under this embedding, µGglobal = nf
which is only even when the anomaly identically vanishes.
2.6
Review of the w2 w3 Anomaly of SU (2) with 2 Adjoint Weyl Fermions
Consider SU (2)gauge theory with Nf = 2 massless Weyl fermions in the adjoint representation,
i.e. in the (3, 2) rep of SU (2)gauge × SU (2)global . The fields are in faithful representations of
Gtotal =
SU (2)gauge
(1)
Z2
×
SU (2)global × Spin(4) × Z8
,
Z2 × Z2
(2.38)
(1)
where Γg = Z2 is the 1-form center global symmetry of SU (2)gauge with matter in only j ∈ Z
representations, and the Z2 × Z2 quotient is because −1SU(2)global = (−1)Z8 = (−1)F . The
U (1)A global symmetry that acts on both fermions is broken by instantons to a Z8 subgroup,
with generator g satisfying g 4 = (−1)F . This theory admits a SpinSU (2)global (4) twisting,
which can be used to put the theory on a non-spin manifold via background gauge field flux
with w2 (SU (2)global ) = w2 (T M ); this is also used in the Donaldson-Witten twist of N = 2.
The various ’t Hooft anomalies of the theory are discussed in detail in [23].
The SpinSU (2)global (4) theory on M4 = CP2 does not have a purely gravitational ’t Hooft
anomaly, κT,T = 0, since SU (2)global has fermions in the j = 21 representation whereas only
j = 32 + 4r with r ∈ Z has the κT,T anomaly of [24]. The flux background for κT,T is the
minimal w2 (SU (2)global ) = w2 (CP2 ) background of the SpinSU (2)global (4) theory on CP2 , and
one can then take ϕ
b = ϕc.c. ◦ Wglobal , which preserves these fluxes. The U (1)cQ flux charges
1
are qi = ± 2 (the SU (2)gauge charges do not enter in this U (1)Q ) so there are I = 0 fermion
zero modes of the Dirac operator, and the anomaly indicator for κT,T is σI = 0.
(1)
The κΓg ,T = 1 mixed ’t Hooft anomaly between the 1-form center symmetry Z2 and
geometry was deduced in [23] by interpreting its relation to an anomaly found earlier in [43]
and in the context of the twisted N = 2 theory of the Donaldson-Witten partition function
15
For non-simply laced G, a factor in restricting G → SU (2) can compensate for an even index of embedding,
µG , so that the anomaly is not trivialized. This occurs in the Sp(Nc ) theories considered in Section 3.6.
– 18 –
(1)
(1)
ZDW [38, 39, 41, 42, 55]. The Γg = Z2 global symmetry of the SU (2) gauge theory is
coupled to a background gauge field B2 and the choice spinc structure is a lift of w2 (T M )
to an integral (vs mod 2) cohomology class w̃2 (T M ), with w2 (T M ) recovered by mod 2
reduction under shifts w̃2 (T M ) → w̃2 (T M ) + 2y with y an integral cohomology class. Under
this transformation, however, ZDW is not invariant but transforms as ZDW → ZDW (−1)B2 ∪y
[43]. As discussed in [23], one can add a local counterterm to remove the y dependence, at
e2 of B2 , with B
e2 → B
e2 + 2x with
the expense of introducing dependence on the integral lift B
x∪w
(T
M
)
2
x an integer 2-cochain leading to ZDW → ZDW (−1)
; as usual for mixed anomalies,
one can preserve one or the other symmetry, shifting between them by a local counterterm,
but not both. As noted in [23], this anomaly is described by inflow from the 5d action
Z
B2 ∪ w3 (M5 ) .
S5 = iπ
(2.39)
M5
We now illustrate using the methods of the previous subsections to compute this anomaly.
(1)
The SpinSU (2)global theory on CP2 with Z2 background B2 = w2 (SU (2)gauge ) has both
(1)
Γ : (−1)F ∼ −1SU (2)global flux, w2 (SU (2)global ) = w2 (CP2 ), and Γg
former can be activated by a U (1)cglobal ⊂ SU (2)cglobal connection
!
fˆZ2
1
âZ2 0
,
Aglobal =
= [H]∨ ,
2π
2
0 −âZ2
flux for B2 6= 0. The
(2.40)
where H is the generator [H] ∈ H2 (CP2 ; Z). To exhibit the κΓg ,T anomaly, we activate the
(1)
Γg flux of the background gauge field B2 = w2 (SU (2)gauge ) via the U (1)cΓg connection
!
âZ2 0
= Aglobal .
(2.41)
Agauge =
0 −âZ2
The total flux is along a diagonal U (1)cQ ⊂ SU (2)cgauge × SU (2)cglobal . The κΓg ,T anomaly is
exhibited by a mapping torus with ϕ
b acting on the S 1 that preserves this total flux:
ϕ
b = ϕc.c. ◦ Wgauge ◦ Wglobal ,
(2.42)
with ϕc.c. complex conjugation on CP2 . Again, ϕc.c. reverses the orientation of the gauge and
global fluxes and acts as a π rotation, so ϕ2c.c. = (−1)F . The Weyl transformations are
!
0 1
Wgauge/global =
.
(2.43)
−1 0
gauge/global
2
These act as π-rotations in SU (2)gauge/global around the ŷ axis, so Wglobal
= (−1)F since
2
the fermions are in the 2 and Wgauge
= 1 since the fermions are in the 3; the upshot is
2
F
F
ϕ
b = (−1) · (−1) = 1, so ϕ
b is Z2 -valued as needed for a mod 2 anomaly κΓg ,T .
The fermion zero modes transform as doublets under the O(2) symmetry group generated
by flat U (1)Q ⊂ U (1)gauge × U (1)global transformations and ϕ
b and consequently, κΓg ,T = σI
– 19 –
mod2 will be given by half the number of zero modes of the 4d Dirac operator in the U (1)cQ
flux background. To compute this, we will apply the Atiyah-Singer index theorem. Under
the restriction of SU (2)gauge/global → U (1)gauge/global , normalized to give integer charges, the
fermions have charges under U (1)gauge × U (1)global 7→ U (1)Q given by
!
3 1 −1
3 ⊗ 2 7−→ (2, 0, −2) ⊗ (1, −1) =
(2.44)
1 −1 −3
Since we have 12 -integral background fluxes in the U (1)s, we apply the CP2 flux background
P
Atiyah Singer index theorem I = i (4qi2 − 1)/8 with 12 of the above charges, i.e.
!
1
1
3
−
(2.45)
qi = 21 21 32
2 −2 −2
The result is I(3 ⊗ 2) = 2. We can also obtain
this result using (2.28) with I2 (3) = 2, Q1 = 21
and Nf = 2: I(3 ⊗ 2) = Nf I2 (3) 2(Q1 )2 = 2 × 2 × 2 × 14 = 2 . The anomaly indicator is
thus non-trivial, σI = I(3⊗2)
mod2 = 1, re-deriving the anomaly of [23]
2
Z
A ⊂ πi B2 ∪ w3 (T M ) ,
(2.46)
that was obtained via the Donaldson-Witten anomaly in the shifts of the integer lifts of B2 or
w2 (T M ) [38, 39, 41, 42, 55]. This can be easily generalized to Nf = 2nf adjoint Weyl fermions,
simply by restricting Gglobal = SU (Nf ) 7→ SU (2)global . The fermions then transform under
(3 ⊗ 2)⊕nf so the index is given by I ((3 ⊗ 2)⊕nf ) = 2nf = Nf , which leads to the anomalous
phase of the action (−1)nf . The anomaly indicator is thus given by σI = nf mod2 . So SU (2)
gauge theory with Nf = 2nf adjoint fermions has the ’t Hooft anomaly
Z
A ⊂ πi nf B2 ∪ w3 (T M ) .
(2.47)
This anomaly was also obtained in [30] in connection with symmetry fractionalization.
The anomaly found in [23] for SU (2) with Nf = 2 adjoint fermions can also be derived
by reducing to the WWW SU (2) anomaly [24]. Generalizing to Nf = 2nf adjoint fermions,
the fields transform faithfully under
Gtotal =
SU (2nf )global × Spin(4) × Z8nf
SU (2)gauge
×
,
Z2
Z2 × Z2
(2.48)
(1)
where there is a Z2 global center symmetry and Z8nf has generating group element g that
satisfies g4 = CSU (2nf ) where CSU (2nf ) is the generator of the Z2nf center of the flavor group
n
and g4nf = CSUf (2nf ) = (−1)F are the Z2 quotients indicated in Gtotal . Neither SU (2)gauge
nor SU (2)global separately have a WWW SU (2) anomaly since they have j ∈
/ 23 + 4Z representations. For nf = 1 the SU (2)global has an ’t Hooft anomaly analog of the original Witten
– 20 –
anomaly of SU (2)global , since there are an odd number (3) of SU (2)global doublets; for nf > 1,
this becomes simply a perturbative, non-zero ’t Hooft anomaly: TrSU (2nf )3global = 3. The
anomaly (2.47) can be exhibited by reducing SU (2)gauge × SU (2nf )global → SU (2)diag . In the
restriction Gglobal → SU (2)global , we can choose the embedding 2nf 7→ 2⊕nf to not trivialize
the anomaly, so the fermions transform under SU (2)gauge × SU (2)global as the (3, 2)⊕nf representation. We now couple the fermions to SpinSU (2)diag bundles where the fermions transform
with respect to SU (2)diag ⊂ SU (2)gauge × SU (2)global as
(3 ⊗ 2)⊕nf 7−→ 4⊕nf ⊕ 2⊕nf .
(2.49)
When nf is odd, the theory has both a WWW SU (2) anomaly, due to the odd number of
fermions in the 4 representation of SU (2)diag , and an SU (2) Witten anomaly [28] (which is
here a ’t Hooft anomaly) from the odd number of fermions in the 2. The WWW anomaly of
SU (2)diag implies that the κΓg ,T anomaly (2.47) is indeed non-trivial for nf odd.
(1)
2.7
Gauging Γg
subgroups, e.g. SO(3)gauge with 2 Adjoint Weyl Fermions
In the following sections, we will mostly consider simply connected Ggauge theories. One can
(1)
obtain other cases by gauging the one-form global symmetry Γg or subgroups, i.e. subgroups
of the original Ggauge symmetry. For example, the SO(3)± gauge theories can be obtained
(1)
from SU (2) gauge theories by gauging the Z2 one-form electric global symmetry [56]. If the
original SU (2) theory has partition function ZQ [B], one replaces the background gauge field
B2 = w2 (SU (2)) with a dynamical Z2 gauge field, B2 → b2 , to obtain the SO(3)+ partition
(1)
function with dual magnetic Z2 global symmetry with background gauge field B2′ [56, 57]
R ′
X
ZSQ [B2′ , w̃2 (T M )] =
(−1) B2 ∪b2 ZQ [b2 , w̃2 (T M )].
(2.50)
b2
For SO(3)gauge theory with Nf = 2 Weyl fermions in the adjoint representation, it suffices to
consider the SO(3)+ theory since the SO(3)± cases are related by θ → θ + 2π.
Mixed anomalies of the original ZQ theory involving B2 can lead to 2-group extensions
upon gauging16 , as illustrated in e.g. [51, 57, 64, 65]. Consider the effect of the anomaly
term (2.47) of the SU (2)gauge theory in the SO(3)gauge version. By choice of counterterm,
we can choose the presentation of the anomaly where ZQ [b̃2 , w̃2 (T M )] is invariant under
b̃2 → b̃2 + 2x, so that we can unambiguously sum over the gauged Z2 -valued b2 . The anomaly
affects the shift of w̃2 (T M ) as ZQ [b2 , w̃2 (T M ) + 2y] = (−1)κΓg ,T b2 ∪y ZQ [b2 , w̃2 (T M )]. So the
SO(3) theory has ZSQ [B̃2′ + 2x, w̃2 (T M )] = ZSQ [B̃2 , w̃2 (T M )] and
ZSQ [B2′ , w̃2 (T M ) + 2y] = ZSQ [B2′ + κΓg ,T y, w̃2 (T M )].
(2.51)
If the original theory had vanishing ’t Hooft anomaly, κΓg ,T = 0, then the theory after gauging
(1)
Z2 also has vanishing ’t Hooft anomaly. If the original theory had ’t Hooft anomaly κΓg ,T = 1
then the theory after gauging has a Z4 extension of the Z2 × Z2 shift symmetries of B̃2′ and
w̃2 , as in (2.51), which implies Z4 symmetry w̃2 (T M ) → w2 (T M ) + 4y.
16
Other mixed anomaly terms, e.g. κrΓ in (5.2), lead to non-invertible symmetries see e.g. [47, 48, 58–63]
– 21 –
Determining the w2 w3 Anomalies of Other Theories
3
In this section we exhibit the w2 w3 ’t Hooft anomalies for a variety of gauge theories.
3.1
SU (2)gauge with 2nf Weyl Fermions in the (2j + 1)-Representation
in this family of theories, the case j = 1, Nf = 2nf = 2 is the adjoint representation
case considered in [23], reviewed in the previous section. We here generalize to other spinj SU (2)-representations for the fermions,17 first for Nf = 2. For j ∈ Z + 21 there is a
SU (2)gauge × SU (2)flavor symmetry-preserving mass term for the fermions, and thus there can
be no anomaly. We thus consider j ∈ Z, and then the gauge invariant fermion bilinear is in
the adjoint of SU (2)flavor so there is no symmetry-preserving mass term. The fields couple to
Gtotal -bundles where:18
Gtotal =
SU (2nf )global × Spin(4)
SU (2)gauge
×
.
Z2
Z2
(3.1)
The theory classically admits a SpinSU (2nf )global (4) structure, allowing us to consider the
theory on CP2 upon twisting by SU (2nf ) flux, taking w2 (SU (2Nf )) = w2 (CP2 ). Since j ∈ Z
(1)
the theory has unbroken Z2 one-form center symmetry, which we can couple to background
field B2 .
There is no purely gravitational anomaly, κT,T = 0, and the κΓg anomaly is non-zero if
there are 2 mod 4 fermion zero modes of the Dirac operator on CP2 in the flux background
with w2 (SU (2Nf )) = w2 (CP2 ) and B2 6= 0. This can be computed from the Atiyah Singer
index by decomposing the fermions under the diagonal U (1)Q ⊂ U (1)gauge × U (1)global , with
U (1)gauge ⊂ SU (2)gauge and U (1)global ⊂ SU (2)flavor and half-integral flux in each for the
P
non-zero w2 (SU (2)global ) and w2 (SU (2)gauge ). The Dirac index is I = i (4qi2 − 1)/8, where
P
for each nf the qi run over m ± 12 with m = j, j − 1, · · · − j, so I = nf jm=0 m2 = nf T2 (Rj )
the Dynkin index of the j-representation:
I(Rj ⊗ 2nf ) = nf ×
j(j + 1)(2j + 1)
.
3
(3.2)
The Z2 anomaly coefficient is σI = 12 I mod 2, i.e.
σI = n f
j(j + 1)(2j + 1)
mod2 .
6
It is easily seen that this implies that the anomaly indicator is given by
(
nf mod2 j = 1, 2 mod4
σI =
0
j = 0, 3 mod4
17
(3.3)
(3.4)
For general Nf and j the theory is not asymptotically free in the UV, but could arise as a low-energy
effective field theory in the IR.
18
We omit writing the Z4nf I2 (Rj ) ⊂ U (1)A , where 2I2 (Rj ) = 2j(j +1)(2j +3)/3 is the number of fermion zero
modes for each flavor in representation Rj in an instanton background. Its generator g has g 2I2 (Rj ) = CSU (2nf )
where C generates the Z2nf center of SU (2nf ), with Cfn = −1 = (−1)F .
– 22 –
Thus the anomaly polynomial is non-trivial iff nf is odd and j = 1, 2, mod4 :
A⊂
(
R
πi nf B2 ∪ w3 (T M )
0
j = 1, 2 mod4
j = 0, 3 mod4
(3.5)
The result (3.5) can also be derived by reducing to the WWW SU (2) anomaly, as discussed for the j = 1 case in [24] and reviewed in the previous section. We project onto the
diagonal subgroup SU (2)diag ⊂ SU (2)gauge ×SU (2nf ) corresponding to B2 = w2 (SU (2nf )) =
w2 (T M ). The fermions decompose under the SU (2)diag via the SU (2) rep tensor product
decomposition:
(2j + 1) ⊗ 2nf → ((2j + 1) ⊗ 2)⊕nf = (2j + 2)⊕nf ⊕ (2j)⊕nf .
(3.6)
The WWW anomaly [24] is non-zero iff there are an odd number of Weyl fermions in the
representations with j = 23 mod 4, i.e. of dimension 4 mod8 . The original SU (2)gauge matter
content is anomaly free and the SU (2)global ⊂ SU (2nf )global has an odd number of doublets
for nf odd (since 2j + 1 is taken to be odd), so it has the original SU (2) anomaly, but not the
WWW anomaly. On the other hand, SU (2)diag has the WWW anomaly for j = 1, 2 mod4
and nf is odd, reproducing the anomaly (3.5).
3.2
SU (Nc )gauge theory with 2nf Weyl Fermions in the Adjoint Representation
In these theories, the fields couple to Gtotal bundles (we suppress the Z2Nc Nf ⊂ U (1)A 19 ):
Gtotal =
SU (Nc ) SU (2nf ) × Spin(4)
×
.
Z Nc
Z2
(3.7)
We take Nf = 2nf here for a generalized SpinSU (2nf ) (4) twisting, with non-zero w2 (SU (2nf )) =
(1)
w2 (T M ) flux background for the theory on CP2 . There is a ZNc = Z(SU (Nc )) 1-form symmetry associated with the center of SU (Nc ). As we discussed, the w2 w3 type ’t Hooft anoma(1)
lies under consideration can only be Z2 -valued and such anomalies involving the ZNc and
(1)
(1)
SpinGglobal structure can only be non-trivial if ZNc has a Z2 subgroup. The w2 w3 ’t Hooft
anomalies thus only arise if Nc is even, so we write Nc = 2nc and we denote the background
(1)
(1)
gauge field coupled to the Z2 ⊂ Z2nc as B2 .
To exhibit the Z2 -valued anomaly, let us turn on the B2 background gauge field. This
can be activated by a U (1)cΓg connection which is generated by
1
QΓg = diag(1, ..., 1, −1, ..., −1) ,
| {z } | {z }
2
nc
(3.8)
nc
acting on the fundamental representation. The Atiyah Singer index theorem in the w2 (SU (2nf )) =
w2 (T M ) flux background on CP2 then gives for the number of adjoint fermion zero modes:
19
Its generator g has g 2Nc = C, the center generator of SU (2nf ), and g 2nc nf = C nf = (−1)F .
– 23 –
P
2
2
I =
i (4qi − 1)/8 = Nf Nc Trfund Q where we used I2 (adj) = 2Nc I2 (fund). The w2 w3
anomaly indicator is thus
σI =
1
I mod2 = nf nc mod2 .
2
(3.9)
Hence adjoint SU (2nc ) QCD has an anomaly term
Z
A ⊂ πi nf nc B2 ∪ w3 (T M ) .
(3.10)
The anomaly is thus non-trivial only if nc and nf are both odd, i.e. if Nc = 4n + 2 for n ∈ Z
and Nf = 4m + 2 for m ∈ Z.
The w2 w3 anomaly in the SpinGglobal (4) twisted theory can be seen from the WWW
SU (2) anomaly [24] in a SU (2)diag ⊂ SU (2)gauge × SU (2)global with appropriately chosen SU (2)gauge ⊂ Ggauge and SU (2)global ⊂ Gglobal so that the Z2 ⊂ Z(Ggauge/global ) 7→
Z(SU (2)gauge/global ). Such embeddings will will only be non-trivial for nc odd. We choose the
embedding SU (2)gauge/global ⊂ SU (2nc )gauge/global where 2nc 7→ 2⊕nc (and similarly for nf ).
Note that the index of the embedding of SU (2nc )gauge → SU (2)gauge is µ = nc , so this would
trivialize the anomaly for nc even. Restricting to SU (2)diag ⊂ SU (2)gauge × SU (2)global gives
the restriction
(3.11)
B2 = w2 (SU (nf )) = w2 (T M ) .
The SU (2nc ) adjoint fermions decompose under the SU (2) ⊂ SU (2nc ) as
2
2
adj = 2nc ⊗ 2nc − 1 −→ 2⊕nc ⊗ 2⊕nc − 1 = 3⊕nc ⊕ 1⊕nc −1 .
(3.12)
So the fermions decompose under SU (2)diag as
adj ⊗ 2nf −→
h
i⊕nf
2
2
2
2
= 4⊕nf nc ⊕ 2⊕nf (2nc −1) .
3⊕nc ⊕ 1⊕nc −1 ⊗ 2
(3.13)
So SU (2)diag has the w2 w3 ’t Hooft anomaly for nf n2c odd, i.e. for both nf and nc odd,
reproducing the anomaly (3.10). The odd number of SU (2)diag doublets implies that the
theory also has a Z2 ’t Hooft anomaly associated with the original SU (2)global Witten anomaly.
3.3
Spin(2nc + 1)gauge with 2nf Weyl Fermions in the Vector Representation
In these theories, the fields couple to Gtotal -bundles with (suppressing the Z4nf ⊂ U (1)A )
Gtotal =
Spin(2nc + 1) SU (2nf ) × Spin(4)
×
.
Z2
Z2
(3.14)
We can put the theory on CP2 via twisting w2 (SU (2nf )) = w2 (T M ) and can turn on back(1)
(1)
ground field B2 for the Z2 center symmetry. This Z2 can be chosen to be generated by
the Cartan element QΓg = 21 Hnc = α∗nc where Hnc is the co-root that is twice the dual of
– 24 –
the short root of so(2nc + 1). The Atiyah Singer index theorem in this flux background then
gives20
1 2
I = nf Tr Hnc = nf (αnc , αnc ) = 2nf =⇒ σI = nf .
(3.15)
4
This leads to the anomaly
A ⊂ πi nf
Z
B2 ∪ w3 (T M ) .
(3.16)
For nc = 1, this matches the SU (2) with adjoints case reviewed in Section 3.2.
We can alternatively demonstrate the anomaly by reducing to the WWW SU (2) anomaly
in some appropriately chosen SU (2)diag . Let us restrict to SU (2)diag ⊂ Spin(2nc + 1) ×
SU (2nf ) where the vector representation of Spin(2nc + 1) maps to the (2nc + 1) representation of SU (2) and SU (2) is embedded diagonally in SU (2nf ). This restriction identifies
w2 (Γg ) = w2 (Γ) = w2 (T M ) .
(3.17)
Under this restriction, the fermions decompose as
[(2nc + 1) ⊗ 2]⊕nf = [2nc ⊕ (2nc + 2)]⊕nf
(3.18)
which demonstrates that this theory has the WWW SU (2) anomaly for nc = 1, 2 mod4 and
nf odd. The index of embedding, µ, is the ratio of the Dynkin index in the (2nc + 1) irrep
of SU (2) to the (2nc + 1) of SO(2nc + 1):
µ=
nc (nc + 1)(2nc + 1)
.
6
(3.19)
The anomaly is trivialized by the embedding if µ is even, i.e. if nc = 0, 3 mod4 . To
avoid trivializing the anomaly for nc = 0, 3 mod4 , we can choose a different embedding
of Spin(2nc + 1) → SU (2), e.g. we can take (2nc + 1) → (2nc − 3) ⊕ 1⊕4 . This embedding
effectively replaces nc → nc − 2 as compared with the previous one, and in particular the
index of embedding differs from that above by this replacement. So the anomaly with this
new embedding is no longer trivialized for nc = 0, 3 mod4 (it is instead trivialized for nc = 1, 2
mod4 ). Indeed, the fermions decompose under the new embedding as
⊕nf
⊕n
,
2nc − 3 ⊕ 1⊕4 ⊗ 2 f = (2nc − 4) ⊕ (2nc − 2) ⊕ 2⊕4
(3.20)
which exhibits WWW SU (2) anomaly for nc = 0, 3 mod4 and nf odd. Together these two
embeddings exhibit the anomaly of (3.16) for all nc .
20
so(2N + 1) is a non-simply laced Lie algebra. This means that the formula relating the trace to inner
products on the root space via the Cartan matrix differs along short roots. For so(2N + 1), this differs for the
∨
2
coroot HN
= 2αN . So that Trvec [HN
] = 4(αN , αN ) = 4CNN = 8.
– 25 –
3.4
Spin(2nc )gauge with 2nf Weyl Fermions in the Vector Representation
The fields in this case couple to Gtotal -bundles where (suppressing the Z4nf ⊂ U (1)A )
Spin(2nc )
(1)
Gtotal =
Z2
× SU (2nf ) × Spin(4)
.
ZF2 × ZC
2
(3.21)
F
F
C
F
The ZF2 × ZC
2 identify Z2 : − 12nf ∼ (−1) and Z2 : (−1) ∼ z ∈ Z(Spin(2nc )) where
2
2
here z = 1Spin(2nc ) for nc even and z = −1Spin(2nc) ∈ Γg for nc odd. The ZC
2 identification
F
implies that (−1) is effectively gauged, so the theory is bosonic and admits a SpinSpin(2nc ) (4)
structure. The ZF2 identification implies that there is also a SpinSU (2nf ) (4) structure. The
theory on CP2 can be realized with either of these generalized spin structures. For odd nc ,
(1)
the Z2 center symetry is actually in a 2-group with (−1)F because a Wilson line W in the
spinor rep of the gauge group has W 2 which is charged under ZC
2 (while being uncharged under
F
Z2 ), which can only be screened by the dynamical fermions at the expense of transforming
21
non-trivially under (−1)F ; this implies that δw2 (Γg ) = βw2 (ZC
2 ) [57, 65].
We first consider SpinSU (2nf ) (4) backgrounds, with w2 (SU (2nf )) = w2 (T M ) flux and
(1)
turn on a background flux for the Z2 global symmetry, B2 . We can restrict to Gtotal -bundles
that lift to
G′ =
Spin(2nc ) SU (2nf ) × Spin(4)
.
×
Z2
ZF2
(3.22)
We use the index theorem as in (2.28) to compute the anomaly. The B2 flux is embedded
along U (1)Γg which is generated by


⌈ n2c ⌉−1
X
1
(3.23)
H2I−1  .
QΓg = Hnc −1 + (−1)nc Hnc + 2
2
I=1
So for the index (2.28) we compute

j n k
 Trvec (Hn −2 + 1 Hn −1 − 1 Hn )2 nc odd
c
c
c
c
2
2
2
h
Trvec [QΓg ] = 2
−1 +
i
 Trvec 1 Hnc −1 + 1 Hnc 2
2
nc even
2
2
(3.24)
This can be computed via an Spin(6) ⊂ Spin(2nc )gauge in terms of its Cartan matrix. The
result for the number of fermion zero modes and the anomaly indicator for nc odd is
I = 2nf nc
=⇒
σI = nf mod2 .
(3.25)
For nc even the index is I = 2nf (nc − 1) and the anomaly indicator is again σI = nf mod2 .
We conclude that Spin(2nc ) gauge theory with 2nf fermions, for all nc , has an anomaly
Z
A ⊃ πi nf B2 ∪ w3 (T M ) .
(3.26)
(1)
Alternatively, there is a 2-group between the Z2 and SU (2nf ) since W 2 is also charged under −1SU (2nf ) :
F
this is equivalent to the above-mentioned 2-group due to the quotient by ZF
2 which relates −1SU (2nf ) ∼ (−1) .
21
– 26 –
We can additionally show that there are no other anomalies when we consider SpinSU (2nf ) (4)
structure. The allowed, independent anomalies (without activating w2 (ZC
2 ) are
Z
Z
A ⊂ πi nf B2 ∪ w3 (T M ) + πiκT,T w2 (ZF2 ) ∪ w3 (ZF2 ) .
(3.27)
Again, there is no independent κΓg ,Γg anomaly because it is redundat with the B2 w3 -anomaly.
Now using the fact that w2 (ZF2 ) = w2 (SU (2nf )) = w2 (T M ), we see that the only additional
anomaly is the purely gravitational anomaly for the SpinSU (2nf ) (4) structure. We can compute κT,T by putting the the theory on CP2 , with flux along w2 (SU (2nf )) = w2 (T M ). In
this background, the fermion charges are
1 ⊕2nc nf
.
,−
2nc ⊗ 2nf 7−→
2 2
1
(3.28)
The Atiyah Singer index theorem then shows that there are no fermion zero-modes. Therefore,
there is no anomaly in this background and κT,T = 0.
Now let us consider the anomalies for SpinSpin(2nc ) (4) backgrounds. First take the case
where nc is even. For these bundles we can again turn on fluxes to activate the anomaly
(3.26) – the computation follows directly as above. The allowed, independent anomalies are
Z
Z
′
C
A ⊂ πi nf B2 ∪ w3 (T M ) + πiκT,T w2 (ZC
(3.29)
2 ) ∪ w3 (Z2 ) ,
F
which is the same as the SpinSU (2nf ) (4) case except that ZC
2 has replaced the role of Z2 .
C
Again, we can identify the w2 (ZC
2 )w3 (Z2 ) anomaly as the purely gravitational anomaly since
′
2
w2 (ZC
2 ) = w2 (T M ). We can compute κT,T directly by putting the theory on CP and turning
on 1-form flux along generated by
nc
2
1X
H2I−1 .
Q=
2
(3.30)
I=1
Now the fermions decompose into U (1)Q representations as 2nc ⊗ 2nf →
1
1
2, −2
the Atiyah Singer index theorem leads to I = 0 zero-modes and hence κ′T,T = 0.
⊕2nc nf
, so
For odd nc , the SpinSpin(2nc ) (4) structure does not permit discrete flux in SU (2nf ).
(1)
Additionally, the fact that Γg forms a 2-group with (−1)F forces the anomalies to be that
of a Z4 -valued discrete flux where we cannot use our mod-2 index to probe the anomalies.
For completeness, we compute these anomalies using the fractionalization technique [30] in
Appendix A and show that there is no anomaly for the SpinSU (2nc ) (4) background.
For all nc , we find for these theories that there is an anomaly
Z
A ⊂ πi nf B2 ∪ w3 (ZF2 ) .
(3.31)
– 27 –
3.5
Spin(Nc )gauge with 2nf Weyl Fermions in the Adjoint Representation
The fields couple to Gtotal -bundles with (suppressing the Z4h∨G
gauge
Gtotal =
nf ⊂ U (1)A )
Spin(Nc ) SU (2nf ) × Spin(4)
×
,
Γg
Z2
(3.32)
(1)
where the one-form symmetry group Γg is the center symmetry


Nc = 2nc + 1

Z 2
Γg = Z2 × Z2 Nc = 4nc


Z
N = 4n + 2 .
4
c
(3.33)
c
Again, our ϕ
b flux background computation can only activate a Z2 ⊂ Γg flux. As discussed in
Section 2.1 the anomaly for Spin(4nc +2) adjoint QCD vanishes due to the fact that the center
of Spin(4nc + 2) is Z4 . We check this by the use of fractionalization techniques developed in
[30] in Appendix A and indeed find that there is no additional anomaly associated with Z4
flux. The Z2 flux obstructs lifting SO(Nc ) to Spin(Nc ) and can be embedded along U (1)Γg
which is generated by


 21 Hnc , Nc = 2nc + 1 ,
QΓg = 1
(3.34)
P nc2−1

H
−
H
+
2
H
Nc = 4nc + 2 ,
2
nc −1
nc
2I−1
I=1
L
and, for Nc = 4nc , we choose to embed the Z2 flux along U (1)L
Γg which is generated by QΓg
where
nc
2
1X
H2I−1
QL
=
Γg
2
nc
−1
2
1 X
1
H
+
QR
H2I−1 .
=
n
Γg
2 c 2
,
I=1
(3.35)
I=1
We can write the above charges in the fundamental representation and then use Tradj =
(Nc − 2) Trvec in computing the index for the adjoint fermions. The Atiyah Singer index
theorem then gives


Nc = 2nc + 1

2nf (Nc − 2)
2
(3.36)
I = Nf Tradj [QΓg ] = 2nf (4nc − 2)(2nc − 1) Nc = 4nc


2n (4n )(2n + 1)
N = 4n + 2
f
c
c
c
c
The anomaly indicator σI = 21 I mod2 coefficient thus gives for the anomaly theory
A⊂
(
R
nf πi B2 ∪ w3 (T M ) ,
0,
– 28 –
Nc = 2nc + 1 ,
Nc = 2nc .
(3.37)
We can also check the anomalies for Spin(Nc ) gauge theory by reducing to the WWW
anomaly in a SU (2)diagonal ⊂ SU (2)gauge × SU (2)global , where we choose a restriction of
SU (2nf ) → SU (2)global and Spin(Nc ) → SU (2)gauge that ideally does not trivialize the
anomaly. For SU (2nf ) → SU (2)global , we choose the embedding where 2nf 7→ 2⊕nf . We will
here only discuss the case of Nc = 2nc + 1 odd; similar considerations verify that there is no
anomaly for Nc even, in agreement with (3.37). We can take the Spin(2nc + 1) → SU (2)gauge
restriction where Nc → Nc , since for Nc = 2nc + 1 this restriction preserves the center. The
trivialization of the anomaly is controlled by the index of embedding which is here
µ=
nc (nc + 1)(2nc + 1)
Nc (Nc2 − 1)
=
24
6
,
Nc = 2nc + 1 .
(3.38)
So the anomaly is trivialized for Nc = 1, 7 mod8 and detectable for Nc = 3, 5 mod8 . For the
trivialized case, we take another embedding below. The adjoint representation of Spin(Nc )
b of vector representations, so it decomposes under
is the anti-symmetric tensor product (⊗)
the Nc → Nc embedding of Spin(Nc ) → SU (2)gauge as
b c 7−→ j = 1 ⊕ 3 ⊕ ... ⊕ Nc − 2
Adjoint = Nc ⊗N
(3.39)
Using the results from Section 3.1, we indeed find that the theory has the anomaly for
Nc = 3, 5 mod8 , matching the result (3.37) for those cases where the index of the embedding
does not trivialize the anomaly.
For the cases Nc = 1, 7 mod8 , we can exhibit the anomaly (3.37) by choosing different
embeddings, to avoid trivializing the anomaly. For Nc = 8nc +1, we can pick the SU (2)gauge ⊂
Spin(Nc ) sub-bundle where 8nc + 1 → (8nc − 3) ⊕ 1⊕4 , which has odd index of embedding
µ = (2nc −1)(4nc3−1)(8nc −3) ∈ 2Z + 1 . Decomposing the adjoint representation, we verified that
for nf odd there are indeed an odd number of SU (2)gauge representations that contribute to
the anomaly, reproducing the result (3.37). Similarly, for Nc = 8nc + 7, we can choose the
embedding SU (2)gauge ⊂ Spin(Nc ) where 8nc − 1 7→ (8nc − 5) + 1⊕4 , which has odd index
of embedding, µ = (2nc −1)(4nc3−3)(8nc −5) ∈ 2Z + 1 , so the anomaly is not trivialized. Again, we
verify that the decomposition of the adjoint yields SU (2)gauge representations with anomaly
matching the results of (3.37).
3.6
Other Ggauge Theories with 2nf Weyl Fermions in the Adjoint Representation
The theories can be put on M4 = CP2 using the SpinSU (2nf ) (4) structure, with w2 (SU (2nf )) =
(1)
(1)
w2 (T M ). The gauge groups with a Z2 ⊂ Γg subgroup of the one-form center symmetry
can have the ’t Hooft anomaly under discussion. In particular, Ggauge = Sp(Nc ), for Nc = 1, 2
mod4 , and Ggauge = E7 have the anomaly
Z
(3.40)
A ⊃ πi nf B2 ∪ w3 (T N ) .
We can obtain these results for Ggauge = Sp(Nc ) by computing the Index of the Dirac operator
(1)
on CP2 with a background B2 gauge field turned on for the Z2 center symmetry. Applying
– 29 –
the Atiyah-Singer Index theorem, one finds
I = nf Nc (Nc + 1)
=⇒
σI =
(
nf mod2
Nc = 1, 2 mod4
0
Nc = 0, 3 mod4
(3.41)
reproducing the anomaly above. This matches the results from Sections 3.1 and 3.5 for the
case of Sp(1) = SU (2) and Sp(2) = Spin(5).
For E7 , we can take the restriction E7 → SU (2) where
56 7−→ 2⊕5 + 4⊕7 + 6⊕3 ,
(3.42)
which maps the center Z2 = Z(E7 ) 7→ Z2 = Z(SU (2)). This restriction has an index of
embeddeing µE7 = 15. Under this restriction, the adjoint representation restricts
133 7−→ 1⊕3 + 3⊕15 + 5⊕10 + 7⊕5 .
(3.43)
Using our results from Section 3.1, we see that this does indeed have an anomaly (3.40).
4
C Symmetry, Higgsing, and Abelian Gauge Theory
The w2 w3 ’t Hooft anomalies must of course match upon continuous, symmetry-preserving
deformations. For example, we can preserve the generalized SpinG (4) structure via added
scalars in real representations of Ggauge . Giving the added scalars masses RG flows to our
original theories, with the added scalars decoupled. Alternatively, the added scalars can have
a potential that leads them to have non-zero expectation values, Higgsing the gauge group.
The ’t Hooft anomalies must be matched in the Higgsed theory. For scalars in the adjoint
representation of the gauge group, the deformed theory can flow to an abelian gauge theory in
the IR. Such deformations can also be used to give masses to the fermions by adding Yukawa
couplings to the scalar field that gets condenses. We will here consider cases without such
Yukawa couplings, with massless fermions in the low-energy theory.
4.1
Cases where ϕ
b Includes Charge Conjugation Symmetry C
The WWW method [24] of determining the w2 w3 type anomaly involved the classical global
symmetry ϕ
b = ϕc.c. ◦ W , with e.g. W ∈ SU (2) to preserve the flux background. In the theory
with added adjoint scalar field(s), we can Higgs SU (2) → U (1) and as discussed in [24]
the anomaly of the SU (2) theory then connects to that of all fermion electrodynamics. Upon
Higgsing SU (2) → U (1), the element W ∈
/ U (1) and instead it is similar to charge conjugation.
We will here discuss some general aspects about cases when the analog of the ϕ
b map needs to
incorporate an element that is not a Weyl symmetry but, instead, the outer automorphism of
charge conjugation, C, to obtain a symmetry of the bosonic flux background. We will apply
the discussion to cases where the gauge group is Higgsed to an Abelian subgroup. We thus
consider ϕ
b = ϕc.c. ◦ Wglobal ◦ C that preserves discrete flux backgrounds for Gtotal -bundles:
Gtotal =
Ggauge
× Gglobal × Spin(4)
Γg
Γ
– 30 –
.
(4.1)
Suppose e.g. that the fermions transform as Rgauge ⊗ Rglobal and are pseudo-real and
Rgauge is real. To probe for an anomaly involving B2 ∪w3 (T M ) we turn on B2 flux background
in addition to the flux background needed for the SpinGglobal structure. For the mapping torus
construction, ϕ
b is given by ϕ
b = ϕc.c. ◦ Wgauge ◦ Wglobal , where the Wglobal is needed to preserve
the fluxes associated with the SpinGglobal (4) structure, and Wgauge is needed to preserve the
Γg flux background. Now we suppose that the gauge group does not include such a Wgauge
element and we instead use charge conjugation to preserve the Γg gauge flux. We then consider
the mapping torus based on ϕ
b = ϕc.c. ◦ Wglobal ◦ C, which can preserve the flux background,
2
2
b2 = 1.
and satisfy ϕc.c. = Wglobal = −1 and C 2 = 1, so ϕ
Now, however, the mapping torus construction for ϕ
b = ϕc.c. ◦ Wglobal ◦ C, probes a 5d SPT
phase that includes flux associated with an C background gauge field: the anomaly measured
by the variation of the action is given by
Z
A ⊃ πiσI w1C ∪ w2 (T M ) ∪ B2 ,
(4.2)
where w1C is a Z2 -valued gauge field for ZC2 symmetry. Here, since we are using the action of C,
we are coupling to Gtotal ⋊ZC2 -bundles which allows for the non-trivial ZC2 gauge fields. Since C
acts non-trivially on Γg (we assume it acts trivially on Gglobal ×Spin(4)), the cohomology of the
Γg -fluxes must be modified in the presence of C-background gauge fields: the Γg cohomology
is valued in a ZC2 -twisted cohomology which takes values in (torsion subgroups of) ZC which
is an associated ZC2 -bundle over spacetime where C acts on the Z fiber with the natural Z2
involution, see e.g. [66, 67]. The ZC2 bundle is characterized by a connection w1C ∈ H 1 (M ; Z2 )
which is a background field for the ZC2 symmetry. We will summarize some of the effects of
turning on a non-trivial w1C ; see [66, 67]for more details.
Let us restrict our discussion to the case where Γg = Z2N . Since Z2 ⊂ Γg is invariant
under the action of ZC2 , the action of the ZC2 twisted Bockstein map β̃2 is related that of the
ordinary Bockstein map by
β̃2 (x) = w1C ∪ x + β2 (x) ,
x ∈ H ∗ (M ; Z2 ) ,
(4.3)
where β2 is the Z2 Bockstein map, similar to the structure of P SO(n) gauge bundles as in
[65]. This implies that, for Γg = Z2N , that the anomaly can only be of the form
Z
(4.4)
A = πi σI w2 (T M ) ∪ β2 (B2 ) + w1C ∪ B2 ,
where the two terms correlated and B2 is the flux for Z2 ⊂ Z2N .
In summary, if the Ggauge matter representations are real and the path integral is anomalous under ϕ
b = ϕ ◦ Wglobal ◦ C, then the theory has the w1C ∪ B2 ∪ w2 (T M ) anomaly term
(4.2). The relations (4.3), then imply tha tthe theory also the correlated Z2 -valued B2 ∪ w3
anomaly. The same statement holds for the case when Γg is a product of such finite abelian
groups.
– 31 –
Now consider the case where the Ggauge representation Rgauge is pseudoreal/symplectic.
After Higgsing to U (1), Wgauge is matched by C ◦ U where U generates a Z4 chiral symmetry
with U 2 = (−1)F . Since the fermions transform in a symplectic representation of the gauge
group, the U 2 operation is part of the gauge group. Again, due to the twisting of the U (1)gauge
b
cohomology by ϕ
b (i.e. O(2) = U (1)gauge ⋊Zϕ
2 still acts on fermion zero modes), the 5d anomaly
theory associated with the ϕ
b = ϕc.c. ◦ C ◦ U mapping torus is of the form
Z
A ⊂ iπ σI w2 (T M ) ∪ (β2 B2 + w1ϕb ∪ B2 ) .
(4.5)
In such theories, there can additionally be a seperate anomaly for the axial symmetry.
4.2
Anomalies in Abelian Gauge Theory
Because anomalies are independent of continuous deformations, one can probe the anomalies
involving 1-form global symmetries in non-abelian QCD-like theories by deforming the theory
so that it flows to pure abelian gauge theory and studying the charges of the IR line operators.
This was discussed in e.g. [24, 30]. The anomalies can then be matched by identifying the
UV 1-form global symmetries with discrete subgroups of the electric- and magnetic- 1-form
global symmetries of the IR theory which have their own mixed anomaly
r Z
i X
(m)
(e)
B2,I ∧ dB2,I .
A⊂
(4.6)
2π
I=1
Here we will show how we can use our discussion of C-symmetry to compute the anomalies
in the abelian gauge theory complementary to the fractionalization technique.
Consider the same setup as earlier: Ggauge theory with Nf fermions in representations
Rgauge so that the fields couple to Gtotal -bundles of the form
Gtotal =
GU V × Spin(4)
Ggauge
×
,
Γg
Z2
(4.7)
where here GU V is the 0-form global symmetry group of the fermions and Γg is the UV 1form center symmetry. We now add an adjoint-valued scalar field Φ, and consider how the
anomalies are matched in the low-energy theory where hΦi ∼ v 6= 0, breaking Ggauge → U (1)r .
One option is to include a Yukawa interaction with coupling λY Φψψ to the fermions, so the
fermions get a mass mψ ∼ λv in the low-energy theory. In that case, the anomalies of the
UV theory can be matched by fractionalization of the line operators as in [30]. We will here
instead discuss how the anomalies are matched in the theory with λY → 0, where the IR
theory is a U (1)r gauge theory with a collection of massless fermions {ψi }⊕Nf with charge
vectors ~qi . We will take the spectrum to be invariant under
C : {~
qi }⊕Nf 7−→ {~q̃i = −~qi }⊕Nf ,
(4.8)
up to permutation. As a consequence of C-symmetry there is no gauge anomaly which means:
X
qi,a qi,b qi,c = 0 , ∀a, b, c , ~qi = (qi,1 , qi,2 , ..., qi,r ) .
(4.9)
i
– 32 –
We can couple the fermions of the IR theory to Gtotal -bundles where
Gtotal =
GU V × Spin(4)
Ggauge
×
.
Γg
Z2
(4.10)
(1)
Here, the theory again has a Γg electric 1-form global symmetry which turns on Γg flux
along U (1)Γg ⊂ U (1)r which is generated by QΓg . Additionally, we will restrict our attention
to (in general sub-group of) the 0-form global symmetry group GU V which matches that of
the U V global symmetry group. Because of the identification of (−1)F ∼ (−1U V ), we can
consistently put the theory on a non-spin manifold.
Now we can proceed as before by putting the theory on CP2 and turning on Z2 ⊂ Γg
B2 and w2 (GU V ) flux. Let us write the charge pairing of ψi with the total fractional flux
along U (1)Γg and SU (2)U V as Qi . We can then apply the Atiyah Singer index theorem to
determine the number of zero modes I of the Dirac operator in the flux background to be
I=
X 4Q2 − 1
i
i
8
.
(4.11)
The map ϕ
b that preserves the bosonic background and acts non-trivially on the fermion zeromodes is ϕ
b = ϕc.c. ◦ WU V ◦ C or ϕ
b = ϕc.c. ◦ WU V ◦ C ◦ U as we have previously discussed,
where WU V is the Weyl transformation in an embedded SU (2)global ⊂ GU V that inverts the
ϕ
b
discrete flux w2 (Γ). As before, ϕ
b acts on the path integral measure as D[Ψ] 7→ D[Ψ] (−1)I/2
So the anomaly indicator is the integer σI = I2 mod2 .
As discussed in the previous subsection, non-trivial action of ϕ
b on the path integral
indicates that the anomaly contains a term
Z
A1 = πiσI w1ϕb ∪ w2 (T M ) ∪ B2 ,
(4.12)
which implies the presence of an anomaly
A0 = πiσI
4.2.1
Z
B2 ∪ w3 (T M ) .
(4.13)
Example: SU (2) with 2nf Weyl Fermions in the Adjoint Representation
The fields, including the added adjoint scalar, couple to the total bundle
Gtotal =
SU (2nf )U V × Spin(4)
SU (2)gauge
×
.
Z2
Z2
(4.14)
Giving an expectation value to the added scalar Higgses SU (2)gauge 7→ U (1)gauge and the theory RG flows to QED in the IR. The adjoint fermions decompose as 3⊗2 7−→ (2 ⊕ 0 ⊕ (−2))⊗
(1)
2. The Z2 electric 1-form symmetry can be coupled to a background flux B2 and the
SpinSU (2nf ) (4) structure leads to fractional flux along U (1)gauge and SU (2)global ⊂ SU (2nf )global
⊕nf
. The calculation of the index of the Dirac operator
with Qi given {Qi } = ± 32 , ± 21 , ± 12
– 33 –
P 4Q2i −1
and the anomaly indicator are the same as before: I =
= 2nf and σI = nf .
i
8
The mapping torus has ϕ
b = ϕ ◦ WU V ◦ C where ϕ is charge conjugation on CP2 and
WU V ∈ SU (2nf )U V , with the anomaly now given by
Z
A ⊂ πi nf B2 ∪ (w3 (T N ) + w1C ∪ w2 (T N )).
(4.15)
4.2.2
Example: SU (2) with Weyl Fermion in 4-Representation
Now let us consider the example of Ggauge = SU (2)gauge with a single Weyl fermion in the
j = 3/2 representation. The fermion in this theory couples to the total bundle
Gtotal =
SU (2)gauge × Spin(4) × Z10
,
Z2 × Z2
(4.16)
where here we suppress the Z10 discrete chiral symmetry. We deform this theory by coupling
to an adjoint scalar field and condensing it so that gauge symmetry is broken SU (2)gauge →
U (1)gauge and we flow to an abelian gauge theory in the IR. There the chiral symmetry is
enhanced
Gtotal −→
SU (2)gauge × Spin(4) × Z20
.
Z2 × Z2
(4.17)
With respect to U (1)gauge the fermion decomposes into four Weyl fermions ψqi with charges
4 7−→ qi = ±3, ±1 .
(4.18)
In the IR, there is a SpinC (4) structure and the theory can be put on CP2 by turning on a
fractional flux along U (1)gauge . Additionally, the map ϕ
b of the un-Higgsed theory which is
given by ϕ
bU V = ϕc.c. ◦Wgauge ◦WU V is matched in the Higgsed theory ϕ
bIR = ϕc.c. ◦W ◦C ◦U ◦g
and C, U, g act as
C : ψqi 7−→ ψ−qi
,
2πi
U =e 4 1 ,
2πi
g = e 4 Qgauge ,
(4.19)
where U generates a Z4 ⊂ Z20 chiral rotation and g generates a Z4 ⊂ U (1)gauge gauge
transformation.
We can now compute the index in this background
I=
X 4Q2 − 1
i
i
8
=2
=⇒
σI = 1 ,
and see that there is an anomaly
Z
A ⊃ πi w2 (T M ) ∪ w3 + w1ϕb ∪ w2 (T M ) .
(4.20)
(4.21)
Note that the chiral symmetry Z20 has additional ‘t Hooft anomalies both with itself and
with the SpinC (4) structure.
– 34 –
4.3
Anomalies and Higgsing with Adjoints Fermions
The anomalies must additionally match if we partially Higgs the gauge group by an adjoint
Higgs field,
Ggauge →
G′gauge × U (1)
.
Z
(4.22)
The Z(Ggauge )(1) global symmetry is unbroken by the vev and arises in the IR theory as
Z(Ggauge ) ⊂ Z(G′gauge ) × U (1) .
(4.23)
If there is no Yukawa coupling, the anomaly matching is as discussed in the previous subsection. With Yukawa couplings the IR theory can have a U (1)(1) emergent 1-form global
(1)
symmetry and there are two cases. If Z(Ggauge ) is in the U (1) factor, then Γg embeds into
the emergent U (1)(1) global symmetry and the IR U (1)(1) symmetry matches the anomaly
(1)
(1)
of Γg . If Z(Ggauge ) → Z(G′gauge ) then the anomaly of Γg will instead be matched by an
anomaly of Z (1) , the 1-form center symmetry of the IR model.
As an example, consider SU (Nc ) gauge theory with 2nf adjoint fermions, where
(U V )
Gtotal =
SU (Nc ) SU (2nf ) × Spin(4)
×
.
Z Nc
Z2
(4.24)
Now we can consider Higgsing the theory by an adjoint Higgs field which breaks the gauge
group
SU (Nc ) 7→
SU (Nc − 1) × U (1)
= U (Nc − 1) .
ZNc −1
(4.25)
If we additionally couple the fermions to the Higgs field via a Yukawa coupling, we break
SU (2nf ) → Sp(nf ) and the resulting IR theory will have 2nf fermions in the adjoint representation of U (Nc − 1) which transform faithfully under the group
(IR)
Gtotal =
U (Nc − 1) Sp(nf ) × Spin(4)
×
.
U (1)
Z2
(4.26)
Here we see that the IR theory has an emergent U (1)(1) 1-form global symmetry where
(1)
(1)
ZNc = ZNc ⊂ U (1)(1) . For the case of Nc odd, since Z(SU (Nc )) = ZNc , there is no mixed
(1)
anomaly with Z2 along the entire RG flow. However, since the IR has an emergent U (1)(1)
(1)
global symmetry which contains U (1)(1) ⊃ Z2 , there is the possibility for this emergent
symmetry to be anomalous. Indeed, as we know from the previous section, SU (Nc − 1) QCD
does have a mixed anomaly when Nc = 4nc + 3. For the case of Nc is even, we know that
there is an anomaly for Nc = 4nc + 2. Again, the IR theory has an emergent U (1)(1) global
(1)
symmetry, which contains Z4nc +2 as a subgroup. The κΓg T anomaly of the UV theory is then
matched by κΓg T of the IR theory by a mixed U (1)(1) -gravitational anomaly.
– 35 –
Symmetry Matching in the SSB phase: Nf Adjoint Weyl Fermion QCD
5
Four-dimensional gauge theories have a rich variety of possible IR phases, including IR-free
electric (e.g. if not UV asymptotically free), interacting conformal field theory, exotic IR-free
duals, gapped (including TQFTs), and spontaneous symmetry breaking (SSB). While it is
an unsolved problem to analytically determine the IR phase of general 4d gauge theories
(see e.g. [68] for examples and discussion in the context of supersymmetric gauge theories),
the IR phase is partially constrained by symmetries, ’t Hooft anomaly matching, and the
a-theorem. For example, non-zero ’t Hooft anomalies can rule out IR gapped phases22 and
SSB is constrained by the theorems of [69].
In this section, we will discuss some aspects of symmetries, ’t Hooft anomalies, and
anomaly matching in the IR SSB nonlinear sigma model phase. We will recall that ’t Hooft
anomalies must also match for the spontaneously broken symmetries and how this condition
determines the coefficient of the Wess-Zumino-Witten (WZW) [12, 70] interaction in the lowenergy effective theory. We will illustrate anomaly matching for both standard anomalies
and those associated with SpinG (4) structure in the context of a wide class of theories. See
e.g. [23, 26, 71–78] for a partial list of some recent works on aspects of nonlinear sigma models
and their symmetries and defects.
The class of theories that we will discuss in this section are general Ggauge theories
f =1...N
with Nf Weyl fermions, λα=1,2 f , in the adjoint representation of Ggauge . The theories are
asymptotically free for Nf ≤ 5 and are expected to be in an interacting CFT phase for
Nf∗ ≤ Nf ≤ 5, where the lower bound Nf∗ of the conformal window is not known (and it
could be Ggauge dependent). It is also not known for which Nf values the theories have
SSB. The theories with Nf = 1 are N = 1 supersymmetric Ggauge pure Yang Mills and the
IR phase of is well-understood from a variety of SUSY-based constraints and perspectives:
there are isolated gapped vacua with h Trλλi =
6 0 which leads to SSB of the (discrete) chiral
23
symmetry. The theories for Nf > 1 are non-supersymmetric and less well-understood; this
class of theories has been studied for various values of Nf , and for various choices of Ggauge
(including on the lattice, especially for Nf = 2, i.e. one Dirac flavor, and Ggauge = SU (2)).
See e.g. [8, 9, 23, 25, 31, 81–91] and references therein for a partial list of the large literature
on this class of theories. Some of these works propose novel, IR dual theories, and related
analysis of ’t Hooft anomaly matching constraints. We will not discuss any novel IR duals.
Here we will simply assume the spontaneous symmetry breaking IR phase associated
with the expectation value hO(f g) i ∼ Λ3 6= 0 of the gauge and Lorentz invariant operator
(f g)
O(f g) ≡ Trλα λβ ǫαβ , and discuss how the symmetries and ’t Hooft anomalies are consistent
with that possibility. These checks neither prove nor disprove that the SSB phase is physically
realized, and indeed we expect that the theory is IR-free for Nf > 5 and that the IR CFT
phase conformal window includes Nf = 5 and perhaps also Nf = 4 and Nf = 3.
22
Some anomalies rule out symmetry preserving gapped TQFTs, and others do not; see [8, 9] and therein.
The ’t Hooft anomalies for the broken discrete symmetries are matched by the TQFTs on the domain
walls between the vacua; see [79, 80] and references therein.
23
– 36 –
The vacuum manifold of the theories, with our assumed SSB, consists of h∨
Ggauge (with
∨
adjoint index e.g. hSU (Nc ) = Nc ) disconnected copies of a G/H = SU (Nf )/SO(Nf ) nonlinear
sigma model of Nambu-Goldstone boson pions. The h∨
Ggauge copies are associated with the
spontaneous symmetry breaking of a discrete global symmetry for simply connected Ggauge
theories for all Nf .24 This was discussed in detail in [8] for Ggauge = SU (2) and Nf = 2. Our
discussion will extend some of the analysis and results of [8] to general Ggauge and Nf .
5.1
Ggauge Theory with Nf Massless, Weyl adjoints: UV symmetries
The (simply connected 25 ) Ggauge theory with Nf massless Weyl fermions in the adjoint has
Gtotal =
SU (Nf ) × Spin(4) × Z2n
Ggauge
×
.
Γg
Z Nf × Z 2
(1)
(5.1)
(0)
Γg = Z(Ggauge ) is the 1-form center symmetry. Z2n , with generator g and n ≡ Nf h∨
Gauge ,
26
is the anomaly free subgroup of the ABJ-anomalous U (1)r symmetry. The ZNf quotient
in (5.1) identifies the ZNf center of SU (Nf ) with a subgroup of Z2n generated by g2n/Nf ; the
Z2 quotient identifies gn = (−1)F .
The ’t Hooft anomaly coefficients κ can be computed in the asymptotically free limit
from the anomaly of free fermions. The perturbative anomalies, in the presence of background
b M ), along
gauge fields, are obtained by descent from the anomaly polynomial I = TreF/2π A(T
24
E.g. for Nf = 1 and simply connected Ggauge , this gives the Witten index Tr(−1)F = h∨
Ggauge . Nonsimply connected Ggauge , e.g. SO(3) (see sect. 2.7), can be obtained from the simply connected cases by
(1)
gauging subgroups of the Γg one-form global center symmetries. Then the h∨
Ggauge copies of the vacuum
manifold are then generally not related by SSB and indeed are generally inequivalent, e.g. with different
discrete gauge symmetries.
25
For non-simply connected Ggauge , the Z2n symmetry here is reduced to a subgroup. Non-simply connected
cases (e.g. SO(N )) can be obtained from the simply connected ones by (e.g. Spin(N )) by gauging subgroups
of Γg . For example, as discussed in [92] for Nf = 1 and G = SU (2) ∼
= Z4
= Spin(3), there is a Z2Nf h∨
SU (2)
F
∼
chiral symmetry that is spontaneously broken to Z2 = (−1) , leading to two equivalent vacua which are
related to each other by θY M → θY M + 2π. For the Nf = 1 non-simply connected case of G = SO(3), one
(1)
gauges ΓSU (2) . Then the periodicity is instead θY M ∼
= θY M + 4π and the U (1)A symmetry is instead explicitly
broken to Z2 ∼
= (−1)F and there is no spontaneous symmetry breaking. There are still two vacua, but they
are physically inequivalent: one has an unbroken Z2 gauge theory and the other does not.
26
U (1)r assigns charge +1 to each Weyl fermion, with generator Uφ : λfα → eiφ λfα . This symmetry is broken
by instantons for general φ, and the anomaly free Z2n subgroup is generated by group element g = Uφ=π/n .
The ABJ anomaly implies that the U (1)A transformation λfα → eiφ λfα shifts θY M → θY M + 2Nf h∨
Ggauge φ,
∨
is
the
dual
Coxeter
number
(normalized
to
h
=
N
).
Since
we
assume
that
G
where h∨
c
gauge is
Ggauge
SU (Nc )
(1)
simply connected, instantons preserve θ → θ + 2π (when the background gauge field for Γg is B2 = 0) and
the unbroken Z2n ⊂ U (1)A symmetry has n = Nf h∨
. Non-simply connected
Ggauge , with g = Uφ=π/Nf h∨
G
gauge
cases, e.g. SO(Nc ) instead of Spin(Nc ), can be constructed from this discussion of the simply-connected cases
(1)
by gauging appropriate subgroups of the one-form symmetry Γg , the center of Ggauge .
– 37 –
Ggauge
SU (Nc )
Sp(Nc )
E6
E7
Spin(2nc + 1)
Spin(4nc + 2)
Spin(4nc )
Γg
Z Nc
Z2
Z3
Z2
Z2
Z4
Z2 × Z2
κrΓ
Nc −1
2Nc
Nc
4
2
3
1
2
1
2
2nc +1
8
nc
1
4 , 2
Table 1. In this table we give the anomaly coefficients κrΓ for various choices of Ggauge. For Ggauge =
Spin(4nc) the two anomaly coefficients correspond to the two terms in the anomaly polynomial A ⊃
R
R
2πinc
z ∪ P(B2L + B2R ) + 2πi
z ∪ B2L ∪ B2R .
4
2
with additional anomaly terms for the torsion parts. The anomaly theory terms include
Z
Z
Ar
1
1
A ⊃ κSU (Nf )3 CS5 [SU (Nf )] + κSU (Nf )2 r
∧ ch2 [SU (Nf )]
2πi
3
2π
Z
Z
Z
Ar p1 [T M ] κr3
z
(5.2)
+ κr,T T
∧
+
∪ βz ∪ βz + κr,Γ z ∪ P(B2 )
2π
24
6
2n
Z
Z
1
1
+ κΓ,T
B2 ∪ w3 (T M ) + κT,T
w2 (T M ) ∪ w3 (T M ).
2
2
We denote the Z2n ⊂ U (1)r by r e.g. its background gauge field is Ar and as in [8] we also
Fr
r
denote A
2π → z/2n and 2π → βz where z is a 1-cochain with values in Z2n and β is the Z2n
Bockstein map (β = δ/2n, where 2n = 2Nf h∨
Ggauge ). The anomaly coefficients are
• κSU (Nf )3 = TrSU (Nf )3 = |Ggauge |. This is non-torsion for Nf ≥ 3 (since SU (Nf ≥ 3)
has a cubic Casimir). For Nf = 2, this ’t Hooft anomaly reduces to a Z2 torsion component corresponding to the ’t Hooft anomaly version of the original SU (2) anomaly [28],
with coefficient |Ggauge | mod 2.
• κSU (Nf )2 r = TrSU (Nf )2 Z2n = |Ggauge |.
• κr = κr3 = Nf |Ggauge |.
• κr,Γ is related to the anomalous phase under θ → θ + 2π when B2 6= 0. The κr,Γ
coefficient, for all Nf , coincides with that given as in Table 1 of [93], where we replace
dθ/2π → z. The results are reproduced here in Table 1.
• κΓ,T is as computed in earlier sections, e.g. for SU (Nc ), we found κΓ,T = δNc ,4Z+2 .
• κT,T = 0 for this class of theories.
The discrete symmetry anomaly coefficients κSU (Nf )2 r and κr and κr3 are examples of anomalies of discrete symmetries Z2n symmetries with gn = (−1)F ; such anomalies were discussed
– 38 –
in [49, 94]. The κr and κr3 terms are packaged into anomalies associated with cobordism
∨
group ΩH
5 = Za × Zb where the modularity a and b depends if n = Nf hGgauge is a multiple of
2 and /or a multiple of 3, e.g. for n = 0 mod 6 then a = 24n and b = n/6 – see [49, 94].
5.2
Ggauge Theory with Nf Massless, Weyl adjoints: Possible IR Phase with SSB
For Nf = 1, the theory is N = 1 SUSY Ggauge pure Yang-Mills, and it is known from SUSY
exact results that the theory has SSB by the gaugino condensation expectation value of the
Lorentz and gauge invariant operator O ∼ Trλα λβ ǫαβ , which is the bottom component of
→ Z2 ,
the chiral glueball superfield operator S ∼ TrWα W α . The SSB breaks Z2h∨G
gauge
where if Z2h∨G
gauge
h∨
has generator g, then g Ggauge is identified with Z2 = (−1)F . This leads to
h∨
Ggauge vacua, which are physically equivalent for simply connected Ggauge . These vacua have
unbroken supersymmetry and are trivially gapped for Ggauge simply connected (otherwise they
can have discrete gauge theories, as mentioned in Footnote 25), corresponding to Tr(−1)F =
h∨
Ggauge in these theories. The multiple vacua are connected by BPS domain walls with 3d
TQFTs on their worldvolumes, see e.g. [80] and references therein.
For Nf > 1, we consider the possibility that the IR theory is also in the SSB phase with
(f
hO g) i ∼ Λ3 for the operator O(f g) = Trλfα λgβ ǫαβ . This spontaneously breaks the continuous symmetry as SU (Nf ) → SO(Nf ) and spontaneously breaks the discrete symmetry27 as
→ Z2 = (−1)F , so the vacuum manifold consists of multiple disconnected copies
Z2Nf h∨G
gauge
of the SU (Nf )/SO(Nf ) NG boson target space. This breaking does not lead to Nf h∨
Ggauge
copies of SU (Nf )/SO(Nf ): because g
2h∨
Ggauge
is equal to the generator of the ZNf center of
2h∨
Ggauge
maps a vacuum in a SU (Nf )/SO(Nf ) to another vacuum
SU (Nf ), it follows that g
in the same SU (Nf )/SO(Nf ). The SSB to Z2 then leads to a vacuum manifold consisting of
h∨
Ggauge disconnected copies of SU (Nf )/SO(Nf ) non-linear sigma model for all Nf :
∨
SU (Nf ) ⊕hGgauge
.
Mvac =
SO(Nf )
(5.3)
For Nf = 2, assuming SSB, the vacuum manifold consists of h∨
Ggauge disconnected copies of
1
∨
SU (2)/SO(2) ∼
CP
,
as
in
the
h
=
2
case
discussed
in [8]. For Nf = 3, assuming
=
Ggauge =SU (2)
SSB, the space SU (3)/SO(3) is the 5d Wu manifold, which is discussed further e.g. in [26].
The fact that the vacuum manifold (5.3) consists of h∨
Ggauge disconnected components for
all Nf is nicely compatible with RG flows continuously deforming from Nf > 1 to Nf = 1
by adding mass terms and RG decoupling of the heavy flavors. In the UV gauge theory,
one can add mass terms for the fermions, ∆LU V = m(f,g) O(f g) where O(f g) = Trλfα λgβ ǫαβ
is the operator that gets a vev in the SSB phase. In the IR pion theory, non-zero m(f,g)
lifts the SU (Nf )/SO(Nf ) vacuum degeneracy; some of the pions become pseudo NG-bosons.
In particular, choosing m(f,g) to give non-zero mass to Nf − 1 of the Weyl fermions gives a
27
As discussed in [9], ’t Hooft anomalies obstruct an IR phase where the discrete symmetry is unbroken and
only acts nontrivially on gapped states (it acts trivially on the massless NG bosons). See also [95].
– 39 –
RG flow that reduces to the Nf = 1 case in the IR. The h∨
Ggauge disconnected copies of the
SU (Nf )/SO(Nf ) sigma model is indeed nicely consistent with the h∨
Ggauge isolated vacua of
the Nf = 1 case upon deformation by adding mass terms to Nf −1 of the Weyl fermions: doing
so in the low-energy sigma model lifts the SU (Nf )/SO(Nf ) vacuum degeneracy and leads to
an isolated vacuum for each SU (Nf )/SO(Nf ) sigma model theory, correctly reproducing the
h∨
Ggauge vacua of the Nf = 1 low-energy theory. See [9] for details in the case discussed there.
The continuous part of the vacuum manifold, the G/H ≡ SU (Nf )/SO(Nf ) sigma model,
has been much-studied in connection with SO(Nc ) QCD with Nf fundamental fermions. The
sigma model admits solitonic particles, and also lines, which were matched to to the baryons
and flux tubes of SO(Nc ) in [96]; see also [97, 98] for further consideration in connection with
SO(Nc ) QCD flux tubes and [99, 100] for the flux tubes in connection with SU (Nc ) gauge
theories with adjoint fermions. The π3 and π2 homotopy groups for solitonic particles and
vortices, respectively, have some Nf dependence for low Nf : π3 (MNf >3 ) = Z2 , π3 (MNf =3 ) =
Z4 , π3 (MNf =2 ) = Z; π2 (MNf ≥3 ) = Z2 , π2 (CP1 ) = π3 (CP1 ) = Z. See [8, 72, 74] for discussion
and clarification of the physics and symmetry associated with the Hopf number π3 (CP1 ) = Z,
related to the fact that it is not independent of the π2 (CP1 ) quantum number.
In the remainder of this subsection, we will discuss the sigma model and symmetry and
anomaly matching for Nf ≥ 3. Unlike the case of SO(Nc ) with Nf fundamentals, where
SSB might occur for arbitrary Nf by taking Nc sufficiently large, for the case of Ggauge with
adjoint Weyl fermions the asymptotically free cases have Nf ≤ 5; the theories with Nf = 5
are likely in an interacting CFT phase rather than a SSB, and that is perhaps also the case
for Nf = 4 and Nf = 3. We will here anyway assume SSB, so the remainder of this subsection
is perhaps an academic exercise in how symmetries could be matched in IR phases that are
not physically realized. The exercise also serves as a warmup for the Nf = 2 case in the next
subsection, where SSB is a quite plausible scenario.
The G/H ≡ SU (Nf )/SO(Nf ) sigma model for Nf ≥ 3 admits a WZW interaction [12, 70]
in the low-energy effective theory for the NG boson pions with coefficient k determined by
matching the κSU (Nf )3 = |Ggauge | ’t Hooft anomaly:
−SW ZW
e
2πi∆κG3
=e
R
1 ∗
φ y5
N5 4
,
so
∆κG3
I
1 ∗
φ y5 ∈ Z
Z5 4
(5.4)
with M4 = ∂N5 and the quantization condition is needed for the WZW term to be well-defined
under the ambiguity in N5 → N5′ with closed Z5 = N5 − N5′ . The H 5 (G/H, Z) representative
H
y5 is normalized such that Z5 y5 ∈ Z for general closed Z5 , so the 14 is interesting – see [26]
for a nice, recent discussion. For our present case28 of G/H = SU (Nf )/SO(Nf ) it was
argued in [26] (in the context of SO(Nc ) gauge theory with fundamental fermions) that the
quantization condition is satisfied for all k if M4 has a spin structure, w2 (T M ) = 0: they
28
For the case of SU (Nc ) gauge theory with Nf massless Dirac fundamental fermions, each flavor contributes
2 to the TrSU (Nf )3L−R = 2Nc ’t Hooft anomaly. For Nc odd, the IR theory must have a spin structure,
w2 (T M ) = w2 (T N5 ) = 0, because e.g. the baryon associated with π3 (SU (Nf )) is a fermion. This ensure that
R
the quantization condition is satisfied because then Z5 =spin φ∗ y5 ∈ 2Z [26, 71].
– 40 –
R
show that then Z5 =spin φ∗ (y5 ) ∈ 4Z.29 Recall that k affects the statistics of solitons in the
IR theory [12, 96] and that p-form symmetry charged objects of the UV theory can arise
as skyrmionic solitons30 of the φ field in the IR, with their codimension p + 1 symmetry
operators associated with πd−p−1 (G/H). The more general, proper formulation of pions and
WZW terms in terms of generalized differential cohomology theories [26, 71, 93, 101–103]
shows that the homotopy and cohomology groups should be replaced with corresponding
bordism classes. This formulation also incorporates torsion anomalies, and generalizations of
the solitons and defect operators.
In the present context, k ≡ ∆κG3 = κSU (Nf )3 = |Ggauge |, so
−SW ZW
e
R
2πi|Ggauge | N
=e
1 ∗
φ y5
5 4
,
so
|Ggauge |
I
φ∗ y5 ∈ 4Z.
(5.5)
Z5
The WZW 5-form y5 in (5.5) includes, as usual, additional terms involving the background
gauge fields for the SU (Nf ) global symmetry and also the background gauge field Ar for the
Z2n global symmetry. For the theories with ordinary spin structure, we restrict to M4 with
spin structure, which can be extended to N5 so Z5 = N5 − N5′ = spin. It then follows from
R
the result of [26] that Z5 =spin y5 ∈ 4Z so the quantization condition is satisfied for all Ggauge .
We will discuss the 14 again below in the case with generalized SpinSU (Nf =2) (4) structure. For
Nf = 2, the TrSU (Nf )3 ’t Hooft anomaly reduces to the Witten SU (2) anomaly (see below).
For Ggauge theories with a non-trivial center Γg , we expect that the one-form center
(1)
(1)
symmetry Γg is not spontaneously broken, so the Wilson loops with non-trivial Γg representations will have area (rather than perimeter) law. For the case of Nf = 1, the IR theory
consists of the h∨
Ggauge isolated (supersymmetric) vacua with mass gap, and the Γg symmetry
is unbroken in each of the vacua. See e.g. [79, 104] and references therein for some discussion
of the interplay between the vortices (quanta of the line operators) and the domain walls in
the Nf = 1 case. For Nf = 1, the vortices do not arise as solitons of the IR theory.
For the Nf > 1 theories, on the other hand, the IR theory in the SSB phase has a oneform global symmetry associated with π2 (SU (Nf )/SO(Nf )); this symmetry is independent
(1)
of Ggauge , unlike the Γg center symmetry. For Nf = 2, the SU (2)/SO(2) ∼
= CP1 theory has
1
(1)
π2 (CP ) = Z, corresponding to a U (1) one-form global symmetry of the IR theory [79].
(1)
The U (1)(1) IR symmetry could arise as an enhanced version of the Γg symmetry of the
UV theory in cases31 where the center symmetry is Γg = ZN . Assuming that the ZN center
of the UV theory indeed maps to a subgroup of U (1)(1) , we can ask if the embedding is to
identify the basic, ZN charge 1 string of the UV theory with the basic U (1) charge 1 string
R
R
As discussed in [26] it is easily argued that Z5 φ∗ y5 = Z5 w2 (T N ) ∪ w3 (T N ) mod 2, so w2 (T N ) = 0
R
∗
easily implies Z5 =spin φ (y5 ) ∈ 2Z; the additional factor of 2 requires further analysis in [26].
30
Non-trivial topology of the vacuum manifold is of course not sufficient to ensure stability (e.g. a rubber
band can unwind off the bottom of a wine-bottle potential). Assumptions about higher derivative terms are
needed to ensure stability and also to evade Derrick’s theorem.
31
For the case of Ggauge = SO(4nc ), where Γg = Z2 × Z2 , one of the Z2 factors can be embedded in the
(1)
U (1)IR and the other could require being added separately to the IR description.
29
– 41 –
of the IR theory. We will assume that this is the case. In other words, we assume that the
background gauge field B2IR that couples to n∗ (ω) in the IR is identified with the integer
(N )
lift of the UV B2 background gauge field. For Nf ≥ 3, the IR nonlinear sigma model has
π2 (SU (Nf )/SO(Nf )) = Z2 one-form global symmetry, so it can only directly match the UV
center symmetry Γg for Ggauge with Γg = Z2 .32
For the particle-like solitons, associated with π3 (G/H), we recall that π3 (SU (3)/SO(3)) =
Z4 and π3 (SU (Nf > 3)/SO(Nf > 3)) = Z2 . In the context of SO(Nc ) QCD, this Z2 was
identified with baryon number [12, 96], and this nicely fits with how the coefficient of the
WZW term affects the spin / statistics of the solitons: the κSU (Nf )3 = Nc , WZW term implies that the π3 (G/H) soliton has (−1)F = (−1)Nc , befitting its identification with a baryon
B ∼ ψ Nc . In the context of Ggauge theory with adjoints, on the other hand, the WZW term
κSU (Nf )3 = |Ggauge | suggests that the π3 (G/H) soliton has (−1)F = (−1)|Ggauge | and the identification of such a soliton with a composite from the Ggauge gauge fields and adjoint fermions
of the UV theory is less clear. It was suggested in [99, 105] that the Z2 in the case of adjoint
fermions has an exotic interpretation that we will mention further in the next section.
5.3
Symmetry Matching for Nf = 2 in the CP1 Sigma Model (on Spin Manifolds)
For Nf = 2 massless Weyl fermions in the adjoint representation, the fields couple to
Gtotal =
Ggauge
SU (2)R × Spin(4) × Z2n
×
,
Γg
Z2 × Z2
(5.6)
where we denote SU (Nf = 2)global ≡ SU (2)R . The SU (2)gauge case was discussed in [23, 89],
and we will review some of their discussion and extend to general (simply connected) Ggauge .
In this subsection, we consider the theory on M4 with spin structure on M4 with w2 (T M ) = 0.
The case with SpinSU (2)R (4) structure on M4 = CP2 will be discussed in the next subsection.
As in [23], one can gain some intuition about the IR phase by starting with the N = 2
supersymmetric pure Yang-Mills theory, which also has Nf = 2 Weyl fermions in the adjoint
(but also with an additional complex scalar Φ in the adjoint representation), and the deform
the N = 2 theory by a supersymmetry-breaking mass term for the (unwanted) scalars, ∆V ∼
(1)
M 2 Tr|Φ|2 . For the SU (2)gauge case this analysis in [23] nicely leads to unbroken Γg =
(1)
Z2 (along with confinement of the photon) from monopole / dyon condensation [106] with
1
∼
h∨
SU (2) = 2 copies of the SU (2)R /U (1)R = CP sigma model arising via a linear sigma model
with the photon, with the sigma model scalars coming from the monopole hypermultiplet. The
detailed analysis for analogously deforming from N = 2 for more general Ggauge = SU (Nc ) is
more subtle and required a deeper understanding of the Coulomb branch solution of N = 2
theories for Nc > 2 [107, 108] and will be discussed in [109]. We will here simply assume the
confinement and chiral symmetry breaking scenario in the IR for Nf = 2 Weyl adjoints for
32
For the case where the UV theory is instead Spin(Nc ) with Nf fermions in the fundamental, the UV theory
(1)
instead has a Z2 , which nicely matches the solitonic symmetry associated with π2 (SU (Nf )/SO(Nf )) = Z2
solitonic symmetry for Nf > 2. This theory is generally not bosonic, and then the spin / statistics of lines is
generally not a well-defined, scheme-independent quantity.
– 42 –
all Ggauge , and discuss how the anomalies are then matched in the IR nonlinear sigma model.
We will review the matching for the Ggauge = SU (2) case, following [23], and then generalize
the discussion to general Ggauge theories.
The κSU (Nf )3 ’t Hooft anomaly, which for Nf ≥ 3 is matched by the WZW term (5.5)
in the SSB phase, reduces to the Witten anomaly for Nf → 2: κSU (2)3 → |Ggauge | mod 2.
Correspondingly, the WZW term (5.5) reduces for Nf → 2 to a topological term related to
2
π4 (CP1 ) = Z2 [23, 96]; in fact, it is given by the bordism class Ω̃spin
4 (S ) = Z2 see [26, 71].
The Nf ≥ 3 WZW term that matches the ’t Hooft anomaly leads by transgression [26, 71] to
the Nf = 2 case of the π4 (SU (2)) = Z2 matching anomaly. In particular, for the CP1 sigma
model in the context of Ggauge = SU (2) with adjoints [23], and likewise for SO(Nc ) with
fundamentals [26], the SU (2)R Witten anomaly is matched in the CP1 sigma model by the
coefficient of a Z2 valued theta term corresponding to π4 (CP1 ) = Z2 . For the case of general
Ggauge with Nf = 2 Weyl fermions in the adjoint, we write the WZW term as
2
e−SW ZW = (−1)|Ggauge |[φ:M4 →S ]
(5.7)
which matches for the IR theory the SU (2)R Witten κWitten = |Ggauge | (mod 2) ’t Hooft
anomaly of the UV theory. As in [26], the notation [φ : M4 → S 2 ] denotes the reduced
e spin (S 2 ) = Z2 , which is integral valued so the term in (5.7) is only nonbordism class in Ω
4
trivial for |Ggauge | odd, reflecting the fact that the Witten anomaly is Z2 valued.
For QCD in the SSB IR phase, particle-like solitons of the IR non-linear sigma model,
associated with π3 (G/H), can be skyrmionic realizations of the baryons of the UV theory,
with π3 (G/H) charge identified with U (1)B (or a Z2 version for SO(Nc ) QCD with Nf
fundamentals) [96]. For the present case of G/H = SU (2)/U (1) ∼
= CP1 , the IR sigma-model
is essentially the Fadeev-Hopf model (which has quartic terms to ensure soliton stability),
whose Hopfion solitons are toroidal loops or knots [110]. The fact that the solitons have Hopf
winding number33 π3 (CP1 ) = Z initially suggests an associated U (1)H Hopf number global
symmetry, which is not present in the UV theory. As discussed in [72] in the related context
of the 3d θ angle, and in 4d in [23, 74] there is actually no U (1)H global symmetry of the
IR theory either: only a Z2 ⊂ U (1)H Hopfion number is conserved. As discussed in [74],
e spin (CP1 ) = Z2 , and the Z2 is part of a larger,
this agrees with the spin bordism class Ω
3
non-invertible global symmetry.
The spin / statistics of the Hopf soliton has been much-studied in the literature, see
e.g. [100, 111, 112] and references therein. Although there is no standard WZW to determine
the spin / statistics of the solitons as in the case of the baryonic skyrmions [12, 96], the π4 (S 2 )
Hopf-WZW term (5.7) associated with matching the SU (2)R Witten anomaly is essentially
a WZW term (and related by transgression). This Hopf-WZW term suggests that the basic
Z2 -charged Hopfion soliton has (−1)F = (−1)|Ggauge | , i.e. it is a fermion for |Ggauge | odd
and a boson for |Ggauge | even. This was discussed for Ggauge = SU (2) case in [23]. The
33
It is measured by the Chern-Simons number
model with gauge field a.
R
ada/4π 2 if one obtains SU (2)/U (1) via a U (1) linear sigma
– 43 –
works [81, 99, 100, 105], on the other hand, emphasized that Hopfions admit an alternative,
fermionic quantization [112, 113] which makes use of the noncontractible π1 configuration
space of the Hopfion solutions (the π3 (S 2 ) Hopf fibration solitons are essentially loops of the
π2 (S 2 ) vortices). These works have a speculative interpretation of the Z2 Hopfion 0-form
global symmetry as being associated with a conjectural exotic sector of the Hilbert space
in which the spin-charge relation is violated, rather than identifying the Z2 with (−1)F .
Solutions of equations can of course be in a non-trivial representation of the underlying
symmetry, e.g. the Hopfion toroidal solutions vs the full rotational symmetry. But we do not
(0)
expect that the IR Z2 global symmetry is related to spin-charge violation. For example,
consistency of the Donaldson-Witten twist of N = 2 SYM relies on non-violation of the
generalized spin structure. As far as we are aware, the well-studied BPS soliton spectrum of
N = 2 supersymmetric gauge theories does not exhibit violation of the spin-charge relation.34
The CP1 sigma model has a U (1)(1) one-form global symmetry [79] that assigns charge to
solitonic strings (these are the ANO vortices if CP1 is realized via a linear sigma model of a
U (1) gauge theory). The associated closed 2-form is ∗J (2) = φ∗ (ω), where the 2-form ω is the
CP1 Kähler form, and the charge of the solitonic winding configurations is the π2 (CP1 ) = Z
(1)
winding number. The UV theory has the one-form global symmetry Γg , where Γg is the
(1)
(1)
Ggauge center symmetry, and if Γg = ZN , it can be embedded in the apparent U (1)(1) of the
(N )
IR theory. With this identification, the background gauge field B2 for the UV symmetry
(1)
(1)
Γg = ZN is expected to35 couple in the IR CP1 sigma model’s functional integral as
Z
Z
2πi
(N )
IR
∗
∗
B2 ∧ φ (ω) → exp
B
∪ φ (ω) .
exp i
(5.8)
N M4 2
M4
(N )
(1)
Here, much as in the SU (2) discussion in [23], B2 ∈ H 2 (M4 , ZN ) is the ZN background
(N )
gauge field of the UV theory that can be embedded in U (1)(1) via B2IR = 2π
N B2 . The Kähler
2-form of the CP1 in the presence of the SU (2)R background gauge field ASU (2)R ≡ A is [23]
φ∗ (ω) ≡ n∗ (ω) =
1
(ǫIJK nI dR nJ dR nK − 2nI F I ) ,
8π
(5.9)
where the CP1 ∼
= S 2 sigma model is written in terms of a unit vector36 nI transforming in
the 3 of SU (2)R , with covariant dR nI = dnI + ǫIJK AJ nK and F I = dAI + 21 ǫIJK AJ ∧ AK .
The IR theory also admits a θ term, which multiplies the functional integral by [23]
Z
iθ
∗
∗
n (ω) ∧ n (ω) ,
exp
(5.10)
2 M4
34
The spin-charge relation for N = 2 involves both the gauge charge and the spin charge. E.g. for N = 2
supersymmetric SU (2)gauge QCD, the generalized spin structure is SpinSU (2)R ×SU (2)gauge and the spin charge
relation is (−1)F = (−1)Q (−1)FR where Q is the electric or magnetic charge of the state and FR is the charge
under U (1) ⊂ SU (2)R .
35
In the case of U (1)B in QCD, the analogous coupling between the background gauge field and the G/H
soliton the IR is dictated by matching the TrU (1)B G2 ’t Hooft anomaly.
36
For general G/H, we prefer φ for the pions, but for CP1 we take φ → n to match the notation of [23].
– 44 –
which by the time-reversal symmetry of the UV theory could take the values θ = 0 or θ = π.
For Ggauge = SU (2), it was noted in [23] that RG flow from N = 2 SYM suggests that θ = π.
5.4
Symmetry Matching on M4 = CP2 with Twisted, SpinSU (2)R (4) Structure
The Ggauge theory with adjoint fermions is not a bosonic theory, e.g. Tr(λfα Fµν ) is a gauge
invariant fermionic operator. As emphasized in [30], in fermionic theories neither the spin
nor the statistic of line operators are scheme-independent. For even Nf , on the other
hand, the identification (−1)F ∼ −1SU (Nf ) allows assigning a scheme independent combined
(−1)F ◦ (−1SU (Nf ) ) charge to line operators in addition to admitting the twisted generalized SpinSU (Nf ) (4) structure. As mention in the previous subsection, we expect that the
SpinSU (Nf ) (4) structure is preserved along the RG flow for all Ggauge , and that all of these
theories can thus be placed on M4 = CP2 with w2 (SU (Nf )) = w2 (T M ). In this subsection,
we discuss how the IR CP1 sigma model can match the symmetries, and the κΓ,T and κT,T
’t Hooft anomaly coefficients in (5.2) that we have computed in previous sections, when the
theories are formulated with twisted, SpinSU (2)R (4) structure on non-spin manifolds. The
case Ggauge = SU (2) was discussed in [23] and we will here consider general Ggauge .
(1)
As discussed in [23], the expression (5.9) for n∗ (ω)(AR ) in the presence of SU (2)R
background gauge fields, e.g. in the vacuum configuration with nI = δI3 , shows that 2n∗ (ω)
is an integral cohomology class that, with the SpinSU (2)R (4) structure, satisfies (mod 2):
2n∗ ∼
= w2 (SU (2)R ) ∼
= w2 (T M ). In the IR theory, this is a constraint on the functional
(N )
integral over the pion fields. The coupling (5.8) of n∗ (ω) to the background gauge field B2
for the one-form global symmetry can be modified by a counterterm, replacing n∗ (ω) with
(n∗ (ω)+ 12 w̃2 (T M )), which is integral valued as needed so that the coupling is invariant under
(N )
(1)
ZN background gauge transformations of B2 [23]:
Z
Z
2πi
1
2πi
(N )
(N )
∗
∗
B
∪ n (ω) → exp
B
∪ (n (ω) + w̃2 (T M )) ,
exp
N M4 2
N M4 2
2
(5.11)
where w̃2 (T M ) is an integral lift of w2 (T M ). The θ term (5.10) also requires a modification,
with an additional term −i θ8 σ(M4 ) involving the signature of M4 [23]. For Ggauge = SU (2),
the appearance of w2 (T M ) in (5.11) correctly reproduces the κΓ,T = 1 ’t Hooft anomaly [23].
(1)
(1)
For general Γg = ZN of UV theories, the coupling (5.11), which relies on identifying
(1)
the UV ZN in an enhanced IR U (1)(1) of the CP1 sigma model, naturally leads to anomaly
κΓ,T = 1. Recall that a κΓ,T anomaly implies a variation of the partition function under
e (N ) → B
e (N ) + N x or w̃2 (T M ) → w̃2 (T M ) + 2y:
either shifts of integral lifts B
2
2
Z 7−→ Z × (−1)κΓ,T
R
x∪w2 (T M )
or
Z 7−→ Z × e
2πiκΓ,T R
(N)
B2 ∪y
N
,
(5.12)
where x and y are integral cohomology classes or cochains. These two options differ by a
counterterm, which is the latter term in (5.11); it gives the anomalous variation as the latter
one in (5.12), under w̃2 (T M ) → w̃2 (T M ) + 2y shifts. When N is odd, all anomalous variation
– 45 –
can be cancelled by the local counter term Sc.t. = 2πin
N
R
e (N ) ∪ w̃2 (T M ) where 2n = 1 modN .
B
2
(1)
(1)
As we have discussed, κΓ,T is associated with flux in a Z2 ⊂ Γg , which requires even N .
In summary, the IR analysis readily leads to κΓ,T = 1 for N even. On the other hand,
we have computed the κΓ,T in the UV description and found that κΓ,T = 1 only if Γg = ZN
with N ∈ 4Z + 2. The apparent mismatch for N ∈ 4Z requires a compensating effect. One
option is the addition of a TQFT in the IR sigma model in those cases [8, 10]. Alternatively,
(1)
it could be that matching of the center symmetry Γg of the UV theory to the enhanced
U (1)(1) symmetry of the IR sigma model requires modification.
6
Symmetric Mass Generation
A chiral gauge theory has symmetric mass generation if it generates an IR mass gap despite
not admitting quadratic mass terms; this can happen if ’t Hooft anomalies do not rule out
the IR mass-gapped phase. In some classes of examples, the IR mass gap can be generated
by e.g. 4-fermion interactions [44–46], see also [114–118] for earlier work and references.
We will here consider the SpinG ’t Hooft anomalies are non-zero in variants of these
theories that admit a generalized spin structure. A class theories that have been proposed to
lead to symmetric mass generation are the symmetric with fundamentals examples discussed
in [45]. One starts with SU (N ) gauge theory with a single fermion λ in the 2-index symmetric
representation Sym2 N, and cancels the TrSU (N )3 gauge anomaly via N + 4 fermions ψ in
the N representation. The global SU (N +4) symmetry can be made anomaly free by coupling
to 12 (N + 4)(N + 3) uncharged fermions that transform under SU (N + 4) in the Λ2 N + 4
(two-index antisymmetric). In all, the fermion content is given by
λ
ψ
χ
SU (N )g
Sym2 N
N
1
(0)
SU (N + 4)R
1
N+4
Λ2 N + 4
(0)
U (1)r
−(N + 4)
N +2
−N
Here U (1)r is also a non-anomalous global symmetry. The ’t Hooft anomalies for these
symmetries all-vanish, and they and are all preserved by the 4-fermion interaction term
Lint =
1
ψχψλ + c.c.
Λ2
(6.1)
The theory with this interaction has no obstruction to an IR mass gap, and the proposal
of symmetric mass generation is that the term (6.1) indeed generates a mass gap. This can
be intuitively understood if at some energy scale ΛQCD , SU (N )g confines and IR phase is
described by χ together with the SU (N )g neutral composite fermion
Ψ=
1
ψχψ ,
Λ3QCD
(0)
(6.2)
that transforms under the Λ2 (N + 4) of SU (N + 4)R . The 4-fermion interaction Lint would
then RG flow to a mass term for Ψ, χ, leading to the conjectured trivially gapped IR phase.
– 46 –
We here consider variants of the above theory, with K generations, i.e. K copies, of the
λ, ψ, and χ matter content. The SU (N )g gauge coupling is asymptotically free if K(N + 3) <
11
2 N . For K sufficiently close to the upper limit of the bound will presumably put the theory
in a conformal window, where it RG flows to an IR CFT. We are here more interested in
a possible confining gapped IR phase, with K below the conformal window. We will in
particular focus on K = 2 and can further disfavor a possible IR CFT phase by taking N
sufficiently large. The K = 2 generations leads to a global SU (2)3 symmetry acting on each
matter field, and the four-fermion interaction (6.1) can be chosen to preserve only a diagonal
SU (2)f . We will use SU (2)f to have a SpinSU (2)f (4) structure. We thus consider
λ
ψ
χ
SU (N )g
Sym2 N
N
1
(0)
SU (N + 4)R
1
N+4
Λ2 N + 4
SU (2)f
2
2
2
(0)
U (1)r
−(N + 4)
N +2
−N
Note that SU (2)f has no Witten anomaly, no WWW SU (2) anomaly, and no mixed anomaly
(0)
with U (1)r . Since all anomalies vanish, one might conjecture an IR gapped phase here too.
For N = 2n there is a common Z2 shared between SU (2n)g × SU (2n + 4)R , SU (2)f , and
(−1)F . The fields couple to Gtotal -bundles with
Gtotal =
SU (2)f × Spin(4)
SU (2n)g × SU (2n + 4)R
×
.
Z2
Z2
(6.3)
This allows a generalized SpinSU (2)f (4) structure and a background B2 field with37
w2 (T M ) = w2 (SU (2)f ) ,
B2 = w2 (g) = w2 (R) .
(6.4)
We can now compute the w2 w3 -type ’t Hooft anomaly of this theory on CP2 by computing
the index of the Dirac operator in an appropriate flux background. E.g. for N = 2n = 2,
considering a diagonal U (1)Q ⊂ U (1)Γ × U (1)Γg , the fermions have the charges
3 3
1 1 ⊕2
1 1
,−
,−
,−
,
=
⊕
qλ = (1, 0, −1) ⊗
2 2
2 2
2 2
1 1
1 1 ⊕9
1 1 1 1 1 1
1 1
3 3 ⊕3
qψ =
,−
, , ,− ,− ,−
,−
,−
⊕
,−
, (6.5)
⊗
⊗
=
2 2
2 2 2 2 2 2
2 2
2 2
2 2
3 3 ⊕3 1 1 ⊕12
1 1
⊕3
⊕9
⊕
,−
=
,−
,−
⊗ (1, −1) ⊕ 0
qχ =
2 2
2 2
2 2
This leads to contributions to the index Iλ = 2, Iψ = 6, Iχ = 6 and anomaly indicator
σI =
Iλ + Iψ + Iχ
mod2 = 1 .
2
37
(6.6)
Note that here we are turning on a correlated flux in the dynamical gauge field and background gauge
field for SU (2n + 4)R which does not correspond to a 1-form global symmetry. The interpretation of such flux
backgrounds has been recently discussed in [11, 119, 120].
– 47 –
Therefore, we see that this theory has a non-perturbative anomaly κΓg ,T = 1
Z
A ⊂ πi B2 ∪ w3 (T M ) .
(6.7)
Therefore the two-generation theory with symmetry preserving 4-fermion operator cannot be
trivially gapped. For N = 2 this two-generation case is close to the asymptotic freedom bound,
so it might be expected to be in the conformal window rather than trivially gapped. Twogeneration cases with larger N are farther from the asymptotic freedom bound, so perhaps
not in the conformal window, and similarly can have κΓ,T 6= 0, preventing a trivially gapped
IR phase. Of course, it is still possible that the one-generation case is trivially gapped.
Acknowledgements
We thank Greg Moore, Misha Shifman, and Juven Wang for helpful comments, and especially
Clay Córdova and Thomas Dumitrescu for helpful discussions and related collaborations. DB
and KI are supported in part by the Simons Collaboration on Global Categorical Symmetries,
and by Simons Foundation award 568420. KI is also supported by DOE award DE-SC0009919.
A
Fractionalization and the Example of Spin(4nc + 2) Vector-QCD
In this appendix we will briefly describe the fractionalization technique [24, 30, 50] to compute
discrete anomalies, and detail a few computations that we quoted in the main text.
Consider Ggauge gauge theory with Nf fermions in the representation Rgauge such that
Ggauge is simply connected and the fields couple to Gtotal -bundles of the form
Gtotal =
Ggauge
× Gglobal × Spin(4)
Γg
Γ
(A.1)
,
where Γg = Zn . This theory has a 1-form global symmetry corresponding to Γg and SpinGglobal
structure corresponding to Γ. We now consider adding an adjoint-valued scalar field that also
couples to the fermions via a Yukawa interaction. A vev for the adjoint scalar generically
breaks the gauge group Ggauge 7→ U (1)r , where r is the rank
the fermions
of Ggauge , and gives
(1)
(1)
(1) r
a mass. The IR abelian theory has an emergent GIR = U (1)e × U (1)m
symmetry with mixed ’t Hooft anomalies as in [79]
r
i X
A⊂
2π
I=1
Z
(e)
(m)
B2,I ∧ dB2,I .
1-form global
(A.2)
The 1-form and SpinG symmetries of the UV theory are preserved along this RG flow and
can be tracked to the IR abelian gauge theory by identifying the quantum numbers of the
emergent line operators. The discrete fluxes for Γ, Γg , which we denote w2 (Γ) and B2U V
(1)
respectively, can activate fluxes for subgroups of the emergent GIR . This can be used to
exhibit the existence of w2 w3 and other ’t Hooft anomalies of the UV theory.
– 48 –
Let us explain how to track the Γ, Γg fluxes along the RG flow from Ggauge → U (1)r .
We first discuss how they activate the IR electric 1-form fluxes. Note that it is clear how
Γg , Γ ⊂ Ggauge embed into U (1)r ⊂ Ggauge since U (1)r is identified as the Cartan torus of
Ggauge . Let us focus on a single U (1)gauge factor, and consider a unit charge Wilson line W1 [γ].
(1)
This Wilson line is charged under the corresponding 1-form electric symmetry U (1)e . Let
(1)
(e)
us introduce a 2-form background gauge field for U (1)e : B2 . The statement that W1 [γ] is
(1)
charged under U (1)e means that it transforms under B2 gauge transformations as
(e)
(e)
δB2 = dΛ1
,
R
(e)
W1 [γ] 7−→ W1 [γ] × ei γ Λ1 .
(A.3)
Along the RG flow, the Wilson line can experience charge fractionalization. This means
(Γ)
that, if we introduce a Γ-background gauge field B2 , the Wilson line will transform under
(Γ)
the B2 -background gauge transformation as
(Γ)
δB2
(Γ)
= dΛ1
,
R
(Γ)
W1 [γ] 7−→ W1 [γ] × ei γ Λ1
.
(A.4)
(1)
The only way this can be matched is if we enact the Γ, Γg transformations with the U (1)e
transformation. This means that if the UV operator that flows to the IR Wilson line is
(1)
charged under Γ, Γg , then Γ, Γg flows (at least in part) to Γ, Γg ⊂ U (1)e . So turning on a
(Γ )
(Γ)
(e)
background gauge field B2 , B2 g turns on a B2 depending on the transformation property
of W1 [γ]. Now we wish to determine the charge fractionalization of the Wilson line. This
differs between the two cases:
• Γg -non-trivial: In this case the fundamental Wilson line of Ggauge maps to the minimal
b populates a diagonal subgroup Γ
b ⊂
Wilson line of the IR theory. This means that Γ
(1) r
b is the extension
(U (1)e ) where Γ
b −→ Γg −→ 1 .
1 −→ Γ −→ Γ
(A.5)
This is the case as discussed in Section 3.4 where Γg forms a 2-group with the Gglobal ×
Spin(4).
• Γg trivial: Here the fundamental Wilson line of the IR theory is described by the world
volume of the fundamental fermion in the UV theory. When Γ does not act on Ggauge ,
(e)
the Wilson line will not have any fractionalization and w2 (Γ) does not activate B2 . On
the other hand, when Γ acts on Ggauge , it identifies part of the center of Ggauge with part
of Z(Gglobal × Spin(4)). Then, since the center of the gauge group flows to the diagonal
(1)
subgroup of the 1-form electric symmetry in the IR, the subgroup Z(Gglobal )(1) ⊂ GIR
(e)
would be identified with part of Z(Gglobal ×Spin(4)) and w2 (Γ) activates B2 . However,
if Γ acts trivially on Ggauge , then there can be no such identification.
Consider the special case where Γg = ZN and the Γ = Z2 acts trivially on Ggauge . Let us
pick U (1)Γg ⊂ U (1)r ⊂ Ggauge where Γg embeds into U (1)Γg and let QΓg be the generator of
– 49 –
U (1)Γg which further decomposes as
QΓg =
X
(I)
QΓg HI ,
(A.6)
I
where HI are simple coroots of Ggauge that generate the IR U (1)r . Then the minimal UV
Wilson line will flow to a minimal IR Wilson line and we can identify the IR 1-form electric
fluxes as
(e)
B2,I
2π
=
1 U V (I)
B QΓg
N 2
(e)
=⇒
B2
1
= B2U V QΓg ,
2π
N
(A.7)
where again B2U V is the background gauge field for the UV 1-form global symmetry.
Now let us turn to the IR 1-form magnetic symmetries. The line operators that are
charged under the 1-form magnetic symmetry are the magnetic line operators. Since we
assume that Ggauge is simply connected, the magnetic line operators in the IR come from
smooth magnetic monopoles that arise when the adjoint scalar field condenses. Unlike the
Wilson line operators, the magnetic line operators acquire charge fractionalization from the
fermion zero-modes that localize on its world volume.
The magnetic 1-form global symmetries are generated by weights that are dual to the
generators of the U (1) factors. The background magnetic gauge field to U (1)Γg is valued
(m)
σf
B2
=
w2 (Γ) Q∨
Γg
2π
2
Q∨
Γg =
,
X
I
(Q∨
Γg )I λ
(HI , λJ ) = δI J ,
,
(A.8)
I
where Q∨
Γg is the dual weight to QΓg . In general, QΓg which can be expressed as a (fractional)
sum of the dual weights, λI and ( , ) is the natural pairing co-roots and dual weights. Here
σf = 0, 1 is the spin indicator as in [30] which indicates whether the monopole is a fermion
(σf = 1) or a boson (σf = 0).
We can compute the fermion zero-modes as follows. Consider the monopole MQΓg [γ]
with magnetic charge QΓg . The fermions decompose into representations under U (1)QΓg with
respect to their magnetic charge pairing with QΓg :
R 7−→
M
pi
,
ψ 7−→
i
X
ψi vi ,
(A.9)
i
where again the vi are the basis vectors of the representation R which have corresponding
charges QΓg ·vi = pi vi . When there are Nf fermions in the representation R, which transforms
under a subgroup of Gglobal ⊂ SU (Nf ), the number of fermion zero-modes given by the Callias
Index Theorem [121] is:
X
I(R ⊗ Nf ) = {# of R fermion z.m.} = Nf
pi .
(A.10)
i | pi >0
– 50 –
These zero-modes transform under Gglobal × Spin(3)rot , where Spin(3)rot ⊂ Spin(4) is the
rotation group preserved by the monopole line configuration, as [30, 122]
M
Nf ⊗
pi .
(A.11)
i | pi >0
These fermion zero-modes give rise to a quantum mechanics that is localized on the
monopole which transforms under the Gglobal × Spin(3)rot . The charge fractionalization of
the monopole is given by the representation of the Hilbert space associated to this quantum
mechanics: Hmono . The charge of the monopole with respect to Γ is determined by whether
or not Hmono transforms projectively under the Gsub ⊂ Gtotal subgroup
Gsub =
Gglobal × Spin(3)rot
.
Z2
(A.12)
To any projective representation Rproj of G, we can associate a cohomology class w2 [Rproj ] ∈
H 2 (BGsub ; Z2 ). When Hmono transforms projectively with representation Rproj under Gsub ,
we can identify a cohomology class w2 [Rproj ] ∈ H 2 (BG; Γ) which corresponds to a 1-form
magnetic flux
(m)
B2
1
(A.13)
= w2 [Rproj ]Q∨
Γg .
2π
2
For our case where Γ = Z2 acts trivially on Ggauge , there is a Z2 possibility of charge fractionalization corresponding to whether or not Hmono transform projectively under Gsub . Physically this means the monopole is a fermion, or is in a projective Gglobal /Z2 multiplet. If we
introduce a spin/fractionalization indicator
(
1 Hmono is proj. rep. of Gsub
σf =
(A.14)
0 else
then we can succinctly write the magnetic background flux as
(m)
σf
B2
=
w2 (T M )Q∨
Γg .
2π
2
Plugging these into the anomaly formula (A.2), we find the resulting anomaly
Z
Z
i
(e)
(m)
A⊂
B2 ∧ dB2 = πi σf B2U V ∪ w3 (T M ) .
2π
(A.15)
(A.16)
See [30] for more details on this approach to computing non-perturbative anomalies.
A.1
Anomaly for Spin(4nc + 2) Vector QCD
Now let us compute the anomaly in Spin(4nc + 2) QCD with 2nf vector fermions to complete
the discussion in Section 3.4. The fields couple to Gtotal -bundles:
Gtotal =
Spin(4nc +2)
× SU (2nf ) × Spin(4)
Z2
,
ZF2 × ZC
2
– 51 –
(A.17)
F
where ZF2 identifies (−1)F ∼ −1SU (2nf ) and ZC
2 identifies (−1) ∼ −1Spin(4nc +2) . Here we
will focus on the case of SpinGgauge (4). In this case we can activate the obstruction class for
lifting P SO(4nc + 2) to Spin(4nc + 2)-bundles which is Z4 -valued, but we are not allowed to
activate the flux for w2 (ZF2 ). The 2 independent fluxes for B2 and w2 (ZF2 ) = w2 (T M ) then
parametrize the Z4 flux as
(A.18)
w2 (Z4 ) = 2B2 + w2 (T M ) .
To compute the anomaly, we will deform the theory by adding an adjoint valued scalar field
with Yukawa coupling – which breaks SU (2nf ) → Sp(nf ) – and condense the scalar field so
that we flow to pure abelian gauge theory. First, since the fermions do not screen Wilson
lines in the spin representation of Spin(4nc + 2), we can identify the minimal Wilson lines of
the IR theory with the Wilson lines of the UV theory. These lines are charged under w2 (Z4 )
so that we can identify
(e)
1
B2
= (w2 (T M ) + w2 (Γ) + 2w2 (Γg )) QΓg ,
2π
4
where
QΓg =
2
nX
c −1
H2I−1 − H2nc +1 + H2nc
I=1
!
.
(A.19)
(A.20)
Since 14 QΓg generates Z(Spin(2nc )), we know that every fermion coming from the vector
representation has charge ±2 under QΓg . This means that in the presence of the minimal
monopole of U (1)QΓg the fermion zero-modes transform as 2 ⊗ 2nf ⊕2nc +1 under Spin(3) ×
Sp(nf ). This implies that the monopole Hilbert space decomposes as
Hmono = (Hf und )⊗2nc +1 .
(A.21)
Since there are an odd number of factors and we are looking for Z2 -valued fractionalization,
we see that Hmono has the same fractionalization as Hf und , which is the Dirac spinor representation of Spin(4nf ), Hf und = S0 ⊕ S1 , where S0,1 has an even (odd) number of fermion mode
operators respectively (here S0 is the bare monopole and S1 is the dyon as in [30]). When nf
is odd (even) the fermionic (bosonic) components of Hf und transform as a real (pseudo-real)
representation of Sp(nf ) which implies that there is (possible) charge fractionalization.
Now we can use the fact that the fermion zero-modes transform as spin- 21 under Spin(3).
This means that if the Fock-vacuum of Hf und has charge q under T3 ⊂ Spin(3), then the
maximum state has charge 2nf + q. If we then tune the UV θ-angle so that the IR theory is
T -symmetric, then we see that
−q = 2nf + q
⇒
q = −nf ,
(A.22)
which implies that S0 is a fermion (boson) if nf is odd (even) which implies that
(m)
nf
B2
=
w2 (T M ) Q∨
Γg .
2π
2
– 52 –
(A.23)
This is consistent with the observation [30] that the monopole Hilbert space decomposes as
Hmono = Hmono ⊕ Hdyon ,
(m)
(A.24)
(e)
since the fractionalization class for Hdyon is related to B2 by two times B2 . The anomaly
is thus
Z
Z
2πinf
2πinf
w2 (T M ) ∧ d(2B2 + w2 (T M )) =
w2 (T M ) ∧ (4dB2 )
A⊂
8
8
Z
(A.25)
= 2πi nf B2 ∪ w3 (T M ) = 0 modZ ,
where here we used the 2-group identity dw2 (T M ) = 2dB2 and the identity in Footnote 12.
The upshot is thus that the anomaly vanishes, κΓg ,T = 0, for this case.
A.2
Anomaly for Spin(4nc + 2) Adjoint QCD
For Spin(4nc + 2) adjoint QCD with 2nf fermions, the fields couple to the total bundle
Gtotal =
Spin(4nc + 2) SU (2nf ) × Spin(4)
×
.
Z4
Z2
(A.26)
Using the fractionalization method, we consider coupling to an adjoint fermion with Yukawa
(1)
couplings (breaks SU (2nf ) → Sp(n)f ) and Higgsing Spin(4nc + 2) → U (1)2nc +1 . The Z4
(1)
UV center symmetry is matched to a Z4 ⊂ (U (1)(1) )2nc +1 of the enhanced, IR 1-form electric
symmetries. The IR and UV background gauge fields are thus matched as
(e)
1
B2
= B2U V Q∨
Γg ,
2π
4
(A.27)
where 14 QΓg generates the center of Spin(4nc + 2). The fermion zero-modes of a minimal
U (1)Γg monopole are of the form
4 ⊗ 2nf ⊕2nc +1 .
(A.28)
Again, the monopole Hilbert space decomposes into N tensor factors
Hmono = (Hf und )⊗2nc +1 ,
(A.29)
where Hf und is the Dirac spin representation of Spin(8nf ), Hf und = Sb ⊕ Sf , (Sb,f is the
bosonic, fermionic component). For all nf that the fermion spin representations are real so
that the fermionic (bosonic) components of Hf und transform as a pseudo-real (real) representation of Sp(nf ) and hence there is no fractionalization. We therefore see that adjoint QCD
does not carry the center symmetry anomaly for Nc = 4nc + 2.
This can be checked explicitly in Spin(6) ∼
= SU (4) gauge theory. The fields couple to
Gtotal =
SU (4) SU (2)f × Spin(4)
×
,
Z4
Z2
– 53 –
(A.30)
bundles, where Z4 ⊂ U (1)Γg is generated by
1
1
1
QΓ = H2 + (H1 − H3 ) .
4 g
2
4
(A.31)
(1)
Since this theory has Z4 1-form global symmetry, we can identify the U (1)Γg Wilson line as
the fundamental Wilson line of the UV theory and hence
(e)
B2
1
= B2U V QΓg .
2π
4
(A.32)
We now compute the monopole fractionalization. Upon decomposing the adjoint fermions
under the U (1)3gauge Cartan, the charge coupling has QΓg 15 7−→ ±4⊕3 ⊕ 0⊕9 . So each adjoint
fermion provides 12 real fermion zero-modes. As a representation of Spin(3)rot × SU (2)global
these zero-modes transform as (4, 2)⊕3 . Upon quantization of these zero modes, the monopole
Hilbert space decomposes under SU (2)global × Spin(3)rot as:
⊗3
Hmono = (4, 2) ⊕ (1, 3) ⊕ (5, 1)
.
So the monopole Hilbert space is a faithful representation of
(m)
B2
2π
(A.33)
SU (2)global ×Spin(3)rot
and therefore
Z2
= 0 and there is no anomaly.
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