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One-form symmetries and the 3d $\mathcal{N}=2$ $A$-model:
Topologically twisted indices for any $G$

Cyril Closset, Elias Furrer, Osama Khlaif

School of Mathematics, University of Birmingham,

Watson Building, Edgbaston, Birmingham B15 2TT, UK

We study three-dimensional $\mathcal{N}=2$ supersymmetric Chern–Simons-matter
gauge theories with a one-form symmetry in the $A$-model formalism on
$\Sigma_{g}\times S^{1}$. We explicitly compute expectation values of
topological line operators that implement the one-form symmetry. This allows
us to compute the topologically twisted index on the closed Riemann surface
$\Sigma_{g}$ for any real compact gauge group $G$. All computations are
carried out in the effective $A$-model on $\Sigma_{g}$, whose ground states
are the so-called Bethe vacua. We discuss how the 3d one-form symmetry acts on
the Bethe vacua, and how its ’t Hooft anomaly constrains the vacuum structure.
In the special case of the $SU(N)_{K}$ $\mathcal{N}=2$ Chern–Simons theory, we
obtain results for the $(SU(N)/\mathbb{Z}_{r})^{\theta}_{K}$ $\mathcal{N}=2$
Chern–Simons theories, for all non-anomalous
$\mathbb{Z}_{r}\subseteq\mathbb{Z}_{N}$ subgroups of the center of the gauge
group, and with the associated $\mathbb{Z}_{r}$ $\theta$-angle turned on,
reproducing and extending various results in the literature. In particular, we
find an interesting mixed ’t Hooft anomaly between gravity and the
$\mathbb{Z}_{r}$ one-form symmetry of the $SU(N)_{K}$ theory (for $N$ even,
${N\over r}$ odd and ${K\over r}$ even). This plays a key role in our
derivation of the Witten index, which we explicitly compute for any $N$, $K$
and $r$ in terms of refinements of Jordan’s totient function. Our results lead
to precise conjectures about integrality of indices, which appear to have a
strong number-theoretic flavour. Note: this paper directly builds upon
unpublished notes by Brian Willett from 2020.

###### Contents

  1. 1 Introduction
    1. 1.1 Warm-up example: The $SU(2)_{K}$ supersymmetric Chern–Simons theory
    2. 1.2 Results for general $G$ and for the $SU(N)_{K}$ gauge theory
  2. 2 One-form symmetries in the 3d $A$-model
    1. 2.1 Discrete symmetries in the 2d description
      1. 2.1.1 Hilbert spaces and Bethe vacua for $G=\widetilde{G}$
      2. 2.1.2 ’t Hooft anomalies and vacuum structure
    2. 2.2 Background gauge fields and expectation values of topological operators
    3. 2.3 Gauging the 1-form symmetry $\Gamma^{(1)}$
    4. 2.4 Gauging the 0-form symmetry $\Gamma^{(0)}$
    5. 2.5 Topologically twisted index for the gauge group $\widetilde{G}/\Gamma$
  3. 3 The 3d $\mathcal{N}=2$ $SU(N)_{K}$ Chern–Simons theory
    1. 3.1 The $A$-model for the $SU(N)_{K}$ theory and its higher-form symmetries
      1. 3.1.1 Two-dimensional 1-form symmetry, gauging and decomposition
      2. 3.1.2 Two-dimensional 0-form symmetry and orbits of Bethe vacua
      3. 3.1.3 ’t Hooft anomaly and the allowed $(SU(N)/\mathbb{Z}_{r})_{K}$ CS theories
    2. 3.2 Mixed correlators and a modular anomaly on $T^{3}$
    3. 3.3 The $PSU(N)_{K}$ twisted index for $N$ prime
    4. 3.4 Gauging the 3d one-form symmetry on $T^{3}$
    5. 3.5 Gauging the 3d one-form symmetry on $\Sigma_{g}\times S^{1}$
  4. 4 Further aspects of the 3d $\mathcal{N}=2$ $SU(N)_{K}$ CS theory
    1. 4.1 Gauging the 1-form symmetry in the 3d TQFT
    2. 4.2 Comments on level/rank duality
  5. 5 Including matter: $U(1)$ and $SU(N)$ theories
    1. 5.1 $U(1)_{k}$ with matter
    2. 5.2 $SU(N)_{k}$ with adjoint matter
  6. 6 Conclusion and outlook
  7. A 3d $A$-model for $\widetilde{G}$: a lightning review
    1. A.1 The Coulomb branch parameters
    2. A.2 The effective twisted superpotential and the effective dilaton
    3. A.3 The 3d topologically twisted index
  8. B Lattices associated with Lie groups
  9. C Mixed ’t Hooft anomaly from dimensional reduction
  10. D Proofs for $SU(N)_{K}$ CS theory
    1. D.1 Fixed points under the full group
    2. D.2 Fixed points under subgroups
    3. D.3 Sum over mixed correlators
    4. D.4 Theta angle for general $N$
  11. E Number theory tidbits
    1. E.1 Glaisher’s theorem
    2. E.2 Totient functions

##  1 Introduction

Three-dimensional $\mathcal{N}=2$ supersymmetric gauge theories are free
theories in the ultraviolet (UV) but flow to strong coupling in the infrared
(IR). Over the past 15 years, a huge amount of non-perturbative results were
obtained for such 3d theories that also have a $U(1)_{R}$ $R$-symmetry, using
powerful supersymmetric localisation techniques, starting with the exact
computations of the $S^{2}\times S^{1}$ (so-called) superconformal index
Kim:2009wb ; Imamura:2011su  and of the three-sphere partition function
Kapustin:2009kz ; Jafferis:2010un  – see e.g. Pestun:2016zxk ; Willett:2016adv
; Closset:2019hyt  for reviews.

Almost all of these 3d $\mathcal{N}=2$ supersymmetric partition functions can
be understood in the framework of the 3d $A$-model, which allows us to
evaluate supersymmetric path integrals on any Seifert 3-manifold
$\mathcal{M}_{3}$ Closset:2017zgf ; Closset:2018ghr ; Closset:2019hyt . The 3d
$A$-model is a two-dimensional cohomological topological field theory (Coh-
TQFT) obtained as the topological $A$-twist of the 2d $\mathcal{N}=(2,2)$
supersymmetric effective field theory description of the 3d $\mathcal{N}=2$
gauge theory compactified on a circle. In the simplest instance, the 3d
$A$-model computes the twisted index on a genus-$g$ Riemann surface,
$\Sigma_{g}$ Nekrasov:2014xaa ; Benini:2015noa ; Benini:2016hjo ;
Closset:2016arn :

$Z_{\Sigma_{g}\times S^{1}}={\rm Tr}_{\Sigma_{g}}\left((-1)^{\rm
F}y^{Q_{F}}\right)\\\
=\sum_{\hat{u}\in\mathcal{S}_{\text{BE}}}\mathcal{H}(\hat{u},\nu)^{g-1}\leavevmode\nobreak\
.$ (1.1)

Here the trace in the first equality is over the supersymmetric ground states
on $\Sigma_{g}$ with a topological twist, where $(-1)^{\rm F}$ is the fermion
number and $y=e^{2\pi i\nu}$ denote fugacities for some flavour charges
$Q_{F}$. The second equality of (1.1) computes this twisted flavoured index as
a sum over the so-called Bethe vacua $\\{\hat{u}\\}$. These are the ground
states of the 3d theory compactified on $T^{2}$. More generally, for any
Seifert-fibered three-manifold $\mathcal{M}_{3}$, there exists a similar
formula for the supersymmetric partition function Closset:2018ghr . Let

$\mathcal{H}_{S^{1}}\cong{\rm Span}_{\mathbb{C}}\\{|\hat{u}\rangle\\}$ (1.2)

denote the Hilbert space of $A$-model ground states, spanned by the Bethe
vacua, where the states $|\hat{u}\rangle$ are orthonormal. We then have:

$Z_{\mathcal{M}_{3}}={\rm
Tr}_{\mathcal{H}_{S^{1}}}\left(\mathcal{H}^{g-1}\mathcal{G}_{\mathcal{M}_{3}}\right)=\sum_{\hat{u}\in\mathcal{S}_{\text{BE}}}\langle\hat{u}|\mathcal{H}^{g-1}\mathcal{G}_{\mathcal{M}_{3}}|\hat{u}\rangle=\sum_{\hat{u}\in\mathcal{S}_{\text{BE}}}\mathcal{H}(\hat{u},\nu)^{g-1}\,\mathcal{G}_{\mathcal{M}_{3}}(\hat{u},\nu)\leavevmode\nobreak\
.$ (1.3)

Here, $\mathcal{H}$ and $\mathcal{G}_{\mathcal{M}_{3}}$ are called the handle-
gluing operator and the Seifert fibering operator, respectively. They are both
diagonalised by the Bethe vacua (with $\mathcal{O}(\hat{u})$ denoting the
eigenvalue of $\mathcal{O}$). The fibering operator
$\mathcal{G}_{\mathcal{M}_{3}}$ induces a non-trivial fibration of $S^{1}$
over $\Sigma_{g}$ Closset:2017zgf ; Closset:2018ghr .

While this 3d $A$-model formalism is very powerful, it was originally
developed under the assumption that the gauge group $\widetilde{G}$ be a
product of simply-connected and of unitary compact gauge groups – that is, the
compact Lie group $\widetilde{G}$ is such that $\pi_{1}(\widetilde{G})$ is a
free abelian group. In this paper, we are interested in more general cases
with a gauge group of the form:

$G=\widetilde{G}/\widetilde{\Gamma}\leavevmode\nobreak\
,\qquad\qquad\widetilde{\Gamma}\subseteq Z(\widetilde{G})\leavevmode\nobreak\
,$ (1.4)

where the discrete abelian group $\widetilde{\Gamma}$ is a subgroup of the
center of $\widetilde{G}$. This group is also a subgroup of the one-form
symmetry group $\Gamma_{\rm 3d}^{(1)}$ of the $\widetilde{G}$ gauge theory:

$\Gamma_{\rm 3d}^{(1)}\cong\Gamma\leavevmode\nobreak\
,\qquad\qquad\widetilde{\Gamma}\subseteq\Gamma\leavevmode\nobreak\ ,$ (1.5)

which is the group under which Wilson lines can be charged Kapustin:2005py ;
Aharony:2013hda . Supersymmetric observables in the gauge theory with gauge
group $G$ can be obtained from the original $\widetilde{G}$ gauge theory by
gauging a non-anomalous subgroup $\widetilde{\Gamma}$ of the one-form symmetry
Gaiotto:2014kfa . Schematically, this corresponds to summing over all
background gauge fields $B$ for the one-form symmetry
$\widetilde{\Gamma}^{(1)}_{\rm 3d}\cong\widetilde{\Gamma}$, namely:

$Z_{\mathcal{M}_{3}}[\mathcal{T}/\widetilde{\Gamma}^{(1)}_{\rm
3d}]=\sum_{B}Z_{\mathcal{M}_{3}}[\mathcal{T}](B)\leavevmode\nobreak\ .$ (1.6)

Here, $Z_{\mathcal{M}_{3}}[\mathcal{T}](B)$ denotes the path integral of the
original $\widetilde{G}$ gauge theory, $\mathcal{T}$, in the presence of a
background gauge field $B\in H^{2}(\mathcal{M}_{3},\widetilde{\Gamma})$.

Turning on a background gauge field $B$ for $\Gamma_{\rm 3d}^{(1)}$ is
equivalent to inserting a web of topological line operators
$\mathcal{U}^{\gamma}$ Gaiotto:2014kfa . In this paper, we discuss such
insertions in the 3d $A$-model formalism. More generally, we discuss the
action of the one-form symmetry on the Bethe vacua. In particular, we explain
how ’t Hooft anomalies for the one-form symmetry $\Gamma_{\rm 3d}^{(1)}$
usefully constrain the structure of the Bethe vacua. We then discuss the
gauging of non-anomalous subgroups of $\Gamma_{\rm 3d}^{(1)}$ as explicitly as
possible.

The core ideas explored in this work were first discussed by Brian Willett in
unpublished notes willett ; they were also briefly explained in
Eckhard:2019jgg . The gauging of non-anomalous one-form and zero-form
symmetries in 2d TQFTs was also discussed in detail by Gukov, Pei, Reid and
Shehper Gukov:2021swm , and we will recover many of their results. In this
paper, we focus on explicit computations of topologically-twisted
supersymmetric indices for 3d $\mathcal{N}=2$ gauge theories with general
gauge groups $G$ through a careful discussion of the physics of the 3d
theories with one-form symmetries when compactified on $S^{1}$. For
concreteness, after discussing the formalism in general terms, our main
explicit computations will focus on the case of theories with
$\widetilde{G}=SU(N)$ and a $\mathbb{Z}_{N}$ one-form symmetry. As we will
see, the story can become particularly intricate depending on the arithmetic
properties of $N$ – here, $N$ prime will be the easy case, but for general $N$
a number of interesting phenomena will arise, including a subtle 3d modular
anomaly.

We hope to discuss further aspects of generalised symmetries of 3d
$\mathcal{N}=2$ supersymmetric field theories, including a more thorough
discussion of supersymmetric partition functions for general Seifert three-
manifolds, in future work. The present work is also related in various way to
many recent papers on 3d field theory, including e.g. Creutzig:2021ext ;
Hosseini:2022vho ; Comi:2023lfm ; Gang:2023rei ; Dedushenko:2023cvd ;
Argurio:2024oym ; Sharpe:2024ujm .

###  1.1 Warm-up example: The $SU(2)_{K}$ supersymmetric Chern–Simons theory

Before discussing our general result, let us first discuss the example of the
3d $\mathcal{N}=2$ supersymmetric Chern–Simons (CS) theory $SU(2)_{K}$.
Gauging the $\mathbb{Z}_{2}$ one-form symmetry, one obtains the $SO(3)_{K}$ CS
theory. This is an interesting example which already involves many of the
intricacies of the general $SU(N)_{K}$ case to be discussed below.

Upon integrating out the gauginos in the vector multiplet, the $\mathcal{N}=2$
$SU(2)_{K}$ CS theory is equivalent to the bosonic CS theory $SU(2)_{k}$ with
$k=K-2$. The theory has $K-1$ Bethe vacua, corresponding to the allowable
solutions to the Bethe equations:

$\Pi(u)\equiv x^{2K}=1\leavevmode\nobreak\ ,\qquad x\equiv e^{2\pi
iu}\leavevmode\nobreak\ ,\qquad u\rightarrow u+1\leavevmode\nobreak\ .$ (1.7)

Here, $u$ is the 2d Coulomb-branch parameter, with the $\mathbb{Z}_{2}$ Weyl
group acting as $u\rightarrow-u$. The operator $\Pi$ is called the gauge flux
operator, and it is given in terms of the effective twisted superpotential
$\mathcal{W}$ of the $A$-model:

$\Pi(u)=\exp\left(2\pi i{\partial\mathcal{W}\over\partial
u}\right)\leavevmode\nobreak\ ,\qquad\mathcal{W}=ku^{2}\leavevmode\nobreak\ .$
(1.8)

The Bethe vacua corresponds to pairs of solutions $\pm\hat{u}_{l}$ related by
the Weyl symmetry, with:

$\hat{u}_{l}=\frac{l}{2K}\leavevmode\nobreak\ ,\qquad
l=1,\cdots,K-1\leavevmode\nobreak\ .$ (1.9)

The handle-gluing operator of this theory reads:

$\mathcal{H}(u)=\frac{K}{2\sin^{2}(2\pi u)}\leavevmode\nobreak\ ,$ (1.10)

and therefore the $\Sigma_{g}$ twisted index (1.1) is given by:

$Z_{\Sigma_{g}\times S^{1}}[SU(2)_{K}]\equiv\langle
1\rangle_{\Sigma_{g}}=\sum_{l=1}^{K-1}\left[\sqrt{2\over K}\sin\left(\frac{\pi
l}{K}\right)\right]^{2-2g}\leavevmode\nobreak\ .$ (1.11)

Setting $K=k+2$, this is the well-known Verlinde formula for the pure CS
theory $SU(2)_{k}$ – equivalently, it gives the number of conformal blocks of
the $SU(2)_{k}$ WZW model Verlinde:1988sn  (see also beauville1996conformal ;
alekseev2000formulas ; Blau:1993tv ; Dowker1992 ; Schottenloher2008  and
references therein).

Higher-form symmetries and their anomalies. The 1-form symmetry $\Gamma_{\rm
3d}^{(1)}=\mathbb{Z}_{2}$ of the 3d CS theory is realised as
$\mathbb{Z}_{2}^{(1)}\oplus\mathbb{Z}_{2}^{(0)}$ in the 2d description. Let us
now explain how these two symmetries are manifested in the $A$-model. First of
all, the non-trivial charge operator for $\mathbb{Z}_{2}^{(1)}$ is a local
topological operator. As we will explain later on, it is given by a square
root of the gauge flux operator, which we thus denote as:

$\Pi(u)^{\frac{1}{2}}=-x^{K}\leavevmode\nobreak\ .$ (1.12)

Here, the sign of the square root is chosen to match know results. Inserting
this operator on $\Sigma_{g}$, we obtain:

$\langle\Pi^{\frac{1}{2}}\rangle_{\Sigma_{g}}=\sum_{l=1}^{K-1}(-1)^{l+1}\left[\sqrt{2\over
K}\sin\left(\frac{\pi l}{K}\right)\right]^{2-2g}\leavevmode\nobreak\ .$ (1.13)

Note that this vanishes if $K$ is odd. In particular, the insertion on the
torus gives us:

$\langle\Pi^{\frac{1}{2}}\rangle_{T^{2}}=\begin{cases}1\qquad&\text{for}\,K\,\text{even}\leavevmode\nobreak\
,\\\ 0\qquad&\text{for}\,K\,\text{odd}\leavevmode\nobreak\ .\end{cases}$
(1.14)

Let $\mathcal{T}$ denote the $A$-model for this theory. The gauging of
$\mathbb{Z}_{2}^{(1)}$ consists simply of summing over insertions of the flux
operator:

$Z_{\Sigma_{g}}[SU(2)_{K}/\mathbb{Z}_{2}^{(1)}]={\frac{1}{2}}\left(\langle
1\rangle_{\Sigma_{g}}+\langle\Pi^{\frac{1}{2}}\rangle_{\Sigma_{g}}\right)=\sum_{j=0}^{{\left\lfloor{K-2\over
2}\right\rfloor}}\left[\sqrt{2\over
K}\sin\left(\frac{\pi(2j+1)}{K}\right)\right]^{2-2g}\leavevmode\nobreak\ .$
(1.15)

When gauging, we have the freedom of introducing a coupling of the
$\mathbb{Z}_{2}^{(1)}$ gauge field $B$ to a background gauge field $\theta$
for the so-called quantum $(-1)$-form symmetry, as we will review; here we
chose $\theta=0$ for simplicity of presentation.

The zero-form symmetry $\mathbb{Z}_{2}^{(0)}$ is generated by topological line
operators $\mathcal{U}^{\gamma}$, for $\gamma\in\mathbb{Z}_{2}^{(0)}$. To
study the action of the $0$-form symmetry on $\mathcal{H}_{S^{1}}$, we first
consider the $A$-model on a cylinder. We thus have a single generator
$\mathcal{U}^{\gamma}(\mathcal{C})$ wrapping the cylinder, with the fusion
$\mathcal{U}^{2}={\bf 1}$, and we find that:

$\mathcal{U}(\mathcal{C})\lvert\hat{u}_{l}\rangle=\lvert\hat{u}_{K-l}\rangle\leavevmode\nobreak\
.$ (1.16)

Note that $\mathcal{U}(\mathcal{C})$ has a single fixed point,
$\lvert\hat{u}_{K\over 2}\rangle$, if and only if $K$ is even. The $SU(2)_{K}$
3d $\mathcal{N}=2$ CS theory has a ’t Hooft anomaly ${K\over 2}$ (mod $1$) for
$\Gamma_{\rm 3d}^{(1)}=\mathbb{Z}_{2}$ – that is, the $\mathbb{Z}_{2}$ one-
form symmetry is anomalous if $K$ is odd Gaiotto:2014kfa . In the 3d
$A$-model, this manifests itself as a mixed
$\mathbb{Z}_{2}^{(1)}$-$\mathbb{Z}_{2}^{(0)}$ anomaly. The topological
operators acting on $\mathcal{H}_{S^{1}}$ satisfy the twisted commutation
relation:

$\mathcal{U}(\mathcal{C})\,\Pi^{\frac{1}{2}}=(-1)^{K}\Pi^{\frac{1}{2}}\,\mathcal{U}(\mathcal{C})\leavevmode\nobreak\
,$ (1.17)

which follows from (1.16) and the fact that
$\Pi^{\frac{1}{2}}\lvert\hat{u}\rangle=(-1)^{l+1}\lvert\hat{u}\rangle$.

Let us now set $g=1$, with the $\mathcal{C}$ being one generator of
$H_{1}(T^{2},\mathbb{Z})\cong\mathbb{Z}^{2}$ and $\widetilde{\mathcal{C}}$
denoting the other generator (the Euclidean time direction). Inserting a
topological line along $\mathcal{C}$, we find:

$\langle\mathcal{U}(\mathcal{C})\rangle_{T^{2}}=\sum_{l=1}^{K-1}\langle\hat{u}_{l}\rvert\mathcal{U}(\mathcal{C})\lvert\hat{u}_{l}\rangle=\sum_{l=1}^{K-1}\langle\hat{u}_{l}|\mathopen{}\hat{u}_{K-l}\rangle=\begin{cases}1\quad&\text{if}\;K\;\text{is
even}\leavevmode\nobreak\ ,\\\ 0\quad&\text{if}\;K\;\text{is
odd}\leavevmode\nobreak\ .\end{cases}$ (1.18)

Finally, the insertion of a topological line operator along
$\widetilde{\mathcal{C}}\subset T^{2}$ corresponds to a trace over the non-
trivial twisted-sector Hilbert space, which arises from the Bethe vacuum
$\hat{u}_{K\over 2}$ that preserves $\mathbb{Z}_{2}^{(0)}$ – such a twisted
sector only exists for $K$ even. We can compute:

$\langle\mathcal{U}(\widetilde{\mathcal{C}})\rangle_{T^{2}}=\langle\mathcal{U}(\mathcal{C})\rangle_{T^{2}}=\langle\Pi^{\frac{1}{2}}\rangle_{T^{2}}\leavevmode\nobreak\
.$ (1.19)

The first equality is expected from modular invariance in 2d. Moreover, the
second equality is also a consequence of modular invariance for the 3d
Chern–Simons theory on $T^{3}$, and it is one way to fix the sign in (1.12).
Similarly, if we insert both the $\mathbb{Z}_{2}^{(1)}$ operator
$\Pi^{{\frac{1}{2}}}$ and $\mathbb{Z}_{2}^{(0)}$ operator
$\mathcal{U}(\mathcal{C})$ on $T^{2}$, we find:

$\langle\Pi^{\frac{1}{2}}\mathcal{U}(\mathcal{C})\rangle_{T^{2}}=\sum_{l}\Pi^{\frac{1}{2}}(\hat{u}_{l})\langle\hat{u}_{l}|\mathopen{}\hat{u}_{l}+\tfrac{1}{2}\rangle=\begin{cases}(-1)^{K-2\over
2}\quad&\text{if}\;K\;\text{is even\leavevmode\nobreak\ ,}\\\
0\quad&\text{if}\;K\;\text{is odd\leavevmode\nobreak\ .}\end{cases}$ (1.20)

Note that this vanishes if and only if the theory is anomalous, which is the
expected result given the commutator (1.17). Finally, we note that:

$\langle\Pi^{\frac{1}{2}}\mathcal{U}(\mathcal{C})\rangle_{T^{2}}=(-1)^{K-2\over
2}\langle\mathcal{U}(\mathcal{C})\rangle_{T^{2}}\leavevmode\nobreak\ ,$ (1.21)

for $K$ even. From the point of view of the 3d CS theory on $T^{3}$, we would
naively have expected that relation to hold with a trivial sign, but this only
happens if ${K\over 2}$ is odd. For ${K\over 2}$ even, we have a modular
anomaly on $T^{3}$, which is a consequence of a mixed anomaly between the 3d
one-form symmetry and gravity. We will discuss this subtle and important point
in section 3.2, in the context of the $SU(N)_{K}$ $\mathcal{N}=2$ CS theory.

The $SO(3)_{K}$ Witten index. Let us now assume that $K$ is even, so that we
can gauge both $\mathbb{Z}_{2}^{(1)}$ and $\mathbb{Z}_{2}^{(0)}$. Let us first
consider $\Sigma_{g}=T^{2}$. This should give us the Witten index of the
$SO(3)_{K}$ $\mathcal{N}=2$ supersymmetric CS theory. The gauging corresponds
to inserting all possible topological operators for
$\mathbb{Z}_{2}^{(1)}\times\mathbb{Z}_{2}^{(0)}$ on $T^{2}$. We thus find:

$\displaystyle Z_{T^{2}}[SO(3)_{K}]$ $\displaystyle=$
$\displaystyle\;\frac{1}{4}\sum_{n,n^{\prime},n^{\prime\prime}\in\mathbb{Z}_{2}}\left\langle\mathcal{U}(\widetilde{\mathcal{C}})^{n}\mathcal{U}(\widetilde{\mathcal{C}})^{n^{\prime}}\Pi^{n^{\prime\prime}\over
2}\right\rangle_{T^{2}}\leavevmode\nobreak\ $ (1.22) $\displaystyle=$
$\displaystyle\;{1\over 4}\left(K-1+3+1+3(-1)^{K-2\over
2}\right)=\begin{cases}{K\over 4}+{3\over 2}\qquad&\text{for }\,{K\over
2}\,\text{ odd\leavevmode\nobreak\ ,}\\\ {K\over 4}\qquad&\text{for }\,{K\over
2}\,\text{ even\leavevmode\nobreak\ .}\\\ \end{cases}$

It is illuminating to compare this result to the non-supersymmetric
$SO(3)_{k}$ CS theory (with $k=K-2$). The above Witten index should give us
the number of states of the $SO(3)_{k}$ theory on $T^{2}$, which then reads:

$Z_{T^{3}}[\mathcal{N}=0\;SO(3)_{k}]=\begin{cases}{k\over 4}+2\qquad&\text{for
}\,{k\over 2}\,\text{ even\leavevmode\nobreak\ ,}\\\ {k\over 4}+{1\over
2}\qquad&\text{for }\,{k\over 2}\,\text{ odd\leavevmode\nobreak\ .}\\\
\end{cases}$ (1.23)

We have a bosonic 3d TQFT for $k\in 4\mathbb{Z}$, while $SO(3)_{k}$ with
$k+2\in 4\mathbb{Z}$ is a spin-TQFT Dijkgraaf:1989pz . Then (1.23) should also
correspond to the number of $SO(3)_{k}$ Wilson lines. The 3d one-form gauging
$SU(2)_{k}\rightarrow SO(3)_{k}=SU(2)_{k}/\mathbb{Z}_{2}$ can be performed
directly as follows. There are $k+1$ $SU(2)_{k}$ Wilson lines, or anyons, for
the $SU(2)$ representations of spin $j=0,\cdots,{k\over 2}$, namely:

$\mathcal{N}=0\;\;SU(2)_{k}\quad:\quad\\{W_{j}\\}=\\{{\bf
1}\leavevmode\nobreak\ ,\,W_{\frac{1}{2}}\leavevmode\nobreak\
,\,W_{1}\leavevmode\nobreak\ ,\,\cdots\leavevmode\nobreak\ ,W_{k\over
2}\\}\leavevmode\nobreak\ ,$ (1.24)

with the fusion rules Gepner:1986wi :

$W_{j_{1}}W_{j_{2}}=\sum_{j=|j_{1}-j_{2}|}^{{\rm
min}(j_{1}+j_{2},k-j_{1}-j_{2})}W_{j}\leavevmode\nobreak\ .$ (1.25)

The lines (1.24) include exactly two abelian anyons, ${\bf 1}$ and $a\equiv
W_{k\over 2}$, with fusion $a^{2}={\bf 1}$, which are therefore the
$\mathbb{Z}_{2}^{(1)}$ topological lines in 3d. We have:

$(\mathbb{Z}_{2}^{(1)})_{\rm
3d}\,:\,W_{j}\rightarrow(-1)^{2j}W_{j}\leavevmode\nobreak\ ,\quad\qquad
aW_{j}=W_{{k\over 2}-j}\leavevmode\nobreak\ ,$ (1.26)

for the action of $a$ on $W_{j}$ by linking and for the fusion, respectively.
The gauging of $(\mathbb{Z}_{2}^{(1)})_{\rm 3d}$ consists of three steps
Hsin:2018vcg . First, we discard all lines which are not invariant under
$(\mathbb{Z}_{2}^{(1)})_{\rm 3d}$, leaving us with $\\{W_{j}\\}$,
$j\in\mathbb{Z}$. Secondly, we identify the lines $W$ and $aW$. Thirdly, any
line that is a fixed point of the fusion with $a$ gives rise to two distinct
lines in the gauged theory. The fixed-point line is $W_{k\over 4}$, and thus
only survives the first step of gauging for ${k\over 2}$ even. We thus find
the $SO(3)_{k}$ lines:

$\mathcal{N}=0\;\;SO(3)_{k}\quad:\quad\begin{cases}\\{{\bf
1}\leavevmode\nobreak\ ,\,W_{1}\leavevmode\nobreak\
,\,W_{2}\leavevmode\nobreak\ ,\cdots\leavevmode\nobreak\ ,\,W_{{k\over
4}-1}\leavevmode\nobreak\ ,\,W_{{k\over 4};(1)}\leavevmode\nobreak\
,\,W_{{k\over 4};(2)}\\}\qquad&\text{for }{k\over 2}\text{ even,}\\\ \\{{\bf
1}\leavevmode\nobreak\ ,\,W_{1}\leavevmode\nobreak\
,\,W_{2}\leavevmode\nobreak\ ,\cdots\leavevmode\nobreak\ ,\,W_{{k\over
4}-{1\over 2}}\\}\qquad&\text{for }{k\over 2}\text{ odd\leavevmode\nobreak\
.}\\\ \end{cases}$ (1.27)

This precisely reproduces (1.23).

The higher-genus twisted index for $SO(3)_{K}$. We can generalise the
computation (LABEL:SO(3)-index) to obtain the $SO(3)_{K}$ twisted index for
any $\Sigma_{g}$, for $K$ even. First, we find that the insertion of the non-
trivial $\mathbb{Z}_{2}^{(0)}$ line along any generator $\mathcal{C}_{i}$ of
$H_{1}(\Sigma_{g},\mathbb{Z})\cong\mathbb{Z}^{2g}$ gives us:

$\langle\mathcal{U}(\mathcal{C}_{i})\rangle_{\Sigma_{g}\times
S^{1}}=\sum_{l=1}^{K-1}\langle\hat{u}_{l}\rvert\mathcal{U}(\mathcal{C}_{i})\mathcal{H}^{g-1}\lvert\hat{u}_{l}\rangle=\mathcal{H}(\hat{u}_{K\over
2})^{g-1}=\left({K\over 2}\right)^{g-1}\leavevmode\nobreak\ .$ (1.28)

Moreover, the insertion of any non-trivial set of $\mathbb{Z}_{2}^{(0)}$ lines
on $\Sigma_{g}$ is equal to (1.28), due to invariance under large
diffeomorphism. We similarly find that:

$\langle\mathcal{U}(\mathcal{C}_{i})\Pi^{\frac{1}{2}}\rangle_{\Sigma_{g}\times
S^{1}}=(-1)^{K-2\over 2}\left({K\over 2}\right)^{g-1}\leavevmode\nobreak\ .$
(1.29)

The $\Sigma_{g}\times S^{1}$ partition function for $SO(3)_{K}$ is obtained
as:

$Z_{\Sigma_{g}}[SO(3)_{K}]={1\over
2^{2g}}\sum_{n^{\prime}\in\mathbb{Z}_{2}}\sum_{(n_{i})\in\mathbb{Z}_{2}^{2g}}\left\langle\Pi^{n^{\prime}\over
2}\,\prod_{i=1}^{2g}\mathcal{U}(\mathcal{C}_{i})^{n_{i}}\right\rangle_{\Sigma_{g}\times
S^{1}}\leavevmode\nobreak\ ,$ (1.30)

which gives us:

$Z_{\Sigma_{g}}[SO(3)_{K}]={K^{g-1}\over 2^{3g-1}}\left(2\sum_{j=0}^{{K\over
2}-1}\left[\sin\left(\pi(2j+1)\over
K\right)\right]^{2-2g}+(2^{2g}-1)\left(1+(-1)^{K-2\over
2}\right)\right)\leavevmode\nobreak\ .$ (1.31)

This matches known results. In the case where ${K\over 2}$ is odd, so that
$k=K-2\in 4\mathbb{Z}$, (1.30) is in perfect agreement with the number of
conformal blocks for the $SO(3)_{k}$ WZW model – see e.g.
beauville1996verlinde  for a mathematical derivation. In the spin-TQFT case,
$k+2\in 4\mathbb{Z}$, our result can also be recovered from the corresponding
fermionic WZW model; for instance, WZW$[SO(3)_{2}]$ is a free CFT
Dijkgraaf:1989pz  (equivalent to three Majorana fermions), which explains why
we find $Z_{\Sigma_{g}\times S^{1}}=1$ for any $g$ when setting $K=4$ in
(1.31).

###  1.2 Results for general $G$ and for the $SU(N)_{K}$ gauge theory

In the rest of this introduction, we summarise the main results of this paper,
essentially in the order in which they will be presented.

One-form symmetries in the 3d $\bm{A}$-model. Given a 3d $\mathcal{N}=2$ gauge
theory with gauge group $\widetilde{G}$ compactified on $S^{1}$, as discussed
above, its one-form symmetry $\Gamma_{\rm 3d}^{(1)}$ descends to both a one-
form symmetry and a zero-form symmetry in 2d, denoted by $\Gamma^{(1)}$ and
$\Gamma^{(0)}$, respectively. Thus, we must first understand how these two
distinct symmetries act on the $A$-model Hilbert space. For any 2d TQFT, this
question was recently addressed in Gukov:2021swm . The one-form symmetry
$\Gamma^{(1)}$ acts on the ground states by a phase, and the zero-form
symmetry $\Gamma^{(0)}$ acts by permutation. Indeed, we have willett ;
Eckhard:2019jgg :

$\Gamma^{(1)}\,:\,|\hat{u}\rangle\longrightarrow\Pi(\hat{u})^{\gamma}|\hat{u}\rangle\leavevmode\nobreak\
,\qquad\qquad\qquad\Gamma^{(0)}\,:\,|\hat{u}\rangle\longrightarrow|\hat{u}+\gamma\rangle\leavevmode\nobreak\
,$ (1.32)

where $\gamma\in\Gamma$ is a group element. Here $\Pi(\hat{u})^{\gamma}\in{\rm
Hom}(\Gamma^{(1)},U(1))$ is a phase, and $|\hat{u}+\gamma\rangle$ denotes the
new vacuum under the action of $\gamma\in\Gamma^{(0)}$. We will explain these
transformations and their consequences in much detail throughout this work.

Focusing on $\mathcal{M}_{3}=\Sigma\times S^{1}$, the 0-form symmetry
$\Gamma^{(0)}$ along $\Sigma$ is implemented by topological lines, while the
1-form symmetry $\Gamma^{(1)}$ is implemented by the topological point
operators that arise from the 3d topological lines wrapping $S^{1}$. The Bethe
vacua diagonalise the $\Gamma^{(1)}$ topological operators $\Pi^{\gamma}$, as
shown in (1.32), while the 0-form symmetry $\Gamma^{(0)}$ is (partially)
spontaneously broken in most 2d vacua. In particular, the Bethe vacua organise
themselves into orbits of $\Gamma^{(0)}$.

’t Hooft anomalies. The three-dimensional one-form symmetry can be anomalous,
as the topological lines can themselves be charged under $\Gamma_{\rm
3d}^{(1)}$. This results in a mixed $\Gamma^{(1)}$-$\Gamma^{(0)}$ anomaly in
the $A$-model on $\Sigma$. This ’t Hooft anomaly implies general constraints
on the structure of the orbits of Bethe vacua under the action of
$\Gamma^{(0)}$. For instance, some Bethe vacua can be fixed under the whole of
$\Gamma^{(0)}$ if and only if the anomaly vanishes. This follows from the fact
that $\Gamma^{(1)}$ cannot be spontaneously broken in the 2d vacuum
Gaiotto:2014kfa , and thus the mixed anomaly must be matched by distinct 2d
symmetry-protected topological (SPT) phases for $\Gamma^{(1)}$ in the
$A$-model description.

We will also see that the 3d ’t Hooft anomaly is entirely determined by the
bare Chern–Simons levels $K$ of the UV gauge theory. We further comment on the
structure of the $(d+1)$-dimensional anomaly theory, in both the $d=3$ and
$d=2$ descriptions, in appendix C.

In the explicit $\widetilde{G}=SU(N)_{K}$ example at the core of this paper, a
key role will also be played by a mixed ’t Hooft anomaly between gravity and
the 1-form symmetry in 3d.

Background gauge fields. The partition function of the $\widetilde{G}$ gauge
theory on $\Sigma_{g}\times S^{1}$ can be found by inserting powers of the
handle-gluing operator $\mathcal{H}$ on $\Sigma_{g}=T^{2}$ Nekrasov:2014xaa .
We wish to compute the topologically twisted index in the presence of
topological operators for the discrete symmetry $\Gamma$, which is equivalent
to turning on some background gauge fields for
$\Gamma^{(1)}\times\Gamma^{(0)}$. Schematically, we have:

$Z_{\Sigma_{g}\times
S^{1}}(B,C)=\left\langle\Pi^{\gamma^{(1)}}\mathcal{U}^{\gamma^{(0)}}\right\rangle_{\Sigma_{g}}\leavevmode\nobreak\
.$ (1.33)

Here, $B\in H^{2}(\Sigma_{g},\Gamma^{(1)})$ and $C\in
H^{1}(\Sigma_{g},\Gamma^{(0)})$ are the background gauge fields, while
$\Pi^{\gamma^{(1)}}$ for $\gamma^{(1)}\in\Gamma^{(1)}$ and
$\mathcal{U}^{\gamma^{(0)}}$ for $\gamma^{(0)}\in\Gamma^{(0)}$ denote the
point and line operators, respectively, corresponding to the background gauge
fields. We explicitly evaluate (1.33) using the 2d TQFT point of view. This
simply corresponds to using the pair-of-pants decomposition of the Riemann
surface with appropriate lines inserted. This prescription gives unambiguous
results for insertions of the $\Gamma^{(0)}$ lines. The insertion of the
$\Gamma^{(1)}$ operator is slightly ambiguous, as the flux operator
$\Pi^{\gamma^{(1)}}$ is only defined up a certain root of unity. In all
examples we will consider, we will be able to fix this ambiguity by demanding
3d modularity for the expectations values of elementary topological lines at
$g=1$, namely on $\mathcal{M}_{3}=T^{3}$.

Discrete gauging. Gauging the zero-form and one-form symmetries corresponds to
summing over all possible background gauge fields Gaiotto:2014kfa . Note that,
in the $A$-model description, we can gauge $\Gamma^{(1)}$ and $\Gamma^{(0)}$
separately. We also keep track of the background gauge fields for the dual
$(-1)$-form and $0$-form symmetries, respectively. We are careful to fix the
overall normalisation of the partition functions for the gauged theories in
order to preserve the 2d Hilbert-space interpretation in the gauged theories.

The $\Gamma^{(1)}$ symmetry induces a so-called decomposition Hellerman:2006zs
; Sharpe:2022ene  of the $A$-model into disconnected sectors, also called
‘universes’, and the gauging of the $1$-form symmetry projects us onto one
particular sector. These sectors are indexed by $\theta$-angles which are the
background gauge fields for the dual $(-1)$-form symmetry. The gauging of the
$\Gamma^{(0)}$ symmetry on $\Sigma_{g}$ is a standard orbifolding procedure
familiar from string theory: in canonical quantisation on $T^{2}$, we identify
the states related by $\Gamma^{(0)}$ while also adding in the twisted sectors
states induced by non-trivial stabiliser subgroups. Overall, our general
analysis closely follows previous discussions obtained in the 2d TQFT
framework willett ; Gukov:2021swm ; the main improvement in our analysis is
that we carefully study the Hilbert spaces spanned by the Bethe states before
and after the gauging, and that we elucidate how ’t Hooft anomalies constrain
the structure of 2d ground states. We also keep track of all the dual
symmetries that arise upon gauging.

Symmetries of the $\bm{\mathcal{N}=2}$ $\bm{SU(N)_{K}}$ CS theory. We apply
the general descriptions of the higher-form symmetries of the 3d $A$-model to
the pure $\mathcal{N}=2$ supersymmetric CS theory with gauge group $SU(N)$ at
CS level $K$. We study this theory in depth in section 3. In this example, we
can compare the results obtained using our formalism to many previous results
in the literature, and we find perfect agreement. We also obtain a number of
results that are new, to the best of our knowledge.

The Bethe equations for $SU(N)_{K}$ have $\binom{K-1}{N-1}$ solutions, which
gives us the Witten index $\textbf{I}_{\text{W}}[SU(N)_{K}]$ Ohta:1999iv .
This 3d CS theory has a $\mathbb{Z}_{N}^{(1)}$ one-form symmetry, the center
symmetry, with a ’t Hooft anomaly given by $K$ modulo $N$. The 3d $1$-form
symmetry descends to $\Gamma^{(1)}=\mathbb{Z}_{N}^{(1)}$ and
$\Gamma^{(0)}=\mathbb{Z}_{N}^{(0)}$ in the $A$-model. While the
$\mathbb{Z}_{N}^{(1)}$ symmetry acts on the Hilbert space by $N$-th roots of
unity, the $\mathbb{Z}_{N}^{(0)}$ symmetry organises the Bethe vacua into
orbits whose lengths are divisors of $N$. There exists a unique fixed point if
and only if the t’ Hooft anomaly vanishes.

The gauging pattern of the $\mathbb{Z}_{N}$ centre symmetry is determined by
the subgroups of $\mathbb{Z}_{N}$, which are labelled by the divisors $d$ of
$N$. For instance, for $N$ a prime number and with $K\in N\mathbb{Z}$, so that
the non-anomalous $\mathbb{Z}_{N}$ has no non-trivial subgroups, we easily
determine the 3d Witten index of the fully gauged theory $PSU(N)_{K}\equiv
SU(N)_{K}/\mathbb{Z}_{N}$ to be:

$\textbf{I}_{\text{W}}[PSU(N)_{K}]=\frac{1}{N^{2}}\left[\binom{K-1}{N-1}-1+N^{3}\right]\leavevmode\nobreak\
.$ (1.34)

This index is always an integer, since it can be written as a trace over the
$PSU(N)_{K}$ Hilbert space on $T^{2}$, and we can prove this fact using
number-theoretic identities. We find a much richer structure for general
values of $N$, since we can consider the discrete gauging of any non-anomalous
subgroup $\mathbb{Z}_{r}\subset\mathbb{Z}_{N}$. We study in detail the
insertion of all topological operators on $\Sigma_{g}\times S^{1}$, for all
values of $N$ and $K$.

$\bm{SU(N)_{K}}$ at genus 1. The Witten index for the gauged theory can be
obtained by enumerating the fixed points under all non-anomalous subgroups
$\mathbb{Z}_{d}\subseteq\mathbb{Z}_{N}$. Three-dimensional modularity implies
many relations amongst the expectation values of topological lines inserted on
1-cycles of $T^{3}$. These relations can be affected by a 3d modular anomaly
which we determine explicitly. This anomaly is a sign that only depends on the
arithmetic properties of $N$, $K$ and $d$, we discuss its the relation to spin
structures, quantisation conditions of WZW models and mixed 1-form/gravity
anomalies. For instance, this allows us to determine explicitly the
$PSU(N)_{K}$ Witten index for any $N$ and for $K\in N\mathbb{Z}$, as a sum
over divisors of $N$:

$\textbf{I}_{\text{W}}[PSU(N)_{K}]=\frac{1}{N^{2}}\sum_{d|N}\mathscr{J}_{3}^{N,K}(d)\binom{\frac{K}{d}-1}{\frac{N}{d}-1}\leavevmode\nobreak\
.$ (1.35)

Here, $\mathscr{J}_{3}^{N,K}$ is a refinement of Jordan’s totient function
$J_{3}$ which takes into account the 3d modular anomaly.111The Jordan’s
totient function itself is a generalisation of the more familiar Euler totient
$\varphi$, also known as Euler’s $\varphi$ function. The functions $J_{k}$
(for some integer $k$) can be understood as enumeration functions of
interactions of $k$ instances of $\mathbb{Z}_{N}$ groups, similar to how
$\varphi(d)$ counts the number of generators of $\mathbb{Z}_{d}$.

A generalisation to the discrete gauging of a non-anomalous subgroup
$\mathbb{Z}_{r}\subset\mathbb{Z}_{N}$ is straightforward, by truncating the
sum (1.35) to the corresponding divisors. This thus computes the Witten index
of the 3d $\mathcal{N}=2$ CS theories for all groups $SU(N)/\mathbb{Z}_{r}$
whose simply-connected cover is $SU(N)$. It can be checked that the index
$\textbf{I}_{\text{W}}[SU(N)_{K}/\mathbb{Z}_{r}]$ is indeed an integer, as
expected. Including a $\theta$-angle for $\Gamma^{(-1)}\cong\mathbb{Z}_{r}$
provides an interesting refinement Gaiotto:2014kfa ; Aharony:2013hda ;
Hsin:2020nts ; Gaiotto:2017yup , and probes additional arithmetic properties
of the integer $N$. We find that this modifies (1.35) by a further
generalisation of Jordan’s totient.

$\bm{SU(N)_{K}}$ at genus $\bm{g}$. Riemann surfaces of arbitrary genus $g$
admit $2g$ elementary topological operators associated with the zero-form
symmetry $\Gamma^{(0)}\cong\mathbb{Z}_{N}$. By inserting the handle-gluing
operator $\mathcal{H}$, we find that the genus-$g$ partition function for
$K\in N\mathbb{Z}$ can be written as:

$Z_{\Sigma_{g}\times
S^{1}}[PSU(N)_{K}^{\theta_{s}}]=\frac{1}{N^{2g-1}}\sum_{d|N}J_{2g}(d)\\!\\!\\!\sum_{\hat{u}\in\mathcal{S}_{\text{BE}}^{\left(\frac{N}{d}\gamma_{0}\right)}\cap\,\mathcal{S}_{\text{BE}}^{\vartheta_{s}}}\mathcal{H}(\hat{u})^{g-1}\leavevmode\nobreak\
,$ (1.36)

where $\gamma_{0}$ is a generator of $\mathbb{Z}_{N}$,
$\mathcal{S}_{\text{BE}}^{(\gamma)}$ is the set of Bethe vacua fixed under the
action of $\gamma\in\mathbb{Z}_{N}$, and
$\mathcal{S}_{\text{BE}}^{\vartheta_{s}}$ is the set of Bethe vacua that span
the smaller ‘universe’ indexed by the $\theta$-angle $\theta_{s}$. The
genus-$g$ Jordan totient $J_{2g}$ has previously appeared in the study of
Verlinde bundles over curves Oprea+2011+181+217 ; alekseev2000formulas ;
beauville1998picard ; osti_10067008 ; oprea2010note . Our result (1.36)
generalises various partial results in the literature Oprea+2011+181+217 ;
Gukov:2021swm ; alekseev2000formulas ; Meinrenken2018 .

Comparison with abelian anyon condensation. In the $SU(2)_{K}$ warm-up example
of section 1.1, it was demonstrated that the one-form gauging can be performed
directly in three dimensions using the 3d TQFT approach, and that the result
on $T^{3}$ agrees with the $SO(3)_{K}$ Witten index obtained by gauging both
zero-form and one-form symmetries in 2d. In section 4.1, we revisit the
calculation of the $PSU(N)_{K}$ Witten index from one-form gauging in the 3d
TQFT description, also known as abelian anyon condensation. The $SU(N)_{K}$
$\mathcal{N}=2$ supersymmetric CS theory has a spectrum of Wilson lines
$W_{\bm{\lambda}}$ indexed by Young tableaux $\bm{\lambda}$. We discuss their
fusion rules and identify the abelian anyons (i.e. $\mathbb{Z}_{N}$
topological lines), which allows us to implement the three-step gauging
procedure discussed in Hsin:2018vcg . For $N$ prime, one can understand each
step as a simple modification of the $SU(N)_{K}$ index, and one recovers
(1.34) as a result. For arbitrary values of $N$, the fusion with the abelian
anyons furnishes a group action on the $SU(N)_{K}$ Wilson lines, and the 3d
gauging partitions them into orbits. We express the $PSU(N)_{K}$ index as a
sum over those orbits, and find precise numerical agreement with (1.35) for
small values of $N$ and $K$.222The exact equality between the results obtained
using the two distinct methods is left as a conjecture.

Including matter. While we focussed on the pure $SU(N)_{K}$ CS theories in
this work, the inclusion of chiral multiplets is straightforward. In general,
the one-form symmetry of the $\widetilde{G}$ theory is the discrete subgroup
of the center of $\widetilde{G}$ that is preserved by the matter content. In
section 5, we briefly study $U(1)_{k}$ theories with matter, as well as the
$SU(N)_{k}$ Chern–Simons theory with adjoint matter. Models with an adjoint
chiral multiplet are of particular interest since they provide equivariant
generalisations of the Verlinde formula Gukov:2015sna ; Gukov:2021swm , as has
been studied in the past from various perspectives – see e.g. Okuda:2013fea ;
Nekrasov:2014xaa ; Gukov:2015sna ; Benini:2016hjo ; Closset:2016arn ;
Kanno:2018qbn ; Ueda:2019qhg ; halpern2016equivariant .

This paper is organised as follows. In section 2, we present the general
results on one-form symmetries of 3d $\mathcal{N}=2$ supersymmetric gauge
theories on $\Sigma_{g}\times S^{1}$. In sections 3 and 4, we study in much
detail the 3d $\mathcal{N}=2$ $SU(N)_{K}$ Chern–Simons theory. In section 5,
we briefly discuss $U(1)$ and $SU(N)$ $\mathcal{N}=2$ gauge theories with
matter. Section 6 presents our conclusions and outlook. Some useful review
material and the more technical computations are relegated to several
appendices.

##  2 One-form symmetries in the 3d $A$-model

Consider a 3d $\mathcal{N}=2$ supersymmetric field theory with a discrete one-
form symmetry $\Gamma_{\rm 3d}^{(1)}$. In this work, we will specifically
study 3d $\mathcal{N}=2$ gauge theories, and the one-form symmetry will be
(part of) the center symmetry for some gauge group $G$. For instance, a gauge
theory with gauge group $SU(N)$ and matter fields transforming in the adjoint
representation will have a one-form symmetry $\Gamma_{\rm
3d}^{(1)}=\mathbb{Z}_{N}$.

###  2.1 Discrete symmetries in the 2d description

The 3d $A$-model (in the terminology of Closset:2017zgf ) is a two-dimensional
topologically-twisted $\mathcal{N}=(2,2)$ supersymmetric field theory that
captures all the information about the twisted chiral ring $\mathcal{R}^{\rm
3d}$ of the 3d $\mathcal{N}=2$ theory compactified on a circle, $S^{1}_{A}$,
including twisted indices Nekrasov:2014xaa ; Benini:2015noa ; Benini:2016hjo ,
correlation functions of half-BPS lines wrapping $S^{1}_{A}$ Kapustin:2013hpk
; Closset:2016arn , and supersymmetric partition functions on Seifert
3-manifolds Closset:2018ghr ; Blau:2018cad ; Closset:2019hyt ; Fan:2020bov .

Let us start by explaining how the 3d one-form symmetry manifests itself in
the two-dimensional description. We start with the 3d theory on $\Sigma\times
S^{1}_{A}$, with the topological $A$-twist along a two-manifold $\Sigma$. We
will mostly consider $\Sigma=\Sigma_{g}$, a compact closed Riemann surface of
genus $g$. Here, translation along the circle $S^{1}_{A}$ is part of the 3d
$A$-model supersymmetry algebra, and the corresponding momentum is the Kaluza-
Klein (KK) charge Closset:2017zgf . In the 2d description on $\Sigma$, it
manifests itself as a $U(1)_{\rm KK}$ symmetry. In three dimensions, the
operators charged under $\Gamma^{(1)}_{\text{3d}}$ are line operators, which
are acted on by topological line operators Gaiotto:2014kfa .333Hence, the
invertible topological lines that implement the one-form symmetry are a subset
of the set of all lines operators of the 3d theory, and symmetry operators can
themselves be charged, as we will review. The 3d one-form symmetry
$\Gamma_{\rm 3d}^{(1)}\cong\Gamma$, for $\Gamma$ a finite abelian group,
manifests itself as two distinct symmetries in 2d, namely a one-form and a
zero-form symmetry:

$\Gamma_{\rm
3d}^{(1)}\quad\qquad\longrightarrow\qquad\quad\Gamma^{(1)}\cong\Gamma\leavevmode\nobreak\
,\qquad\Gamma^{(0)}\cong\Gamma\leavevmode\nobreak\ .$ (2.1)

Here, the abelian zero-form symmetry $\Gamma^{(0)}$ action is generated by
topological lines with support on one-cycles $\mathcal{C}\subset\Sigma$, which
act on local operators on $\Sigma$. These 2d local operators correspond to the
3d lines wrapping $S^{1}_{A}$. We will be particularly interested in such
operators that also commute with the $A$-model supercharge – these are the
twisted chiral operators ${\mathscr{L}}\in\mathcal{R}^{\rm 3d}$. The 2d one-
form symmetry $\Gamma^{(1)}$, on the other hand, is generated by topological
point operators that arise as the 3d topological lines wrapping $S^{1}_{A}$.
We shall denote the topological line operators that implement $\Gamma^{(0)}$
by:

$\mathcal{U}^{\gamma}(\mathcal{C})\leavevmode\nobreak\ ,$ (2.2)

and the topological point operators that implement $\Gamma^{(1)}$ by:

$\Pi^{\gamma}(p)\equiv\mathcal{U}^{\gamma}(S^{1}_{A})\leavevmode\nobreak\ ,$
(2.3)

for every $\gamma\in\Gamma$. These topological operators are depicted in
figure 1. The support of $\Pi^{\gamma}$ at a point $p\in\Sigma$ will be kept
implicit in the following.

(a)

(b)

Figure 1: Depiction of the $\Gamma^{(1)}$ and $\Gamma^{(0)}$ symmetry
operators $\mathcal{U}^{\gamma}(S_{A}^{1})\equiv\Pi^{\gamma}$ and
$\mathcal{U}^{\gamma}(\mathcal{C})$, respectively, on $\Sigma\times
S^{1}_{A}$. Here, $\mathcal{U}^{\gamma}(S_{A}^{1})$ wraps the $S^{1}_{A}$
factor in the 3d geometry, depicted here by a small circle, and it is thus a
local operator on $\Sigma$. The topological line
$\mathcal{U}^{\gamma}(\mathcal{C})$ is supported on a cycle $\mathcal{C}$ on
$\Sigma$, in general. Here $\Sigma$ is the cylinder, which is the relevant
configuration to discuss the Hilbert space $\mathcal{H}_{S^{1}}$ of the
$A$-model.

####  2.1.1 Hilbert spaces and Bethe vacua for $G=\widetilde{G}$

In the $A$-model description, we are mainly concerned with the supersymmetric
ground states of the 2d $\mathcal{N}=(2,2)$ effective field theory. These are
indexed by the so-called Bethe vacua, $\hat{u}$, which should be viewed as
states in a (cohomological) 2d topological quantum field theory (TQFT) that
assigns the (ground-state) Hilbert space $\mathcal{H}_{S^{1}}$ to the spatial
circle:

$\mathcal{H}_{S^{1}}\cong{\rm
Span}_{\mathbb{C}}\Big{\\{}\,|\hat{u}\rangle\;\Big{|}\;\hat{u}\in\mathcal{S}_{\text{BE}}\Big{\\}}\leavevmode\nobreak\
.$ (2.4)

In order to discuss the Bethe vacua more explicitly, let us focus on a 3d
$A$-model that arises from a 3d $\mathcal{N}=2$ gauge theory with compact
gauge group $G$. For now, let us assume that the fundamental group of
$G=\widetilde{G}$ is a free abelian group:

$\pi_{1}(\widetilde{G})\cong\mathbb{Z}^{n_{T}}\leavevmode\nobreak\ ,$ (2.5)

for some non-negative integer $n_{T}$. More concretely, $\widetilde{G}$ is a
product of simply-connected groups and of $U(N)$ factors, so that each $U(N)$
factor contributes a $\mathbb{Z}$ factor in (2.5), and $n_{T}$ is the number
of topological $\mathfrak{u}(1)$ symmetries of the 3d gauge theory. Then the
Bethe vacua are essentially given by the critical point of some effective
twisted superpotential $\mathcal{W}(u)$ in some Coulomb-branch description,
where $u$ denote 2d scalars in abelianised vector multiplets. We are
interested in the one-form symmetry of this 3d gauge theory, which is an
ordinary center symmetry:

$\Gamma\subseteq Z(\widetilde{G})\leavevmode\nobreak\ .$ (2.6)

Namely, $\Gamma$ is the maximal subgroup of the center of $\widetilde{G}$
which is preserved by the chiral multiplets and by the CS interactions.

The basic construction of the 3d $A$-model with a gauge group $\widetilde{G}$
is summarised in appendix A, which also spells out our conventions in more
details. For our present purpose, let us note that:444See appendix B for a
definition of the relevant lattices.

$u\in\mathfrak{t}_{\mathbb{C}}\leavevmode\nobreak\
,\qquad\qquad\mathfrak{m}\in\Lambda_{\rm
mw}^{\widetilde{G}}\subset\mathfrak{t}\leavevmode\nobreak\ .$ (2.7)

Namely, the Coulomb-branch scalar $u$ is valued in the complexified Cartan
algebra of $\widetilde{G}$, $\mathfrak{t}_{\mathbb{C}}$, and the magnetic
fluxes $\mathfrak{m}$ are valued in the magnetic weight lattice of
$\widetilde{G}$, which is a discrete sublattice of the Lie algebra. Large
gauge transformations along $S^{1}_{A}$ lead to the identifications:

$u\sim u+\mathfrak{m}\leavevmode\nobreak\
,\qquad\qquad\forall\mathfrak{m}\in\Lambda_{\rm
mw}^{\widetilde{G}}\leavevmode\nobreak\ ,$ (2.8)

hence the classical Coulomb branch is given by
$\mathfrak{t}_{\mathbb{C}}/\Lambda_{\rm
mw}^{\widetilde{G}}\cong(\mathbb{C}^{\ast})^{\text{rank}(\widetilde{G})}$. We
pick a basis $\\{e^{a}\\}$ of $\mathfrak{t}$ such that $u=u_{a}e^{a}$ and
$u_{a}\sim u_{a}+n_{a}$, $n_{a}\in\mathbb{Z}$, under any large gauge
transformation (2.8). The gauge flux operators are defined in terms of the
effective twisted superpotential of the gauge theory on $\mathbb{R}^{2}\times
S^{1}$:

$\Pi_{a}(u,\nu)\equiv\exp\left(2\pi i{\partial\mathcal{W}(u,\nu)\over\partial
u_{a}}\right)\leavevmode\nobreak\ ,\qquad a=1\leavevmode\nobreak\
,\cdots\leavevmode\nobreak\ ,\text{rank}(\widetilde{G})\leavevmode\nobreak\ ,$
(2.9)

with our conventions for $u_{a}$ as outlined above and in appendix A. Then,
the Bethe vacua correspond to elements of the set of allowable Bethe
solutions:

$\mathcal{S}_{\text{BE}}\equiv\left\\{\hat{u}\in\mathfrak{t}/\Lambda_{\rm
mw}^{\widetilde{G}},\Big{|}\;\Pi_{a}(\hat{u},\nu)=1\leavevmode\nobreak\
,\forall a\quad\text{and}\quad w\cdot\hat{u}\neq\hat{u}\leavevmode\nobreak\
,\forall w\in
W_{\widetilde{G}}\right\\}/{W_{\widetilde{G}}}\leavevmode\nobreak\ ,$ (2.10)

where $W_{\widetilde{G}}$ is the Weyl group of $\widetilde{G}$. Here, recall
that we need to exclude putative solutions to the Bethe equations,
$\\{\Pi_{a}=1\leavevmode\nobreak\ ,\forall a\\}$, which are not acted on
freely by $W_{\widetilde{G}}$. These Bethe equations can be written more
covariantly as:

$\Pi(u,\nu)^{\mathfrak{m}}\equiv\exp\left(2\pi
i\mathfrak{m}{\partial\mathcal{W}\over\partial u}\right)=1\leavevmode\nobreak\
,\qquad\forall\mathfrak{m}\in\Lambda_{\rm
mw}^{\widetilde{G}}\leavevmode\nobreak\ .$ (2.11)

We denote the Bethe vacua by $\hat{u}$, corresponding to the allowed solutions
to the Bethe equations modulo the action of the Weyl group.

Let us now discuss how the symmetries (2.1) are realised in the $A$-model
description.

Action of $\Gamma^{(1)}$ on $\mathcal{H}_{S^{1}}$. The Bethe vacua diagonalise
the topological operator $\mathcal{U}^{\gamma}[S^{1}_{A}]$ defined in (2.3).
Indeed, the insertion of the 3d topological line along $S^{1}_{A}$ is
equivalent to the insertion of a background flux $\gamma$ along $\Sigma$ –
that is, a non-trivial background gauge field for $\Gamma^{(1)}$ corresponds
to a non-trivial $\widetilde{G}/\Gamma$ bundle over $\Sigma$ which cannot be
lifted to a $\widetilde{G}$ bundle. In the $A$-model description, this
operator is given explicitly by a flux operator:

$\Pi(u,\nu)^{\gamma}\equiv\exp\left(2\pi
i\gamma{\partial\mathcal{W}\over\partial u}\right)\leavevmode\nobreak\
,\qquad\quad\gamma\in\Lambda_{\rm
mw}^{\widetilde{G}/\Gamma}\subseteq\Lambda_{\rm
mw}^{\mathfrak{g}}\leavevmode\nobreak\ ,$ (2.12)

where $\gamma$ corresponds to a magnetic flux valued in the larger GNO lattice
$\Lambda_{\rm mw}^{\widetilde{G}/\Gamma}$, with:

$\Lambda_{\rm mw}^{\widetilde{G}/\Gamma}\supset\Lambda_{\rm
mw}^{\widetilde{G}}\leavevmode\nobreak\ ,\qquad\quad\Gamma\cong{\Lambda_{\rm
mw}^{\widetilde{G}/\Gamma}/\Lambda_{\rm
mw}^{\widetilde{G}}}\leavevmode\nobreak\ .$ (2.13)

See also appendix B for a definition of these lattices. On Bethe vacua, the
Bethe equations (2.11) ensure that:

$\Pi(\hat{u})^{\gamma+\mathfrak{m}}=\Pi(\hat{u})^{\gamma}\leavevmode\nobreak\
,\qquad\forall\mathfrak{m}\in\Lambda_{\rm
mw}^{\widetilde{G}}\leavevmode\nobreak\ .$ (2.14)

The one-form symmetry operators then act on Bethe vacua as:

$\Pi^{\gamma}|\hat{u}\rangle=\Pi(\hat{u})^{\gamma}|\hat{u}\rangle\leavevmode\nobreak\
,\qquad\quad\Pi(\hat{u})^{\gamma}\in\mathbb{C}^{\ast}\leavevmode\nobreak\ ,$
(2.15)

at fixed flavour parameters, for $\gamma\in\Gamma^{(1)}$. In fact, since we
also know that, due to the Bethe equations, $\Pi(\hat{u})^{\gamma n}=1$ for
some positive integer $n\leq|\Gamma|$, the one-form symmetry acts on Bethe
vacua by multiplication by a root of unity. This is in agreement with the
general discussion of 2d TQFTs in Gukov:2021swm .

It is important to note that the definition for $\Pi^{\gamma}$ given in (2.12)
is ambiguous, because the twisted superpotential suffers from the ambiguity
Closset:2017zgf :

$\mathcal{W}\rightarrow\mathcal{W}+\rho(u)\leavevmode\nobreak\
,\qquad\quad\rho\in\Lambda^{\widetilde{G}}_{\text{w}}\leavevmode\nobreak\ .$
(2.16)

Any such linear shift corresponds to multiplying the flux operator (2.12) by a
phase:

$\Pi^{\gamma}\rightarrow\chi_{\rho}(\gamma)\,\Pi^{\gamma}\leavevmode\nobreak\
,\qquad\chi_{\rho}\equiv e^{2\pi
i\rho}\in\hat{\Gamma}^{(1)}\leavevmode\nobreak\ ,$ (2.17)

where $\hat{\Gamma}^{(1)}$ denotes the Pontryagin dual:

$\hat{\Gamma}^{(1)}\equiv{\rm Hom}(\Gamma^{(1)},U(1))\leavevmode\nobreak\ .$
(2.18)

We will discuss how to fix this ambiguity in section 2.2.

The action of the 1-form symmetry splits the Hilbert space into sectors:

$\mathcal{H}_{S^{1}}=\bigoplus_{\chi\in\hat{\Gamma}^{(1)}}\mathcal{H}_{S^{1}}^{\chi}\leavevmode\nobreak\
,\qquad\qquad\mathcal{H}_{S^{1}}^{\chi}\equiv{\rm
Span}_{\mathbb{C}}\Big{\\{}\,|\hat{u}\rangle\;\Big{|}\;\hat{u}\in\mathcal{S}_{\text{BE}}^{\chi}\Big{\\}}\leavevmode\nobreak\
,$ (2.19)

where each sector corresponds to the Bethe vacua that return a specific phase
$\chi(\gamma)$ for $\Pi^{\gamma}$ in (2.15). More precisely, for each
$\chi\in\hat{\Gamma}^{(1)}$, we define:

$\mathcal{S}_{\text{BE}}^{\chi}\equiv\Big{\\{}\hat{u}\in\mathcal{S}_{\text{BE}}\;\Big{|}\;\Pi(\hat{u},\nu)^{\gamma}=\chi(\gamma)\leavevmode\nobreak\
,\;\forall\gamma\in\Gamma^{(1)}\Big{\\}}\leavevmode\nobreak\ ,$ (2.20)

and we then have that
$\mathcal{S}_{\text{BE}}=\oplus_{\chi}\mathcal{S}_{\text{BE}}^{\chi}$. This
decomposition of the Bethe vacua will be useful below. Let us also note that
the existence of a 1-form symmetry in a 2d QFT always implies a so-called
decomposition Hellerman:2006zs ; Sharpe:2022ene  of the theory into disjoint
sectors.

Action of $\Gamma^{(0)}$ on $\mathcal{H}_{S^{1}}$. Let
$\mathcal{U}^{\gamma}(\mathcal{C})$ be a a topological line operator wrapping
the cylinder, as shown in figure 1. To understand its action on the Bethe
vacua, recall that $x_{a}\equiv e^{2\pi iu_{a}}$ can be seen as complexified
holonomies of a maximal torus $\prod_{a}U(1)_{a}\subset\widetilde{G}$ along
$S^{1}_{A}$, and that any Wilson loop in the representation $\mathfrak{R}$ of
$\widetilde{G}$ is represented in the $A$-model by:

$W_{\mathfrak{R}}=\sum_{\rho\in\mathfrak{R}}e^{2\pi
i\rho(u)}\leavevmode\nobreak\ .$ (2.21)

Then, the insertion of $\mathcal{U}^{\gamma}(\mathcal{C})$ acts on Wilson
loops wrapping $S^{1}_{A}$ as the 3d center symmetry, which precisely shifts
$u$ to $u+\gamma$, for $\gamma\in\Lambda_{\rm mw}^{\widetilde{G}/\Gamma}$ –
this only depends on $\gamma\in\Gamma$ because of (2.8). The action:

$\mathcal{U}^{\gamma}(\mathcal{C})\;:\;\mathfrak{t}_{\mathbb{C}}\rightarrow\mathfrak{t}_{\mathbb{C}}\leavevmode\nobreak\
:\;u\mapsto u+\gamma$ (2.22)

is a symmetry of the Bethe equations for any $\gamma\in\Gamma^{(0)}$, by
assumption (since $\Gamma$ is precisely the subgroup of $Z(\widetilde{G})$
that is preserved by the matter content and by the CS levels). Hence we have:

$\mathcal{U}^{\gamma}(\mathcal{C})|\hat{u}\rangle=|\hat{u}+\gamma\rangle\leavevmode\nobreak\
.$ (2.23)

Therefore, the one-form symmetry element $\gamma\in\Gamma^{(0)}$ acts as a
permutation of the Bethe vacua, as expected on general grounds Gukov:2021swm .

Twisted sector Hilbert spaces $\mathcal{H}_{S^{1}}^{(\delta)}$. The twisted
sector for any $\delta\in\Gamma^{(0)}$ corresponds to adding a topological
line along the time direction on the cylinder
$\Sigma\cong\mathbb{R}_{\tau}\times S^{1}$. The twisted-sector ground states
are Bethe vacua which satisfy the additional twist condition:

$u+\delta=w\cdot u\leavevmode\nobreak\ ,$ (2.24)

for some $w\in W_{\widetilde{G}}$. These states, denoted by
$|\hat{u};\delta\rangle$, form a basis for the twisted Hilbert space:

$\mathcal{H}_{S^{1}}^{(\delta)}\cong{\rm
Span}_{\mathbb{C}}\Big{\\{}\,|\hat{u};\delta\rangle\;\Big{|}\;\hat{u}\in\mathcal{S}_{\text{BE}}\leavevmode\nobreak\
,\hat{u}+\delta\sim\hat{u}\Big{\\}}\leavevmode\nobreak\ .$ (2.25)

Of course, we have $\mathcal{H}_{S^{1}}^{(0)}=\mathcal{H}_{S^{1}}$ for
$\delta=0$ the zero element. For future reference, let use define the set of
Bethe solutions that enter in (2.25) as:

$\mathcal{S}_{\text{BE}}^{(\gamma)}\equiv\Big{\\{}\hat{u}\in\mathcal{S}_{\text{BE}}\;\Big{|}\;\hat{u}+\gamma\sim\hat{u}\Big{\\}}\leavevmode\nobreak\
,\qquad\gamma\in\Gamma^{(0)}\leavevmode\nobreak\ ,$ (2.26)

where $\hat{u}\sim\hat{u}^{\prime}$ means that the respective Weyl-group
orbits of the solutions $\hat{u}$ and $\hat{u}^{\prime}$ are equal. Note that
(2.26) should not to be confused with (2.20).

(a) Intersection of two $\Gamma^{(0)}$ lines.

(b) Resolution into trivalent vertices.

Figure 2: The 2d SPT phase arising from the intersection of the topological
lines $\mathcal{U}^{\gamma}$ and $\mathcal{U}^{\delta}$ on the cylinder, see
(2.30). The phase is obtained by resolving the intersection into two trivalent
junctions, where we assign the phase $b_{2}(\gamma,\delta)$ to each junction
(with some customary orientation). That $b_{2}$ is a group cohomology class
follows from the associativity of fusion and from gauge invariance.

The discrete group $\Gamma^{(0)}$ acts on the set of all Bethe vacua,
$\mathcal{S}_{\text{BE}}$, as in (2.23). Let ${\rm
Orb}(\hat{u})\subseteq\mathcal{S}_{\text{BE}}$ denote the orbit of the vacuum
$\hat{u}$ under the action of $\Gamma^{(0)}$, and let ${\rm
Stab}(\hat{u})\subseteq\Gamma^{(0)}$ denote the stabiliser of $\hat{u}$ by
$\Gamma^{(0)}$. The orbit-stabiliser theorem gives us the cardinality of ${\rm
Stab}(\hat{u})$:

$\left|{\rm Stab}(\hat{u})\right|={\left|\Gamma^{(0)}\right|\over\left|{\rm
Orb}(\hat{u})\right|}\leavevmode\nobreak\ ,$ (2.27)

which is the number of twisted sectors in which $\hat{u}$ appears:

$|\hat{u};\delta\rangle\in\mathcal{H}_{S^{1}}^{(\delta)}\leavevmode\nobreak\
,\qquad\forall\delta\in{\rm Stab}(\hat{u})\leavevmode\nobreak\ .$ (2.28)

Finally, let us discuss the action of $\Gamma^{(0)}$ on the twisted sector
$\mathcal{H}_{S^{1}}^{(\delta)}$. In general, the action of
$\mathcal{U}^{\gamma}(\mathcal{C})$ on the twisted-sector states may take the
form:

$\mathcal{U}^{\gamma}(\mathcal{C})|\hat{u};\delta\rangle=e^{iS_{\rm
SPT}(\gamma,\delta)}\,|\hat{u}+\gamma;\delta\rangle\leavevmode\nobreak\ ,$
(2.29)

where $S_{\rm SPT}$ is a 2d SPT phase for $\Gamma^{(0)}$ (see e.g.
Bhardwaj:2023kri ; Dumitrescu:2023hbe  for recent discussions). More
concretely, the phase this introduces in (2.29) is given in terms of a
$U(1)$-valued group 2-cocycle $b_{2}$, as follows:

$e^{iS_{\rm SPT}(\gamma,\delta)}={b_{2}(\gamma,\delta)\over
b_{2}(\delta,\gamma)}\leavevmode\nobreak\ ,\qquad\qquad[b_{2}]\in
H^{2}(\Gamma^{(0)},U(1))\leavevmode\nobreak\ .$ (2.30)

In the two-dimensional description, this phase arises because of the
intersection of the topological lines $\mathcal{U}^{\gamma}$ and
$\mathcal{U}^{\delta}$ on the cylinder, as shown in figure 2; see e.g.
Bhardwaj:2022lsg  for a recent discussion. Note that, in the 3d description
wherein these lines arise as topological operators for a 3d one-form symmetry,
they could be separated along $S^{1}_{A}$. In this work, we will only consider
examples where the phase (2.30) is necessarily trivial, $e^{iS_{\rm SPT}}=1$;
indeed, we will briefly focus on $\Gamma=\mathbb{Z}_{N}$, in which case
$H^{2}(\mathbb{Z}_{N},U(1))\cong 1$ (see also Gaiotto:2020iye ). We hope to
discuss cases where this 2d SPT phase can be non-trivial in future work.

####  2.1.2 ’t Hooft anomalies and vacuum structure

The one-form symmetry $\Gamma_{\rm 3d}^{(1)}$ of the 3d theory can have a non-
trivial ’t Hooft anomaly captured by a four-dimensional anomaly theory:

$S^{\rm anom}_{\rm
4d}[B]=2\pi\int_{\mathfrak{M}_{4}}\mathcal{P}_{\mathfrak{a}}(B)\leavevmode\nobreak\
,$ (2.31)

where $\mathcal{P}_{\mathfrak{a}}(B)$ is an anomaly-dependent modification of
the Pontryagin square $\mathcal{P}(B)$ of the one-form background gauge field
in 3d extended onto the four-manifold $\mathfrak{M}_{4}$ with boundary
$\Sigma\times S^{1}$, $B\in H^{2}(\mathfrak{M}_{4},\Gamma)$. The construction
of $\mathcal{P}(B)$ is reviewed in appendix C. More concretely, we pick a set
of generator for $\Gamma$ such that
$\Gamma\cong\bigoplus_{i}\mathbb{Z}_{N_{i}}$ and $B$ decomposes as
$B=\sum_{i}B_{i}$ with $B_{i}\in H^{2}(\mathfrak{M}_{4},\mathbb{Z}_{i})$, in
which case the anomaly theory (2.31) takes the concrete form Hsin:2018vcg :

$S^{\rm anom}_{\rm 4d}[B]=2\pi\sum_{i}{\mathfrak{a}_{ii}\over
2N_{i}}\int_{\mathfrak{M}_{4}}\mathcal{P}(B_{i})+2\pi\sum_{i<j}{\mathfrak{a}_{ij}\over{\rm
gcd}(N_{i},N_{j})}\int_{\mathfrak{M}_{4}}B_{i}\cup B_{j}\leavevmode\nobreak\
,$ (2.32)

where the anomaly is captured by the symmetric matrix $\mathfrak{a}_{ij}$ with
integer coefficients:

$\mathfrak{a}_{ij}\in\mathbb{Z}_{{\rm gcd}(N_{i},N_{j})}\leavevmode\nobreak\
.$ (2.33)

Such an anomaly arises because some of the topological lines can themselves be
charged under $\Gamma^{(1)}_{\rm 3d}$ Gaiotto:2014kfa . Note also that we
consider the 3d $\mathcal{N}=2$ theory on 3-manifolds with an explicit choice
of spin structure, which then extends to a choice of spin structure on
$\mathfrak{M}_{4}$ – this is important for (2.32) to be well-defined with the
periodicities (2.33) Hsin:2018vcg . In the two-dimensional description on
$\Sigma$, the ’t Hooft anomaly is realised as a mixed anomaly between
$\Gamma^{(1)}$ and $\Gamma^{(0)}$, corresponding to the 3d anomaly theory on
the three-manifold $\mathfrak{M}_{3}$ with boundary $\Sigma$:

$S^{\rm anom}_{\rm 3d}[B,C]=2\pi\sum_{i,j}{\mathfrak{a}_{ij}\over{\rm
gcd}(N_{i},N_{j})}\int_{\mathfrak{M}_{3}}C_{i}\cup B_{j}\leavevmode\nobreak\
,$ (2.34)

where now $C\in H^{1}(\mathfrak{M}_{3},\Gamma)$, $B\in
H^{2}(\mathfrak{M}_{3},\Gamma)$, and we have expanded $C=\sum_{i}C_{i}$ and
$B=\sum_{i}B_{i}$ as above. In appendix C, we obtain this mixed ’t Hooft
anomaly by considering the continuum version of the anomaly theory
Kapustin:2014gua ; Hsin:2018vcg  and dimensionally reducing along $S^{1}$ on
$\mathfrak{M}_{4}=\mathfrak{M}_{3}\times S^{1}$, with
$\partial\mathfrak{M}_{3}=\Sigma_{g}$.

In two space-time dimensions, the one-form symmetry $\Gamma^{(1)}$ can never
be spontaneously broken Gaiotto:2014kfa , and it is therefore preserved in
each vacuum. On the other hand, the zero-form symmetry $\Gamma^{(0)}$ in the
vacuum $\hat{u}$ is spontaneously broken to the subgroup ${\rm
Stab}(\hat{u})\subseteq\Gamma^{(0)}$. The ’t Hooft anomaly must be matched by
the 2d low-energy description, which constrains the structure of the
$\Gamma^{(0)}$-orbits of Bethe vacua – we will discuss this momentarily.

The mixed anomaly (2.34) arises because the one-form symmetry operators
$\Pi^{\gamma_{(1)}}$ can be charged under $\Gamma^{(0)}$, and vice versa. (In
the 3d description, two topological lines that implement $\Gamma_{\rm
3d}^{(1)}$ can link non-trivially.) At the level of the Hilbert space
$\mathcal{H}_{S^{1}}$, the $\Gamma^{(0)}$ and $\Gamma^{(1)}$ charge operators
need not commute. Instead, we have a projective representation of
$\Gamma^{(0)}\times\Gamma^{(1)}$ on $\mathcal{H}_{S^{1}}$, with the twisted
commutation relations:

$\Pi^{\gamma_{(1)}}\mathcal{U}^{\gamma_{(0)}}=e^{2\pi
i\mathcal{A}(\gamma_{(0)},\gamma_{(1)})}\,\mathcal{U}^{\gamma_{(0)}}\Pi^{\gamma_{(1)}}\leavevmode\nobreak\
,$ (2.35)

determined by the anomaly:

$\mathcal{A}\,:\,\Gamma^{(0)}\times\Gamma^{(1)}\rightarrow\mathbb{R}/\mathbb{Z}\leavevmode\nobreak\
.$ (2.36)

Given a choice of generators $\gamma_{(0),i}$ and $\gamma_{(1),j}$ for
$\Gamma^{(0)}\cong\bigoplus_{i}\mathbb{Z}_{N_{i}}^{(0)}$ and of
$\Gamma^{(1)}\cong\bigoplus_{j}\mathbb{Z}_{N_{j}}^{(j)}$, respectively, we can
expand any group element as:

$\gamma_{(0)}=\sum_{i}n_{i}\gamma_{(0),i}\leavevmode\nobreak\
,\qquad\quad\gamma_{(1)}=\sum_{j}m_{j}\gamma_{(1),j}\leavevmode\nobreak\ ,$
(2.37)

for $n_{i},m_{i}\in\mathbb{Z}_{N_{i}}$. Then the anomaly (2.36) is related to
the coefficients $\mathfrak{a}_{ij}$ appearing in the 3d anomaly theory (2.34)
according to:

$\mathcal{A}(\gamma_{(0)},\gamma_{(1)})=\sum_{i,j}{n_{i}m_{j}\mathfrak{a}_{ij}\over{\rm
gcd}(N_{i},N_{j})}\mod 1\leavevmode\nobreak\ .$ (2.38)

For a given 3d $\mathcal{N}=2$ gauge theory, this anomaly is easily computed
using the $A$-model description. Indeed, from (2.15) and (2.23), we
find:555Here $[a,b]$ denotes the multiplicative commutator,
$[a,b]=aba^{-1}b^{-1}$.

$[\Pi^{\gamma_{(1)}},\mathcal{U}^{\gamma_{(0)}}]|\hat{u}\rangle={\Pi(\hat{u}+\gamma_{(0)})^{\gamma_{(1)}}\over\Pi(\hat{u})^{\gamma_{(1)}}}\left|\hat{u}+\gamma_{(0)}\right\rangle=e^{2\pi
i\mathcal{A}(\gamma_{(0)},\gamma_{(1)})}\left|\hat{u}+\gamma_{(0)}\right\rangle\leavevmode\nobreak\
,$ (2.39)

and therefore:

$\mathcal{A}(\gamma_{(0)},\gamma_{(1)})=\gamma_{(1)}\left({\partial\mathcal{W}(u+\gamma_{(0)})\over\partial
u}-{\partial\mathcal{W}(u)\over\partial u}\right)\mod 1\leavevmode\nobreak\ ,$
(2.40)

which in turn is independent of $u$ and only depends on the bare CS levels of
the UV theory. We will discuss this point further in section 5.2, based on the
explicit form of the effective twisted superpotential which we review in
appendix A.

Given the anomaly (2.36), we may also define a homomorphism
$\phi_{\mathcal{A}}$ from $\Gamma^{(0)}$ to the Pontryagin dual (2.18) of
$\Gamma^{(1)}$, given by:

$\phi_{\mathcal{A}}\,:\,\Gamma^{(0)}\rightarrow\hat{\Gamma}^{(1)}\,:\,\gamma\mapsto
e^{2\pi i\mathcal{A}(\gamma,-)}\leavevmode\nobreak\ .$ (2.41)

Now, consider coupling the $A$-model to background gauge fields $(C,B)$ for
$\Gamma^{(0)}\times\Gamma^{(1)}$, which is equivalent to inserting the
topological symmetry operators on $\Sigma$. The action of $\Gamma^{(0)}$ on
the 2d vacua comes with a gauge transformation $C\rightarrow C+\delta\lambda$,
which generates the anomalous term $\lambda\mathfrak{a}\int_{\Sigma}B$,
schematically. This anomaly can only be matched if distinct vacua in a given
$\Gamma^{(0)}$ orbit are stacked with distinct SPT phases for $\Gamma^{(1)}$.
Fixing some vacuum $\hat{u}$ to have the trivial phase, the other vacua
$\hat{u}+\gamma$ in ${\rm Orb}(\hat{u})$ will have the SPT phases:

$e^{iS_{\rm SPT}^{(\gamma)}(B)}=\phi_{\mathcal{A}}(\gamma)\leavevmode\nobreak\
.$ (2.42)

Note that which vacua is set to be the trivial SPT phase is immaterial,
because we can always shift all vacua by a common SPT phase by adding the
corresponding counterterm in the UV. Thus, we explicitly see how the anomaly
constrains the vacuum structure.

The homomorphism $\phi_{\mathcal{A}}$ defines a subgroup
$\ker{\phi_{\mathcal{A}}}\leq\Gamma^{(0)}$. The associated quotient group:

$\Xi^{(0)}\equiv{\Gamma^{(0)}/\ker{\phi_{\mathcal{A}}}}$ (2.43)

labels the distinct SPT phases. In general, $\Gamma^{(0)}$ is a direct sum of
cyclic groups, and the distinct SPT phases may not correspond to distinct
Bethe vacua in each orbit, which is due to the mixing of the various cyclic
subgroups of $\Gamma^{(0)}$. Nevertheless, we can classify all possible
orbits. Let $\hat{u}$ be a vacuum with trivial SPT phase, and let
$\gamma\in\Gamma^{(0)}$ be in the stabiliser group $\text{Stab}(\hat{u})$,
which means in particular that $\hat{u}+\gamma\sim\hat{u}$. The vacuum
$\hat{u}+\gamma$ has a SPT phase (2.42), but since we started with a vacuum
$\hat{u}$ that has a trivial phase, so does $\hat{u}+\gamma$. Therefore,
$\phi_{\mathcal{A}}(\gamma)=1$ and thus $\gamma\in\ker{\phi_{\mathcal{A}}}$.
This argument is valid for any element $\gamma\in\text{Stab}(\hat{u})$, and
thus every stabiliser of a given Bethe vacuum $\hat{u}$ is a subgroup of the
kernel:

$\text{Stab}(\hat{u})\leq\ker{\phi_{\mathcal{A}}}\leavevmode\nobreak\ .$
(2.44)

This restricts the allowed orbit dimensions according to the orbit-stabiliser
theorem (2.27). Given a subgroup $H\leq\ker{\phi_{\mathcal{A}}}$, there is a
corresponding orbit of length $\Gamma^{(0)}/|H|$. Note that any such orbit
dimension is an integer multiple of $|\Xi^{(0)}|$.

The extremal cases are the non-anomalous and the maximally anomalous case. The
latter case corresponds to $\ker{\phi_{\mathcal{A}}}=0$, so that
$\text{Stab}(\hat{u})$ is trivial for every vacuum $\hat{u}$. Hence all orbits
must have maximal dimension $|\Gamma^{(0)}|$. At the other extreme, if the
anomaly $\mathcal{A}$ vanishes identically, then
$\ker{\phi_{\mathcal{A}}}=\Gamma^{(0)}$, and $\text{Stab}(\hat{u})$ can be any
of the subgroups of $\Gamma^{(0)}$. For the trivial stabiliser, this results
in orbits of length $|\Gamma^{(0)}|$. Meanwhile, for
$\text{Stab}(\hat{u})=\Gamma^{(0)}$ we find orbits of length 1 – these are of
course the fixed points under $\Gamma^{(0)}$, which can only occur in a theory
with trivial ’t Hooft anomaly. The generic case occurs when the full
$\Gamma^{(0)}$ is anomalous but contains some non-anomalous subgroups.666That
is, non-anomalous subgroups $\widetilde{\Gamma}^{(0)}\subset\Gamma^{(0)}$ with
respect to $\Gamma^{(1)}$, i.e. such that the mixed ’t Hooft anomaly for
$\widetilde{\Gamma}^{(0)}$-$\Gamma^{(1)}$ vanishes. We could more generally
consider non-anomalous subgroups
$\widetilde{\Gamma}^{(p)}\subset\Gamma^{(p)}$, $p=0,1$ (with
$\widetilde{\Gamma}^{(0)}$ and $\widetilde{\Gamma}^{(1)}$ not necessarily
isomorphic), such that their mixed anomaly vanishes, by an obvious
generalisation of the discussion above. Since $\ker{\phi_{\mathcal{A}}}$ is a
subgroup of $\Gamma^{(0)}$, the allowed orbits are therefore those associated
with the non-anomalous stabilisers.

For concreteness, let us consider the parameterisation
$\Gamma\cong\bigoplus_{i}\mathbb{Z}_{N_{i}}$ as above. Then we have the SPT
phases:

$S_{\rm SPT}^{(\gamma)}(B_{\gamma_{(1)}})=2\pi\left(\sum_{i,j}n_{i}{\rm
A}_{ij}m_{j}\quad{\rm mod}\,1\right)\leavevmode\nobreak\ ,\qquad{\rm
A}_{ij}\equiv{\mathfrak{a}_{ij}\over{\rm
gcd}(N_{i},N_{j})}\leavevmode\nobreak\ ,$ (2.45)

for $\gamma=\sum_{i}n_{i}\gamma_{(0),i}\in\Gamma^{(0)}$ and a background gauge
field $B_{\gamma_{(1)}}$ corresponding to
$\gamma_{(1)}=\sum_{j}m_{j}\gamma_{(1),j}\in\Gamma^{(1)}$. Thus we have:

$\ker{\phi_{\mathcal{A}}}\cong\left\\{\gamma=\sum_{i}n_{i}\gamma_{(0),i}\,\Bigg{|}\,\sum_{i}n_{i}{\rm
A}_{ij}\in\mathbb{Z}\right\\}\leavevmode\nobreak\ .$ (2.46)

This allows for explicit computations in any given example.

The case $\Gamma=\mathbb{Z}_{N}$. Let us further discuss the special case of a
cyclic group, $\Gamma=\mathbb{Z}_{N}$, on which we shall focus in the
following sections. Denoting by the integers $n\in\mathbb{Z}_{N}^{(0)}$ and
$m\in\mathbb{Z}_{N}^{(1)}$ the elements $\gamma_{(0)}$ and $\gamma_{(1)}$,
respectively, we have the anomaly:

$\mathcal{A}(\gamma_{(0)},\gamma_{(1)})={\mathfrak{a}nm\over N}\;{\rm
mod}\,1\leavevmode\nobreak\
,\qquad\qquad\mathfrak{a}\in\mathbb{Z}_{N}\leavevmode\nobreak\ ,$ (2.47)

determined by the integer $\mathfrak{a}$ (mod $N$). We also have the
$\Gamma^{(1)}$ SPT phases:

$e^{iS_{\rm SPT}^{(n)}(B)}=\phi_{\mathcal{A}}(n)=e^{2\pi i{\mathfrak{a}n\over
N}\int_{\Sigma}B}=e^{2\pi i{\mathfrak{a}nm\over N}}\leavevmode\nobreak\ ,$
(2.48)

for $B$ corresponding to $m\in\mathbb{Z}_{N}^{(1)}$. We then find that
$\ker{\phi_{\mathcal{A}}}\cong\mathbb{Z}_{{\rm gcd}(\mathfrak{a},N)}$ and
therefore:

$\Xi^{(0)}\cong\mathbb{Z}_{d(\mathfrak{a},N)}\leavevmode\nobreak\
,\qquad\qquad d(\mathfrak{a},N)\equiv{N\over{\rm
gcd}(\mathfrak{a},N)}\leavevmode\nobreak\ .$ (2.49)

The $\mathbb{Z}_{N}^{(0)}$-orbits spanned by the Bethe vacua therefore have
dimensions that are integer multiples of $d(\mathfrak{a},N)$. Due to (2.44),
the orders of the stabilisers and hence the orbits will be divisors of $N$. In
the non-anomalous case, the orbit dimensions can be any divisor of $N$, while
in the maximally anomalous case all orbits are of length $N$. We will discuss
this more in detail in section 3.

###  2.2 Background gauge fields and expectation values of topological
operators

Recall that the partition function of the $\widetilde{G}$ gauge theory
$\mathcal{T}$ on $\Sigma_{g}\times S^{1}$ corresponds to the insertion of the
handle-gluing operator $\mathcal{H}$ in the $A$-model, so that:

$Z_{\Sigma_{g}\times S^{1}}^{[\mathcal{T}]}=\langle
1\rangle_{\Sigma_{g}}=\sum_{\hat{u}\in\mathcal{S}_{\text{BE}}}\langle\hat{u}\rvert\mathcal{H}^{g-1}\lvert\hat{u}\rangle=\sum_{\hat{u}\in\mathcal{S}_{\text{BE}}}\mathcal{H}(\hat{u})^{g-1}\leavevmode\nobreak\
.$ (2.50)

We are interested in computing the insertion of topological operators for
$\Gamma^{(1)}\times\Gamma^{(0)}$ on $\Sigma_{g}$. This is equivalent to
turning on background gauge fields for these discrete symmetries.

Insertion of the 1-form symmetry operators. Let us first consider the
insertion of the local operator $\Pi^{\gamma}$ on $\Sigma_{g}$. We simply
have:

$\langle\Pi^{\gamma}\rangle_{\Sigma_{g}}=\sum_{\hat{u}\in\mathcal{S}_{\text{BE}}}\langle\hat{u}\rvert\Pi^{\gamma}\mathcal{H}^{g-1}\lvert\hat{u}\rangle=\sum_{\hat{u}\in\mathcal{S}_{\text{BE}}}\Pi(\hat{u})^{\gamma}\mathcal{H}(\hat{u})^{g-1}\leavevmode\nobreak\
.$ (2.51)

Using the decomposition (2.19)-(2.20), we can write this as:

$\langle\Pi^{\gamma}\rangle_{\Sigma_{g}}=\sum_{\chi\in\hat{\Gamma}^{(1)}}\chi(\gamma)\sum_{\hat{u}\in\mathcal{S}_{\text{BE}}^{\chi}}\mathcal{H}(\hat{u})^{g-1}\leavevmode\nobreak\
.$ (2.52)

This insertion is equivalent to turning on a background gauge field
$B_{\gamma}$,

$Z_{\Sigma_{g}\times
S^{1}}^{[\mathcal{T}]}(B_{\gamma})=\langle\Pi^{\gamma}\rangle_{\Sigma_{g}}\leavevmode\nobreak\
,\qquad B_{\gamma}\in H^{2}(\Sigma_{g},\Gamma^{(1)})\leavevmode\nobreak\ ,$
(2.53)

with:

$\int_{\Sigma_{g}}B_{\gamma}=\gamma\leavevmode\nobreak\ .$ (2.54)

For later purpose, it will be useful to define the discrete $\theta$-angle and
its pairing to $B_{\gamma}$:

$\vartheta(\gamma)\equiv e^{-i(\theta,B_{\gamma})}\leavevmode\nobreak\
,\qquad\quad(\theta,B_{\gamma})\equiv\theta\int_{\Sigma_{g}}B_{\gamma}\leavevmode\nobreak\
,\qquad\quad\vartheta\in\hat{\Gamma}^{(1)}\leavevmode\nobreak\ .$ (2.55)

Here, $\theta$ is the background gauge field for a $(-1)$-form symmetry, which
will appear as a dual symmetry once we gauge $\Gamma^{(0)}$, and $(\theta,B)$
is a the canonical pairing on $\Sigma_{g}$. For instance, for
$\Gamma^{(1)}=\mathbb{Z}_{N}$, we have $\theta\in{2\pi\over N}\mathbb{Z}_{N}$.

Insertion of the 0-form symmetry operators. Next, we consider the insertion of
0-form symmetry operators along the Riemann surface $\Sigma_{g}$. These
topological line operators can wrap any of the generators of
$H_{1}(\Sigma_{g},\mathbb{Z})\cong\mathbb{Z}^{2g}$, which we denote by
$\mathcal{C}^{i}\;(i=1,\cdots,2g)$. Thus, an insertion is specified by a
$\Gamma^{(0)}$-valued 1-cycle $\bm{\gamma}$ corresponding to:

$[\bm{\gamma}]=\sum_{i=1}^{2g}\gamma_{i}\,[\mathcal{C}_{i}]\in
H_{1}(\Sigma_{g},\Gamma^{(0)})\leavevmode\nobreak\ ,$ (2.56)

which is determined by the $2g$ group elements $\gamma_{i}\in\Gamma^{(0)}$. We
will also use the notation $\bm{\gamma}=(\gamma_{i})\in\Gamma^{2g}$, by a
slight abuse of notation. We will then denote the topological operator as:

$\mathcal{U}^{\bm{\gamma}}\equiv\prod_{i=1}^{2g}\mathcal{U}^{\gamma_{i}}(\mathcal{C}_{i})\leavevmode\nobreak\
.$ (2.57)

This insertion is equivalent to turning on a background gauge field
$C_{\bm{\gamma}}$ for $\Gamma^{(0)}$, with:

$[C_{\bm{\gamma}}]\equiv{\rm PD}[\bm{\gamma}]\in
H^{1}(\Sigma_{g},\Gamma^{(0)})\leavevmode\nobreak\ .$ (2.58)

Here ${\rm PD}[\bm{\gamma}]$ denotes the cohomology class Poincaré dual to
(2.56). For future reference, let us also choose a symplectic basis of $a$\-
and $b$-cycles as shown in figure 3, with the standard intersection pairing:

$[a_{k}]\leavevmode\nobreak\ ,[b_{l}]\in
H_{1}(\Sigma_{g},\mathbb{Z})\leavevmode\nobreak\ ,\qquad
k,l=1,\cdots,g\leavevmode\nobreak\
,\qquad[a_{k}]\cdot[b_{l}]=-[b_{l}]\cdot[a_{k}]=\delta_{kl}\leavevmode\nobreak\
,$ (2.59)

and $[a_{k}]\cdot[a_{l}]=[b_{k}]\cdot[b_{l}]=0$. In this basis, we have
$\bm{\gamma}\equiv(\gamma_{a,k},\gamma_{b,l})$.

Figure 3: $a$\- and $b$-cycles on the Riemann surface $\Sigma_{g}$, here
depicted for genus $g=3$.

Figure 4: Operators in 2d TQFT corresponding to the cap, cylinder, and pair of
pants. We can think as $\mathcal{H}^{{\frac{1}{2}}}$ as a formal square root
of the handle-gluing operator $\mathcal{H}$, keeping in mind that we always
obtain integer powers of $\mathcal{H}$ when computing observables on a closed
$\Sigma$.

As reviewed in appendix A, the 3d $A$-model is a 2d TQFT. From this
perspective, the twisted index (2.50) can be computed by basic surgery
operations on the Riemann surface. The three basic ingredients are the cap,
the cylinder and the pair of pants, to which the TQFT functor assigns states
as summarised in figure 4. In the $\widetilde{G}$ theory with the
$\Gamma^{(0)}$ symmetry, this TQFT prescription can be extended to compute the
correlation functions of the topological line operators (2.57). The modified
dictionary is shown in figure 5.

Let us first consider the case $g=1$, the torus, with the notation
$a_{1}=\mathcal{C}$ and $b_{1}=\widetilde{\mathcal{C}}$. Here $\mathcal{C}$ is
considered as the spatial direction and $\widetilde{\mathcal{C}}$ is the
Euclidean time direction. Setting
$\bm{\gamma}=(\gamma_{a,1},\gamma_{b,1})=(\gamma,\delta)$, we have the general
insertion:

$Z_{T^{2}\times
S^{1}}^{[\mathcal{T}]}(C_{\bm{\gamma}})=\left\langle\mathcal{U}^{\gamma}(\mathcal{C})\;\mathcal{U}^{\delta}(\widetilde{\mathcal{C}})\right\rangle_{T^{2}}\leavevmode\nobreak\
,$ (2.60)

which is of the form considered in figure 2(a). We thus have the trace over
the $\delta$-twisted sector:

$Z_{T^{2}\times
S^{1}}^{[\mathcal{T}]}(C_{\bm{\gamma}})=\sum_{\hat{u}\in\mathcal{S}_{\text{BE}}}\langle\hat{u};\delta\rvert\mathcal{U}^{\gamma}(\mathcal{C})\lvert\hat{u};\delta\rangle=\sum_{\hat{u}\in\mathcal{S}_{\text{BE}}}\langle\hat{u};\delta|\mathopen{}\hat{u}+\gamma;\delta\rangle=\sum_{\hat{u}\in\mathcal{S}_{\text{BE}}^{(\gamma,\delta)}}1=\left|\mathcal{S}_{\text{BE}}^{(\gamma,\delta)}\right|\leavevmode\nobreak\
,$ (2.61)

with $\mathcal{S}_{\text{BE}}^{(\gamma)}$ defined in (2.26), and we also
defined
$\mathcal{S}_{\text{BE}}^{(\gamma,\delta)}\equiv\mathcal{S}_{\text{BE}}^{(\gamma)}\cap\mathcal{S}_{\text{BE}}^{(\delta)}$.
This result is modular invariant, as expected for a 2d TQFT on $\Sigma_{g}$.

Figure 5: Operators in the TQFT corresponding to the cylinder and pair of
pants with topological lines inserted. We could also consider the insertion of
a line at the boundary of a cap, but this is topologically trivial; in this
formalism, this follows from the fact that
$\mathcal{H}(\hat{u}+\gamma)=\mathcal{H}(\hat{u})$, so that formally
$\sum_{\hat{u}}\mathcal{H}^{-{\frac{1}{2}}}\mathcal{U}^{\gamma}\lvert\hat{u}\rangle=\sum_{\hat{u}}\mathcal{H}^{-{\frac{1}{2}}}\lvert\hat{u}\rangle$.

The generalisation to any $\Sigma_{g}$ with the operator (2.57) inserted can
be derived using the 2d TQFT perspective sketched above willett ;
Gukov:2021swm . Following the decomposition of the Riemann surface
$\Sigma_{g}$ with the insertions of $\Gamma^{(0)}$ symmetry operators as shown
in figure 6, we find:

$Z_{\Sigma_{g}\times
S^{1}}^{[\mathcal{T}]}(C_{\bm{\gamma}})=\left\langle\mathcal{U}^{\bm{\gamma}}\right\rangle_{\Sigma_{g}}=\sum_{\hat{u}\in\mathcal{S}_{\text{BE}}^{(\bm{\gamma})}}\mathcal{H}(\hat{u})^{g-1}\leavevmode\nobreak\
.$ (2.62)

Here, for any set $(\gamma_{i})$ of $n$ elements of $\Gamma^{(0)}$, we define
the set of Bethe vacua
$\mathcal{S}_{\text{BE}}^{(\gamma_{1},\cdots,\gamma_{n})}\equiv\bigcap_{i=1}^{n}\mathcal{S}_{\text{BE}}^{(\gamma_{i})}$.
Equivalently, let us denote by ${\rm
H}^{(0)}_{\bm{\gamma}}\subseteq\Gamma^{(0)}$ the smallest subgroup of
$\Gamma^{(0)}$ that contains the $\gamma_{i}$’s – it is the smallest subgroup
such that $[\bm{\gamma}]\in H_{1}(\Sigma_{g},H_{\bm{\gamma}})$. Then, we can
define:

$\mathcal{S}_{\text{BE}}^{(\bm{\gamma})}\equiv\\{\hat{u}\in\mathcal{S}_{\text{BE}}\;|\;\hat{u}+\gamma^{(0)}\sim\hat{u}\leavevmode\nobreak\
,\leavevmode\nobreak\ \forall\gamma^{(0)}\in{\rm
H}^{(0)}_{\bm{\gamma}}\\}\cong\mathcal{S}_{\text{BE}}^{(\gamma_{1},\cdots,\gamma_{2g})}\leavevmode\nobreak\
.$ (2.63)

Note that the expression (2.62) is invariant under large diffeomorphisms of
$\Sigma_{g}$, which act as $Sp(2g,\mathbb{Z})$ transformations of
$\bm{\gamma}=(\gamma_{a,k},\gamma_{b,l})$. Moreover, it is important to note
that the correlator (2.62) only depends on the subgroup ${\rm
H}^{(0)}\subseteq\Gamma^{(0)}$ and not on the specific insertion, namely:

$\left\langle\mathcal{U}^{\bm{\gamma}_{1}}\right\rangle_{\Sigma_{g}}=\left\langle\mathcal{U}^{\bm{\gamma}_{2}}\right\rangle_{\Sigma_{g}}\leavevmode\nobreak\
,\qquad\text{if}\quad{\rm H}^{(0)}_{\bm{\gamma}_{1}}={\rm
H}^{(0)}_{\bm{\gamma}_{2}}\leavevmode\nobreak\ ,$ (2.64)

since
$\mathcal{S}_{\text{BE}}^{(\bm{\gamma}_{1})}=\mathcal{S}_{\text{BE}}^{(\bm{\gamma}_{2})}$
in this case. In section 3, we will discuss these subgroups ${\rm
H}^{(0)}_{\bm{\gamma}}$ more explicitly in the case of
$\Gamma^{(0)}\cong\mathbb{Z}_{N}$ a cyclic group.

Figure 6: Decomposition of a handle-gluing operator with
$\mathcal{U}^{\bm{\gamma}}$ operator inserted

Mixed correlators. Finally, we may consider the simultaneous insertion of
$\Gamma^{(1)}$ and $\Gamma^{(0)}$ background gauge fields, which gives us:

$Z_{\Sigma_{g}\times
S^{1}}^{[\mathcal{T}]}(B_{\gamma^{(1)}},C_{\bm{\gamma}^{(0)}})=\left\langle\Pi^{\gamma^{(1)}}\mathcal{U}^{\bm{\gamma}^{(0)}}\right\rangle_{\Sigma_{g}}=\sum_{\hat{u}\in\mathcal{S}_{\text{BE}}^{(\bm{\gamma}^{(0)})}}\Pi(\hat{u})^{\gamma^{(1)}}\mathcal{H}(\hat{u})^{g-1}\leavevmode\nobreak\
.$ (2.65)

Note that such mixed correlation functions necessarily vanish when the anomaly
is non-trivial:

$\left\langle\Pi^{\gamma^{(1)}}\mathcal{U}^{\bm{\gamma}^{(0)}}\right\rangle_{\Sigma_{g}}=0\qquad\text{if
}\quad e^{2\pi i\mathcal{A}(\gamma^{(0)},\gamma^{(1)})}\neq
1\leavevmode\nobreak\ .$ (2.66)

The expression (2.65) can also be written as:

$\left\langle\Pi^{\gamma^{(1)}}\mathcal{U}^{\bm{\gamma}^{(0)}}\right\rangle_{\Sigma_{g}}=\sum_{\chi\in\hat{\Gamma}^{(1)}}\chi(\gamma^{(1)})\sum_{\hat{u}\in\mathcal{S}_{\text{BE}}^{\chi}\cap\mathcal{S}_{\text{BE}}^{(\bm{\gamma}^{(0)})}}\mathcal{H}(\hat{u})^{g-1}\leavevmode\nobreak\
,$ (2.67)

similarly to (2.52).

3d modularity. Correlation functions on $\Sigma_{g}\times S^{1}$ are not
completely fixed by the above prescription, due to the ambiguity
$\Pi^{\gamma}\rightarrow\chi_{\rho}(\gamma)\Pi^{\gamma}$ of the gauge flux
operator from linear shifts of the twisted prepotential (2.17). A practical
way to resolve this ambiguity is to demand modularity when inserting 3d lines
on the three-torus. As before, for $g=1$ we can choose $\mathcal{C}$ to be the
spatial direction on $T^{2}$, and compare the correlator
$\langle\chi_{\rho}(\gamma)\Pi^{\gamma}\rangle_{T^{2}}$ with
$\langle\mathcal{U}^{\gamma}(\mathcal{C})\rangle_{T^{2}}$. The ambiguity
$\chi_{\rho}(\gamma)$ of the flux operator can be fixed by identifying those
correlators (2.52) and (2.61),

$\chi_{\rho}(\gamma)\sum_{\chi\in\hat{\Gamma}^{(1)}}\chi(\gamma)|\mathcal{S}_{\text{BE}}^{\chi}|=|\mathcal{S}_{\text{BE}}^{(\gamma)}|\leavevmode\nobreak\
,\qquad\forall\gamma\in\Gamma\leavevmode\nobreak\ .$ (2.68)

Note that the RHS is a positive integer. If a solution
$\chi_{\rho}\in\hat{\Gamma}^{(1)}$ exists, then it fixes a normalisation of
the gauge flux operator for a given theory.

On physical grounds, we expect that a solution to (2.68) exist.
Mathematically, the existence of a solution to (2.68) is not clear at all. As
we show in section 3 for the case of the $SU(N)_{K}$ $\mathcal{N}=2$ CS
theory, the positive integers $|\mathcal{S}_{\text{BE}}^{(\gamma)}|$ depend
intricately on the elements $\gamma$, and merely solving the equation for a
generator $\gamma_{0}$ of $\Gamma$ – if one exists – may not be enough to
determine $\chi_{\rho}$.777On the other hand, we may constrain the set of
possible solutions somewhat, as follows. The _exponent_ of the finite abelian
group $\Gamma\cong\bigoplus_{i}\mathbb{Z}_{N_{i}}$ $\Gamma$, denoted by
$\exp(\Gamma)$, is the least common multiple of the orders of all elements of
the group. We then have ${\exp(\Gamma)}\gamma=0$ for any $\gamma\in\Gamma$.
Since $\chi_{\rho}$ is a group homomorphism, this forces
$\chi_{\rho}(\gamma)^{\exp(\Gamma)}=1$ and thus $\chi_{\rho}$ is necessarily
an $\exp(\Gamma)$-th root of unity.  In section 3, we study in some detail the
3d modularity for the $\mathcal{N}=2$ $SU(N)_{K}$ CS theory, and we find that
a normalisation $\chi_{\rho}$ exists for all values of $N$ and $K$. We also
encounter an interesting modular anomaly in this case; this will be discussed
in section 3.2.

###  2.3 Gauging the 1-form symmetry $\Gamma^{(1)}$

Let us now gauge the $\Gamma^{(1)}$ symmetry – such a symmetry is always non-
anomalous in 2d. This simply corresponds to summing over background gauge
fields, as follows:

$Z_{\Sigma_{g}\times
S^{1}}^{[\mathcal{T}/\Gamma^{(1)}]}(\theta)={1\over|\Gamma|}\sum_{B\in
H^{2}(\Sigma_{g},\Gamma^{(1)})}e^{i(\theta,B)}Z_{\Sigma_{g}\times
S^{1}}^{[\mathcal{T}]}(B)\leavevmode\nobreak\ .$ (2.69)

Here, we weighted the contributions with the $\theta$-angle for the dual
$(-1)$-form symmetry, as discussed in (2.55). This gives us:

$Z_{\Sigma_{g}\times
S^{1}}^{[\mathcal{T}/\Gamma^{(1)}]}(\theta)={1\over|\Gamma|}\sum_{\gamma\in\Gamma^{(1)}}\bar{\vartheta}(\gamma)\,\langle\Pi^{\gamma}\rangle_{\Sigma_{g}}\leavevmode\nobreak\
,$ (2.70)

where $\bar{\vartheta}$ denotes the complex conjugate of $\vartheta$. Plugging
in (2.52) and using the orthogonality of the characters,888Namely, for any
$\Gamma$ a discrete abelian group, and $\vartheta,\chi\in\hat{\Gamma}$, we
have that
$\sum_{\gamma\in\Gamma}\bar{\vartheta}(\gamma)\chi(\gamma)=|\Gamma|\delta_{\vartheta,\chi}$.
we find that:

$Z_{\Sigma_{g}\times
S^{1}}^{[\mathcal{T}/\Gamma^{(1)}]}(\theta)=\sum_{\hat{u}\in\mathcal{S}_{\text{BE}}^{\chi=\vartheta}}\mathcal{H}(\hat{u})^{g-1}=\operatorname{Tr}_{\mathcal{H}_{S^{1}}^{\vartheta}}(\mathcal{H}^{g-1})\leavevmode\nobreak\
.$ (2.71)

That is, gauging the one-form symmetry ‘undoes decomposition’ by projecting us
onto a given $\chi=\vartheta$ sector Sharpe:2019ddn . The process is
completely reversible, since gauging the $\Gamma^{(-1)}$ symmetry simply
corresponds to summing over all possible $\theta$-angles:

$Z_{\Sigma_{g}\times
S^{1}}^{[\mathcal{T}]}(B)=\sum_{\theta}e^{-i(\theta,B)}\,Z_{\Sigma_{g}\times
S^{1}}^{[\mathcal{T}/\Gamma^{(1)}]}(\theta)\leavevmode\nobreak\ .$ (2.72)

It is worth commenting on the overall normalisation of the sum in (2.69) and
in (2.72), respectively. Here we sum over all insertions with the overall
normalisation ${1/|\Gamma|}$ when gauging $\Gamma^{(1)}$, and therefore we
have a unit normalisation in (2.72) when gauging $\Gamma^{(-1)}$. This
prescription is such that the result (2.71) for the gauged theory has a
standard Hilbert space interpretation – it is simply given as a trace over the
Hilbert space $\mathcal{H}_{S^{1}}^{\chi=\vartheta}$ defined in (2.19), with
the local operator $\mathcal{H}$ being unaffected by the 1-form gauging.999Any
other consistent normalisation, such as the one used in Gukov:2021swm , would
differ from ours by a factor of the form $\alpha^{g-1}$, corresponding to a
rescaling of the handle-gluing operator (which is equivalent to adding a
supersymmetry-preserving counterterm $\propto\int_{\Sigma}R$ – see e.g.
Closset:2014pda ).

Note also that, in the gauged theory, the ambiguity (2.17) in defining the
gauge flux operator $\Pi^{\gamma}$ is equivalent to a redefinition of angle
$\theta$ according to $\theta\to\theta+2\pi\rho$ (that is,
$\bar{\vartheta}(\gamma)\to\chi_{\rho}(\gamma)\bar{\vartheta}(\gamma)$), which
is simply a relabelling of the distinct universes in the decomposition (2.19).

###  2.4 Gauging the 0-form symmetry $\Gamma^{(0)}$

The discrete symmetry $\Gamma^{(0)}$ is non-anomalous in the 3d $A$-model for
$\widetilde{G}$. Thus it can be gauged by summing over all $\Gamma^{(0)}$
gauge fields on $\Sigma_{g}$:

$Z_{\Sigma_{g}\times
S^{1}}^{[\mathcal{T}/\Gamma^{(0)}]}(C^{D})={1\over|\Gamma|^{2g-1}}\sum_{C\in
H^{1}(\Sigma_{g},\Gamma^{(0)})}e^{2\pi i(C^{D},C)}Z_{\Sigma_{g}\times
S^{1}}^{[\mathcal{T}]}(C)\leavevmode\nobreak\ .$ (2.73)

Note the normalisation constant in front, which differs from the ‘symmetric’
normalisation used in Gukov:2021swm . Here, $C^{D}$ is a background gauge
field for the dual 0-form symmetry, $\Gamma^{(0)}_{D}\cong\Gamma$, which non-
trivially permutes the twisted sectors. (This is sometimes known as the
‘quantum symmetry.’) We also defined the pairing:

$(C^{D},\,C)=\int_{\Sigma_{g}}C^{D}\cup C\leavevmode\nobreak\ .$ (2.74)

In the gauged theory, the topological line operator for $\bm{\gamma}_{D}\in
H_{1}(\Sigma_{g},\Gamma^{(0)}_{D})$ can be written as:

$\mathcal{U}_{D}^{\bm{\gamma}_{D}}(\mathcal{C})=e^{i\int_{\mathcal{C}}C}\leavevmode\nobreak\
,\qquad\qquad[C^{D}_{\bm{\gamma}_{D}}]={\rm
PD}[\bm{\gamma}_{D}]\leavevmode\nobreak\ ,$ (2.75)

where $C$ is the dynamical gauge field for $\Gamma^{(0)}$. Note that we can
view $\mathcal{U}_{D}^{\bm{\gamma}_{D}}$ as a character:

$\mathcal{U}_{D}^{\bm{\gamma}_{D}}\in{\rm
Hom}(H_{1}(\Sigma_{g},\Gamma^{(0)}),U(1))\leavevmode\nobreak\
,\qquad\qquad\mathcal{U}_{D}^{\bm{\gamma}_{D}}(\bm{\gamma})\equiv e^{2\pi
i(C^{D}_{\bm{\gamma}_{D}},\,C_{\bm{\gamma}})}\leavevmode\nobreak\ .$ (2.76)

The partition function (2.73) can thus be written as:

$Z_{\Sigma_{g}\times
S^{1}}^{[\mathcal{T}/\Gamma^{(0)}]}(\bm{\gamma}_{D})={1\over|\Gamma|^{2g-1}}\sum_{[\bm{\gamma}]\in
H_{1}(\Sigma_{g},\Gamma^{(0)})}\mathcal{U}_{D}^{\bm{\gamma}_{D}}(\bm{\gamma})\,Z_{\Sigma_{g}\times
S^{1}}^{[\mathcal{T}]}({\bm{\gamma}})\leavevmode\nobreak\ .$ (2.77)

Conversely, we have:

$Z_{\Sigma_{g}\times
S^{1}}^{[\mathcal{T}]}({\bm{\gamma}})={1\over|\Gamma|}\sum_{[\bm{\gamma}_{D}]\in
H_{1}(\Sigma_{g},\Gamma^{(0)}_{D})}e^{2\pi
i(C_{\bm{\gamma}},\,C^{D}_{\bm{\gamma}_{D}})}\,Z_{\Sigma_{g}\times
S^{1}}^{[\mathcal{T}/\Gamma^{(0)}]}(\bm{\gamma}_{D})\leavevmode\nobreak\ ,$
(2.78)

wherein gauging the dual 0-form symmetry gives us back the theory we started
with. The partition function (2.77) can be written as a trace over the gauged
Hilbert space Gukov:2021swm . The Hilbert space of the 2d theory
$\mathcal{T}/\Gamma^{(0)}$ takes the form:

$\mathcal{H}_{S^{1}}^{[\mathcal{T}/\Gamma^{(0)}]}\cong{\rm
Span}_{\mathbb{C}}\Big{\\{}\,|\hat{\omega};s_{\hat{\omega}}\rangle\;\Big{|}\;\hat{\omega}\in\mathcal{S}_{\text{BE}}/\Gamma^{(0)}\leavevmode\nobreak\
,\;s_{\hat{\omega}}=1,\cdots,|{\rm
Stab}(\hat{\omega})|\Big{\\}}\leavevmode\nobreak\ ,$ (2.79)

where $\hat{\omega}\equiv{\rm Orb}(\hat{u})$ denote the distinct
$\Gamma^{(0)}$-orbits of Bethe vacua of the $\widetilde{G}$ gauge theory, and
$s_{\hat{\omega}}$ indexes the twisted sectors. If we first consider
$C_{D}=0$, for definiteness:

$Z_{\Sigma_{g}\times
S^{1}}^{[\mathcal{T}/\Gamma^{(0)}]}\equiv{1\over|\Gamma|^{2g-1}}\sum_{[\bm{\gamma}]\in
H_{1}(\Sigma_{g},\Gamma^{(0)})}\left\langle\mathcal{U}^{\bm{\gamma}}\right\rangle_{\Sigma_{g}}\leavevmode\nobreak\
,$ (2.80)

we find that:

$Z_{\Sigma_{g}\times
S^{1}}^{[\mathcal{T}/\Gamma^{(0)}]}=\sum_{\hat{\omega}\in\mathcal{S}_{\text{BE}}/\Gamma^{(0)}}|{\rm
Stab}(\hat{\omega})|\left({|{\rm
Stab}(\hat{\omega})|^{2}\over|\Gamma|^{2}}\mathcal{H}(\hat{\omega})\right)^{g-1}\leavevmode\nobreak\
,$ (2.81)

where we used the notation:

$\mathcal{H}(\hat{\omega})\equiv\mathcal{H}(\hat{u})\leavevmode\nobreak\
,\quad\forall\hat{u}\in\hat{\omega}\leavevmode\nobreak\ ,$ (2.82)

which is well defined since
$\mathcal{H}(\hat{u}+\gamma)=\mathcal{H}(\hat{u})$,
$\forall\gamma\in\Gamma^{(0)}$. Indeed, starting with the expression for the
partition function of the gauged theory (2.80), we have:

$\displaystyle\begin{split}Z_{\Sigma_{g}\times
S^{1}}^{[\mathcal{T}/\Gamma^{(0)}]}&=\frac{1}{|\Gamma|^{2g-1}}\sum_{[\bm{\gamma}]\in
H_{1}(\Sigma_{g},\Gamma^{(0)})}\sum_{\hat{u}\in\mathcal{S}_{\text{BE}}^{(\bm{\gamma})}}\mathcal{H}(\hat{u})^{g-1}\leavevmode\nobreak\
,\\\
&=\frac{1}{|\Gamma|^{2g-1}}\sum_{\begin{subarray}{c}(\hat{u},\bm{\gamma})\in\mathcal{S}_{\text{BE}}\times\Gamma^{2g}\\\
\hat{u}\in\mathcal{S}_{\text{BE}}^{(\bm{\gamma})}\end{subarray}}\mathcal{H}(\hat{u})^{g-1}\leavevmode\nobreak\
,\\\
&=\frac{1}{|\Gamma|^{2g-1}}\sum_{\hat{u}\in\mathcal{S}_{\text{BE}}}\sum_{\bm{\gamma}\in\text{Stab}(\hat{u})^{2g}}\mathcal{H}(\hat{u})^{g-1}\leavevmode\nobreak\
,\\\
&=\frac{1}{|\Gamma|^{2g-1}}\sum_{\hat{\omega}\in\mathcal{S}_{\text{BE}}/\Gamma^{(0)}}\mathcal{H}(\hat{\omega})^{g-1}|\hat{\omega}|\;|\text{Stab}(\hat{\omega})|^{2g}\leavevmode\nobreak\
,\end{split}$ (2.83)

where in the third equality we exchanged the order of the two sums. Moreover,
in the last step we used (2.82) and
$|\text{Stab}(\hat{\omega})|\equiv|\text{Stab}(\hat{u})|,\forall\hat{u}\in\hat{\omega}$.
Then, using the stabiliser-orbit theorem (2.27), the final expression above
simplifies to (2.81).

Note that we chose the overall normalisation in (2.73) such that the
expression (2.81) can be interpreted as a trace over the Hilbert space (2.79)
of the $\Gamma^{(0)}$-gauged theory, including all twisted-sector states. Note
also that, for $g=0$, this implies that we have:

$Z_{S^{2}\times S^{1}}^{[\mathcal{T}/\Gamma^{(0)}]}=|\Gamma|Z_{S^{2}\times
S^{1}}^{[\mathcal{T}]}\leavevmode\nobreak\ .$ (2.84)

More generally, if we turn on a background gauge field for $\Gamma^{(0)}_{D}$
as in (2.77), a similar computation gives us:

$Z_{\Sigma_{g}\times
S^{1}}^{[\mathcal{T}/\Gamma^{(0)}]}(\bm{\gamma}_{D})=\sum_{\hat{\omega}\in\mathcal{S}_{\text{BE}}/\Gamma^{(0)}}|\text{Stab}(\hat{\omega})|^{-1}\left(\frac{\mathcal{H}(\hat{\omega})}{|\Gamma|^{2}}\right)^{g-1}\sum_{\bm{\gamma}\in\text{Stab}(\hat{\omega})^{2g}}\mathcal{U}_{D}^{\bm{\gamma}_{D}}(\bm{\gamma})\leavevmode\nobreak\
.$ (2.85)

In the special case $\bm{\gamma}_{D}=0$ (so that
$\mathcal{U}_{D}^{\bm{\gamma_{D}}}(\bm{\gamma})=1$), this reduces to (2.81).

###  2.5 Topologically twisted index for the gauge group
$\widetilde{G}/\Gamma$

Finally, we can combine the above results to gauge the full $\Gamma^{(1)}_{\rm
3d}$ in the $A$-model description, assuming the symmetry is non-anomalous. We
can then gauge the symmetries in whichever order, as indicated in figure 7.

Figure 7: Gaugings of 0-form and 1-form symmetries of a 2d theory
$\mathcal{T}$. In the absence of a ’t Hooft anomaly, the diagram commutes. If
$\Gamma^{(1)}_{\rm 3d}$ is anomalous, the bottom theory does not exist and the
diagram truncates.

The general formula for the 3d theory with gauge group
$G=\widetilde{G}/\Gamma$ on $\Sigma_{g}\times S^{1}$ is obtained by summing
over all insertions of topological operators for $\Gamma^{(1)}_{\rm 3d}$:

$Z^{\mathcal{T}/\Gamma_{\rm
3d}^{(1)}}(\theta,C^{D})={1\over|\Gamma|^{2g}}\sum_{\delta\in\Gamma^{(1)}}\,\sum_{[\bm{\gamma}]\in
H_{1}(\Sigma_{g},\Gamma^{(0)})}e^{i(\theta,B_{\delta})}e^{2\pi
i(C^{D},C_{\bm{\gamma}})}\left\langle\Pi^{\delta}\,\mathcal{U}^{\bm{\gamma}}\right\rangle_{\Sigma_{g}}\leavevmode\nobreak\
.$ (2.86)

Conceptually, it is simplest to consider gauging the symmetries one after the
other. For instance, consider gauging $\Gamma^{(0)}$ first. The intermediate
2d theory on the left-hand corner of figure 7 still has a $\Gamma^{(1)}$
symmetry, and it therefore enjoys decomposition. The subsequent gauging of
$\Gamma^{(1)}$ is then a projection on a particular universe. If we first
gauge $\Gamma^{(1)}$ instead, we first project onto one universe and then
consider the $\Gamma^{(0)}$ gauging inside that universe. Either way, we end
up considering the Bethe vacua determined by the $\vartheta$-twisted Bethe
equations (2.20), which we can suggestively rewrite as:

$\mathcal{S}_{\text{BE}}^{\vartheta}\equiv\left\\{\hat{u}\in\mathfrak{t}/\Lambda_{\rm
mw}^{\widetilde{G}}\;\Big{|}\;\Pi^{\gamma}(\hat{u})=\vartheta(\gamma)\leavevmode\nobreak\
,\forall\gamma\in\Lambda^{\widetilde{G}/\Gamma}_{\rm mw}\quad\text{and}\quad
w\cdot\hat{u}\neq\hat{u}\leavevmode\nobreak\ ,\forall w\in
W_{\widetilde{G}}\right\\}/{W_{\widetilde{G}}}\leavevmode\nobreak\ .$ (2.87)

Note that $\vartheta(\gamma)=1$ for any $\gamma\in\Lambda^{\widetilde{G}}_{\rm
mw}\subset\Lambda^{\widetilde{G}/\Gamma}_{\rm mw}$. We then find:

$Z_{\Sigma_{g}\times S^{1}}^{\mathcal{T}/\Gamma_{\rm
3d}^{(1)}}(\theta,C^{D}_{\bm{\gamma}_{D}})=\sum_{\hat{\omega}\in\mathcal{S}_{\text{BE}}^{\vartheta}/\Gamma^{(0)}}|\text{Stab}(\hat{\omega})|^{-1}\left(\frac{\mathcal{H}(\hat{\omega})}{|\Gamma|^{2}}\right)^{g-1}\sum_{\bm{\gamma}\in\text{Stab}(\hat{\omega})^{2g}}\mathcal{U}_{D}^{\bm{\gamma}_{D}}(\bm{\gamma})\leavevmode\nobreak\
.$ (2.88)

Finally, let us consider the special case $\theta=\bm{\gamma}_{D}=0$. The
twisted index for the $G=\widetilde{G}/\Gamma$ gauge theory then takes the
simple form:

$Z_{\Sigma_{g}\times S^{1}}^{\mathcal{T}/\Gamma_{\rm 3d}^{(1)}}={\rm
Tr}_{\mathcal{H}_{S^{1}}^{[\widetilde{G}/\Gamma]}}\left(\mathcal{H}_{G}^{g-1}\right)\leavevmode\nobreak\
,$ (2.89)

where the trace is over the Hilbert space of the $A$-model for the
$G=\widetilde{G}/\Gamma$ gauge theory:

$\mathcal{H}_{S^{1}}^{[\widetilde{G}/\Gamma]}\cong{\rm
Span}_{\mathbb{C}}\Big{\\{}\,\lvert\hat{\omega};s_{\hat{\omega}}\rangle\;\Big{|}\;\hat{\omega}\in\mathcal{S}_{\text{BE}}^{\vartheta=1}/\Gamma^{(0)}\leavevmode\nobreak\
,\;s_{\hat{\omega}}=1,\cdots,|{\rm
Stab}(\hat{\omega})|\Big{\\}}\leavevmode\nobreak\ .$ (2.90)

Here, we have introduced the handle-gluing operator $\mathcal{H}_{G}$, which
acts on the Bethe vacua of the $G$ gauge theory as:

$\mathcal{H}_{G}\lvert\hat{\omega};s_{\hat{\omega}}\rangle={\mathcal{H}(\hat{\omega})\over|\hat{\omega}|^{2}}\lvert\hat{\omega};s_{\hat{\omega}}\rangle\leavevmode\nobreak\
.$ (2.91)

where $\mathcal{H}=\mathcal{H}_{\widetilde{G}}$ is the ordinary handle-gluing
operator of the $\widetilde{G}$ gauge theory, and we used the notation (2.82).

This completes our general discussion of the topologically twisted index for
3d $\mathcal{N}=2$ supersymmetric gauge theories with a general gauge group
$G$ and a $U(1)_{R}$ symmetry. The same caveats that held for $\widetilde{G}$
apply here Closset:2017zgf  – in particular, we implicitly assumed that the
set of Bethe vacua is discrete. Finally, we note that it would be
straightforward to generalise our discussion to ‘non-Lagrangian’ 3d
$\mathcal{N}=2$ field theories with one-form symmetries for which the
$A$-model formalism is available. For the $T[\mathcal{M}_{3}]$ theories of the
3d/3d correspondence Dimofte:2011ju ; Gukov:2016gkn ; Gukov:2017kmk , the
computation of $\Sigma=T^{2}$ was first discussed in Eckhard:2019jgg .

##  3 The 3d $\mathcal{N}=2$ $SU(N)_{K}$ Chern–Simons theory

In the rest of this paper, we mainly set $\widetilde{G}=SU(N)$. This is a
simply-connected gauge group with centre $Z(SU(N))=\mathbb{Z}_{N}$, and we can
thus study all the possible quotients $SU(N)/\mathbb{Z}_{r}$, where $r$ is a
divisor of $N$. This gives us a rich structure of gauge theories obtained by
discrete gauging of subgroups, which depends crucially on the arithmetic
properties of the integer $N$ – see e.g. Aharony:2013hda ; Delmastro:2019vnj ;
Gaiotto:2014kfa  for closely related discussions.

In this section, we specifically study the $\mathcal{N}=2$ supersymmetric
Chern–Simons theory $SU(N)_{K}$ (with supersymmetric CS level $K\geq N$). Upon
integrating out the gauginos, this is equivalent to the bosonic CS theory
$SU(N)_{k}$ with $k=K-N\geq 0$. Since many explicit results are available for
this 3d TQFT, this serves as a useful testing ground for the general formalism
of the previous section. We also obtain some seemingly new results.

###  3.1 The $A$-model for the $SU(N)_{K}$ theory and its higher-form
symmetries

The $SU(N)_{K}$ CS theory has a one-form symmetry $\Gamma^{(1)}_{\rm
3d}=\mathbb{Z}_{N}$, which descends to the higher-form symmetry
$\mathbb{Z}_{N}^{(0)}\oplus\mathbb{Z}_{N}^{(1)}$ in the 2d description. In
this subsection, we give an explicit parametrisation of the Bethe vacua for
this theory and we study how the higher-form symmetries act on them.

Effective twisted superpotential and dilaton. Although $SU(N)=N$ has rank
$N-1$, it is convenient to use a slightly redundant description with $N$
variables $u_{a}$, $a=1,\cdots,N$, together with the tracelessness condition:

$\sum_{a=1}^{N}u_{a}=0\leavevmode\nobreak\ .$ (3.1)

The effective twisted superpotential is given by:

$\mathcal{W}(u)=\frac{K}{2}\sum_{a=1}^{N}u_{a}^{2}+u_{0}\sum_{a=1}^{N}u_{a}\leavevmode\nobreak\
,$ (3.2)

where $u_{0}$ is a Lagrange multiplier for the constraint (3.1). The effective
dilaton potential is given by:

$e^{2\pi i\Omega(u)}=-\prod_{\begin{subarray}{c}a,b=1\\\ a\neq
b\end{subarray}}^{N}\left(1-\frac{x_{a}}{x_{b}}\right)^{-1}\leavevmode\nobreak\
.$ (3.3)

Here, for later convenience, we turned on an effective CS level $K_{RR}\in
2\mathbb{Z}+1$ for the R-symmetry $U(1)_{R}$. We also use the notation
$x_{a}\equiv e^{2\pi iu_{a}}$ and $q\equiv e^{2\pi iu_{0}}$.

Parametrising the Bethe vacua. The Bethe equations that follow from the
effective twisted superpotential (3.2) are:

$\Pi_{a}(u)\equiv qx_{a}^{K}=1\leavevmode\nobreak\ ,\qquad
a=1,\cdots,N\leavevmode\nobreak\
,\qquad\quad\Pi_{0}(u)\equiv\prod_{a=1}^{N}x_{a}=1\leavevmode\nobreak\ .$
(3.4)

By taking the product of the $N$ flux operators $\Pi_{a}$, we see that $q$
must be an $N$-th root of unity:

$\hat{q}=e^{2\pi i\frac{\ell}{N}}\leavevmode\nobreak\
,\qquad\quad\ell\in\mathbb{Z}_{N}\leavevmode\nobreak\ .$ (3.5)

Substituting $q=\hat{q}$ into $\Pi_{a}$, one finds the Bethe vacuum solutions:

$\hat{u}_{a}=\left(\frac{l_{a}}{K}-\frac{\ell}{KN}\right)\;{\rm
mod}\,1\leavevmode\nobreak\ ,\qquad a=1,\cdots,N\leavevmode\nobreak\ ,$ (3.6)

for $l_{a}\in\mathbb{Z}_{K}$, and with the tracelessness constraint:

$\sum_{a=1}^{N}l_{a}-\ell\in K\mathbb{Z}\leavevmode\nobreak\ .$ (3.7)

We index the Bethe vacua by the $(N+1)$-tuples $\underline{l}$ defined as:

$\underline{l}=(l_{1},\cdots,l_{N};\ell)\leavevmode\nobreak\
,\qquad\quad\hat{u}_{\underline{l}}\equiv\hat{u}_{a}e^{a}\leavevmode\nobreak\
.$ (3.8)

The Weyl symmetry $W_{\widetilde{G}}=S_{N}$ permutes the $u_{a}$ variables,
and we can therefore choose the ordered $N$-tuples
$\\{l_{1},\cdots,l_{N}\\}\subset\\{0,1\cdots,K-1\\}$ by fixing a gauge. That
is, we must have $l_{a}>l_{b}$ if $a>b$. Before imposing the condition (3.7),
there are thus $N\binom{K}{N}$ possibilities for $\underline{l}$. Evaluating
the trace $\sum_{a=1}^{N}l_{a}-\ell$ (mod $K$) partitions this set of tuples
into $K$ sets of equal size. Selecting the traceless tuples thus gives us the
total number of Bethe vacua, which reproduces the well-known Witten index
Witten:1999ds ; Ohta:1999iv :

$\textbf{I}_{\text{W}}[SU(N)_{K}]\equiv Z_{T^{3}}[SU(N)_{K}]={N\over
K}\binom{K}{N}=\binom{K-1}{N-1}\leavevmode\nobreak\ .$ (3.9)

In summary, the Bethe vacua
$\hat{u}_{\underline{l}}\in\mathcal{S}_{\text{BE}}$ are in one-to-one
correspondence with the elements $\underline{l}$ in the indexing set:

$\mathcal{J}_{N,K}\equiv\left\\{(l_{1},\cdots,l_{N};\ell)\in\mathbb{Z}_{K}^{N}\oplus\mathbb{Z}_{N},\Big{|}\,0\leq
l_{1}<\cdots<l_{N}\leq K\leavevmode\nobreak\ ,\,\sum_{a=1}^{N}l_{a}-\ell\in
K\mathbb{Z}\right\\}\leavevmode\nobreak\ ,$ (3.10)

with $\hat{u}_{\underline{l}}$ given in (3.6). An equivalent determination of
this set is obtained by first looking at the $\binom{K}{N}$ ordered sets
$(l_{1},\cdots,l_{N})\in\mathbb{Z}_{K}^{N}$, defining
$\ell\equiv\sum_{a=1}^{N}l_{a}\,{\rm mod}\,K$, and retaining the resulting
sets $\underline{l}$ if and only if $\ell<N$.101010As an example, consider the
$SU(3)_{6}$ $\mathcal{N}=2$ theory, which has $10$ vacua. Using the
presentation (3.10), we can list them as:
$\displaystyle(0,1,5;0)\leavevmode\nobreak\
,\quad(1,2,3;0)\leavevmode\nobreak\ ,\quad(3,4,5;0)\leavevmode\nobreak\
,\quad(0,2,5;1)\leavevmode\nobreak\ ,\quad(1,2,4;1)\leavevmode\nobreak\ ,$
$\displaystyle(0,3,4;1)\leavevmode\nobreak\
,\quad(0,3,5;2)\leavevmode\nobreak\ ,\quad(1,2,5;2)\leavevmode\nobreak\
,\quad(1,3,4;2)\leavevmode\nobreak\ ,\quad(0,2,4;0)\leavevmode\nobreak\ .$

Handle-gluing operator and the twisted index. The twisted index can be
computed as in (2.50). Using the parametrisation (3.10) for the Bethe vacua,
the handle-gluing operator evaluated on-shell gives us:

$\mathcal{H}(\hat{u}_{\underline{l}})=N\left(\frac{K}{2^{N}}\right)^{N-1}\prod_{1\leq
a<b\leq N}\sin^{-2}\left(\frac{\pi(l_{a}-l_{b})}{K}\right)\leavevmode\nobreak\
,$ (3.11)

and the topologically twisted index of the $SU(N)_{K}$ theory is then given
by:

$Z_{\Sigma_{g}\times
S^{1}}[SU(N)_{K}]=N^{g-1}\left(\frac{K}{2^{N}}\right)^{(g-1)(N-1)}\sum_{\underline{l}\in\mathcal{J}_{N,K}}\prod_{1\leq
a<b\leq
N}\left(\sin\frac{\pi(l_{a}-l_{b})}{K}\right)^{2-2g}\leavevmode\nobreak\ ,$
(3.12)

where the sum is over the elements of the set (3.10). This is the well-known
Verlinde formula for the $SU(N)_{k}$ bosonic CS theory with $k=K-N$
Verlinde:1988sn ; Blau:1993tv .

Let us consider a few explicit examples. For $K=N$ and $K=N+1$, the Verlinde
formula simplifies to:

$\displaystyle Z_{\Sigma_{g}\times S^{1}}[SU(N)_{N}]=1\leavevmode\nobreak\
,\qquad Z_{\Sigma_{g}\times S^{1}}[SU(N)_{N+1}]={N^{g}}\leavevmode\nobreak\ ,$
(3.13)

This is can be easily understood from the level/rank duality
$SU(N)_{K}\leftrightarrow U(K-N)_{-K,-N}$ (see e.g. Hsin:2016blu ;
Closset:2023jiq ). With some more effort, a few more complicated examples can
be evaluated explicitly as a function of $g$. For instance, we find:

$\displaystyle Z_{\Sigma_{g}\times S^{1}}[SU(3)_{6}]$
$\displaystyle=\frac{1}{12}\left(3\cdot 4^{g}+8\cdot
9^{g}+36^{g}\right)\leavevmode\nobreak\ ,$ (3.14) $\displaystyle
Z_{\Sigma_{g}\times S^{1}}[SU(4)_{6}]$
$\displaystyle=6^{g-1}\left(4^{g}+2\right)+2^{3g-1}\leavevmode\nobreak\ .$

Note that these are integers for any $g$, and that they give $1$ at $g=0$, as
expected for any 3d TQFT. See cvijovic2012 ; fonseca2017basic ; Dowker1992
for comments on closed-form summations of trigonometric sums of these kinds.

####  3.1.1 Two-dimensional 1-form symmetry, gauging and decomposition

In the $A$-model description, we have a two-dimensional one-form symmetry
whose topological operator is the flux operator that inserts some magnetic
flux $\gamma$ along $\Sigma$, corresponding to a $PSU(N)\equiv
SU(N)/\mathbb{Z}_{N}$ bundle that cannot be lifted to an $SU(N)$ bundle:

$\gamma\in{\Lambda_{\rm mw}^{PSU(N)}/\Lambda_{\rm
mw}^{SU(N)}}\cong\mathbb{Z}_{N}\leavevmode\nobreak\ .$ (3.15)

In our parameterisation of $u\in\mathfrak{t}\subset\mathfrak{su}(N)$ with
$u=u_{a}e^{a}$ as above, the generator $\gamma_{0}$ of $\mathbb{Z}_{N}$ in
(3.15) can be chosen to be:111111The shift by the integer $\delta_{a,1}$ is
such that $\sum_{a=1}^{N}\gamma_{0,a}=0$. Note also that
$\mathfrak{m}\in\Lambda_{\rm mw}^{SU(N)}$ is similarly defined as
$\mathfrak{m}=\mathfrak{m}_{a}e^{a}$ with $\mathfrak{m}_{a}\in\mathbb{Z}$ and
$\sum_{a=1}^{N}\mathfrak{m}_{a}=0$.

$\gamma_{0}=\gamma_{0,a}e^{a}\leavevmode\nobreak\
,\qquad\gamma_{0,a}=-\frac{1}{N}+\delta_{a,1}\leavevmode\nobreak\ ,\qquad
a=1,\cdots,N\leavevmode\nobreak\ .$ (3.16)

We then write the elements (3.15) as $\gamma=n\gamma_{0}$ for
$n\in\mathbb{Z}_{N}$ an integer modulo $N$. The corresponding
$\mathbb{Z}_{N}^{(1)}$ operators are
$\Pi^{\gamma}\equiv\mathcal{U}^{\gamma}(S^{1}_{A})$ with:

$\Pi^{\gamma_{0}}(u)\equiv\prod_{a=1}^{N}\Pi_{a}^{\gamma_{0,a}}(u)\leavevmode\nobreak\
.$ (3.17)

On-shell, this evaluates to:

$\Pi^{\gamma_{0}}(u_{\underline{l}})=(-1)^{N-1}\hat{q}^{-1}=(-1)^{N-1}e^{-{2\pi
i\ell\over N}}\leavevmode\nobreak\ ,$ (3.18)

where we have fixed the overall phase by hand for future convenience, as we
will discuss momentarily. Hence $\mathbb{Z}_{N}^{(1)}$ acts on the Bethe vacua
as:

$\Pi^{m\gamma_{0}}\lvert\hat{u}_{\underline{l}}\rangle=(-1)^{m(N-1)}e^{-2\pi
i{m\ell\over N}}\lvert\hat{u}_{\underline{l}}\rangle\leavevmode\nobreak\
,\qquad m=0,1,\cdots,N-1\leavevmode\nobreak\ ,$ (3.19)

where recall that $\ell=\sum_{a}l_{a}\;{\rm mod}\,K$. We can then directly
evaluate the expectation value of this flux operator on $\Sigma_{g}$:

$\langle\Pi^{m\gamma_{0}}\rangle_{\Sigma_{g}}=N^{g-1}\left(\frac{K}{2^{N}}\right)^{(g-1)(N-1)}\\\
\times\sum_{\underline{l}\in\mathcal{J}_{N,K}}(-1)^{m(N-1)}e^{-2\pi
i\frac{m\ell}{N}}\prod_{1\leq a<b\leq
N}\left(\sin\frac{\pi(l_{a}-l_{b})}{K}\right)^{2-2g}\leavevmode\nobreak\ .$
(3.20)

For $g=1$, we find the following explicit evaluation formula:

$\displaystyle\langle\Pi^{m\gamma_{0}}\rangle_{T^{2}}$ $\displaystyle=$
$\displaystyle\,\sum_{\underline{l}\in\mathcal{J}_{N,K}}(-1)^{m(N-1)}e^{-2\pi
i\frac{m\ell}{N}}$ (3.21) $\displaystyle=$ $\displaystyle\,\delta_{K\;{\rm
mod}\;d,0}\binom{{K\over d}-1}{{N\over d}-1}\qquad\text{with}\quad
d\equiv{N\over{\rm gcd}(m,N)}\leavevmode\nobreak\ .$

This is an ‘experimental’ result that follows from the physical expectation of
modularity on $T^{3}$, as we will explain below.121212We checked this identity
on a computer for a large number of values of $N$ and $K$. We leave finding a
mathematical proof as a challenge for the interested reader. In particular,
(LABEL:genus-1-A-ops) is always a non-negative integer.

Gauging of $\mathbb{Z}_{N}^{(1)}$. To gauge $\mathbb{Z}_{N}^{(1)}$ in the
$A$-model description, we simply sum over the insertions (3.20) for all
possible values of $m$, as in (2.69):

$Z_{\Sigma_{g}\times
S^{1}}\left[SU(N)_{K}/\mathbb{Z}^{(1)}_{N}\right](\theta_{s})={1\over
N}\sum_{m=0}^{N-1}e^{i(\theta_{s},m\gamma_{0})}\left\langle\Pi^{m\gamma_{0}}\right\rangle_{\Sigma_{g}}\leavevmode\nobreak\
.$ (3.22)

The discrete $\theta$-angle for the dual $(-1)$-form symmetry takes values:

$\theta_{s}\equiv 2\pi\frac{s}{N}\leavevmode\nobreak\ ,\qquad
s=0,\cdots,N-1\leavevmode\nobreak\ ,$ (3.23)

so that $(\theta_{s},m\gamma_{0})=2\pi\frac{sm}{N}$ in (3.22). Let us define
the geometric series:

$\Delta^{N}_{\ell}(s)\equiv{1\over N}\sum_{m=0}^{N-1}(-1)^{m(N-1)}e^{2\pi
im\frac{s-\ell}{N}}=\begin{cases}\,\delta_{s-\ell+\frac{N}{2}\text{ mod
}N,0}\quad&\text{if }N\text{ is even}\leavevmode\nobreak\ ,\\\
\,\delta_{s-\ell\text{ mod }N,0}\quad&\text{if }N\text{ is
odd}\leavevmode\nobreak\ .\end{cases}$ (3.24)

We then have:

$Z_{\Sigma_{g}\times
S^{1}}\left[SU(N)_{K}/\mathbb{Z}^{(1)}_{N}\right](\theta_{s})=N^{g-1}\left(\frac{K}{2^{N}}\right)^{(g-1)(N-1)}\times\\\
\sum_{\underline{l}\in\mathcal{J}_{N,K}}\Delta^{N}_{\ell}(s)\prod_{1\leq
a<b\leq
N}\left(\sin\frac{\pi(l_{a}-l_{b})}{K}\right)^{2-2g}\leavevmode\nobreak\ .$
(3.25)

This result is in agreement with (2.71) and with the decomposition
$\mathcal{S}_{\text{BE}}\cong\bigoplus_{s}\mathcal{S}_{\text{BE}}^{\vartheta_{s}}$
induced by the $1$-form symmetry on the Bethe vacua of the $SU(N)_{K}$ theory,
with:

$\hat{u}_{\underline{l}}\in\mathcal{S}_{\text{BE}}^{\vartheta_{s}}\;\longleftrightarrow\;\underline{l}\in\mathcal{J}_{N,K}^{s}\equiv\left\\{\underline{l}\in\mathcal{J}_{N,K}\;\Big{|}\;(-1)^{N-1}e^{2\pi
i{\ell\over N}}=e^{2\pi i{s\over N}}\right\\}\leavevmode\nobreak\ ,$ (3.26)

and thus (3.25) is simply a sum over the Bethe vacua
$\hat{u}\in\mathcal{S}_{\text{BE}}^{\vartheta_{s}}$ indexed by
$\mathcal{J}_{N,K}^{s}$:

$Z_{\Sigma_{g}\times
S^{1}}\left[SU(N)_{K}/\mathbb{Z}^{(1)}_{N}\right](\theta_{s})=\sum_{\hat{u}\in\mathcal{S}_{\text{BE}}^{\vartheta_{s}}}\mathcal{H}(\hat{u})^{g-1}\leavevmode\nobreak\
.$ (3.27)

We thus see that the 2d universes indexed by $\theta_{s}$ are equivalently
indexed by $\ell$, as:

$\ell=\begin{cases}\,s+{N\over 2}\;{\rm mod}\,N\quad&\text{if }N\text{ is
even\leavevmode\nobreak\ ,}\\\ \,s\quad&\text{if }N\text{ is
odd\leavevmode\nobreak\ .}\end{cases}$ (3.28)

For $N=2$ and $s=0$, (3.25) of course reproduces (1.15). In the special case
$g=1$, we have:

$Z_{T^{3}}\left[SU(N)_{K}/\mathbb{Z}^{(1)}_{N}\right](\theta_{s})=\left|\mathcal{J}_{N,K}^{s}\right|\leavevmode\nobreak\
.$ (3.29)

Interestingly, ‘experimentally’ we find that, if $N$ is odd and prime, then
$|\mathcal{J}_{N,K}^{s}|$ is given by:

$\left|\mathcal{J}_{N,K}^{s}\right|={1\over N}\binom{K-1}{N-1}+\delta_{K\text{
mod }N,0}\left(\delta_{s,0}-{1\over N}\right)\leavevmode\nobreak\ ,$ (3.30)

which is always an integer. We will derive this below using 3d modularity. Of
course, we see that
$\sum_{s=0}^{N-1}|\mathcal{J}_{N,K}^{s}|=|\mathcal{J}_{N,K}|$.

Gauging of a subgroup $\mathbb{Z}_{r}^{(1)}$. For later purpose, we can also
consider the gauging of a subgroup
$\mathbb{Z}_{r}^{(1)}\subseteq\mathbb{Z}_{N}^{(1)}$, for $r$ any divisor of
$N$ and $K$:

$Z_{\Sigma_{g}\times
S^{1}}\left[SU(N)_{K}/\mathbb{Z}^{(1)}_{r}\right](\theta_{s}^{(\mathbb{Z}_{r})})={1\over
r}\sum_{m=0}^{r-1}e^{2\pi i{ms\over r}}\left\langle\Pi^{m{N\over
r}\gamma_{0}}\right\rangle_{\Sigma_{g}}\leavevmode\nobreak\ ,$ (3.31)

where $\theta_{s}^{(\mathbb{Z}_{r})}=2\pi{s\over r}$ with
$s\in\mathbb{Z}_{r}$. Note the normalisation of the sum in (3.31). The sum
over $m$ generalises (3.24) to:

$\displaystyle\Delta^{N,r}_{\ell}(s)$ $\displaystyle\equiv$
$\displaystyle\,\,\,{1\over r}\sum_{m=0}^{r-1}(-1)^{m{N\over r}(N-1)}e^{2\pi
im\frac{s-\ell}{s}}$ (3.32) $\displaystyle=$
$\displaystyle\,\,\,\begin{cases}\,\delta_{s-\ell-{N(N-1)\over 2}\text{ mod
}r,0}\quad&\text{if }N\text{ is even and }{N\over r}\text{ is
odd\leavevmode\nobreak\ ,}\\\ \,\delta_{s-\ell\text{ mod
}r,0}\quad&\text{otherwise\leavevmode\nobreak\ .}\end{cases}$

Hence, the sum (3.31) over the $\mathbb{Z}_{r}^{(1)}$ topological operators
projects us onto universes indexed by $\theta_{s}^{(\mathbb{Z}_{r})}$, and
(LABEL:theta_symbol_Zr) tells us that we can equivalently parameterise these
universes by $\ell\text{ mod }r$, with the exact relation being:

$s=\begin{cases}\,\ell+{N(N-1)\over 2}\;{\rm mod}\,r\quad&\text{if }N\text{ is
even and }{N\over r}\text{ is odd\leavevmode\nobreak\ ,}\\\ \,\ell\;{\rm
mod}\,r\quad&\text{otherwise\leavevmode\nobreak\ .}\end{cases}$ (3.33)

We then have:

$Z_{\Sigma_{g}\times
S^{1}}\left[SU(N)_{K}/\mathbb{Z}^{(1)}_{r}\right](\theta_{s}^{(\mathbb{Z}_{r})})=\sum_{\hat{u}\in\mathcal{S}_{\text{BE}}^{\vartheta_{s}^{(\mathbb{Z}_{r})}}}\mathcal{H}(\hat{u})^{g-1}\leavevmode\nobreak\
,$ (3.34)

where we defined:

$\hat{u}_{\underline{l}}\in\mathcal{S}_{\text{BE}}^{\vartheta_{s}^{(\mathbb{Z}_{r})}}\;\longleftrightarrow\;\underline{l}\in\mathcal{J}_{N,K}^{s,r}\equiv\left\\{\underline{l}\in\mathcal{J}_{N,K}\;\Big{|}\;(-1)^{{N\over
r}(N-1)}e^{2\pi i{\ell\over r}}=e^{2\pi i{s\over
r}}\right\\}\leavevmode\nobreak\ .$ (3.35)

This obviously generalises (3.27). Note that, for a divisor $r<N$, the
‘universes’ $\mathcal{S}_{\text{BE}}^{\vartheta_{s}^{(\mathbb{Z}_{r})}}$
permuted by the dual $(-1)$-form symmetry $\mathbb{Z}_{r}^{(-1)}$ are
generally larger than the universes
$\mathcal{S}_{\text{BE}}^{\vartheta_{s}^{(\mathbb{Z}_{N})}}=\mathcal{S}_{\text{BE}}^{\vartheta_{s}}$
permuted by $\mathbb{Z}_{N}^{(-1)}$.

####  3.1.2 Two-dimensional 0-form symmetry and orbits of Bethe vacua

Let us now consider the action of the $0$-form symmetry $\mathbb{Z}_{N}^{(0)}$
on the Bethe vacua. On the cylinder, the $\mathcal{U}^{\gamma}$ operator acts
as:

$\mathcal{U}^{\gamma}\lvert\hat{u}_{\underline{l}}\rangle=\lvert\hat{u}_{\underline{l}}+\gamma\rangle\leavevmode\nobreak\
.$ (3.36)

Given the choice of generator $\gamma_{0}$ in (3.16), we see that, for
$n\in\mathbb{Z}_{N}$:

$\hat{u}_{\underline{l},a}\rightarrow\hat{u}_{\underline{l}^{\prime},a}=(\hat{u}_{\underline{l}}+n\gamma_{0})_{a}=\frac{1}{NK}(N\;l_{a}-\ell-
nK)\,{\rm mod}\,1\leavevmode\nobreak\ ,$ (3.37)

where $\hat{u}_{\underline{l}}=(\hat{u}_{\underline{l},a})$ is defined up to
permutations of the $u_{a}$’s. The corresponding action
$\underline{l}\rightarrow\underline{l}^{\prime}$ on the labels
$\underline{l}=(l_{a};\ell)$ reads:

$\ell\rightarrow\ell^{\prime}=(\ell+nK)\;\text{mod}\;N\leavevmode\nobreak\
,\qquad l_{a}\rightarrow l_{a}^{\prime}=l_{a}-{nK+\ell-\ell^{\prime}\over
N}\;\text{mod}\;K\leavevmode\nobreak\ .$ (3.38)

Note that $\ell$ is preserved by the action of $\mathbb{Z}_{N}^{(0)}$ if and
only if $K\in N\mathbb{Z}$. Since $\ell$ indexes the decomposition of the
Hilbert space induced by $\mathbb{Z}_{N}^{(1)}$, as in (3.28), this means that
the action of $\mathbb{Z}_{N}^{(0)}$ generally does not preserve
decomposition.131313As an aside, note that $\mathbb{Z}_{N}^{(0)}$ acts by
permutation on the Bethe vacua, hence there exists a homomorphism:
$\varphi\,:\;\mathbb{Z}_{N}^{(0)}\longrightarrow
S_{\binom{K-1}{N-1}}\;:\;\gamma\mapsto\varphi_{\gamma}\leavevmode\nobreak\ .$
For the generator $\gamma_{0}$, the permutation $\varphi_{\gamma_{0}}$ is
implicitly given by the action (3.37). It can sometimes be instructive to
study the permutation $\varphi_{\gamma_{0}}$ in cycle notation. As an example,
consider the permutation of the Bethe vacua for $N=3$ and $K=6$,
$\displaystyle SU(3)_{6}:\quad\varphi_{\gamma_{0}}$
$\displaystyle=(1\,2\,3)(4\,5\,6)(7\,8\,9)(10)\leavevmode\nobreak\ ,$ where
the vacua are as in footnote 10. There are three cycles of length 3, while
there is a unique fixed point in this case – this is the vacuum
$\underline{l}=(0,2,4;0)$. This is a manifestation of the mixed ’t Hooft
anomaly between $\mathbb{Z}_{N}^{(1)}$ and $\mathbb{Z}_{N}^{(0)}$, as we will
explain momentarily.

Let $\mathcal{C}$ denote a basis element of
$H_{1}(\Sigma_{g},\mathbb{Z})\cong\mathbb{Z}^{2g}$, and let us wrap a single
topological line of charge $\gamma=n\gamma_{0}$ along $\mathcal{C}$. Its
expectation value is given by (2.62), namely:

$\displaystyle\begin{split}\langle\mathcal{U}^{\gamma}(\mathcal{C})\rangle_{\Sigma_{g}}=\sum_{\hat{u}\in\mathcal{S}_{\text{BE}}^{(\gamma)}}\mathcal{H}(\hat{u})^{g-1}\leavevmode\nobreak\
,\end{split}$ (3.39)

where $\mathcal{S}_{\text{BE}}^{(\gamma)}$ denotes the subset of the Bethe
vacua that are fixed under the action of $\gamma\in\mathbb{Z}_{N}^{(0)}$. In
particular, for $g=1$ this counts the number of fixed points:

$\langle\mathcal{U}^{\gamma}(\mathcal{C})\rangle_{T^{2}}=\left|\mathcal{S}_{\text{BE}}^{(\gamma)}\right|\leavevmode\nobreak\
.$ (3.40)

By modularity on $T^{3}$, we expect that:

$\langle\Pi^{\gamma}\rangle_{T^{2}}=\langle\mathcal{U}^{\gamma}(\mathcal{C})\rangle_{T^{2}}\leavevmode\nobreak\
,$ (3.41)

where $\langle\Pi^{\gamma}\rangle_{T^{2}}$ is given in (LABEL:genus-1-A-ops).
Indeed, this is simply the constraint (2.68) described in section 2.2.

We will now calculate (3.40) explicitly, for any $\gamma=n\gamma_{0}$ and any
value of $N$ and $K$. As previously alluded to, the flavour of the discussion
of fixed points and orbits depends crucially on the subgroup structure of the
zero-form symmetry $\mathbb{Z}_{N}^{(0)}$. The simplest case is where $N$ is a
prime number. Then, $\mathbb{Z}_{N}$ has the special property that every
element except the trivial element generates the full group. As a consequence,
$\mathbb{Z}_{N}$ has $N-1$ generators, and the result of the fixed point
counting will be independent on the choice of non-zero element
$\gamma\in\mathbb{Z}_{N}^{(0)}$. This is in strong contrast to the case where
$N$ has divisors, in which case elements $\gamma\in\mathbb{Z}_{N}^{(0)}$ can
lie in proper subgroups of $\mathbb{Z}_{N}$, which are labelled by the
divisors of $N$.

Fixed points for $N$ a prime integer. Let us first consider the case where $N$
is a prime number. Since there are no nontrivial subgroups of
$\mathbb{Z}_{N}$, there are only two different cases depending on the CS level
$K$. If $K$ is a multiple of $N$, the theory is non-anomalous, while if $K$ is
not a multiple of $N$ then the theory is maximally anomalous. In section
2.1.2, we described how the mixed anomaly constrains the orbit structure. In
the maximally anomalous case, all orbits are of ‘maximal’ length $N$. As a
consequence, there cannot be any fixed points. If $K$ is a multiple of $N$ on
the other hand, the orbit lengths are all divisors of $N$ – in our case where
$N$ is prime, these are just $1$ and $N$ itself.

We can easily find all fixed points. Assume that the generator $\gamma_{0}$
leaves invariant a particular Bethe vacuum, $\hat{u}=\hat{u}+\gamma$. By Weyl
transformations of the associated tuples $(l_{1},\dots,l_{N};\ell)$, (3.37)
implies that the values $(l_{1},\dots,l_{N})$ can at most be permuted, while
$\ell$ has to be invariant. From (3.38) it is clear that $\ell$ is preserved
if and only if $K\in N\mathbb{Z}$. Thus, this is a necessary condition to have
a fixed point. Moreover, the fixed points satisfy $(l_{a})\sim(l_{a}+\kappa)$,
with $\kappa\equiv{K\over N}\in\mathbb{Z}$. The equivalence above is up to
permutation. Any solution must consequently take the form:

$(l_{a};\ell)=\Big{(}s,s+\kappa,s+2\kappa,\cdots,s+(N-1)\kappa;\,Ns+\kappa{N(N-1)\over
2}\;\text{mod}\;\kappa N\Big{)}\leavevmode\nobreak\ ,$ (3.42)

for an integer $s$ such that $0\leq s<\kappa$, and with the constraint that
$\ell<N$. For $N$ odd, we see that there is a unique solution with $s=0$,
since the constraint on $\ell$ reads $\ell=Ns<N$, and thus $\ell=0$. In
appendix D.1, we prove the existence and uniqueness of the fixed point under
the generator $\gamma_{0}\in\mathbb{Z}_{N}$ for all integers $N$, with $K$ a
multiple of $N$.

Returning to the case $N\geq 3$ prime, every nontrivial element $\gamma\neq 0$
is a generator, and thus we have shown:

$\langle\mathcal{U}^{\gamma}(\mathcal{C})\rangle_{T^{2}}=\delta_{K\text{ mod
}N,0}$ (3.43)

for all $\gamma\in\mathbb{Z}_{N}\setminus\\{0\\}$. For the identity, we
trivially have $\mathcal{U}^{0}(\mathcal{C})=\mathbbm{1}$ and thus
$\langle\mathcal{U}^{0}(\mathcal{C})\rangle_{T^{2}}=\textbf{I}_{\text{W}}[SU(N)_{K}]$.

In order to determine the expectation value
$\langle\mathcal{U}^{\gamma}(\mathcal{C})\rangle_{\Sigma_{g}}$ for arbitrary
genus, we need to evaluate the handle-gluing operator (3.11) at the fixed
point. This is done by combining (3.39) with (D.5), such that:141414In the
second equation, we use the identity  $\prod_{1\leq a<b\leq
N}\sin^{2}\left(\frac{\pi(a-b)}{N}\right)=(2^{1-N}N)^{N}\leavevmode\nobreak\
,$ which in fact holds for any $N$. To prove this, note that:  $\prod_{1\leq
a<b\leq
N}\sin^{2}\left(\frac{\pi(a-b)}{N}\right)=\prod_{d=1}^{N-1}\sin\left(\frac{\pi
d}{N}\right)^{2(N-d)}.$ Using $\sin(\pi-\theta)=\sin(\theta)$, we can
rearrange the factors, such that it gives $N$ products of
$\prod_{d=1}^{N-1}\sin\left(\frac{\pi d}{N}\right)=2^{1-N}N$, which is a
standard identity.

$\mathcal{H}(\hat{u}_{\text{fixed}})=N\left(\frac{K}{2^{N}}\right)^{N-1}\prod_{1\leq
a<b\leq
N}\sin^{-2}\left(\frac{\pi(a-b)}{N}\right)=\left(\frac{K}{N}\right)^{(N-1)},$
(3.44)

which is an integer by the assumption that $N|K$. Since
$\hat{u}_{\text{fixed}}$ is fixed under the generator $\gamma_{0}$ of
$\mathbb{Z}_{N}$, it is also fixed under the action of
$\mathcal{U}^{n\gamma_{0}}$ for $n=1,2,\dots,N-1$. Therefore, we have

$\langle\mathcal{U}^{n\gamma_{0}}(\mathcal{C})\rangle_{\Sigma_{g}}=\mathcal{H}(\hat{u}_{\text{fixed}})^{g-1}=\left(\frac{K}{N}\right)^{(g-1)(N-1)}\leavevmode\nobreak\
,$ (3.45)

for all $n=1,\dots,N-1$, while for $n=0$ it is simply $\langle
1\rangle_{\Sigma_{g}}$.

Fixed points for general $N$. For each divisor $d$ of $N$, we have a cyclic
subgroup $\mathbb{Z}_{d}$ of $\mathbb{Z}_{N}$. If the $SU(N)_{K}$ theory is
non-anomalous, we can discretely gauge the full centre $\mathbb{Z}_{N}$, or
indeed any of its subgroups $\mathbb{Z}_{d}$. If the $SU(N)_{K}$ theory is
anomalous, there can still be a non-anomalous subgroup $\mathbb{Z}_{d}$ that
we can discretely gauge – we will spell it out more explicitly in section
3.1.3 below. Especially including background fields for the higher form
symmetries, this picture has been shown in other contexts to lead to a rich
web of theories related by gaugings of discrete symmetries Aharony:2013hda ;
Tachikawa:2017gyf .

It is easy to see that the order of the group generated by
$\gamma=n\gamma_{0}$ is the smallest integer $d$ such that $N|dn$, which is
given by $d=\frac{N}{\gcd(n,N)}$. Therefore:

$\langle n\gamma_{0}\rangle\cong\mathbb{Z}_{d(n)}\leavevmode\nobreak\ ,\qquad
d(n)\equiv\tfrac{N}{\gcd(n,N)}\leavevmode\nobreak\ ,$ (3.46)

which gives an explicit relation between the integers $n$ and divisors $d$ of
$N$. See figure 8 for an explicit example.

Figure 8: The full list of subgroups of the cyclic group $\mathbb{Z}_{12}$ is
$\mathbb{Z}_{1}$, $\mathbb{Z}_{2}$, $\mathbb{Z}_{3}$, $\mathbb{Z}_{4}$,
$\mathbb{Z}_{6}$ and $\mathbb{Z}_{12}$. They are generated by elements
$n\in\mathbb{Z}_{12}$ as visualised on this clock – for instance, $n=3$
generates $\mathbb{Z}_{4}\subset\mathbb{Z}_{12}$.

Before studying the orbits in general, let us first count the number of fixed
points under the subgroup $\langle n\gamma_{0}\rangle$ – that is, the orbits
of length 1. Clearly, this number can only depend on the divisor $d(n)$
(3.46). For $N$ prime, the only divisors are $d=1$ and $d=N$, which gives the
previous result (3.43). When $N$ has a nontrivial divisor $d\not\in\\{1,N\\}$,
then there can be multiple fixed points under a $\mathbb{Z}_{d}$ subgroup, as
long as the CS level $K$ is a multiple of $d$ as well. In appendix D.2, we
prove that the number of fixed points under the action of a $\mathbb{Z}_{d}$
subgroup is given by the simple formula:151515This number is simply the Witten
index (3.9) of the $SU(\tfrac{N}{d})_{\frac{K}{d}}$ theory, but this appears
to be a coincidence, i.e. there does not seem to be any deeper meaning to this
fact.

$\langle\mathcal{U}^{n\gamma_{0}}(\mathcal{C})\rangle_{T^{2}}=\binom{\frac{K}{d(n)}-1}{\frac{N}{d(n)}-1}\delta_{K\text{
mod }d(n),0}\leavevmode\nobreak\ .$ (3.47)

Inserting the explicit $n$-dependence from (3.46), the number of fixed points
(3.47) reads:

$\langle\mathcal{U}^{n\gamma_{0}}(\mathcal{C})\rangle_{T^{2}}=\binom{\frac{K}{N}\gcd(n,N)-1}{\gcd(n,N)-1}\,\delta_{K\text{
mod }\frac{N}{\gcd(n,N)},0}\leavevmode\nobreak\ .$ (3.48)

Note that when $n=1$, then $\frac{N}{\gcd(n,N)}=N$, such that the expression
(3.48) is zero unless $N|K$, for which the binomial coefficient evaluates to
1, giving back (3.43) which holds for any value of $N$ if $\gamma=\gamma_{0}$.
Moreover, if $N$ is prime and $n=1,\dots,N-1$ is arbitrary, then
$\frac{N}{\gcd(n,N)}=N$, while $\gcd(n,N)=1$ such that we arrive at (3.43)
again, in agreement with the fact that $n\gamma_{0}$ is a generator of
$\mathbb{Z}_{N}$. Finally, (3.48) is valid as well for $n=0$, for which we get
the trivial subgroup $\langle 0\gamma_{0}\rangle\cong\mathbb{Z}_{1}$. All
Bethe vacua are fixed under the trivial subgroup, and we get back the
$SU(N)_{K}$ Witten index (3.9).

The result (3.47) now allows us to perform an exact test of the 3d modularity
expectation (3.41). Indeed, the insertions of
$\mathcal{U}^{\gamma}(\mathcal{C})$ (3.47) and $\Pi^{\gamma}$
(LABEL:genus-1-A-ops) on $T^{2}$ agree precisely for all $\gamma$, as
expected. This is due to the convention of the gauge flux operator (3.18),
where the overall sign $(-1)^{N-1}$ has been inserted to eliminate any
relative minus signs. Thus, assuming 3d modularity, we have now established
(LABEL:genus-1-A-ops) as well.

Orbit structure. As outlined in section 2.2, the orbit structure under
$\mathbb{Z}_{N}$ is constrained by the mixed ’t Hooft anomaly $\mathcal{A}$.
In particular, any anomaly-free subgroup $\mathbb{Z}_{d}$ admits an orbit of
dimension $N/d$. However, this does not answer the questions of how many
orbits exist or, more specifically, of how many orbits of dimension $N/d$
exist.

The first question can be answered using the Cauchy–Frobenius lemma.161616This
lemma is also known as the orbit-counting theorem, or the _lemma that is not
Burnside’s_ neumann1979lemma . It asserts that the total number of orbits,
$|\mathcal{S}_{\text{BE}}/\mathbb{Z}_{N}^{(0)}|$, is given by the sum over
$\gamma\in\mathbb{Z}_{N}^{(0)}$ of the fixed points under the $\gamma$-action,
divided by the order of the group.171717Recall the lemma: Let $G$ be a finite
group that acts on a set $X$. The lemma asserts that
$|X/G|=\frac{1}{|G|}\sum_{g\in G}|X^{g}|$, where $X^{g}$ are the $g$-fixed
points in $X$. The proof goes as follows: write $\sum_{g\in
G}|X^{g}|=\sum_{x\in X}|G_{x}|$, where $|G_{x}|\subset G$ is the stabiliser
subgroup of $x\in X$. This identity simply re-expresses the sum over all
$(g,x)\in G\times X$ with $g\cdot x=x$. Using the orbit-stabiliser theorem
$|G_{x}|=|G|/|G\cdot x|$, we write this as a sum over orbits $G\cdot x$.
Finally, every element $x$ in a given orbit has the same orbit length,
$\sum_{g\in G}|X^{g}|=\sum_{x\in X}|G_{x}|=|G|\sum_{x\in X}\frac{1}{|G\cdot