CALT-TH-2021-041

UTTG 23-2021

1Walter Burke Institute for Theoretical Physics, California Institute of
Technology, Pasadena, CA 91125, USA

2Center of Mathematical Sciences and Applications, Harvard University,
Cambridge, MA 02138, USA

3Department of Mathematics, University of Texas, Austin, TX 78712, USA

4Department of Physics, University of Texas, Austin, TX 78712, USA

5Department of Mathematics, Yale University, New Haven, CT 06520, USA

###### Contents

  1. 1 Introduction
    1. 1.1 Generalized symmetries of 2D semisimple TQFTs
    2. 1.2 Quantization of moduli space of Higgs Bundles
    3. 1.3 Quantization of moduli space of parabolic Higgs bundles
  2. 2 Finite gauging of 2d TQFTs
    1. 2.1 Symmetries of two-dimensional TQFTs
    2. 2.2 Gauging of 0-form and 1-form symmetries
  3. 3 Equivariant Verlinde formula for any simple group
    1. 3.1 Review: the simply connected case
    2. 3.2 0-form symmetry and its gauging
      1. 3.2.1 Examples
    3. 3.3 1-form symmetry and its gauging
      1. 3.3.1 Examples
    4. 3.4 Quantization of $\mathcal{M}(G)$
      1. 3.4.1 Examples
  4. 4 Equivariant Verlinde algebra for $PSL(2,\mathbb{C})$
    1. 4.1 $PSL(2,\mathbb{C})$ Verlinde algebra from geometry
    2. 4.2 Categorification and generalization
  5. 5 Quantization of ${\mathcal{M}}(PSL(2,\mathbb{C}))$ for once-punctured torus
    1. 5.1 Computation from 2d TQFT
    2. 5.2 Equivariant trace: a geometric computation
    3. 5.3 Hitchin characters for $\mathcal{M}_{w_{2}}(PSL(2,\mathbb{C}))$
  6. A Partition function of $\mathcal{T}/\mathbb{Z}_{n}^{(0)}$
    1. A.1 Computation of $\mathcal{Z}_{h}$
    2. A.2 Computation of $\mathcal{Z}_{{\mathcal{T}}/\mathbb{Z}_{n}^{(0)}}$
    3. A.3 The case of product of cyclic groups
  7. B Fusion rules of $\mathcal{T}(PSL(2,\mathbb{C})_{k})$
    1. B.1 Fusion rules of $\mathcal{T}(SL(2,\mathbb{C})_{k})/\mathbb{Z}_{2}^{(1)}$
    2. B.2 Further gauging the $\mathbb{Z}_{2}^{(0)}$ symmetry

##  1 Introduction

2d TQFTs are ubiquitous in mathematics and physics, and they often serve as
bridges between different subjects and fields. One classical example is the
Verlinde formula [1], which not only counts conformal blocks of certain
rational conformal field theories known as Wess–Zumino–Witten models, gives
the dimension of the space of generalized theta functions on a Riemann
surface, but also enjoys a hidden connection with the quantum cohomology of
Grassmannians [2] (see also [3]). To see this hidden connection, one will need
to realize that the Verlinde formula is in fact a partition function of a two-
dimensional topoogical quantum field theory (2d TQFT), which is also equipped
with an algebra of local operators, also known as the Verlinde algebra. This
is illustrated in Figure 1.

Another benefit of working with the fully-fledged 2d TQFT, instead of just its
partition function, is that 2d TQFTs can have symmetries, which not only
impose strong constraints on the algebraic structures that the TQFT encodes,
but also enable one to produce families of distinct TQFTs by gauging these
symmetries.

As we will see in this paper, such perspective naturally unifies different
versions of the Verlinde formula associated with the same Lie algebra, and
explain several observations made in the mathematical study of the generalized
theta functions, including an observation by Beauville that there are variants
of the Verlinde formula which are not directly associated with conformal field
theories. To achieve such unification at the maximal extent, one will have to
incorporate generalized symmetries such as 1-form and $(-1)$-form symmetries.

In this paper, we will systematically study generalized symmetries and their
gauging in a class of 2d TQFTs that are “semisimple,” which obeys the original
Atiyah–Segal axioms for TQFTs.111This is to ensure that we won’t run into
potential pathologies that sometime are present in theory that are topological
in the boarder sense. One example is the 2d Yang–Mills theory in the zero
coupling limit (see e.g. [4, 5] for a detailed study of this theory), which
has an infinite-dimensional Hilbert spaces. However many results in the
present paper still apply to these cases. Such approach via generalized
symmetries is the most useful when the 2d TQFTs come in family which all enjoy
the same symmetry. This is indeed the case for the Verlinde formula, for which
there exists a 1-parameter deformation known as the equivariant Verlinde
formula [6, 7, 8] that comes from a 1-parameter family of 2d TQFTs. They
encode very interesting information about the geometry of the moduli space of
Higgs bundles on a Riemann surface, just like the original formula can be used
to probe the geometry of the moduli space of bundles. And one of the main goal
of this paper is to apply the general results about symmetries of 2d TQFTs to
investigate the problem of quantization of the moduli space of $G$-Higgs
bundles when $G$ is not necessarily simply connected.

Figure 1: Symmetry of a 2D TQFT is reflected in its partition function as well
as its algebra of local operators.

###  1.1 Generalized symmetries of 2D semisimple TQFTs

Global symmetry is one of the most important tools in the study of quantum
field theories. In the past decade, the classical notion of symmetries has
been generalized to the case of “p-form” symmetries [9]. These symmetries
share many of their properties with ordinary symmetries known to the
practitioners of QFT since the twentieth century. For example, they can have
anomalies, and when non-anomalous they can be gauged. Subsequent years have
seen a vast amount of applications of these new symmetries in improving our
understanding of strongly coupled systems.

Our focus in this paper will be on a class of QFTs in two dimensions known as
“semisimple topological QFTs”. These QFTs are topological in the sense that
their observables depend only on the topology of the underlying spacetime
manifold. The algebra of the local operators in such a topological theory will
turn out to be a commutative Frobenius algebra, and a well-known result
regarding 2d TQFTs is that the information they contain is fully captured by
the commutative Frobenius algebra,

$\text{2d TQFTs}\quad\Longleftrightarrow\quad\text{commutative Frobenius
algebras}.$ (1.1)

Taking the product of two operators is sometimes referred to as “fusion,”
while the bilinear form (also known as the Frobenius form, or, sometimes, the
“metric”) is defined by the $S^{2}$-partition function with insertion of two
local operators.

In this paper we will focus on 2d TQFTs that are semisimple, in the sense that
the corresponding commutative Frobenius algebra is semisimple.

This is equivalent to the condition that this algebra has a basis
${\mathcal{B}}=\\{e_{i}\\}$, sometime referred to as the “idempotent basis,”
such that the fusion is diagonalized, $e_{i}e_{j}=\delta_{ij}N_{j}e_{j}$; and
the pairing is orthonormal, $(e_{i},e_{j})=\delta_{ij}$. Such a basis is
subject to the ambiguity of $e_{j}\mapsto-e_{j}$ for each $j$, which can be
fixed by assuming that the $N_{j}$’s all have positive real part.222$N_{j}$’s
are often referred to as “fusion eigenvalues” or “quantum dimensions.” When
the TQFT is unitary, $N_{j}$ take real values and we can further assume that
they are positive[10].

A global symmetry of a two-dimensional QFT is generated by a topological
defect of dimension 0, 1 or 2, which corresponds respectively to a $1$-form, a
$0$-form or a $(-1)$-form symmetry. While $(-1)$-form symmetries are not
symmetries in the traditional sense as they alter the theory, 0- and 1-form
symmetries must act in a very restricted way. We will show in this paper that
for a semisimple TQFT,

  * •

a 0-form symmetry necessarily acts as a permutation on ${\mathcal{B}}$,
permuting the basis elements with the same fusion eigenvalue $N_{i}$, and

  * •

a 1-form symmetry acts on the elements of ${\mathcal{B}}$ by multiplication
with a phase.

These statements have interesting consequences. For example, it follows from
the first statement that the action of any connected Lie group on a 2d
semisimple TQFT must be trivial. Furthermore, if a 2d TQFT has $m$ identical
eigenvalues, it will have an $S_{m}$ symmetry, and any symmetry of a 2d TQFT
must be enhanced to a product of permutation groups.

Gauging generalized symmetries is often a useful way of constructing new QFTs.
In particular, gauging a $p$-form symmetry of a semisimple TQFT produces
another semisimple TQFT. So we can ask how the data of the new TQFT is
determined in terms of the data of the original TQFT. In this paper, we give
an algorithm to fully determined the commutative Frobenius algebra of the
resulting theory from the data of the original theory.

For example, consider the case of gauging a $\mathbb{Z}_{n}^{(0)}$ symmetry of
a TQFT ${\mathcal{T}}$ with diagonal basis ${\mathcal{B}}$. We find that the
gauged TQFT has a diagonal basis $\\{e_{i}^{(a)}\\}$ with
$i\in{\mathcal{B}}/\mathbb{Z}_{n}$ — the set of all orbits of ${\mathcal{B}}$
under the action of $\mathbb{Z}_{n}$, and
$a\in\\{1,2,\cdots,|\text{Stab}(i)|\\}$ — the stabilizer of a representative
of an orbit $i$. The fusion of $\\{e_{i}^{(a)}\\}$ is

$\displaystyle
e_{i}^{(a)}e_{j}^{(b)}=\delta_{ij}\delta^{(a)(b)}\left(\frac{|\text{Stab}(i)|}{n^{\frac{1}{2}}}N_{i}\right)e_{i}^{(a)}.$
(1.2)

Note that the fusion eigenvalues are identical for each fixed $i$. Therefore,
the gauged TQFT has a product 0-form symmetry, $\prod\limits_{i}S_{m_{i}}$ for
$m_{i}=|\text{Stab}(i)|$. In particular, the familiar “dual”
$\mathbb{Z}_{n}^{(0)}$ symmetry of the gauged TQFT is necessarily enhanced to
this much larger group, and this fact is true about any semisimple TQFT.

###  1.2 Quantization of moduli space of Higgs Bundles

As an application of the general discussion of finite symmetries of two-
dimensional TQFTs and their gaugings, we study the problem of quantization of
the moduli space of Higgs bundles in this paper. This problem has already been
studied in the recent years in the case of moduli space
${\mathcal{M}}:={\mathcal{M}}(\widetilde{G},\Sigma)$ of $\widetilde{G}$-Higgs
bundles on a genus-$g$ Riemann surface $\Sigma$ when $\widetilde{G}$ is either
simply connected, or with a free $\pi_{1}$, such as $SU(N)$ or $U(N)$ [6, 11,
7, 8]. The moduli space is non-compact, and the procedure of geometric
quantization leads to an infinite-dimensional Hilbert space identified with
$H^{0}({\mathcal{M}},{\mathcal{L}}^{k})$, the space of holomorphic sections of
the “pre-quantum line bundle” ${\mathcal{L}}^{k}$. However, there exists a
$\mathbb{C}^{\star}$-action on ${\mathcal{M}}$ that lifts to an action on this
infinite-dimensional space, giving a decomposition

$\displaystyle
H^{0}({\mathcal{M}},{\mathcal{L}}^{k})=\oplus_{n=0}^{\infty}H^{0}_{n}({\mathcal{M}},{\mathcal{L}}^{k}).$
(1.3)

The $\mathbb{C}^{\star}$-character,

$\displaystyle\text{dim}_{t}\
H^{0}({\mathcal{M}},{\mathcal{L}}^{k})=\sum\limits_{n=0}^{\infty}\text{dim}\
H^{0}_{n}({\mathcal{M}},{\mathcal{L}}^{k})t^{n},$ (1.4)

also known as the Hitchin character, was computed in [6, 7, 8]. When $\Sigma$
is a closed surface of genus $g>1$, it takes the value,

$\displaystyle\text{dim}_{t}H^{0}({\mathcal{M}},{\mathcal{L}}^{k})=\sum\limits_{\lambda\in
P_{k}}\theta_{t}(f_{\lambda,t})^{1-g}.$ (1.5)

The quantities appearing on the right hand side were defined in [7, 8], and
are reviewed in Section 3.1. It was proved in [7] that the Hitchin character
is the partition function of a two dimensional semisimple TQFT, which we
denote by ${\mathcal{T}}(\widetilde{G}_{k})$.

In this paper, we will study the quantization of the moduli space
${\mathcal{M}}(G,\Sigma)$ of $G$-Higgs bundles when $G$ is not simply-
connected. Let $\pi_{1}(G)=Z$, then these moduli spaces are identified with
the quotient of ${\mathcal{M}}$ by $H_{1}(\Sigma,Z)$. This action and the
$\mathbb{C}^{\star}$-action on ${\mathcal{M}}$ commute, thus it leads to a
natural question: what are the Hitchin characters $\text{dim}_{t}\
H^{0}({\mathcal{M}}(G),{\mathcal{L}}^{k})$? The approach used in [7, 8] cannot
be directly applied to this case (at least not without any modifications). On
the other hand, $H^{0}({\mathcal{M}}(G),{\mathcal{L}}^{k})$ is identified with
the subspace of sections in $H^{0}({\mathcal{M}},{\mathcal{L}}^{k})$ that are
invariant under the action of ${\mathcal{G}}:=H_{1}(\Sigma,Z)$, giving

$\displaystyle\text{dim}_{t}\
H^{0}({\mathcal{M}}(G),{\mathcal{L}}^{k})=\frac{1}{|{\mathcal{G}}|}\sum\limits_{h\in{\mathcal{G}}}\text{Tr}_{t}(h|H^{0}({\mathcal{M}},{\mathcal{L}}^{k})).$
(1.6)

Thus the problem of quantizing ${\mathcal{M}}(G)$ is exchanged with the
problem of finding the equivariant trace,
$\text{Tr}_{t}(h|H^{0}({\mathcal{M}},{\mathcal{L}}^{k}))=\sum\limits_{n=0}^{\infty}{\rm
Tr\,}(h|H^{0}_{n}({\mathcal{M}},{\mathcal{L}}^{k}))t^{n}$ for each
$h\in{\mathcal{G}}$.

Our understanding of finite symmetries and their gaugings presented in Section
1.1 helps solve this problem. In particular, the TQFT
${\mathcal{T}}(\widetilde{G}_{k})$ has the center symmetry as (a subgroup of
its) 0-form and 1-form symmetry groups. We argue in Section 3.4 that the
equivariant trace $\text{Tr}_{t}(h|H^{0}({\mathcal{M}},{\mathcal{L}}^{k}))$ is
equal to the partition function of ${\mathcal{T}}(G_{k})$ in the presence of
certain 0-form and 1-form symmetry backgrounds. Here we will present some
examples.

Let us consider the simplest case $\widetilde{G}=SL(2,\mathbb{C})$, $k\equiv
0\pmod{4}$ and a non-zero $h\in H_{1}(\Sigma,\mathbb{Z}_{2})$.333The reason we
restrict to $k\equiv 0\pmod{4}$ is explained in Section 3.1. See the
discussion around equation (3.14). We find

$\displaystyle\text{Tr}_{t}(h|H^{0}({\mathcal{M}},{\mathcal{L}}^{k}))=\left(\frac{\frac{k}{2}+1+\left(\frac{k}{2}-1\right)t}{(1-t)(1+t)^{3}}\right)^{g-1}.$
(1.7)

This readily gives the Hitchin characters for
${\mathcal{M}}(PSL(2,\mathbb{C}))$ using (1.6). For another example, consider
the case $\widetilde{G}=SL(4,\mathbb{C})$ and $k\equiv 0\pmod{8}$. If
${\mathcal{M}}_{d}\equiv{\mathcal{M}}_{d}(SL(4,\mathbb{C}))$ denotes the
moduli space of semi-stable $SL(4,\mathbb{C})$-Higgs bundles of degree $d$
(where $d\in\\{0,1,2,3\\}$), we find that for any non-zero $h\in
H_{1}(\Sigma,\mathbb{Z}_{2})\subset H_{1}(\Sigma,\mathbb{Z}_{4})$,

$\displaystyle\text{Tr}_{t}(h|H^{0}({\mathcal{M}}_{d},{\mathcal{L}}^{k}))=\sum\limits_{\begin{subarray}{c}\lambda=(a,\frac{k}{2}-a,a)\\\
a\leq\frac{k}{2}\end{subarray}}(-1)^{ad}\theta_{t}(f_{\lambda,t}).$ (1.8)

The quantities appearing on the right-hand side are defined in Section 3. Here
we remark that it depends only on the parity of $d$, and not on its actual
value. This is a surprising result as the geometry of the moduli space
${\mathcal{M}}_{d}$ for different values of degree $d$ are vastly different,
but our results show that the equivariant traces some times turn out to be the
same.

Using the formulae for equivariant trace, we will find the Hitchin characters
$\text{dim}_{t}\ H^{0}({\mathcal{M}}(G),{\mathcal{L}}^{k})$ for an arbitrary
complex simple Lie group $G$. For example, when $G=PSL(2,\mathbb{C})$, and the
underlying surface $\Sigma$ is a genus-$2$ curve, we find

$\displaystyle\text{dim}_{t}\
H^{0}({\mathcal{M}}_{w_{2}=0}(PSL(2,\mathbb{C})),{\mathcal{L}}^{k})=\frac{1}{(1-t)^{3}}\left(c_{3}k^{3}+c_{2}k^{2}+c_{1}k+c_{0}\right)$
(1.9)

where

$\displaystyle c_{3}$ $\displaystyle=$ $\displaystyle\frac{1}{96},$
$\displaystyle c_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{16}\frac{1+t^{2}}{1-t^{2}},$ $\displaystyle c_{1}$
$\displaystyle=$
$\displaystyle\frac{7-27t+34t^{2}-27t^{3}+7t^{4}}{12(1-t^{2})^{2}},$
$\displaystyle c_{0}$ $\displaystyle=$
$\displaystyle\frac{1-16t+15t^{2}-19t^{3}+15t^{4}-6t^{5}+t^{6}-t^{k+3}}{(1-t^{2})^{3}}.$

Finally, we remark that as in [6], the zeroth piece in the decomposition (1.3)
(and its analogue in the non-simply-connected case) is identified with the
Hilbert space of compact group Chern–Simons theory on $\Sigma$. The $t\to 0$
limit of our formulae give quantization of the moduli space of semi-stable
holomorphic $G$-bundles. In special cases, these formulas have been obtained
in the literature using more geometric methods [12, 13, 14]. In each case, the
$t\to 0$ limit of our results matches with the formulae found there.

###  1.3 Quantization of moduli space of parabolic Higgs bundles

We also discuss the quantization of the moduli space of parabolic semi-stable
$G$-Higgs bundles in this paper.444When the genus is small, the right
geometric interpretation of the equivariant Verlinde formula is often in terms
of the moduli _stack_ of (parabolic) Higgs bundles. However, we will not be
concerned with this distinction in the present paper. For more details, see
e.g. Section 5 in [7]. When $G=\widetilde{G}$, i.e. the group is simply-
connected, the quantization of these spaces was discussed in [6, 11, 7]. The
Hitchin character is captured by the partition functions of
${\mathcal{T}}(\widetilde{G}_{k})$ on punctured surfaces. In particular,
${\mathcal{T}}(\widetilde{G}_{k})$ has a distinguished basis called the
“parabolic basis” in which the fusion rules and the metric compute the Hitchin
characters for the moduli spaces of parabolic Higgs bundles on three-punctured
and two-punctured spheres respectively. The algebra given by the fusion rules
of ${\mathcal{T}}(\widetilde{G}_{k})$ in this basis is sometimes referred to
as the “equivariant Verlinde algebra” [6].

As an example, consider the case of $\widetilde{G}=SL(2,\mathbb{C})$ where a
choice of parabolic structure is determined by an integrable weight
$j\in\\{0,1,\cdots,k\\}$. As we will review in greater detail in Section 4, in
the corresponding parabolic basis, ${\mathcal{T}}(SL(2,\mathbb{C})_{k})$
assigns

$\displaystyle f^{a_{1}a_{2}a_{3}}=\begin{cases}1&\mbox{if
}a_{1}+a_{2}+a_{3}\in 2\mathbb{Z}\ \text{and}\Delta\leq 0\\\
t^{\Delta/2}&\mbox{if }a_{1}+a_{2}+a_{3}\in 2\mathbb{Z}\ \text{and}\Delta>0\\\
0&\mbox{otherwise}\end{cases}$ (1.10)

to the three-punctured sphere labelled by $a_{i}$. This is also the Hitchin
character of the moduli space of semi-stable parabolic
$SL(2,\mathbb{C})$-Higgs bundles on the surface [6, 7].

In this paper, we extend the results of [6, 11, 7] to the case when $G$ is not
simply-connected. Our main example in Section 4 is that of
$G=PSL(2,\mathbb{C})$. The corresponding TQFT
${\mathcal{T}}(PSL(2,\mathbb{C})_{k})$ is obtained by gauging a
$\mathbb{Z}_{2}^{(0)}\times\mathbb{Z}_{2}^{(1)}$ symmetry of
${\mathcal{T}}(SL(2,\mathbb{C})_{k})$. We propose that there exists a
distinguished basis of ${\mathcal{T}}(PSL(2,\mathbb{C})_{k})$ in which the
fusion rules and the Frobenius bilinear form compute the Hitchin characters
for the moduli space of parabolic $PSL(2,\mathbb{C})$-Higgs bundles on three-
punctured and two-punctured spheres respectively.

An interesting fact that might be surprising from the purely geometric point
of view is that ${\mathcal{T}}(PSL(2,\mathbb{C})_{k})$ now has an extra
“twisted state,” which has to be included to fully characterize the 2d TQFT.
In other words, the corresponding Frobenius algebra now has basis elements,

$\displaystyle
x^{0},x^{2},\cdots,x^{\frac{k}{2}-2},x^{{\frac{k}{2}}^{(1)}},x^{{\frac{k}{2}}^{(2)}},$
(1.11)

where the superscript denotes an integrable highest weight of
$SL(2,\mathbb{C})$. The appearance of an additional operators labeled by the
highest weight $\frac{k}{2}$ is essential to ensure that the Hitchin
characters of ${\mathcal{M}}(PSL(2,\mathbb{C}))$ is well behaved under cutting
and gluing. It will be interesting to better understand its geometric
interpretation in terms of the moduli space of Higgs bundles, which we hope to
further investigate in future work.

We check the physics prediction against geometry in Section 5, where we
compute the Hitchin character for ${\mathcal{M}}(PSL(2,\mathbb{C}))$ in the
case of once-punctured torus in two ways. We carry out this computation first
through TQFT methods, and then through a Lefschetz trace formula due to Atiyah
and Singer [15], and show that the two match. In either case, we find

$\displaystyle\text{Tr}_{t}(h|H^{0}({\mathcal{M}}_{d},{\mathcal{L}}^{k}))=\frac{(-1)^{a/2}}{1+t}$
(1.12)

which gives the following expressions for the Hitchin character,

$\displaystyle\text{dim}_{t}H^{0}({\mathcal{M}}_{w_{2}=0}(PSL(2,\mathbb{C}),{\mathcal{L}}^{k})$
$\displaystyle=$
$\displaystyle\frac{1}{4}\left(\frac{k-a+1}{1-t}+\frac{2t}{(1-t)^{2}}+\frac{4t^{a/2}}{(1-t^{-1})(1-t^{2})}+\frac{3(-1)^{a/2}}{1+t}\right)$

and

$\displaystyle\text{dim}_{t}H^{0}({\mathcal{M}}_{w_{2}=1}(PSL(2,\mathbb{C}),{\mathcal{L}}^{k})$
$\displaystyle=$
$\displaystyle\frac{1}{4}\left(\frac{a+1}{1-t}+\frac{2t}{(1-t)^{2}}+\frac{4t^{(k-a)/2}}{(1-t^{-1})(1-t^{2})}+\frac{3(-1)^{a/2}}{1+t}\right).$

##  2 Finite gauging of 2d TQFTs

In this section, we will discuss finite symmetries of two-dimensional
semisimple TQFTs and study their gauging. In Section 2.1, we will discuss the
action of 0-form and 1-form symmetries on the TQFT in the basis that
diagonalizes the fusion rule. We will show that a 0-form symmetry necessarily
acts as a permutation on the basis elements, and that a 1-form symmetry acts
by multiplication by a root of unity. In Section 2.2, we will study the
gauging of abelian 0-form and 1-form symmetries. As a result, we will find the
partition functions and the Frobenius algebras of the new theories obtained
after gauging.

###  2.1 Symmetries of two-dimensional TQFTs

A symmetry of a two-dimensional QFT is generated by a topological defect of
dimension 0, 1, or 2 which respectively corresponds to 1-form, 0-form and
$(-1)$-form symmetries. Among these, both 1-form and 0-form symmetries can act
on the Hilbert space on $S^{1}$. This is depicted in Figure 2. In the language
of commutative Frobenius algebras, having a 0-form symmetry means that the
bilinear form is invariant

$(v_{1},v_{2})=(g\cdot v_{1},g\cdot v_{2}),$ (2.1)

where $g$ is a 0-form symmetry transformation, and that the fusion of two
operators satisfies

$(g\cdot v_{1})\times(g\cdot v_{2})=g\cdot(v_{1}\times v_{2}).$ (2.2)

Similarly, when $g$ is a 1-form symmetry transformation, the bilinear form is
invariant in the sense that

$(v_{1},v_{2})=(g\cdot v_{1},g^{-1}\cdot v_{2}),$ (2.3)

and the fusion product satisfies

$(g\cdot v_{1})\times(g^{-1}\cdot v_{2})=v_{1}\times v_{2}.$ (2.4)

The geometric meanings of these equalities are illustrated in Figure 3.

(a) The action of 0-form symmetry (b) The action of 1-form symmetry

Figure 2: The action of 0-form and 1-form symmetry transformations on a state
in Hilbert space.

(a) The action of 0-form symmetry (b) The action of 1-form symmetry

Figure 3: The action of 0-form and 1-form symmetry transformations on surfaces
in a two-dimensional TQFT.

A “semisimple” 2d TQFT—meaning the commutative Frobenius algebra is
semisimple—can be decomposed into ones with one-dimensional Hilbert space.
This amounts to diagonalizing the Frobenius algebra into a direct sum of one-
dimensional subalgebras. In other words, there exists a basis
${\mathcal{B}}=\\{e_{i}\\}$ that diagonalizes both the pairing
$(e_{i},e_{j})=\delta_{ij}$ and the fusion $e_{i}e_{j}=\delta_{ij}N_{i}e_{i}$.
The trace map sends $e_{i}$ to $\frac{1}{N_{i}}$. We will refer to $N_{i}$’s
as the“fusion eigenvalues,” and write the decomposition as
${\mathcal{A}}=\bigoplus_{i}\mathbb{C}_{N_{i}}$.555There is an ambiguity in
the choice of a diagonal basis that satisfies the stated conditions:
$\\{-e_{i}\\}$ gives another basis but with eigenvalues $-N_{i}$. We will fix
this ambiguity by imposing $\text{arg}(N_{i})\in[0,\pi)$.

We will now proceed to show that

  * •

a 0-form symmetry transformation acts on the set of $e_{i}$’s by permutation
where it can only permute the elements with the same fusion eigenvalue, and

  * •

a 1-form symmetry acts on $e_{i}$ by a phase.

To see the first, assume that a 0-form symmetry transformation labeled by $g$
acts on $e_{i}$ as

$g\cdot e_{i}=\sum_{j}c_{ij}e_{j}.$ (2.5)

As symmetry preserves the inner product,

$(g\cdot e_{i},g\cdot e_{\ell})=\delta_{i\ell},$ (2.6)

so we have

$\sum\limits_{j}c_{ij}c_{\ell j}=\delta_{i\ell}.$ (2.7)

In other words, $c_{ij}$ can be assembled into an orthogonal matrix. On the
other hand, we also have the equality

$(g\cdot e_{i})\times(g\cdot e_{\ell})=g\cdot(e_{i}\times e_{\ell}),$ (2.8)

which leads to

$c_{ij}c_{\ell j}N_{j}=\delta_{i\ell}c_{ij}N_{i}.$ (2.9)

To see what this equation implies, first choose $\ell=i$. Then we have
$N_{j}c_{ij}^{2}=N_{i}c_{ij}$, which tells us that

$c_{ij}=0\text{ or }N_{i}/N_{j}.$ (2.10)

But for $\ell\neq i$, Equation (2.9) becomes

$c_{ij}c_{lj}=0$ (2.11)

for all $j$. Therefore, if a $c_{ij}$ is non-zero, it will be the only non-
zero entry in the $j$-th column. Furthermore, it also has to be the only non-
zero entry in the $i$-th row, as $c^{T}$ represents the action of $g^{-1}$.
Since the choice of $j$ is unique given $i$, we can define a function $\sigma$
such that $j=\sigma(i)$. It will be single-valued and hence is a permutation.
The constraint (2.7) now implies that $N_{i}=N_{j}$ if $j=\sigma(i)$.666 Note
that (2.7) really says $N_{i}^{2}=N_{j}^{2}$ if $j=\sigma(i)$. However, the
other possibility $N_{i}=-N_{j}$ is not consistent with footnote 5, i.e.
$\text{arg}(N_{i})\in[0,\pi)$. Therefore, the action of $g$ is by permutation
$g\cdot e_{i}=e_{\sigma(i)}$ and it can only permute $e_{i}$’s with the same
fusion eigenvalues.

This simple fact has many interesting consequences. For example, the action of
any connected Lie group on a 2d TQFT has to be trivial. On the other hand, if
a 2d TQFT has $m$ identical eigenvalues, it will have an $S_{m}$ symmetry, and
any symmetry group of a 2d TQFT will be enhanced to a product of permutation
groups.

Now we show the second point mentioned above, i.e. the action of 1-form
symmetry on the elements in the diagonal basis must be as multiplication by a
phase. Let $g$ be a 1-form symmetry transformation, then

$(g\cdot e_{i})\times e_{j}=e_{i}\times(g\cdot e_{j})$ (2.12)

tells us that $g\cdot e_{i}=\chi_{i}(g)e_{i}$ for some
$\chi_{i}(g)\in\mathbb{C}^{\times}$. When $g$ is an element of finite order,
$\chi_{i}(g)$ is in fact a root of unity.

###  2.2 Gauging of 0-form and 1-form symmetries

In this subsection we will consider gauging abelian 0-form and 1-form
symmetries. A finite abelian group is a product of cyclic groups so we may
gauge one factor at a time. Therefore, we will focus on gauging a
$\mathbb{Z}_{n}^{(0)}$ and a $\mathbb{Z}_{n}^{(1)}$ symmetry of a TQFT
${\mathcal{T}}$. As a result we will find the partition functions and the
commutative Frobenius algebras associated to the TQFTs
${\mathcal{T}}/\mathbb{Z}_{n}^{(0)}$ and
${\mathcal{T}}/\mathbb{Z}_{n}^{(1)}$.777We will assume throughout this section
that these symmetries are non-anomalous. Of course, if these symmetries were
anomalous, they could not be gauged.

Let us first start with the case of ${\mathcal{T}}/\mathbb{Z}_{n}^{(0)}$. The
partition function ${\mathcal{Z}}_{{\mathcal{T}}/\mathbb{Z}_{n}^{(0)}}$ on a
closed surface $\Sigma$ of genus $g$ is obtained by summing over principal
$\mathbb{Z}_{n}$ bundles. These are parameterized by elements of
$H_{1}(\Sigma,\mathbb{Z}_{n})$, and we write888Our choice of normalization
factor is such that gauging the $\mathbb{Z}_{n}^{(0)}$ symmetry twice gives
back the partition function of ${\mathcal{T}}$ on the nose, and not up to the
partition function of an Euler TQFT.

$\displaystyle{\mathcal{Z}}_{{\mathcal{T}}/\mathbb{Z}_{n}^{(0)}}=\frac{1}{|H^{1}(\Sigma,\mathbb{Z}_{n})|^{\frac{1}{2}}}\sum\limits_{h\in
H_{1}(\Sigma,\mathbb{Z}_{n})}{\mathcal{Z}}_{h}.$ (2.13)

For $h=0$, ${\mathcal{Z}}_{h}$ is the ordinary partition function of
${\mathcal{T}}$. It is well known that in terms of the fusion eigenvalues
$N_{i}$ introduced in the previous subsection,

$\displaystyle{\mathcal{Z}}_{{\mathcal{T}}}=\sum\limits_{i\in{\mathcal{B}}}N_{i}^{2g-2}.$
(2.14)

We will show in Appendix A that

$\displaystyle{\mathcal{Z}}_{{\mathcal{T}}/\mathbb{Z}_{n}^{(0)}}$
$\displaystyle=$
$\displaystyle\sum\limits_{i\in{\mathcal{B}}/\mathbb{Z}_{n}}|\text{Stab}(i)|\left(\frac{|\text{Stab}(i)|}{|\mathbb{Z}_{n}|^{\frac{1}{2}}}N_{i}\right)^{2g-2}$
(2.15)

where ${\mathcal{B}}/\mathbb{Z}_{n}$ is the set of all orbits of
${\mathcal{B}}$ under the $\mathbb{Z}_{n}$ action. For an orbit $i$, we denote
the cardinality of the stabilizer of any of its representatives in
${\mathcal{B}}$ as $|\text{Stab}(i)|$. A crucial ingredient in the derivation
of this result will be the partition function ${\mathcal{Z}}_{h}$ which we
also compute in Appendix A,

$\displaystyle{\mathcal{Z}}_{h}=\sum\limits_{i\in{\mathcal{B}}^{\mathbb{Z}_{m}}}N_{i}^{2g-2}.$
(2.16)

Here $\mathbb{Z}_{m}$ is the smallest subgroup of $\mathbb{Z}_{n}$ such that
$H_{1}(\Sigma,\mathbb{Z}_{m})$ contains $h$, and
${\mathcal{B}}^{\mathbb{Z}_{m}}\subset{\mathcal{B}}$ is the subset of fixed
points of $\mathbb{Z}_{m}$.

The commutative Frobenius algebra associated to the gauged TQFT
${\mathcal{T}}/\mathbb{Z}_{n}^{(0)}$ can be read off from equation (2.15).
This algebra has a diagonal basis $\\{e_{i}^{(a)}\\}$ for
$i\in{\mathcal{B}}/\mathbb{Z}_{n}$ and $a\in|\text{Stab}(i)|$. The fusion of
these operators is

$\displaystyle
e_{i}^{(a)}e_{j}^{(b)}=\delta_{ij}\delta^{(a)(b)}\left(\frac{|\text{Stab}(i)|}{|\mathbb{Z}_{n}|^{\frac{1}{2}}}N_{i}\right)e_{i}^{(a)}.$
(2.17)

We note that the fusion eigenvalues are independent of the value of $a$, i.e.
the gauged TQFT has a $\prod\limits_{i}S_{m_{i}}$ symmetry for
$m_{i}=|\text{Stab}(i)|$. This is an enhancement of the familiar “dual”
$\mathbb{Z}_{n}^{(0)}$ symmetry obtained after gauging a
$\mathbb{Z}_{n}^{(0)}$ symmetry. We emphasize that this enhancement occurs for
all two-dimensional semisimple TQFTs.

We turn now to the case of ${\mathcal{T}}/\mathbb{Z}_{n}^{(1)}$. As discussed
in Section 2.1, a 1-form symmetry transformation acts on ${\mathcal{B}}$ by
multiplication with roots of unity, $z.e_{i}=\chi_{i}(z)e_{i}$. Here,
$z\in\mathbb{Z}_{n}$ denotes a local operator in ${\mathcal{T}}$. As
$\mathbb{Z}_{n}$ is gauged, an operator $e_{i}$ with $\chi_{i}(z)\neq 1$ for
any $z\in\mathbb{Z}_{n}$ is projected out. The diagonal basis of
${\mathcal{T}}/\mathbb{Z}_{n}^{(1)}$ is the subset of operators
$\\{e_{i}|\chi_{i}(z)=1\ \forall z\in\mathbb{Z}_{n}\\}$ with fusion rules

$\displaystyle e_{i}e_{j}=\delta_{ij}N_{i}n^{-\frac{1}{2}}e_{i}.$ (2.18)

The normalization factor $n^{-\frac{1}{2}}$ has been fixed by comparison with
the partition function that we now compute.

The partition function of ${\mathcal{T}}/\mathbb{Z}_{n}^{(1)}$ is obtained by
summing over various $\mathbb{Z}_{n}$ gerbes which are parameterized by
$H^{2}(\Sigma,\mathbb{Z}_{n})=\mathbb{Z}_{n}$. In particular,

$\displaystyle{\mathcal{Z}}_{{\mathcal{T}}/\mathbb{Z}_{n}^{(1)}}=\frac{1}{|H^{1}(\Sigma,\mathbb{Z}_{n})|^{\frac{1}{2}}}\sum\limits_{z\in\mathbb{Z}_{n}}{\mathcal{Z}}_{z}.$
(2.19)

Here, ${\mathcal{Z}}_{z}$ is the partition function of ${\mathcal{T}}$ in the
presence of the local operator $z$. The equation $z.e_{i}=\chi_{i}(z)e_{i}$
gives

$\displaystyle{\mathcal{Z}}_{z}=\sum\limits_{i\in{\mathcal{B}}}\chi_{i}(z)N_{i}^{2g-2}.$
(2.20)

Summing over $\mathbb{Z}_{n}$ backgrounds, we obtain

$\displaystyle{\mathcal{Z}}_{{\mathcal{T}}/\mathbb{Z}_{n}^{(1)}}=\sum\limits_{\\{e_{i}|\chi_{i}(z)=1\
\forall
z\in\mathbb{Z}_{n}\\}}\left(\frac{N_{i}}{|\mathbb{Z}_{n}|^{\frac{1}{2}}}\right)^{2g-2}$
(2.21)

which gives the result in (2.18).

We will end this section with two remarks. The first concerns gauging
$(-1)$-form symmetries. The presence of such a symmetry implies that there is
a parameter $\theta$ and a family of theories ${\mathcal{T}}_{\theta}$
parametrized by it [16, 17]. Gauging a $(-1)$-form symmetry at the level of
Frobenius algebra corresponds to taking direct products. One way that
$(-1)$-form symmetry in 2d naturally arises is via gauging 1-form symmetries,
where the $\theta$ parameter often undergoes by the name of “discrete torsion”
(see e.g. [18] for a review, and also [19] and references therein for
discussions about “decomposition,” which is a closely related phenomenon).

The second comment is about gauging at the level of extended 2d TQFTs. These
are theories in which one is allowed to have boundary conditions. Just as
commutative Frobenius algebras are equivalent to 2d (non-extended) TQFTs, 2d
extended TQFTs are fully described by Calabi–Yau categories (see [20] for a
detailed review of this subject). As the simple objects of a Calabi–Yau
category are in one-to-one correspondence with idempotent basis vectors that
diagonalize the fusion rule, it is straightforward to translate the
prescription of gauging at the level of Frobenius algebra to that at the level
of Calabi–Yau categories.

##  3 Equivariant Verlinde formula for any simple group

In this section we will use the machinery developed in section 2 to compute
the equivariant Verlinde formula for ${\mathcal{M}}(G)$ — the moduli space of
semi-stable $G$-Higgs bundles for a connected simple complex Lie group $G$. We
denote the universal cover of $G$ by $\widetilde{G}$. As $\widetilde{G}$ is
simply connected, the answer for it is known [6, 7, 8], and is shown to be the
partition function of a two-dimensional semisimple TQFT [6, 7]. We will study
symmetries of this TQFT and their gaugings. As a consequence, we will obtain
the equivariant Verlinde formula for ${\mathcal{M}}(G)$ for any simple Lie
group $G$.

###  3.1 Review: the simply connected case

Let $\widetilde{G}$ be a simple and simply connected complex Lie group, and
let ${\mathcal{N}}$ be the moduli space of semi-stable $\widetilde{G}$-vector
bundles of degree zero on a genus-$g$ surface $\Sigma$. For
$k\in\mathbb{Z}_{\geq 0}$, the Verlinde formula computes the dimension of
$H^{0}({\mathcal{N}},{\mathcal{L}}^{k})$ where ${\mathcal{L}}$ is the
determinant line bundle over ${\mathcal{N}}$. This formula is described as
follows.

We denote by $\widetilde{T}$ a maximal torus of $\widetilde{G}$ and by $W$ the
Weyl group. Also, let $\mathfrak{t}\subset\mathfrak{g}$ be the Cartan
subalgebra of $\mathfrak{g}$ associated to $\widetilde{T}$. There exists a
unique inner product $\langle-,-\rangle$ on $\mathfrak{g}$ such that the
highest root has length squared equal to 2. If $h$ is the dual coxeter number
of $\mathfrak{g}$ we define $i_{k}:\mathfrak{t}\to\mathfrak{t}^{\star}$ as

$\displaystyle i_{k}(a)=(k+h)\langle a,-\rangle.$ (3.1)

This map can be exponentiated to a map between tori,

$\displaystyle\chi_{k}:\widetilde{T}\to\widetilde{T}^{\star}.$ (3.2)

Now let $\rho=\frac{1}{2}\sum\limits_{\alpha\in\mathfrak{R}_{+}}\alpha$ be the
Weyl vector defined with a choice of positive roots $\mathfrak{R}_{+}$, and
consider the equation $\chi_{k}(f)=e^{2\pi i\rho}$. We denote by
$\widetilde{T}_{k}$ the set of all solutions to this equation, and by
$\widetilde{T}^{\text{reg}}_{k}$ the subset of regular solutions. We define

$\displaystyle\theta(f)=\frac{\Delta(f)^{2}}{|\widetilde{T}_{k}|}$ (3.3)

where

$\displaystyle\Delta=\prod\limits_{\alpha\in\mathfrak{R}_{+}}2\sin\left(\frac{i\alpha}{2}\right).$
(3.4)

The Verlinde formula computes the dimension of the space of sections of
${\mathcal{L}}^{k}$ where ${\mathcal{L}}$ is the determinant line bundle over
${\mathcal{M}}$ [21],

$\displaystyle\text{dim}\
H^{0}({\mathcal{M}}_{v},{\mathcal{L}}^{k})=\sum\limits_{f\in\widetilde{T}^{\text{reg}}_{k}/W}\theta(f)^{1-g}.$
(3.5)

There exists a bijection between the set of conjugacy classes
$\widetilde{T}^{\text{reg}}_{k}/W$ and the set $P_{k}$ of highest integrable
weights of $\widetilde{G}$ at level $k$. The identification is
$P_{k}\ni\lambda\to f_{\lambda}=e^{2\pi i(\lambda+\rho)/(k+h)}$. The
expression (3.5) is equivalently written as999 See, for example, [22] for an
argument.

$\displaystyle\text{dim}\
H^{0}({\mathcal{M}}_{v},{\mathcal{L}}^{k})=\sum\limits_{\lambda\in
P_{k}}\theta(f_{\lambda})^{1-g}.$ (3.6)

#####

In [6, 7, 8], a similar formula was derived for the moduli space
${\mathcal{M}}:={\mathcal{M}}(\widetilde{G})$ of semi-stable
$\widetilde{G}$-Higgs bundles. It is well known that ${\mathcal{M}}$ is non-
compact, and hence its quantization $H^{0}({\mathcal{M}},L^{k})$ must be
infinite dimensional. However, ${\mathcal{M}}$ admits a
$\mathbb{C}^{\star}$-action which naturally lifts to a
$\mathbb{C}^{\star}$-action on ${\mathcal{L}}$, giving a decomposition

$\displaystyle
H^{0}({\mathcal{M}},{\mathcal{L}}^{k})=\oplus_{n=0}^{\infty}H^{0}_{n}({\mathcal{M}},{\mathcal{L}}^{k}).$
(3.7)

The character of $\mathbb{C}^{\star}$, also known as the Hitchin character

$\displaystyle\text{dim}_{t}\
H^{0}({\mathcal{M}},{\mathcal{L}}^{k})=\sum\limits_{n=0}^{\infty}\text{dim}\
H^{0}_{n}({\mathcal{M}},{\mathcal{L}}^{k})t^{n}$ (3.8)

is given by a $t$-deformation of the Verlinde formula. In particular, we
consider a deformation of the map (3.2),

$\displaystyle\chi_{k,t}=\chi_{k}\prod\limits_{\alpha\in\mathfrak{R}_{+}}\left(\frac{1-te^{\alpha}}{1-te^{-\alpha}}\right)^{\alpha}:\widetilde{T}\to\widetilde{T}^{\star}$
(3.9)

and consider solutions to the equation $\chi_{k,t}(f)=e^{2\pi i\rho}$. Each
solution can be expanded in powers of $t$ as $f_{0}+tf_{1}+t^{2}f_{2}+\cdots$
where $f_{0}\in\widetilde{T}_{k}$. We denote the set of $t$-deformed solutions
for which $f_{0}$ is regular as $\widetilde{T}^{\text{reg}}_{k,t}$. Then the
Hitchin character is given as

$\displaystyle\text{dim}_{t}H^{0}({\mathcal{M}},{\mathcal{L}}^{k})=\sum\limits_{f\in\widetilde{T}^{\text{reg}}_{k,t}/W}\theta_{t}(f)^{1-g}.$
(3.10)

Here $\theta_{t}$ is a $t$-deformation of $\theta$ in (3.4),

$\displaystyle\theta_{t}=\frac{(1-t)^{\text{rank}(G)}}{|\widetilde{T}_{k}|\
\text{det}\
H_{t}^{\dagger}}\prod\limits_{\alpha}(1-e^{\alpha})(1-te^{\alpha})$ (3.11)

where the product is taken over all roots, and $H_{t}^{\dagger}$ is the
endomorphism of $\mathfrak{t}$ corresponding to the Hessian of the function
$D_{t}:\mathfrak{t}\to\mathbb{C}$,

$\displaystyle D_{t}(a)=\frac{1}{2}(k+h)\langle a,a\rangle-{\rm
tr\,}_{\mathfrak{g}}(\text{Li}_{2}(te^{a}))+\langle\rho,a\rangle.$ (3.12)

Equivalently we may use the bijection between
$\widetilde{T}^{\text{reg}}_{k}/W$ and $P_{k}$ to write

$\displaystyle\text{dim}_{t}H^{0}({\mathcal{M}},{\mathcal{L}}^{k})=\sum\limits_{\lambda\in
P_{k}}\theta_{t}(f_{\lambda,t})^{1-g}$ (3.13)

where $f_{\lambda,t}\in\widetilde{T}^{\text{reg}}_{k,t}/W$ satisfies
$\lim_{t\to 0}f_{\lambda,t}=f_{\lambda}$.

Equation (3.10) is what the authors of [6] called the equivariant Verlinde
formula. It was argued in [6, 7] that, for each level $k$, it is the partition
function of a two-dimensional semisimple TQFT called the equivariant Verlinde
TQFT. As this TQFT depends on the choice of $\widetilde{G}$ and $k$, we will
denote it as ${\mathcal{T}}(\widetilde{G}_{k})$. In the limit $t\to 0$, it
reduces to the usual Verlinde TQFT which encodes the fusion algebra of Wilson
lines of a compact group Chern–Simons theory. Its partition function (3.5)
computes the partition function of compact group Chern–Simons theory on
$S^{1}\times\Sigma$.

Similarly, ${\mathcal{T}}(\widetilde{G}_{k})$ is also an $S^{1}$ reduction of
a three-dimensional QFT — “the $\beta$-deformed complex Chern-Simons theory”
which is now only partially topological [6] . This QFT can be obtained by
having 3d ${\mathcal{N}}=2$ Chern–Simons theory with an adjoint hypermultiplet
on $S^{1}\times\Sigma$, with a partial topological twist along the $\Sigma$
direction (see also [23, 24, 25, 26, 27, 28, 29, 30] for closely related
analysis of similar systems and [31] for a review).101010Here $\beta$ is
related to the equivariant parameter $t$ as $t=e^{-\beta}$. In the limit
$\beta\to\infty$ , this QFT reduces to the usual Chern–Simons theory with
compact gauge group. On the other hand, when $\beta\to 0$, its Hilbert space
is identified with that of complex Chern–Simons theory. This QFT has a one-
form $Z(\widetilde{G})$ symmetry, inherited from that of the supersymmetric
Chern–Simons–matter theory. This symmetry can in general have anomaly, which
vanishes for a subgroup $Z\subset Z(\widetilde{G})$ when $k$ is a multiple of
the smallest integer $k_{\widetilde{G}/Z}$ satisfying [32, 33]

$\displaystyle\frac{k_{\widetilde{G}/Z}}{2}\langle\mu_{i}^{\vee},\mu_{i}^{\vee}\rangle\in\mathbb{Z}.$
(3.14)

Here $\mu_{i}^{\vee}$ is a coweight such that $e^{2\pi i\mu_{i}^{\vee}}$
generates $Z$.111111We recall that $Z(\widetilde{G})$ is identified with
$\Lambda_{w}^{\vee}/\Lambda_{R}^{\vee}$ where $\Lambda_{w}^{\vee}$ and
$\Lambda_{R}^{\vee}$ are coweight and coroot lattices of $\mathfrak{g}$
respectively.  As $\widetilde{G}_{k}$ Chern–Simons theory is reduced on a
circle, ${\mathcal{T}}(\widetilde{G}_{k})$ inherits both a 0-form and a 1-form
center symmetry. The anomaly in three dimensions manifests itself as a mixed
anomaly between the two symmetries. In the rest of this section, we will
discuss the symmetries of ${\mathcal{T}}(\widetilde{G}_{k})$ and their
gaugings.

###  3.2 0-form symmetry and its gauging

The 0-form center symmetry of the equivariant Verlinde TQFT acts as a
permutation of elements of $\widetilde{T}^{\text{reg}}_{k,t}$. In order to
establish this statement we need to show that if $f$ is a solution satisfying
$\chi_{k,t}(f)=e^{2\pi i\rho}$, then $gf$ is a solution for all $g\in
Z(\widetilde{G})$. Equivalently, we need to show that $\chi_{k,t}(g)=1$.
Furthermore, the fusion eigenvalue $\theta_{t}(f)^{-1/2}$ must be invariant
under $f\to gf$. We will derive these results now.

First, let us consider the case $t\to 0$. Here we want to show $\chi_{k}(g)=1$
and $\theta(gf)=\theta(f)$. We recall that
$Z(\widetilde{G})\equiv\Lambda_{w}^{\vee}/\Lambda_{R}^{\vee}$ where
$\Lambda_{w}^{\vee}$ and $\Lambda_{R}^{\vee}$ are coweight and coroot lattices
respectively. Following [34], we consider a representative $\mu_{i}^{\vee}$ of
each $\Lambda_{w}^{\vee}/\Lambda_{R}^{\vee}$-coset defined to be the dual of
each simple root $\alpha_{i}$ in the expansion
$\vartheta=\sum_{i}m_{i}\alpha_{i}$ that appears with coefficient $1$. Since
any such $\alpha_{i}$ is a long root, the basic inner product identifies
$\mu_{1}^{\vee}$ with a fundamental weight $w_{i}$, and we get

$\displaystyle\chi_{k}(g)=e^{2\pi i(k+h)w_{i}}=e^{2\pi i(k+h)w_{i}}$ (3.15)

which is the identity element in
$\widetilde{T}^{\star}\equiv\mathfrak{t}^{\star}/2\pi i\Lambda_{w}$. This
establishes the result $\chi_{k}(g)=1$. To show $\theta(gf)=\theta(f)$ we note
that $e^{\alpha}(e^{2\pi i\mu_{i}^{\vee}})=1$ for all roots $\alpha$, and
hence $\theta(gf)=\theta(f)$. This establishes the presence of
$Z(\widetilde{G})^{(0)}$ symmetry in the $t\to 0$ limit.

As $t$ is deformed to a non-zero value, we need to show that the deformation
of $\chi_{k}$ in equation (3.9) does not spoil the result. In particular, we
want $\left(\frac{1-te^{\alpha}(g)}{1-te^{-\alpha}(g)}\right)^{\alpha}=1$ for
all $\alpha\in\mathfrak{R}_{+}$. This is straightforward as
$e^{\alpha}(e^{2\pi i\mu_{i}^{\vee}})=1$, giving the required result
$\chi_{k,t}(g)=1$. Finally, in order to show that
$\theta_{t}(f)=\theta_{t}(gf)$ we need
$\det(H_{t}^{\dagger}(gf))=\det(H_{t}^{\dagger}(f))$. In particular, using the
expression for the Hessian of $D_{t}$ given in Lemma 7.7 of [21] we need that
for each root $\alpha$,

$\displaystyle\frac{e^{\alpha}}{1-te^{\alpha}}(gf)=\frac{e^{\alpha}}{1-te^{\alpha}}(f)$
(3.16)

which is of course true again by the use of $e^{\alpha}(e^{2\pi
i\mu_{i}^{\vee}})=1$.

The orbits of $Z(\widetilde{G})$ action on $T^{\text{reg}}_{k,t}/W$ are easy
to compute in the $t\to 0$ limit, where $\widetilde{T}^{\text{reg}}_{k}/W$ is
in bijection with $P_{k}$. $P_{k}$ admits an action of $Z(\widetilde{G})$
which is summarized in terms of Dynkin labels in Table 1. As $t$ is deformed
to a small non-zero value, the deformation of elements in
$\widetilde{T}^{\text{reg}}_{k}/W$ does not cross each other. Therefore, the
orbits of the center action stay the same.

The partition function of ${\mathcal{T}}(\widetilde{G}_{k})$ in the presence
of a 0-form symmetry background $h\in H_{1}(\Sigma,Z)$ can be computed using
equation (2.16),

$\displaystyle{\mathcal{Z}}_{h}=\sum\limits_{\lambda\in
P_{k}^{\widetilde{Z}}}\theta_{t}(f_{\lambda,t})^{1-g}.$ (3.17)

Here $\widetilde{Z}$ is the smallest subgroup of $Z$ such that
$H_{1}(\Sigma,\widetilde{Z})$ contains $h$, and $P_{k}^{\widetilde{Z}}$ is the
set of fixed points of $P_{k}$ under the $\widetilde{Z}$ action. Also, the
partition function of ${\mathcal{T}}(\widetilde{G}_{k})/Z^{(0)}$ TQFT is given
through equation (2.15),121212 Strictly speaking, we stated the results (2.15)
and (2.16) in Section 2 for the case of cyclic groups only. Here when
$\widetilde{G}=Spin(4n,\mathbb{C})$ and
$Z=\mathbb{Z}_{2}\times\mathbb{Z}_{2}$, $Z$ is not cyclic. However, the
results (3.17) and (3.18) still hold. This is discussed in Appendix A.3.

$\displaystyle{\mathcal{Z}}_{{\mathcal{T}}(\widetilde{G}_{k})/Z^{(0)}}=\sum\limits_{\lambda\in
P_{k}/Z}|\text{Stab}(\lambda)|\left(\frac{|Z|}{|\text{Stab}(\lambda)|^{2}}\theta_{t}(f_{\lambda,t})\right)^{1-g}.$
(3.18)

$\widetilde{G}$ |  $Z(\widetilde{G})$ |  Action of $Z(\widetilde{G})$ generators
---|---|---
$SL(n+1,\mathbb{C})$, $n\geq 1$ |  $\mathbb{Z}_{n+1}$ |  $z(\lambda_{1},\cdots,\lambda_{n-1},\lambda_{n})=(\lambda_{0},\lambda_{1},\cdots,\lambda_{n-2},\lambda_{n-1})$
$Spin(2n+1,\mathbb{C})$, $n\geq 2$ |  $\mathbb{Z}_{2}$ |  $z(\lambda_{1},\cdots,\lambda_{n-1},\lambda_{n})=(\lambda_{0},\lambda_{2},\lambda_{3},\cdots,\lambda_{n-1},\lambda_{n})$
$Sp(n,\mathbb{C})$, $n\geq 1$ |  $\mathbb{Z}_{2}$ |  $z(\lambda_{1},\cdots,\lambda_{n-1},\lambda_{n})=(\lambda_{n-1},\lambda_{n-2},\cdots,\lambda_{1},\lambda_{0})$
$Spin(4n,\mathbb{C})$, $n\geq 2$ |  $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ |  $z_{1}(\lambda_{1},\cdots,\lambda_{2n-1},\lambda_{2n})=(\lambda_{0},\lambda_{2},\lambda_{3},\cdots,\lambda_{2n-2},\lambda_{2n},\lambda_{2n-1})$
|  |  $z_{2}(\lambda_{1},\cdots,\lambda_{2n-1},\lambda_{2n})=(\lambda_{2n-1},\lambda_{2n-2},\cdots,\lambda_{1},\lambda_{0})$
$Spin(4n+2,\mathbb{C})$, $n\geq 1$ |  $\mathbb{Z}_{4}$ |  $z(\lambda_{1},\cdots,\lambda_{2n},\lambda_{2n+1})=(\lambda_{2n+1},\lambda_{2n-1},\lambda_{2n-2},\cdots,\lambda_{1},\lambda_{0})$
$E_{6}$ |  $\mathbb{Z}_{3}$ |  $z(\lambda_{1},\cdots,\lambda_{5},\lambda_{6})=(\lambda_{5},\lambda_{4},\lambda_{3},\lambda_{6},\lambda_{0},\lambda_{2})$
$E_{7}$ |  $\mathbb{Z}_{2}$ |  $z(\lambda_{1},\cdots,\lambda_{6},\lambda_{7})=(\lambda_{5},\lambda_{4},\lambda_{3},\lambda_{2},\lambda_{1},\lambda_{0},\lambda_{7})$
Table 1:  The action of generators of $Z(G)$ on the Dynkin labels of weights
in $P_{k}$. In each case, $\lambda_{0}=k-\langle\lambda,\vartheta\rangle$
where $\vartheta$ denotes the highest root of $G$.

####  3.2.1 Examples

We will now present a few examples of
${\mathcal{T}}(\widetilde{G}_{k})/Z^{(0)}$ theories. In each case, we will set
$k\equiv 0\pmod{k_{\widetilde{G}/Z}}$ with the foresight of using these
results in subsection 3.4. As our first example, consider the simplest case
$\widetilde{G}=SL(2,\mathbb{C})$, $Z=\mathbb{Z}_{2}$ and $k\equiv 0\pmod{4}$.
The set $\widetilde{T}^{\text{reg}}_{k,t}/W$ is the set of solutions
$\text{diag}(e^{i\varphi},e^{-i\varphi})$ to the equation

$\displaystyle
e^{2i(k+2)\varphi}\left(\frac{1-te^{2i\varphi}}{1-te^{-2i\varphi}}\right)^{2}=1$
(3.19)

with $\varphi\in(0,\pi)$. It has $k+1$ elements:
$\varphi_{j}=\frac{(j+1)\pi}{k+2}+\mathcal{O}(t)$ for $j=0,1,\cdots,k$. The
equivariant Verlinde formula for $SL(2,\mathbb{C})_{k}$ reads

$\displaystyle{\mathcal{Z}}=\sum\limits_{j=0}^{k}\theta_{t}(e^{i\varphi_{j}})^{1-g}$
(3.20)

where

$\displaystyle\theta_{t}(e^{i\varphi})=(1-t)\frac{4(\sin^{2}\varphi)|1-te^{2i\varphi}|^{2}}{|\widetilde{T}_{k}|\
\text{det}H_{t}^{\dagger}}$ (3.21)

with

$\displaystyle|\widetilde{T}_{k}|\
\text{det}H_{t}^{\dagger}=4\left(\frac{k+2}{2}+\frac{2t\cos
2\varphi-2t^{2}}{|1-te^{2i\varphi}|^{2}}\right).$ (3.22)

Under the identification $P_{k}\leftrightarrow\widetilde{T}^{\text{reg}}_{k}$,
$j$ parameterizes the highest weights of an $\mathfrak{su}(2)$ representation.
The action of $\mathbb{Z}_{2}$ on $P_{k}$ is $j\to k-j$; with the only fixed
point being $j=\frac{k}{2}$. The corresponding element of
$\widetilde{T}^{\text{reg}}_{k,t}/W$ is $\text{diag}\ (i,-i)$. We obtain that
for any non-zero $h\in H_{1}(\Sigma,\mathbb{Z}_{2})$,

$\displaystyle{\mathcal{Z}}_{h}=\theta_{t}(i)^{1-g}=\left(\frac{\frac{k}{2}+1+\left(\frac{k}{2}-1\right)t}{(1-t)(1+t)^{3}}\right)^{g-1}$
(3.23)

and the partition function of
${\mathcal{T}}(SL(2,\mathbb{C})_{k})/\mathbb{Z}_{2}^{(0)}$ is

$\displaystyle{\mathcal{Z}}_{{\mathcal{T}}(SL(2,\mathbb{C})_{k})/\mathbb{Z}_{2}^{(0)}}=\sum\limits_{j=0}^{\frac{k}{2}-1}(2\theta_{t}(e^{i\varphi_{j}}))^{1-g}+2\left(\frac{\theta_{t}(i)}{2}\right)^{1-g}.$
(3.24)

This example generalizes easily to the case of $G=SL(p,\mathbb{C})$ for $p\geq
3$ prime, $Z=\mathbb{Z}_{p}$ and $k\equiv 0\pmod{p}$. The only fixed point
under the action action of $\mathbb{Z}_{p}$ on $P_{k}$ is $\frac{k}{p}\rho$;
it exponentiates to the element
$f_{\star}=\text{diag}(1,\eta,\cdots,\eta^{p-1})\in\widetilde{T}^{\text{reg}}_{k,t}/W$
where $\eta=e^{2\pi i/p}$. For each non-zero $h\in
H_{1}(\Sigma,\mathbb{Z}_{p})$,

$\displaystyle{\mathcal{Z}}_{h}=\theta_{t}(f_{\star})^{1-g}$ (3.25)

and the partition function of
${\mathcal{T}}(SL(p,\mathbb{C})_{k})/\mathbb{Z}_{p}^{(0)}$ is

$\displaystyle{\mathcal{Z}}_{{\mathcal{T}}(SL(p,\mathbb{C})_{k})/\mathbb{Z}_{p}^{(0)}}=\sum\limits_{\lambda\in(P_{k}\backslash\frac{k}{p}\rho)/\mathbb{Z}_{p}}(p\
\theta_{t}(f_{\lambda,t}))^{1-g}+p\left(\frac{\theta_{t}(f_{\star})}{p}\right)^{1-g}.$
(3.26)

As another example, we consider $\widetilde{G}=SL(4,\mathbb{C})$,
$Z=\mathbb{Z}_{4}$ and $k\equiv 0\pmod{8}$. This example is different from the
previous cases since $\mathbb{Z}_{4}$ has a non-trivial $\mathbb{Z}_{2}$
subgroup. The integrable weight $\frac{k}{4}\rho$ is fixed under the action of
$\mathbb{Z}_{4}$; the corresponding element of
$\widetilde{T}^{\text{reg}}_{k,t}/W$ is $f_{\star}=e^{i\pi/4}\
\text{diag}(1,i,-1,-i)$. The fixed point set $P_{k}^{\mathbb{Z}_{2}}$ is
$\\{(a,\frac{k}{2}-a,a)|a\leq\frac{k}{2}\\}$. For each non-zero $h\in
H_{1}(\Sigma,\mathbb{Z}_{2})\subset H_{1}(\Sigma,\mathbb{Z}_{4})$,

$\displaystyle{\mathcal{Z}}_{h}=\sum\limits_{\lambda\in
F_{\mathbb{Z}_{2}}(P_{k})}\theta_{t}(f_{\lambda,t})^{1-g}$ (3.27)

and for $h\notin H_{1}(\Sigma,\mathbb{Z}_{2})$,

$\displaystyle{\mathcal{Z}}_{h}=\theta_{t}(f_{\star})^{1-g}.$ (3.28)

The partition function of
${\mathcal{T}}(SL(4,\mathbb{C})_{k})/\mathbb{Z}_{4}^{(0)}$ is

$\displaystyle{\mathcal{Z}}_{{\mathcal{T}}(SL(4,\mathbb{C})_{k})/\mathbb{Z}_{4}^{(0)}}=\sum\limits_{(P_{k}\backslash
P_{k}^{\mathbb{Z}_{2}})/\mathbb{Z}_{4}}(4\theta_{t}(f_{\lambda,t}))^{1-g}+2\sum\limits_{(P_{k}^{\mathbb{Z}_{2}}\backslash\\{\frac{k}{4}\rho\\})/\mathbb{Z}_{2}}\left(\theta_{t}(f_{\lambda,t})\right)^{1-g}+4\left(\frac{\theta_{t}(f_{\star})}{4}\right)^{1-g}.$
(3.29)

Finally, we consider $\widetilde{G}=SL(4,\mathbb{C})$,
$Z=\mathbb{Z}_{2}\subset\mathbb{Z}_{4}$ and $k\equiv 0\pmod{2}$. Here,
${\mathcal{Z}}_{h}$ is the same as in (3.27) for all non-zero $h\in
H_{1}(\Sigma,\mathbb{Z}_{2})$, and the partition function of
${\mathcal{T}}(SL(4,\mathbb{C})_{k})/\mathbb{Z}_{2}^{(0)}$ is

$\displaystyle{\mathcal{Z}}_{{\mathcal{T}}(SL(4,\mathbb{C})_{k})/\mathbb{Z}_{2}^{(0)}}=\sum\limits_{(P_{k}\backslash
P_{k}^{\mathbb{Z}_{2}})/\mathbb{Z}_{2}}(2\theta_{t}(f_{\lambda,t}))^{1-g}+2\sum\limits_{P_{k}^{\mathbb{Z}_{2}}}\left(\frac{\theta_{t}(f_{\lambda,t})}{2}\right)^{1-g}.$
(3.30)

###  3.3 1-form symmetry and its gauging

The 1-form center symmetry of ${\mathcal{T}}(\widetilde{G}_{k})$ acts on the
elements of $\widetilde{T}^{\text{reg}}_{k,t}/W$ by multiplication with roots
of unity. This action can be described as follows. Let $\lambda\in P_{k}$ be a
highest integrable weight of $\widetilde{G}$; the center $Z(\widetilde{G})$
acts on the associated $\widetilde{G}$-representation as $e^{2\pi
i\mu_{i}^{\vee}}\to e^{2\pi i\lambda(\mu_{i}^{\vee})}\mathbb{1}$. We assign
this phase to $f_{\lambda,t}$; thus giving the required $Z(\widetilde{G})$
action.131313This is consistent with the familiar action of one-form symmetry
in the anyon basis for $t=0$. There, the basis elements are labelled by level
$k$ integrable representations and the center symmetry acts by the action of
$Z(\widetilde{G})$ on the Weyl alcove. In particular, for $z\in
Z(\widetilde{G})$ and $\lambda,\chi\in P_{k}$,
$N_{z\lambda}{}^{\chi}=\delta_{z(\lambda)}{}^{\chi}$. The basis transformation
is carried out by the S-matrix and we have
$\displaystyle\sum\limits_{\rho,\sigma}S_{\lambda}{}^{\rho}e^{2\pi
i\rho(\mu_{i}^{\vee})}\delta_{\rho}{}^{\sigma}(S^{-1})_{\sigma}{}^{\chi}=\sum\limits_{\rho}S_{\lambda}{}^{\rho}e^{2\pi
i\rho(\mu_{i}^{\vee})}(S^{-1})_{\rho}{}^{\chi}=\sum\limits_{\rho}S_{z(\lambda)}{}^{\rho}(S^{-1})_{\rho}{}^{\chi}=\delta_{z(\lambda)}{}^{\chi}$
Here, in the second equality we have used the result
$S_{z(\lambda)}{}^{\rho}=S_{\lambda}{}^{\rho}e^{2\pi i\rho(\mu_{i}^{\vee})}$.
See section 14.6.4 of [35] for a derivation.

In the presence of an $e^{2\pi i\mu_{i}^{\vee}}\in Z$ background, the
partition function of ${\mathcal{T}}(\widetilde{G}_{k})$ is (see equation
(2.20)),

$\displaystyle{\mathcal{Z}}_{\mu_{i}^{\vee}}=\sum\limits_{\lambda\in
P_{k}}e^{2\pi i\lambda(\mu_{i}^{\vee})}\theta_{t}(f_{\lambda,t})^{1-g}.$
(3.31)

Gauging the symmetry projects out weights with $e^{2\pi
i\lambda(\mu_{1}^{\vee})}\neq 1$ (for $e^{2\pi i\mu_{1}^{\vee}}$ a generator
of $Z$). The partition function of ${\mathcal{T}}(\widetilde{G}_{k})/Z^{(1)}$
is (see equation (2.21)),

$\displaystyle{\mathcal{Z}}_{{\mathcal{T}}(\widetilde{G}_{k})/Z^{(1)}}=\sum\limits_{\lambda\in
P_{k}(G)}\left(|Z|\theta_{t}(f_{\lambda,t})\right)^{1-g}.$ (3.32)

Here, $P_{k}(G)\subset P_{k}$ is the subset of highest weights that label
representations of $G=\widetilde{G}/Z$.

####  3.3.1 Examples

Now we will study some examples. Consider the case
$\widetilde{G}=SL(2,\mathbb{C})$ and $Z=\mathbb{Z}_{2}$. Here
$\mu_{1}^{\vee}=\frac{1}{2}\ \text{diag}(1,-1)$, and in the notation of
Section 3.2,

$\displaystyle{\mathcal{Z}}_{\mu_{1}^{\vee}}=\sum\limits_{j=0}^{k}(-1)^{j}\theta_{t}(e^{i\varphi_{j}})^{1-g}.$
(3.33)

The partition function of
${\mathcal{T}}(SL(2,\mathbb{C})_{k})/\mathbb{Z}_{2}^{(1)}$ gets contribution
from even weights only, i.e.

$\displaystyle{\mathcal{Z}}_{{\mathcal{T}}(SL(2,\mathbb{C})_{k})/\mathbb{Z}_{2}^{(1)}}=\sum\limits_{\begin{subarray}{c}j=0\\\
j\
\text{even}\end{subarray}}^{k}\left(2\theta_{t}(e^{i\varphi_{j}})\right)^{1-g}.$
(3.34)

This example generalizes easily to the case of $SL(n,\mathbb{C})$ and
$Z=\mathbb{Z}_{m}$. Let $n_{t}(\lambda)=\sum\limits_{j=1}^{n-1}j\lambda_{j}$
be the $n$-ality of the representation with highest weight $\lambda$, then

$\displaystyle{\mathcal{Z}}_{\mu_{1}^{\vee}}=\sum\limits_{\lambda\in
P_{k}}e^{2\pi in_{t}(\lambda)/m}\theta_{t}(f_{\lambda,t})^{1-g}.$ (3.35)

Summing over $\mathbb{Z}_{m}$ backgrounds gives

$\displaystyle{\mathcal{Z}}_{{\mathcal{T}}(SL(n,\mathbb{C})_{k})/\mathbb{Z}_{m}^{(1)}}=\frac{1}{m}\sum\limits_{\lambda\in
P_{k}}\left(\sum\limits_{s=1}^{m}e^{2\pi
isn_{t}(\lambda)/m}\right)\left(m\theta_{t}(f_{\lambda,t})\right)^{1-g}.$
(3.36)

The sum inside the brackets equals $m$ when $n_{t}(\lambda)=0\mod m$, and zero
otherwise. That is, it is non-zero precisely when $\lambda$ gives a
representation of $SL(n,\mathbb{C})/\mathbb{Z}_{m}$. We obtain

$\displaystyle{\mathcal{Z}}_{{\mathcal{T}}(SL(n,\mathbb{C})_{k})/\mathbb{Z}_{m}^{(1)}}=\sum\limits_{\lambda\in
P_{k}(SL(n,\mathbb{C})/\mathbb{Z}_{m})}\left(m\theta_{t}(f_{\lambda,t})\right)^{1-g}.$
(3.37)

###  3.4 Quantization of $\mathcal{M}(G)$

In this section, we will use the results from the previous two sections to
compute the Hitchin character for the moduli space ${\mathcal{M}}(G)$ of
semistable $G$-Higgs bundles. This moduli space is disconnected with
$\pi_{0}({\mathcal{M}}(G))=\pi_{1}(G)=Z$; the second Stiefel–Whitney class
$w_{2}$ of the $G$-Higgs bundle on the surface parameterizes the various
components.

For a fixed $w_{2}$, ${\mathcal{M}}_{w_{2}}(G)$ is the quotient
${\mathcal{M}}_{d}/\mathcal{G}$ of the moduli space ${\mathcal{M}}_{d}$ of
semistable $\widetilde{G}$-Higgs bundles of degree $d=w_{2}$ by
$\mathcal{G}=H_{1}(\Sigma,Z)$. The action of $\mathcal{G}$ on
${\mathcal{M}}_{d}$ lifts naturally to an action on the pre-quantum line
bundle ${\mathcal{L}}^{k}$. This gives the space of sections
$H^{0}({\mathcal{M}}_{d},{\mathcal{L}}^{k})$ the structure of a
${\mathcal{G}}$-module. Classically, the moduli space ${\mathcal{M}}(G)$ is
the phase space of $G$ Chern–Simons theory on $S^{1}\times\Sigma$. As this
theory is well-defined only for $k\equiv 0\pmod{k_{\widetilde{G}/Z}}$ (see
equation (3.14)), we will consider pre-quantum line bundles on
${\mathcal{M}}_{w_{2}}$ whose pullback to ${\mathcal{M}}_{d}$ is
${\mathcal{L}}^{k}$ for these values. (By an abuse of notation we will also
denote the corresponding line bundle on ${\mathcal{M}}_{w_{2}}$ as
${\mathcal{L}}^{k}$.) The space of sections
$H^{0}({\mathcal{M}}_{w_{2}}(G),{\mathcal{L}}^{k})$ is isomorphic to the
subspace of ${\mathcal{G}}$-invariant sections of
$H^{0}({\mathcal{M}}_{d},{\mathcal{L}}^{k})$.

The actions of ${\mathcal{G}}$ and $\mathbb{C}^{\star}$ on ${\mathcal{L}}^{k}$
commute, therefore, the moduli space ${\mathcal{M}}_{w_{2}}(G)$ also admits a
$\mathbb{C}^{\star}$ action.

$\displaystyle
H^{0}({\mathcal{M}}_{w_{2}}(G),{\mathcal{L}}^{k})=\oplus_{n}H^{0}_{n}({\mathcal{M}}_{w_{2}}(G),{\mathcal{L}}^{k}).$
(3.38)

We define the Hitchin character for ${\mathcal{M}}_{w_{2}}(G)$,

$\displaystyle\text{dim}_{t}\
H^{0}({\mathcal{M}}_{w_{2}}(G),{\mathcal{L}}^{k}):=\sum\limits_{n=0}^{\infty}t^{n}\
\text{dim}H^{0}_{n}({\mathcal{M}}_{w_{2}}(G),{\mathcal{L}}^{k}).$ (3.39)

As the space $H^{0}({\mathcal{M}}_{w_{2}}(G),{\mathcal{L}}^{k})$ is isomorphic
to the ${\mathcal{G}}$-invariant subspace of
$H^{0}({\mathcal{M}}_{d},{\mathcal{L}}^{k})$, we have the equality

$\displaystyle\text{dim}H^{0}_{n}({\mathcal{M}}_{w_{2}}(G),{\mathcal{L}}^{k})=\frac{1}{|{\mathcal{G}}|}\sum\limits_{h\in{\mathcal{G}}}{\rm
Tr\,}(h|H^{0}_{n}({\mathcal{M}}_{d=w_{2}}(\widetilde{G}),{\mathcal{L}}^{k})).$
(3.40)

Let the formal series containing all the traces for a fixed $h$ be

$\displaystyle\text{Tr}_{t}(h|H^{0}({\mathcal{M}}_{d},{\mathcal{L}}^{k}))=\sum\limits_{n=0}^{\infty}{\rm
Tr\,}(h|H^{0}_{n}({\mathcal{M}}_{d},{\mathcal{L}}^{k}))t^{n}.$ (3.41)

We will call this trace the equivariant trace. The Hitchin character is just
the average of the equivariant traces.

$\displaystyle\text{dim}_{t}\
H^{0}({\mathcal{M}}_{w_{2}}(G),{\mathcal{L}}^{k})=\frac{1}{|{\mathcal{G}}|}\sum\limits_{h\in{\mathcal{G}}}\text{Tr}_{t}(h|H^{0}({\mathcal{M}}_{d=w_{2}},{\mathcal{L}}^{k})).$
(3.42)

In the rest of this section, we will compute the equivariant trace as well as
the Hitchin character for ${\mathcal{M}}_{w_{2}}(G)$ using methods in 2d
TQFTs. Our main observation is that the equivariant trace (3.41) is equal to
the partition function of ${\mathcal{T}}(\widetilde{G}_{k})$ in the presence
of the 0-form symmetry background $h\in H_{1}(\Sigma,Z)$ and the 1-form
symmetry background $d\in Z$. For $d=0$, this trace is equal to
${\mathcal{Z}}_{h}$ in equation (3.17). We will give an argument for this
identification now.

We recall that the space of sections
$H^{0}({\mathcal{M}}_{d=0},{\mathcal{L}}^{k})$ is identified with the Hilbert
space of complex $\widetilde{G}$ Chern–Simons theory (or alternatively, the
$\beta$-deformed Chern–Simons theory) on the surface $\Sigma$ [6].
$\widetilde{G}$ Chern–Simons theory has a 1-form center symmetry with an
invertible topological line operator labelled by each $d\in Z$. The insertion
of this line along a temporal cycle amounts to creating the defect Hilbert
space $H^{0}({\mathcal{M}}_{d},{\mathcal{L}}^{k})$,141414The insertion of the
symmetry generating line along $S^{1}$ changes the classical phase space to
${\mathcal{M}}_{d}$. Its quantization
$H^{0}({\mathcal{M}}_{d},{\mathcal{L}}^{k})$ gives the defect Hilbert space.
while its insertion along a spatial cycle creates an action of the line
operator on the Hilbert space. The partition function
${\mathcal{Z}}_{CS}(S^{1}\times\Sigma)$ computes the trace
$\text{Tr}_{t}(h|H^{0}({\mathcal{M}}_{d},{\mathcal{L}}^{k}))$ when these line
operators are inserted.

On the other hand, the insertion of these line operators along the spatial
(temporal) cycles amounts to creating a 0-form (1-form) symmetry background of
${\mathcal{T}}(\widetilde{G}_{k})$. ${\mathcal{Z}}_{CS}(S^{1}\times\Sigma)$
now computes the partition function of ${\mathcal{T}}(\widetilde{G}_{k})$ in
the presence of these backgrounds, thus establishing the afore-mentioned
identification.

We will now compute the trace
$\text{Tr}_{t}(h|H^{0}({\mathcal{M}}_{d},{\mathcal{L}}^{k}))$ and the Hitchin
character $\text{dim}_{t}\ H^{0}({\mathcal{M}}_{w_{2}}(G),{\mathcal{L}}^{k})$
using 2d TQFT methods. These computations will build on the computations
presented in Section 3.2 and Section 3.3.

First, consider the case $d=0$. In this case,
${\mathcal{Z}}_{h}=\text{Tr}_{t}(h|H^{0}({\mathcal{M}}_{d=0},{\mathcal{L}}^{k}))$
was already computed in equation (3.17). The Hitchin character can be computed
as follows: it is $\frac{1}{|Z|^{g}}$ times the partition function of
${\mathcal{T}}(\widetilde{G}_{k})/Z^{(0)}$ TQFT (3.18),151515 The extra factor
$\frac{1}{|Z|^{g}}$ appears because of the difference in normalizations of
$\mathcal{Z}_{{\mathcal{T}}(\widetilde{G}_{k})/Z^{(0)}}$ and $\text{dim}_{t}\
H^{0}({\mathcal{M}}_{w_{2}=0}(G),{\mathcal{L}}^{k})$. In particular,
$\displaystyle\mathcal{Z}_{{\mathcal{T}}(\widetilde{G}_{k})/Z^{(0)}}=\frac{1}{|Z|^{g}}\sum\limits_{h\in
H_{1}(\Sigma,Z)}{\mathcal{Z}}_{h}$ while  $\displaystyle\text{dim}_{t}\
H^{0}({\mathcal{M}}_{w_{2}=0}(G),{\mathcal{L}}^{k})=\frac{1}{|Z|^{2g}}\sum\limits_{h\in
H_{1}(\Sigma,Z)}{\mathcal{Z}}_{h}.$ The Hitchin character for
${\mathcal{M}}_{w_{2}}(G)$ is not a partition function of a 2d TQFT. This is
expected as cutting and gluing do not preserve the second Stiefel–Whiney class
of a $G$-Higgs bundle.

$\displaystyle\text{dim}_{t}\
H^{0}({\mathcal{M}}_{w_{2}=0}(G),{\mathcal{L}}^{k})=\frac{1}{|Z|^{g}}\sum\limits_{\lambda\in
P_{k}/Z}|\text{Stab}(\lambda)|\left(\frac{|Z|}{|\text{Stab}(\lambda)|^{2}}\theta_{t}(f_{\lambda,t})\right)^{1-g}.$
(3.43)

For $d\neq 0$, $\text{dim}_{t}H^{0}({\mathcal{M}}_{d},{\mathcal{L}}^{k})$ is
the partition function of ${\mathcal{T}}(\widetilde{G}_{k})$ in the presence
of an $e^{2\pi i\mu_{i}^{\vee}}\in Z$ background (3.31).161616 We remark that
our notation for elements $\\{e^{2\pi i\mu_{i}^{\vee}}\\}$ of $Z$ is
multiplicative, while that for degree $d$ is additive.  The equivariant trace
$\text{Tr}_{t}(h|H^{0}({\mathcal{M}}_{d},{\mathcal{L}}^{k}))$ can be computed
by further introducing a $Z^{(0)}$ background. This restricts the sum in
(3.31) to $\widetilde{Z}$ invariant operators where $\widetilde{Z}$ is the
smallest subgroup of $Z$ such that $H_{1}(\Sigma,\widetilde{Z})\subset
H_{1}(\Sigma,Z)$ contains $h$,

$\displaystyle\text{Tr}_{t}(h|H^{0}({\mathcal{M}}_{d},{\mathcal{L}}^{k}))=\sum\limits_{\lambda\in
P_{k}^{\widetilde{Z}}}e^{2\pi
i\lambda(\mu_{i}^{\vee})}\theta_{t}(f_{\lambda,t})^{1-g}.$ (3.44)

The Hitchin character $\text{dim}_{t}\
H^{0}({\mathcal{M}}_{w_{2}}(G),{\mathcal{L}}^{k})$ is obtained by averaging
the equivariant traces over all $h\in H_{1}(\Sigma,Z)$. In the absence of a
mixed anomaly, i.e. when $k\equiv 0\pmod{k_{\widetilde{G}/Z}}$, the actions of
$Z^{(0)}$ and $Z^{(1)}$ on $T^{\text{reg}}_{k,t}/W$ commute. This implies that
each element of a $Z^{(0)}$ orbit has the same phase $e^{2\pi
i\lambda(\mu_{i}^{\vee})}$ under $Z^{(1)}$ action. Furthermore, a computation
similar to that of Section 2.2 gives

$\displaystyle\text{dim}_{t}\
H^{0}({\mathcal{M}}_{w_{2}=d}(G),{\mathcal{L}}^{k})=\frac{1}{|Z|^{g}}\sum\limits_{\lambda\in
P_{k}/Z}e^{2\pi
i\lambda(\mu_{i}^{\vee})}|\text{Stab}(\lambda)|\left(\frac{|Z|}{|\text{Stab}(\lambda)|^{2}}\theta_{t}(f_{\lambda,t})\right)^{1-g}.$
(3.45)

In the $t\to 0$ limit, the fact that dimension of sections of
${\mathcal{L}}^{k}$ over the individual component labeled by $d$ takes such a
nice form was observed by Beauville [13]. This phenomenon, while not having a
simple CFT interpretation, is very natural from the 2d TQFT perspective, as
the 1-form and 0-form symmetry, although they share the same 3d origin, are
independent symmetries of the theory.

Summing over $w_{2}$ gives the partition function of $\beta$-deformed $G$
complex Chern–Simons theory on $S^{1}\times\Sigma$. We define
${\mathcal{T}}(G_{k})$ to be the circle reduction of this QFT, then

$\displaystyle\mathcal{Z}_{{\mathcal{T}}(G_{k})}=\sum\limits_{\lambda\in
P_{k}(G)/Z}|\text{Stab}(\lambda)|\left(\frac{|Z|^{2}}{|\text{Stab}(\lambda)|^{2}}\theta_{t}(f_{\lambda,t})\right)^{1-g}.$
(3.46)

Here $P_{k}(G)/Z$ is the set of all orbits of $P_{k}(G)$ under the action of
$Z$.

####  3.4.1 Examples

The examples in this section will build on the examples studied in Section 3.2
and Section 3.3. Consider first $G=SL(2,\mathbb{C})$, $Z=\mathbb{Z}_{2}$ and
$k\equiv 0\pmod{4}$. The equivariant trace was computed in Equation (3.23),

$\displaystyle\text{Tr}_{t}(h|H^{0}({\mathcal{M}}_{d},{\mathcal{L}}^{k}))=\theta_{t}(i)^{1-g}.$
(3.47)

It is independent of $d$ as $e^{2\pi i\frac{k}{2}\rho(\mu_{i}^{\vee})}=1$ (cf.
equation (3.44)). The Hitchin character of
${\mathcal{M}}_{w_{2}}(PSL(2,\mathbb{C}))$ for $w_{2}=0,1$ is

$\displaystyle\text{dim}_{t}H^{0}({\mathcal{M}}_{w_{2}}(PSL(2,\mathbb{C}),{\mathcal{L}}^{k})$
$\displaystyle=$
$\displaystyle\frac{1}{2}\sum\limits_{j=0}^{\frac{k}{2}-1}(-1)^{jw_{2}}(4\theta_{t}(e^{i\varphi_{j}}))^{1-g}+\theta_{t}(i)^{1-g}.$
(3.48)

Summing over $w_{2}$ gives the partition function of
${\mathcal{T}}(PSL(2,\mathbb{C})_{k})$, 171717 The sum in
$Z_{{\mathcal{T}}(PSL(2,\mathbb{C})_{k})}$ is over the set of solutions to the
“$PSL(2,\mathbb{C})$ Bethe equation”, i.e.
$e^{i(k+2)\varphi}\left(\frac{1-te^{-2i\varphi}}{1-te^{2i\varphi}}\right)=-1$.
This equation was derived in [6] by studying $U(2)$ Bethe equations and
appropriately decoupling some degrees of freedom. We note that the range of
$\varphi$ is $\left(0,\frac{\pi}{2}\right]$ here; it is halved compared to the
case of $SL(2,\mathbb{C})$ where $\varphi\in(0,\pi)$ due to the quotient
$SL(2,\mathbb{C})/\mathbb{Z}_{2}\equiv PSL(2,\mathbb{C})$. The fixed point of
$\mathbb{Z}_{2}$ i.e. $\varphi=\frac{\pi}{2}$ contributes two operators to the
diagonal basis of ${\mathcal{T}}(PSL(2,\mathbb{C})_{k})$.

$\displaystyle
Z_{{\mathcal{T}}(PSL(2,\mathbb{C})_{k})}=\sum\limits_{\begin{subarray}{c}j=0\\\
j\
\text{even}\end{subarray}}^{\frac{k}{2}-2}(4\theta_{t}(e^{i\varphi_{j}}))^{1-g}+2(\theta_{t}(i))^{1-g}.$
(3.49)

This example generalizes easily to the case of $G=SL(p,\mathbb{C})$,
$Z=\mathbb{Z}_{p}$ for $p\geq 3$ prime and $k\equiv 0\pmod{p}$. Since the only
one fixed point $\frac{k}{p}\rho$ has $n$-ality equal to 0 mod $p$, we have

$\displaystyle\text{Tr}_{t}(h|H^{0}({\mathcal{M}}_{d},{\mathcal{L}}^{k}))=\theta_{t}(f_{\star})^{1-g}$
(3.50)

where $f_{\star}=\text{diag}(1,\eta,\cdots,\eta^{n-1})$ and $\eta=e^{2\pi
i/p}$. The Hitchin character for ${\mathcal{M}}_{w_{2}}(PSL(p,\mathbb{C}))$ is

$\displaystyle\text{dim}_{t}H^{0}({\mathcal{M}}_{w_{2}}(PSL(p,\mathbb{C}),{\mathcal{L}}^{k})=\frac{1}{p}\sum\limits_{\lambda\in(P_{k}\backslash\frac{k}{p}\rho)/\mathbb{Z}_{p}}e^{2\pi
iw_{2}n_{t}(\lambda)/p}(p^{2}\theta_{t}(f_{\lambda,t}))^{1-g}+\theta_{t}(f_{\star})^{1-g}$

where $w_{2}=0,1,\cdots,p-1$. Summing over $w_{2}$ gives the partition
function of ${\mathcal{T}}(PSL(p,\mathbb{C})_{k})$.

$\displaystyle
Z_{{\mathcal{T}}(PSL(p,\mathbb{C})_{k})}=\sum\limits_{\lambda\in(P_{k}(PSL(p,\mathbb{C}))\backslash\frac{k}{p}\rho)/\mathbb{Z}_{p}}\left(p^{2}\theta_{t}(f_{\lambda,t})\right)^{1-g}+p\
\theta_{t}(f_{\star})^{1-g}.$ (3.52)

Finally, we consider the case of $G=PSL(4,\mathbb{C})$ for $k\equiv
0\pmod{8}$. For non-zero $h\in H_{1}(\Sigma,\mathbb{Z}_{2})$,
${\mathcal{Z}}_{h}$ gets contribution from
$P_{k}^{\mathbb{Z}_{2}}=\\{(a,\frac{k}{2}-a,a)|a\leq\frac{k}{2}\\}$; $n$-ality
of the element $(a,\frac{k}{2}-a,a)$ is $k+2a$. This gives

$\displaystyle\text{Tr}_{t}(h|H^{0}({\mathcal{M}}_{d},{\mathcal{L}}^{k}))=\sum\limits_{\begin{subarray}{c}\lambda=(a,\frac{k}{2}-a,a)|\\\
a\leq\frac{k}{2}\end{subarray}}(-1)^{ad}\theta_{t}(f_{\lambda,t})$ (3.53)

where $d=0,1,2,3$. For $h\notin H_{1}(\Sigma,\mathbb{Z}_{2})$,

$\displaystyle\text{Tr}_{t}(h|H^{0}({\mathcal{M}}_{d},{\mathcal{L}}^{k}))=\theta_{t}(f_{\star})$
(3.54)

where $f_{\star}=e^{i\pi/4}\ \text{diag}(1,i,-1,-i)$. The Hitchin characters
$\text{dim}_{t}H^{0}({\mathcal{M}}_{w_{2}}(PSL(n,\mathbb{C}),{\mathcal{L}}^{k}))$
for $w_{2}=0,1,2,3$ are

$\displaystyle\frac{1}{4}\sum\limits_{(P_{k}\backslash
P_{k}^{\mathbb{Z}_{2}})/\mathbb{Z}_{4}}e^{2\pi
iw_{2}n_{t}(\lambda)/4}\left(16\theta_{t}(f_{\lambda,t})\right)^{1-g}+\frac{1}{2}\sum\limits_{\begin{subarray}{c}\lambda=(a,\frac{k}{2}-a,a)|\\\
a\leq\frac{k}{2},a\neq\frac{k}{4}\end{subarray}}(-1)^{aw_{2}}\left(4\theta_{t}(f_{\lambda,t})\right)^{1-g}+\theta_{t}(f_{\star})^{1-g}.$

Finally, the partition function of ${\mathcal{T}}(PSL(4,\mathbb{C})_{k})$ is

$\displaystyle{\mathcal{Z}}_{{\mathcal{T}}(PSL(4,\mathbb{C})_{k})}=\sum\limits_{\begin{subarray}{c}\lambda\in
P_{k}(PSL(4,\mathbb{C}))/\mathbb{Z}_{4}\\\
|\lambda|=4\end{subarray}}\left(16\theta_{t}(f_{\lambda,t})\right)^{1-g}+2\sum\limits_{\begin{subarray}{c}\lambda\in
P_{k}(PSL(4,\mathbb{C}))/\mathbb{Z}_{4}\\\
|\lambda|=2\end{subarray}}(4\theta_{t}(f_{\lambda,t}))^{1-g}+\theta_{t}(f_{\star})^{1-g}.$

Here $\lambda$ denotes a $\mathbb{Z}_{4}$-orbit of $P_{k}(PSL(4,\mathbb{C}))$
and $|\lambda|$ denotes the length of this orbit.

##  4 Equivariant Verlinde algebra for $PSL(2,\mathbb{C})$

In this section, we will study the equivariant Verlinde algebra for
$PSL(2,\mathbb{C})$ Chern–Simons theory. In the case of $SL(2,\mathbb{C})$
Chern–Simons, the equivariant Verlinde algebra was studied in [6]. It was
extended to the case of $SL(3,\mathbb{C})$ in [11], and to arbitrary simple
and simply-connected complex Lie groups in [7]. These algebras have the
geometric interpretation of Hitchin characters for moduli space of parabolic
semi-stable $G$-Higgs bundles.

Here, we extend their results to non-simply connected groups. We will focus
mostly on the case of $PSL(2,\mathbb{C})$ as a concrete example, obtaining our
results by gauging the $\mathbb{Z}_{2}^{(0)}\times\mathbb{Z}_{2}^{(1)}$
symmetry of ${\mathcal{T}}(SL(2,\mathbb{C})_{k})$ in its parabolic basis. Our
methods can be easily generalized to study the equivariant Verlinde algebra
for arbitrary simple complex Lie groups $G$, however this generalization will
be studied elsewhere. Towards the end of this section, we will remark on the
relation of the $PSL(2,\mathbb{C})$ algebra to the quantization of the moduli
space of parabolic semi-stable $PSL(2,\mathbb{C})$-Higgs bundles.

Let us first recall the details of the equivariant Verlinde algebra for
$SL(2,\mathbb{C})$. The basis elements in parabolic basis are labelled by the
integrable highest weights of $\mathfrak{su}(2)$ at level $k$:
$w^{0},w^{1},\cdots,w^{k}$. The fusion rules of these operators are

$\displaystyle f^{a_{1}a_{2}a_{3}}=\begin{cases}1&\mbox{if
}a_{1}+a_{2}+a_{3}\in 2\mathbb{Z}\ \text{and}\Delta\leq 0\\\
t^{\Delta/2}&\mbox{if }a_{1}+a_{2}+a_{3}\in 2\mathbb{Z}\ \text{and}\Delta>0\\\
0&\mbox{otherwise}\end{cases}$ (4.1)

where $\Delta=\text{max}(d_{0},d_{1},d_{2},d_{3})$ and

$\displaystyle d_{0}$ $\displaystyle=$ $\displaystyle a_{1}+a_{2}+a_{3}-2k$
$\displaystyle d_{1}$ $\displaystyle=$ $\displaystyle a_{1}-a_{2}-a_{3}$
$\displaystyle d_{2}$ $\displaystyle=$ $\displaystyle a_{2}-a_{3}-a_{1}$
$\displaystyle d_{3}$ $\displaystyle=$ $\displaystyle a_{3}-a_{1}-a_{2}.$

The identity element of the algebra is $w^{0}-tw^{2}$ and the metric is given
by

$\displaystyle\eta_{a_{1}a_{2}}=\text{diag}\left(\frac{1}{1-t^{2}},\frac{1}{1-t},\cdots,\frac{1}{1-t},\frac{1}{1-t^{2}}\right).$
(4.2)

The algebra has a $\mathbb{Z}_{2}^{(1)}$ symmetry generated by
$w^{k}-tw^{k-2}$. In particular, this operator fuses with itself to the
identity element,

$\displaystyle(w^{k}-tw^{k-2})\times(w^{k}-tw^{k-2})=w^{0}-tw^{2}$ (4.3)

and it acts on local operators as

$\displaystyle(w^{k}-tw^{k-2})\times w^{j}=w^{k-j}.$ (4.4)

The zero form center symmetry $\mathbb{Z}_{2}^{(0)}$ is generated by a line
operator of ${\mathcal{T}}(SL(2,\mathbb{C})_{k})$. As
${\mathcal{T}}(SL(2,\mathbb{C})_{k})$ is circle reduction of a three-
dimensional QFT (the $\beta$-deformed $SL(2,\mathbb{C})$ Chern–Simons theory),
the 0-form symmetry generating line of ${\mathcal{T}}(SL(2,\mathbb{C})_{k})$
is identified with the 1-form symmetry generating line of the $\beta$-deformed
$SL(2,\mathbb{C})$ Chern–Simons theory. In the $t\to 0$ limit (i.e. the case
of compact group $SU(2)$ Chern–Simons theory), the action of this line
operator is well known. It acts via braiding as $w^{k}w^{j}=(-1)^{j}w^{j}$.
The phase is restricted to be a square root of 1, therefore, it does not
change when $t$ is deformed. We get

$\displaystyle(w^{k}-tw^{k-2})\ w^{j}=(-1)^{j}w^{j}.$ (4.5)

We will now discuss gauging these symmetries. In three dimensions, gauging
1-form symmetry of $SL(2,\mathbb{C})$ Chern–Simons theory gives
$PSL(2,\mathbb{C})$ Chern–Simons theory. In two dimensions, gauging
$\mathbb{Z}_{2}^{(0)}\times\mathbb{Z}_{2}^{(1)}$ of
${\mathcal{T}}(SL(2,\mathbb{C})_{k})$ will give the equivariant Verlinde TQFT
${\mathcal{T}}(PSL(2,\mathbb{C})_{k})$ that encodes fusion rules of the
$\beta$-deformed $PSL(2,\mathbb{C})_{k}$ Chern–Simons theory. We will restrict
to the case $k\equiv 0\pmod{4}$ so that the mixed anomaly between
$\mathbb{Z}_{2}^{(0)}$ and $\mathbb{Z}_{2}^{(1)}$ vanishes.

Now consider first gauging the $\mathbb{Z}_{2}^{(1)}$ symmetry.181818The final
result, i.e. the set of fusion coefficients of
${\mathcal{T}}(PSL(2,\mathbb{C})_{k})$ is independent of the order of gauging.
The gauge invariant operators are

$\displaystyle x^{j}=\frac{1}{2}(w^{j}+w^{k-j})$ (4.6)

for $j=0,1,\cdots,\frac{k}{2}$ where the normalization factor is fixed by
comparison with the partition function of ${\mathcal{T}}/\mathbb{Z}_{2}^{(1)}$
gauged theory given in Section 2.2. The fusion rules of
${\mathcal{T}}(SL(2,\mathbb{C})_{k})/\mathbb{Z}_{2}^{(1)}$ are easy to derive.
The derivation is postponed until Appendix B.1; here we give the final result,

$\displaystyle(f^{\prime})^{ab}{}_{c}$ $\displaystyle=$ $\displaystyle
f^{ab}{}_{c}+f^{(k-a)b}{}_{c}\quad;\quad c\neq\frac{k}{2}$ (4.7)
$\displaystyle(f^{\prime})^{ab}{}_{\frac{k}{2}}$ $\displaystyle=$
$\displaystyle f^{ab}{}_{\frac{k}{2}}.$ (4.8)

${\mathcal{T}}(SL(2,\mathbb{C})_{k})/\mathbb{Z}_{2}^{(1)}$ inherits a
$\mathbb{Z}_{2}^{(0)}$ symmetry from its parent TQFT. This symmetry acts as
$x^{j}\to(-1)^{j}x^{j}$. Gauging $\mathbb{Z}_{2}^{(0)}$ projects out $x^{j}$
for odd $j$, while $x^{j}$ with even $j$ survive. $x^{\frac{k}{2}}$ plays a
distinguished role under gauging; it splits into two operators
$x^{{\frac{k}{2}}^{(1)}}$ and $x^{{\frac{k}{2}}^{(2)}}$,191919This fact is
familiar from the point of view of compact group Chern–Simons theory [32, 36].
It is also consistent with the results obtained for
${\mathcal{T}}(PSL(2,\mathbb{C})_{k})$ in Section 3.2. In the diagonal basis
$T^{\text{reg}}_{k,t}/W$, $\mathbb{Z}_{2}^{(0)}$ has a unique fixed point,
$\text{diag}(i,-i)$. The gauged theory
${\mathcal{T}}(SL(2,\mathbb{C})_{k})/\mathbb{Z}_{2}^{(0)}$ inherits two copies
of this operator. The claim here is that in the parabolic basis, the operator
that gives two copies is $x^{\frac{k}{2}}$. An understanding of why it is the
$x^{\frac{k}{2}}$ operator (and not $x^{j}$ for any other value of $j$) that
splits purely from a 2D TQFT point of view would be very interesting.

$\displaystyle
x^{\frac{k}{2}}=x^{{\frac{k}{2}}^{(1)}}+x^{{\frac{k}{2}}^{(2)}}.$ (4.9)

The dual $\mathbb{Z}_{2}^{(0)}$ symmetry of
${\mathcal{T}}(PSL(2,\mathbb{C})_{k})$ exchanges $x^{{\frac{k}{2}}^{(1)}}$ and
$x^{{\frac{k}{2}}^{(2)}}$.Overall, the operators of
${\mathcal{T}}(PSL(2,\mathbb{C})_{k})$ are

$\displaystyle
x^{0},x^{2},\cdots,x^{\frac{k}{2}-2},x^{{\frac{k}{2}}^{(1)}},x^{{\frac{k}{2}}^{(2)}}.$
(4.10)

It is convenient to split the fusion rules of these operators into six kinds
depending on how the $x^{\frac{k}{2}^{(i)}}$ operators appear. Let $a,b,c$
denote the highest weights $0,2,\cdots,\frac{k}{2}-2$, and $\mu,\nu,\rho$
denote $\frac{k}{2}^{(i)}$, then the six different kinds are

$\displaystyle\widetilde{f}^{ab}{}_{c},\quad\widetilde{f}^{ab}{}_{\mu},\quad\widetilde{f}^{a\mu}{}_{b},\quad\widetilde{f}^{a\mu}{}_{\nu},\quad\widetilde{f}^{\mu\nu}{}_{a},\quad\widetilde{f}^{\mu\nu}{}_{\rho}.$
(4.11)

Using equation (4.9) and the dual $\mathbb{Z}_{2}^{(0)}$ symmetry, the first
three sets of these are easily fixed. We postpone the derivation until
Appendix B.2; here we give the final result.

$\displaystyle\widetilde{f}^{ab}{}_{c}$ $\displaystyle=$ $\displaystyle
f^{ab}{}_{c}+f^{(k-a)b}{}_{c}$ (4.12)
$\displaystyle\widetilde{f}^{ab}{}_{\mu}$ $\displaystyle=$ $\displaystyle
f^{ab}{}_{\frac{k}{2}}$ (4.13) $\displaystyle\widetilde{f}^{a\mu}{}_{b}$
$\displaystyle=$ $\displaystyle f^{a\frac{k}{2}}{}_{b}.$ (4.14)

The last three sets in (4.11) require a little more work. In addition to the
$\mathbb{Z}_{2}^{(0)}$ dual symmetry, here we also need to use associativity
of the TQFT and a positivity condition on the fusion coefficients. This
condition is described as follows. A fusion coefficient of
${\mathcal{T}}(PSL(2,\mathbb{C})_{k})$ is the $\mathbb{C}^{\star}$ character
of the Hilbert space of $\beta$-deformed $PSL(2,\mathbb{C})$ Chern–Simons
theory on $S^{1}\times S^{2}$ with three line operators inserted at three
points of $S^{2}$. The character $\sum\limits_{n=0}^{\infty}C_{n}t^{n}$ must
have positive integer coefficients $C_{n}\in\mathbb{Z}_{\geq 0}$. After
applying these constraints, we find that

$\displaystyle\widetilde{f}^{a\mu}{}_{\nu}=\begin{cases}\frac{1}{1-t^{2}}&\mbox{if
}a\equiv 0\pmod{4}\ \&\ \mu=\nu,\ \text{or}\ a\equiv 2\pmod{4}\ \&\
\mu\neq\nu\\\ \frac{t}{1-t^{2}}&\mbox{if }a\equiv 2\pmod{4}\ \&\ \mu=\nu,\
\text{or}\ a\equiv 0\pmod{4}\ \&\ \mu\neq\nu,\end{cases}$ (4.15)
$\displaystyle\widetilde{f}^{\mu\nu}{}_{a}=\begin{cases}0&\mbox{if
}a\equiv\frac{k}{2}+2\pmod{4}\ \&\ \mu=\nu,\ \text{or}\
a\equiv\frac{k}{2}\pmod{4}\ \&\ \mu\neq\nu\\\ \eta_{aa}&\mbox{if
}a\equiv\frac{k}{2}+2\pmod{4}\ \&\ \mu\neq\nu,\ \text{or}\
a\equiv\frac{k}{2}\pmod{4}\ \&\ \mu=\nu\end{cases}$ (4.16)

and

$\displaystyle\widetilde{f}^{\mu\nu}{}_{\rho}=\begin{cases}0&\mbox{if
}\mu=\rho\neq\nu\\\ \frac{t^{l}}{1-t^{2}}&\mbox{if }\mu=\rho=\nu\\\
\frac{t^{1-l}}{1-t^{2}}&\mbox{if }\mu=\nu\neq\rho.\end{cases}$ (4.17)

In (4.16), $\eta_{aa}$ are components of $SL(2,\mathbb{C})$ metric (4.2), and
in (4.17), $l$ is defined to be a new variable: $l=\frac{k}{4}\bmod 2$. These
fusion coefficients match with the fusion coefficients of $SO(3)_{k}$
Chern–Simons theory previously found in the literature [37, 36].

The identity element of ${\mathcal{T}}(PSL(2,\mathbb{C})_{k})$ is
$x^{0}-tx^{2}$, thus giving the following block diagonal form of the metric,

$\displaystyle\widetilde{\eta}_{ab}=\text{diag}\left(\frac{1}{1-t^{2}},\frac{1}{1-t},\cdots,\frac{1}{1-t}\right)$
(4.18)

and

$\displaystyle\widetilde{\eta}_{\mu\nu}=\frac{1}{1-t^{2}}\times\begin{cases}t^{l}&\mbox{if
}\mu=\nu\\\ t^{1-l}&\mbox{if }\mu\neq\nu\end{cases}$ (4.19)

We propose that the fusion coefficients (4.12)–(4.17) give the quantization of
the moduli space of parabolic semi-stable $PSL(2,\mathbb{C})$-Higgs bundles.
We will check this proposal in the next section where we will compute the
Hitchin character for the moduli space of parabolic semi-stable
$PSL(2,\mathbb{C})$-Higgs bundles on a once-punctured torus. This will give a
strong check for our proposal as well as for the fusion coefficients derived
above.

###  4.1 $PSL(2,\mathbb{C})$ Verlinde algebra from geometry

We can compute the metric, and fusion coefficients of the $PSL(2,\mathbb{C})$
Verlinde algebra in a different way by understanding the sign group actions on
moduli stacks of flat $SL(2,\mathbb{C})$ connections on $\Sigma_{0,2}$ and
$\Sigma_{0,3}$, together with their prequantum line bundles. The moduli stack
of flat $SL(2,\mathbb{C})$ connections on $\Sigma_{0,3}$ is just a point, so
the fusion coefficients will be 1 or 0 depending on weather the sign group
action is trivial or not. The moduli stack of flat $SL(2,\mathbb{C})$
connections on $\Sigma_{0,2}$ is the groupoid $pt/\mathbb{C}^{*}$, so to
compute the metric we have to know the sign group action on this stack, as
well as the sign group action on the prequantum line.

#### Equivariance of the prequantum line bundle

For a Riemann surface $\Sigma$, the $SL(2,\mathbb{C})$ prequantum line bundle
assigns to a Higgs bundle $(E,\phi)$ a power of the determinant line
$\mathcal{L}_{(E,\phi)}=Det^{-1}(\overline{\partial}_{E\otimes K^{1/2}})$. The
sign group consists of line bundles $\alpha$ with $\alpha^{\otimes 2}$
identified with the trivial bundle, and acts by $E\mapsto\alpha\otimes E$.
This determinant line bundle constructed from the standard representation of
$SL(2,\mathbb{C})$ is not equivariant for the sign group action, but if we use
the adjoint representation, it will be, because $End(E)$ is not sensitive to
tensoring $E$ with line bundles. The Killing form of $SL(2,\mathbb{C})$ is 4
times the trace form of the standard representation so, when
$k=0\;\text{mod}\;4$, the lines $Det^{-1}(\overline{\partial}_{End(E)\otimes
K^{1/2}})^{k/4}$ will fit together to a sign group equivariant line bundle
isomorphic to $\mathcal{L}^{k}$.

The definition of the prequantum line bundle on parabolic moduli space is more
difficult, but we state here one of its features which will show up later.
Choose marked points $x_{1},\dots,x_{n}\in\Sigma$ and label them with even
integers $a_{1},\dots,a_{n}$ from $0$ to $k$. A parabolic $SL(2,\mathbb{C})$
Higgs bundle is a rank 2 Higgs bundle $(E,\phi)$ with trivialized determinant,
together with a line $L_{i}$ in each fiber $E_{x_{i}}$. There is a fibration
$(E,\phi,L_{1},\dots,L_{n})\mapsto(E,\phi)$ from parabolic moduli space to
ordinary moduli space with fiber
$\mathbb{C}P^{1}\times\dots\times\mathbb{C}P^{1}$. The prequantum line bundle
restricted to a fiber of this fibration is isomorphic to
$\mathcal{O}(a_{1})\boxtimes\dots\boxtimes\mathcal{O}(a_{n})$, which is the
line bundle who’s fiber over $(L_{1},\dots,L_{n})$ is
$L_{1}^{-a_{1}}\otimes\dots\otimes L_{n}^{-a_{n}}$. Note that the evenness of
$a_{i}$ is necessary for sign group equivariance.

#### Geometric interpretation of the additional state

Fixing the conjugacy class of holonomy around each boundary component of
$\Sigma$ is indeed sufficient for defining a symplectic moduli space, but to
define a prequantum line bundle, we need slightly more data in the non-simply
connected case: we need a conjugacy class equipped with a conjugation-
equivariant line bundle of prescribed curvature. The restrictions on conjugacy
class and curvature come from the requirement that the line bundle trivializes
the level $k$ gerbe on $G$ [38]. For $PSL(2,\mathbb{C})$, there is a special
conjugacy class, corresponding to $k/2$, represented by

$J=\begin{bmatrix}i&0\\\ 0&-i\end{bmatrix}$

which has a disconnected point stabilizer group. To specify an equivariant
line bundle on this conjugacy class, we need to choose a sign that the non-
identity component of this stabilizer should act by. The two choices of this
sign give rise to the states $\frac{k}{2}^{(1)}$ and $\frac{k}{2}^{(2)}$ from
above. The implication for geometric quantization is that the action on the
$SL(2,\mathbb{C})$ prequantum line bundle of a $\mathbb{Z}_{2}$ bundle which
is non-trivial around a puncture labeled $\frac{k}{2}^{(i)}$ will be
multiplied by $(-1)^{i}$. An open symmetry generating line ends at two
punctures labeled $\frac{k}{2}^{(i)}$ and $\frac{k}{2}^{(j)}$, so its action
on the prequantum line bundle will be multiplied by $(-1)^{i+j}$. The fact
that these signs only appear in pairs is a manifestation of the dual
$\mathbb{Z}_{2}$ symmetry of the $PSL(2,\mathbb{C})$ theory.

#### Fixed points of the sign group

We can realize $\mathcal{M}_{PSL(2,\mathbb{C})}$ as a quotient of
$\mathcal{M}_{SL(2,\mathbb{C})}$ by
$H^{1}(\Sigma,\mathbb{Z}_{2})=Hom(\pi_{1}(\Sigma),\mathbb{Z}/2)$. A fixed
point of $Hom(\pi_{1}(\Sigma),\mathbb{Z}/2)$ is represented a homomorphism
$\rho\in Hom(\pi_{1}(\Sigma),SL(2,\mathbb{C}))$ such that there exists $g\in
SL(2,\mathbb{C})$ such that $h\rho=g\rho g^{-1}$. There is only one conjugacy
class preserved under multiplication by $-1$, and it contains $J$, so if
$h(\gamma)$ is non-trivial, we can conjugate $\rho$ so that $\rho(\gamma)=J$.
This forces $g$ to be of the form

$g=\begin{bmatrix}0&a\\\ -a^{-1}&0\end{bmatrix}$

which one checks is a square root of $-I$ (and is also conjugate to $J$). In
conclusion, any non-trivial sign group element will act on a bundle
representing a fixed point in moduli space by a gauge transformation that is a
square root of $-I$. The action of a non-trivial sign group element on the
prequantum line over a fixed point in parabolic Higgs bundle moduli space will
pick up a sign $(-1)^{\sum a_{i}/2}$ coming from
$L_{1}^{-a_{1}}\otimes\dots\otimes L_{n}^{-a_{n}}$. This sign shows up when
computing the $PSL(2,\mathbb{C})$ fusion coefficients $\widetilde{f}^{\mu\nu
a}$.

#### Twice punctured sphere

First, let $a$ be an even integer less than $k/2$. The sign group
$\mathbb{Z}_{2}$ swaps boundary conditions $a$ and $k-a$, so the
$PSL(2,\mathbb{C})$ moduli space can be identified with an $SL(2,\mathbb{C})$
moduli space.

$\mathcal{M}_{PSL(2,\mathbb{C})}(\Sigma_{0,2},a,a)\simeq(\mathcal{M}_{SL(2,\mathbb{C})}(\Sigma_{0,2},a,a)\sqcup\mathcal{M}_{SL(2,\mathbb{C})}(\Sigma_{0,2},k-a,k-a))/\mathbb{Z}_{2}$
$\simeq\mathcal{M}_{SL(2,\mathbb{C}}(\Sigma_{0,2},a,a)$

so $\widetilde{\eta}^{aa}=\eta^{aa}$. When $a$ is not $0$, $k$, the stack is
$pt/\mathbb{C}^{*}$. Taking the symmetric algebra of the tangent complex, we
get the Hilbert space $\mathbb{C}\oplus\mathbb{C}_{t}[1]$ (see [7] Theorem
20), so $\eta^{aa}=1-t$.

Now suppose the punctures are labeled by $\mu=\frac{k}{2}^{(i)}$ and
$\nu=\frac{k}{2}^{(j)}$. The unique connection in
$\mathcal{M}_{SL(2,\mathbb{C})}(\Sigma_{0,2},k/2,k/2)$ has holonomy $J$, and
has automorphism group $\mathbb{C}^{*}$ consisting of diagonal matrices. This
connection is stabilized by simultaniously multiplying by $-1$ and conjugating
by an off diagonal matrix. This acts on the moduli space $pt/\mathbb{C}^{*}$,
by inverting $\mathbb{C}^{*}$, so acts on the tangent complex by $-1$, thus
the sign group acts on functions on the moduli space
$\mathbb{C}\oplus\mathbb{C}_{t}[1]$ by negating $\mathbb{C}_{t}[1]$. The sign
group action on the prequantum line comes out to be $(-1)^{i+j+k/4}$. The
$PSL(2,\mathbb{C})$ Hilbert space is the $\mathbb{Z}_{2}$ invariant subspace.

$\widetilde{\eta}^{\mu\nu}=\begin{cases}1&\mbox{if }(-1)^{i+j+k/4}=1\\\
-t&\mbox{else}\end{cases}$

#### Thrice punctured sphere

There are four cases to consider: 0,1,2 or 3 punctures labeled by $k/2$. First
we compute $\widetilde{f}^{abc}$ where $a,b,c\in\\{0,2,..,k/2-2\\}$. There are
two orbits of the sign group $(\mathbb{Z}_{2})^{2}$ action on the 8
equivalence classes of $SL(2,\mathbb{C})$ connections which map to
$\mathcal{M}_{PSL(2,\mathbb{C})}(\Sigma,a,b,c)$. This gives
$\widetilde{f}^{abc}$ in terms of $SL(2,\mathbb{C})$ data.

$\widetilde{f}^{abc}=f^{abc}+f^{(k-a)bc}$

Next, we compute $\widetilde{f}^{ab\mu}$. The sign group acts simply
transitively on the 4 relevant $SL(2,\mathbb{C})$ moduli spaces, so the
$PSL(2,\mathbb{C})$ moduli space is identified with any one of these.

$\widetilde{f}^{ab\mu}=f^{ab\frac{k}{2}}$

Next, we compute $\widetilde{f}^{a\mu\nu}$ where $\mu=\frac{k}{2}^{(i)}$ and
$\nu=\frac{k}{2}^{(j)}$. The sign group acts transitively on the two relevant
$SL(2,\mathbb{C})$ connections, and each is stabilized by the $\mathbb{Z}_{2}$
bundle which is non-trivial around $x_{1}$ and $x_{2}$. The action of the sign
group on the prequantum line is $(-1)^{i+j+a/2+k/4}$.202020Notice that the
$k/4$ factor is necessary here to ensure the consistency in the computation
for $\widetilde{f}^{\mu\nu\rho}$. We hope to better understand the determinant
line bundle on parabolic moduli space so as to derive it directly. The Chern-
Simons Hilbert space is $1$ dimensional if this action is trivial, and zero
dimensional otherwise.

$\widetilde{f}^{\mu\nu a}=\begin{cases}1&\mbox{if }(-1)^{i+j+a/2+k/4}=1\\\
0&\mbox{otherwise.}\end{cases}$

Finally, we compute $\widetilde{f}^{\mu\nu\rho}$ where
$\mu=\frac{k}{2}^{(i)}$, $\nu=\frac{k}{2}^{(j)}$, and
$\rho=\frac{k}{2}^{(l)}$. The moduli space is a single point with an action of
$(\mathbb{Z}/2)^{2}$. The sign group action is given by the same formula as in
the previous case, with $a$ set to $k/4$. For example, the action of the
$\mathbb{Z}_{2}$ bundle which is non-trivial around $x_{1}$ and $x_{2}$ is
$(-1)^{i+j}$. This gives

$\widetilde{f}^{\mu\nu\rho}=\begin{cases}1&\mbox{if }\mu=\nu=\rho\\\
0&\mbox{otherwise.}\end{cases}$

###  4.2 Categorification and generalization

The equivariant Verlinde algebra is not just an ordinary commutative Frobenius
algebra, as there exist a special basis in which the fusion coefficients and
the metric have non-negative integral coefficients. This is a strong hint that
this algebra can be “categorified.”

Indeed, physics predicts that the category of line operators
${\mathcal{C}}_{\text{lines}}$ in the partially twisted 3d ${\mathcal{N}}=2$
Chern–Simons–matter theory provides such a categorification. As the theory is
only partially topological, ${\mathcal{C}}_{\text{lines}}$ is not expected to
a modular tensor category when $t\neq 0$, but one expects that it still
contains more data, such as braiding, compared to the commutative Frobenius
algebra.

We hope to better understand this category in future work. This will provide a
systematic way of generalizing the equivariant Verlinde algebra to other
groups and incorporate additional deformations.

For example, there seems to be an alternative derivation of the fusion
coefficients of $\mathcal{T}(PSL(2,\mathbb{C}))$, that is more readily
generalizable to other Lie groups. This derivation makes use of the notion of
“untwisted” and “twisted” sector operators on which the dual
$\mathbb{Z}_{2}^{(0)}$ acts as $+1$ and $-1$ respectively. In particular, let
us consider a new basis

$\displaystyle x^{0},x^{2},\cdots,x^{\frac{k}{2}-2},x^{+},x^{-}$ (4.20)

where $x^{0},x^{2},\cdots,x^{\frac{k}{2}-2}$, and $x^{+}=x^{\frac{k}{2}}$ are
the “untwisted” sector operators and

$\displaystyle x^{-}=x^{{\frac{k}{2}}^{(1)}}-x^{{\frac{k}{2}}^{(2)}}$ (4.21)

is the “twisted” sector operator. This operator may be thought of as living at
the end of an open $\mathbb{Z}_{2}^{(0)}$ symmetry generating line in the
parent TQFT ${\mathcal{T}}(SL(2,\mathbb{C})_{k})/\mathbb{Z}_{2}^{(1)}$. It
carries charge $-1$ under the dual $\mathbb{Z}_{2}^{(0)}$ symmetry.

The fusion coefficients involving the untwisted sector operators are fixed
entirely through some straightforward algebra. In particular,
$\widetilde{f}^{++}{}_{a}=2f^{\frac{k}{2}\frac{k}{2}}{}_{a}$. Invariance under
the dual $\mathbb{Z}_{2}^{(0)}$ symmetry implies that fusion coefficients
involving $x^{-}$ are non-zero only if $x^{-}$ appears an even number of
times. For example, $\widetilde{f}^{-a}{}_{b}=0$. Then, we have the relation

$\displaystyle\widetilde{f}^{--}{}_{a}=(-1)^{\frac{a}{2}+\frac{k}{4}}\widetilde{f}^{++}{}_{a},$
(4.22)

which can be used to completely fix all the rest of the fusion coefficients
without using positivity and integrality. At the same time, it impose a very
strong condition on the other fusion coefficients as $a$ is arbitrary. The
phase $(-1)^{\frac{a}{2}+\frac{k}{4}}$ in the $t\to 0$ limit can be
interpreted as a combination of R- and F-matrices, and has a generalization to
other Lie groups. Understanding whether and why such structure preserve when
$t$ is non-zero would shed light on the structure of the category
${\mathcal{C}}_{\text{lines}}$, which we hope to investigate in the future.

##  5 Quantization of ${\mathcal{M}}(PSL(2,\mathbb{C}))$ for once-punctured
torus

In this section, we will compute the equivariant trace and the Hitchin
character for the moduli space of parabolic Higgs bundles on a once-punctured
torus. First, we will compute the equivariant trace in Section 5.1 using
methods in 2d TQFT. Then in Section 5.2, we will propose a method for
computing this result directly from geometry using the Atiyah-Singer-Lefschetz
index theorem. Finally, in Section 5.3, we will compute the Hitchin character
of ${\mathcal{M}}_{w_{2}}(PSL(2,\mathbb{C}))$, and show that it is consistent
with $PSL(2,\mathbb{C})$ equivariant Verlinde algebra studied in Section 4.

###  5.1 Computation from 2d TQFT

Let ${\mathcal{M}}^{a}_{d}$ be the moduli space of parabolic semi-stable
$SL(2,\mathbb{C})$-Higgs bundles of degree $d$ on a once-punctured torus where
$a$ denotes a highest weight of $\mathfrak{su}(2)$ that specifies the
parabolic structure at the puncture. We will set $k\equiv 0\pmod{4}$ and
$a\equiv 0\pmod{2}$. ${\mathcal{M}}^{a}_{d}$ admits an action of
$H_{1}(\Sigma_{1,1},\mathbb{Z}_{2})$. The equivariant trace of an element
$h\in H_{1}(\Sigma_{1,1},\mathbb{Z}_{2})$ can be computed using the methods of
Section 2. For any non-zero element $h$,
$\text{Tr}_{t}(h|H^{0}({\mathcal{M}}_{d},{\mathcal{L}}^{k}))$ gives the
partition function of ${\mathcal{T}}(SL(2,\mathbb{C})_{k})$ in the presence of
a zero-form symmetry background. This partition function is the same for all
non-zero $h$.

Let us consider the configuration in Figure 4. Here, the twisted cylinder has
components $\overline{\eta}^{a}{}_{b}=(-1)^{a}\delta^{a}{}_{b}$, and we obtain

$\displaystyle\text{Tr}_{t}(h|H^{0}({\mathcal{M}}_{d=0},{\mathcal{L}}^{k}))=\sum\limits_{b,c,d}f^{a}{}_{bc}\overline{\eta}^{c}{}_{d}\eta^{bd}=\sum\limits_{b}(-1)^{b}f^{abb}\eta_{bb}.$
(5.1)

Using the fusion rules of $SL(2,\mathbb{C})$ (4.1)–(4.2), we get

$\displaystyle f^{abb}=\begin{cases}1&\mbox{if }\frac{a}{2}\leq b\leq
k-\frac{a}{2}\\\ t^{c/2}&\mbox{if }b<\frac{a}{2}\ \text{or}\
b>k-\frac{a}{2}\end{cases}$ (5.2)

where $c=\text{max}(\frac{a}{2}-b,\frac{a}{2}+b-k)$. Therefore,

$\displaystyle\text{Tr}_{t}(h|H^{0}({\mathcal{M}}_{d=0},{\mathcal{L}}^{k}))$
$\displaystyle=$ $\displaystyle\sum\limits_{b=0}^{k}(-1)^{b}f^{abb}\eta_{bb}$
(5.3) $\displaystyle=$
$\displaystyle\frac{2t^{a/2}}{1-t^{2}}+\frac{2t^{\frac{a}{2}}}{1-t}\sum\limits_{b=1}^{\frac{a}{2}-1}(-t)^{-b}+\sum\limits_{b=\frac{a}{2}}^{k-\frac{a}{2}}\frac{(-1)^{b}}{1-t}$
$\displaystyle=$
$\displaystyle\frac{2t^{a/2}}{1-t^{2}}+\frac{2t^{a/2}}{1-t^{2}}\sum\limits_{b=1}^{\frac{a}{2}-1}(-1)^{b}(t^{-b}+t^{-b+1})+\frac{(-1)^{a/2}}{1-t}$
$\displaystyle=$
$\displaystyle\frac{2t(-1)^{a/2-1}}{1-t^{2}}+\frac{(-1)^{a/2}}{1-t}$
$\displaystyle=$ $\displaystyle\frac{(-1)^{a/2}}{1+t}.$

Figure 4: Once-punctured torus

For $d=1$, we introduce a $\mathbb{Z}_{2}^{(1)}$ background labelled by the
generator of $\mathbb{Z}_{2}^{(1)}$, i.e. $(w^{k}-tw^{k-2})$. This operator
fuses with $w^{a}$ to give $w^{k-a}$. The equivariant trace is obtained by
replacing $a$ with $k-a$. Since $k\equiv 0\pmod{4}$, this does not change the
result, and we get

$\displaystyle\text{Tr}_{t}(h|H^{0}({\mathcal{M}}_{d=1},{\mathcal{L}}^{k}))=\frac{(-1)^{a/2}}{1+t}.$
(5.4)

###  5.2 Equivariant trace: a geometric computation

For arbitrary simple complex Lie groups, the equivariant trace and the Hitchin
characters of ${\mathcal{M}}_{w_{2}}(G)$ may be computed using a trace formula
due to Atiyah and Singer [15]. Let $W$ be a holomorphic vector bundle on
$Y=X/{\mathcal{G}}$ where $X$ is a compact complex manifold and
${\mathcal{G}}$ is a finite group. Let $V$ be the pullback of $W$ on $X$, then
the Euler characteristic $\chi(Y,W):=\sum\limits_{p}(-1)^{p}\text{dim}\
H^{p}(Y|{\mathcal{O}}(W))$ satisfies

$\displaystyle\chi(Y,W)=\frac{1}{|{\mathcal{G}}|}\sum\limits_{h\in{\mathcal{G}}}\mu(h|X,V)$
(5.5)

where

$\displaystyle\mu(h|X,V):=\sum\limits_{p}(-1)^{p}{\rm
Tr\,}(h|H^{p}(X|{\mathcal{O}}(V)))$ (5.6)

can be computed as an integral over the fixed point locus $P$ of $h$,

$\displaystyle\mu(h|X,V)=\int_{P}\frac{\text{Td}(T_{P})\widetilde{\text{ch}}(V_{P},h)}{\lambda(N_{P},h)}.$
(5.7)

Here $\text{Td}(T_{P})$ is the Todd class of $P$, and
$\widetilde{\text{ch}}(V_{P},h)$ and $\lambda(N_{P},h)$ are defined as
follows. Let $V_{P}$ be the restriction of $V$ to $P$; and let $\mu_{i}$ be
the eigenvalues of $h$ as an endomorphism of $V_{P}$. Then, with the eigen-
decomposition $V_{P}=\oplus_{i}V_{\mu_{i}}$ we define

$\displaystyle\widetilde{\text{ch}}(V_{P},h):=\sum\limits_{i}\mu_{i}\text{ch}(V_{\mu_{i}}).$
(5.8)

Similarly, the normal bundle $N_{P}$ of $P$ in $X$ also has an eigen-
decomposition, $N_{P}=\oplus_{i}N_{\lambda_{i}}$ with eigenvalues
$\lambda_{i}$ of $h$, and we define

$\displaystyle\lambda(N_{P},h)=\prod\limits_{i}\sum\limits_{p\geq
0}(-\lambda_{i})^{p}\text{ch}(\Lambda^{p}N_{\lambda_{i}}^{\star}).$ (5.9)

For $h=e$, $\mu(h|X,V)$ reduces to the usual Riemann–Roch formula that counts
the dimension of the space of sections of $V$.

We wish to use this general result to compute the equivariant trace
$\text{Tr}_{t}(h|H^{0}({\mathcal{M}}_{d},{\mathcal{L}}^{k}))$ and the Hitchin
character
$\text{dim}_{t}H^{0}({\mathcal{M}}_{w_{2}}(PSL(2,\mathbb{C})),{\mathcal{L}}^{k})$.
Two issues appear at once.

  1. 1.

$\mu(h|X,V)$ computes an Euler characteristic while the equivariant trace
$\text{Tr}_{t}(h|H^{0}({\mathcal{M}}_{d},{\mathcal{L}}^{k}))$ is computed on
the zeroth cohomology, and

  2. 2.

The theorem above required $X$ and $Y$ to be compact manifolds, while
${\mathcal{M}}_{w_{2}}(PSL(2,\mathbb{C}))$ is non-compact.

The first issue is resolved by using the fact that the higher cohomology
$H^{i}({\mathcal{M}}_{d},{\mathcal{L}}^{k})$ for $i>0$ vanishes. This result
was proved in [7] and [8]. The second issue can be cured by considering a
$\mathbb{C}^{\star}$-equivariant version of the theorem. Alternatively, one
may use the identification [7, 8]

$\displaystyle
H^{i}_{n}({\mathcal{M}},{\mathcal{L}}^{k})=H^{i}(\mathfrak{N},S^{n}T\mathfrak{N}\otimes{\mathcal{L}}^{k})$
(5.10)

to transfer the computation to the moduli stack $\mathfrak{N}$ of $G$-vector
bundles. This gives

$\displaystyle\text{Tr}_{t}(h|H^{0}({\mathcal{M}}_{d=w_{2}},{\mathcal{L}}^{k}))=\sum\limits_{n=0}^{\infty}\mu(h|\mathfrak{N},S^{n}T\mathfrak{N}\otimes{\mathcal{L}}^{k})t^{n}.$
(5.11)

Each summand on the right side can be computed using equation (5.7).

In what follows, we will propose that in the case of once-punctured torus, the
difference between the moduli stack and the moduli space ${\mathcal{N}}$ of
semi-stable $G$ vector bundles will not matter. That is,

$\displaystyle\mu(h|\mathfrak{N},S^{n}T\mathfrak{N}\otimes{\mathcal{L}}^{k})=\mu(h|\mathcal{N},S^{n}T\mathcal{N}\otimes{\mathcal{L}}^{k})$
(5.12)

when $G=SL(2,\mathbb{C})$ and the underlying surface is the once-punctured
torus. Our proposal is based on the observation that

$\displaystyle\sum\limits_{n=0}^{\infty}\mu(h|\mathcal{N},S^{n}T\mathcal{N}\otimes{\mathcal{L}}^{k})t^{n}$
(5.13)

is equal to the equivariant trace computed in the previous section. This is
interesting because while the higher cohomology
$H^{i>0}(\mathfrak{N},S^{n}T\mathfrak{N}\otimes{\mathcal{L}}^{k})$ vanishes
for the moduli stack [7, 8], it does not vanish for the moduli space.
Furthermore, the zeroth cohomology on the stack and the moduli space coincide.
Therefore, our proposal implies in particular that ${\rm
Tr\,}(h|H^{i>0}({\mathcal{N}},S^{n}T{\mathcal{N}}\otimes{\mathcal{L}}^{k}))=0$.

In the rest of this section, we will compute (5.13) for once-punctured torus
and $G=SL(2,\mathbb{C})$. Through the Narasimhan–Seshadri theorem,
${\mathcal{N}}$ is also the moduli space of $SU(2)$ flat connections with the
holonomy around the puncture conjugate to $e^{i\pi\frac{a}{k}\sigma_{3}}$.
This moduli space is a closed surface inside $\mathbb{R}^{3}$, and is
described by the equation [39]

$\displaystyle x^{2}+y^{2}+z^{2}-xyz=4\cos^{2}\left(\frac{\pi a}{2k}\right)$
(5.14)

where $x,y,z\in[-2,2]$.
$H_{1}(\Sigma_{1,1},\mathbb{Z}_{2})=\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ acts
on this surface as $(x,y,z)\to(\pm x,\pm y,\pm z)$ with an even number of
minus signs. A non-zero element of this group has two fixed points, which we
will denote by $p_{1}$ and $p_{2}$. The integral in equation (5.7) reduces to
a sum,

$\displaystyle\mu(h|{\mathcal{N}}_{a},S^{n}T{\mathcal{N}}_{a}\otimes{\mathcal{L}}^{k})=\sum\limits_{i=1}^{2}\frac{\text{Td}(T_{p_{i}})\
\widetilde{\text{ch}}(V_{p_{i}},h)}{\lambda(N_{p_{i}},h)}.$ (5.15)

As $h$ acts on the normal bundle $N_{p_{i}}$ with eigenvalue $-1$,
$\lambda(N_{p_{i}},h)=\text{ch}(\Lambda^{0}N^{\star}_{p_{i}})+\text{ch}(\Lambda^{1}N^{\star}_{p_{i}})$.
After restricting to a point, we obtain $\lambda(N_{p_{i}},h)=2$.

To compute $\widetilde{\text{ch}}(V_{p_{i}},h)$ for
$V=S^{n}(T\mathbb{CP}^{1})\otimes{\mathcal{L}}^{k}$, we note that as
$N_{p_{i}}\equiv T\mathbb{CP}^{1}_{p_{i}}$, $h$ acts on the tangent space
$T\mathbb{CP}^{1}_{p_{i}}$ by multiplication by $-1$. It therefore acts on
${\mathcal{L}}^{k}\equiv(T\mathbb{CP}^{1})^{\frac{k-a}{2}}$ by $(-1)^{a/2}$
(as $k\equiv 0\pmod{4}$). Overall,
$\widetilde{\text{ch}}(V_{p_{i}},g)=(-1)^{n+\frac{a}{2}}\text{ch}(V_{p_{i}})$,
and

$\displaystyle\mu(h|\mathcal{N},S^{n}T\mathcal{N}\otimes{\mathcal{L}}^{k})=\sum\limits_{i=1}^{2}\frac{(-1)^{n+\frac{a}{2}}}{2}=(-1)^{n+\frac{a}{2}}.$
(5.16)

This gives

$\displaystyle\sum\limits_{n=0}^{\infty}\mu(h|\mathcal{N},S^{n}T\mathcal{N}\otimes{\mathcal{L}}^{k})t^{n}=\frac{(-1)^{a/2}}{1+t}$
(5.17)

which is precisely the result (5.3) obtained in the previous subsection.

Finally, we remark that the case $d=1$ can be treated by introducing another
puncture on the torus with holonomy $-\mathbb{1}$. This effectively replaces
$a$ with $k-a$ in equation (5.14). Making this transformation in (5.17) leaves
the right hand side invariant, and we get

$\displaystyle\sum\limits_{n=0}^{\infty}\mu(h|\mathcal{N},S^{n}T\mathcal{N}\otimes{\mathcal{L}}^{k})t^{n}=\frac{(-1)^{a/2}}{1+t}.$
(5.18)

which is equal to the result (5.4) computed in the previous subsection.

###  5.3 Hitchin characters for $\mathcal{M}_{w_{2}}(PSL(2,\mathbb{C}))$

In this section, we will use the formulae for equivariant trace derived in the
previous two sections to compute the Hitchin characters for
${\mathcal{M}}_{w_{2}}(PSL(2,\mathbb{C}))$. Let us first consider the case
$a\in\\{0,2,\cdots,\frac{k}{2}-2\\}$. The case of $a=\frac{k}{2}$ will be
discussed shortly. For $w_{2}=0$,

$\displaystyle\text{dim}_{t}H^{0}({\mathcal{M}}^{a}_{w_{2}=0}(PSL(2,\mathbb{C})),{\mathcal{L}}^{k})=\frac{1}{4}\sum\limits_{h\in
H_{1}(\Sigma_{1,1},\mathbb{Z}_{2})}\text{Tr}_{t}(h|H^{0}({\mathcal{M}}^{a}_{d=0},{\mathcal{L}}^{k}))$
(5.19)

For $h=0$, $\text{Tr}_{t}(h|H^{0}({\mathcal{M}}^{a}_{d=0},{\mathcal{L}}^{k}))$
is the Hitchin character for ${\mathcal{M}}_{d=0}$, which can be computed
using the fusion rules in Section 4,

$\displaystyle\text{dim}_{t}H^{0}({\mathcal{M}}^{a}_{d=0},{\mathcal{L}}^{k})=\frac{k-a+1}{1-t}+\frac{2t}{(1-t)^{2}}+\frac{4t^{a/2}}{(1-t)(1-t^{2})}$
(5.20)

This gives

$\displaystyle\text{dim}_{t}H^{0}({\mathcal{M}}_{w_{2}=0}(PSL(2,\mathbb{C}),{\mathcal{L}}^{k})$
$\displaystyle=$
$\displaystyle\frac{1}{4}\left(\frac{k-a+1}{1-t}+\frac{2t}{(1-t)^{2}}+\frac{4t^{a/2}}{(1-t^{-1})(1-t^{2})}+\frac{3(-1)^{a/2}}{1+t}\right).$

The Hitchin character for $w_{2}=1$ component can be obtained by replacing $a$
with $k-a$,

$\displaystyle\text{dim}_{t}H^{0}({\mathcal{M}}_{w_{2}=1}(PSL(2,\mathbb{C}),{\mathcal{L}}^{k})$
$\displaystyle=$
$\displaystyle\frac{1}{4}\left(\frac{a+1}{1-t}+\frac{2t}{(1-t)^{2}}+\frac{4t^{(k-a)/2}}{(1-t^{-1})(1-t^{2})}+\frac{3(-1)^{a/2}}{1+t}\right).$

The union $\cup_{w_{2}}{\mathcal{M}}_{w_{2}}(PSL(2,\mathbb{C}))$ is the
classical phase space of $PSL(2,\mathbb{C})$ Chern–Simons theory. After
$\beta$-deformation, the partition function of this theory on
$S^{1}\times\Sigma_{1,1}$ is the sum of (5.3) and (5.3). Let us denote the sum
by ${\mathcal{Z}}_{{\mathcal{T}}(PSL(2,\mathbb{C}))}$, then it is easy to
check that

$\displaystyle{\mathcal{Z}}_{{\mathcal{T}}(PSL(2,\mathbb{C}))}=\sum\limits_{\begin{subarray}{c}b=0\\\
b\
even\end{subarray}}^{\frac{k}{2}-2}\widetilde{f}^{ab}{}_{b}+\sum\limits_{\mu\in\\{\frac{k}{2}^{(1)},\frac{k}{2}^{(2)}\\}}\widetilde{f}^{a\mu}{}_{\mu}$
(5.23)

where $\widetilde{f}^{ab}{}_{b}$ and $\widetilde{f}^{a\mu}{}_{\mu}$ are the
fusion coefficients of $PSL(2,\mathbb{C})$ Verlinde algebra in Section 4.

Now let $a=\frac{k}{2}$. There are two operators in the TQFT labelled by this
weight, and the dual $\mathbb{Z}_{2}^{(0)}$ symmetry exchanges them. The
geometric interpretation of these two boundary conditions is not clear, so we
will suffice ourselves with the physical interpretation of the partition
functions. Let ${\mathcal{Z}}^{(1)}$ and ${\mathcal{Z}}^{(2)}$ be the
partition functions of
${\mathcal{T}}(SL(2,\mathbb{C})_{k})/\mathbb{Z}_{2}^{0}$, then the dual
$\mathbb{Z}_{2}^{(0)}$ symmetry implies that

$\displaystyle{\mathcal{Z}}^{(1)}={\mathcal{Z}}^{(2)}.$ (5.24)

If the puncture on the torus is labelled by
$w^{\frac{k}{2}}=w^{\frac{k}{2}^{(1)}}+w^{\frac{k}{2}^{(2)}}$, we can use
equation

$\displaystyle{\mathcal{Z}}^{(1)}+{\mathcal{Z}}^{(2)}=\frac{1}{2}\sum\limits_{h\in
H_{1}(\Sigma_{1,1},\mathbb{Z}_{2})}{\mathcal{Z}}^{\frac{k}{2}}_{h}$ (5.25)

where

$\displaystyle{\mathcal{Z}}^{\frac{k}{2}}_{h}=\begin{dcases}\frac{k+2}{2(1-t)}+\frac{2t}{(1-t)^{2}}+\frac{4t^{k/4}}{(1-t^{-1})(1-t^{2})}&\mbox{if
}h=0\\\ \frac{(-1)^{k/4}}{1+t}&\mbox{if }h\neq 0\end{dcases}$ (5.26)

to obtain

$\displaystyle{\mathcal{Z}}^{(1)}={\mathcal{Z}}^{(2)}=\frac{1}{4}\left(\frac{k+2}{2(1-t)}+\frac{2t}{(1-t)^{2}}+\frac{4t^{k/4}}{(1-t^{-1})(1-t^{2})}+\frac{3(-1)^{k/4}}{1+t}\right).$

Introducing a one-form symmetry background replaces $a$ with $k-a$ but for
$a=\frac{k}{2}$, this has no effect. The partition function of
${\mathcal{T}}(PSL(2,\mathbb{C})_{k})$ labelled by the two boundary conditions
$\frac{k}{2}^{(i)}$ is

$\displaystyle{\mathcal{Z}}^{(i)}_{{\mathcal{T}}(PSL(2,\mathbb{C}))}=\frac{1}{4}\left(\frac{k+2}{2(1-t)}+\frac{2t}{(1-t)^{2}}+\frac{4t^{k/4}}{(1-t^{-1})(1-t^{2})}+\frac{3(-1)^{k/4}}{1+t}\right).$
(5.27)

This partition function is also reproduced by the fusion coefficients of
$PSL(2,\mathbb{C})$ algebra studied in Section 4,

$\displaystyle{\mathcal{Z}}^{(i)}_{{\mathcal{T}}(PSL(2,\mathbb{C}))}$
$\displaystyle=$
$\displaystyle\sum\limits_{a}\widetilde{f}^{\frac{k}{2}^{(i)}a}{}_{a}+\sum\limits_{\mu}\widetilde{f}^{\frac{k}{2}^{(i)}\mu}{}_{\mu}.$

giving another check on the formulae obtained there.

### Acknowledgements

We would like to thank Dan Freed, Po-Shen Hsin, Andrew Neitzke, and Sebastian
Schulz for useful discussions. We would also like to thank Simons Center for
Geometry and Physics for generous hospitality during Graduate Summer School on
the Mathematics and Physics of Hitchin Systems and the Simons Summer Workshop
2021. The work of SG is supported by the U.S. Department of Energy, Office of
Science, Office of High Energy Physics, under Award No. DE-SC0011632, and by
the National Science Foundation under Grant No. NSF DMS 1664227. The work of
AS was supported by NSF grants DMS-2005312 and DMS-1711692. The work of DP was
supported by the Center for Mathematical Sciences and Application. This paper
is partly a result of the ERC-SyG project, Recursive and Exact New Quantum
Theory (ReNewQuantum) which received funding from the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
programme under grant agreement No 810573.

##  Appendix A Partition function of $\mathcal{T}/\mathbb{Z}_{n}^{(0)}$

The partition function of ${\mathcal{T}}/\mathbb{Z}_{n}^{(0)}$ is defined as

$\displaystyle{\mathcal{Z}}_{{\mathcal{T}}/\mathbb{Z}_{n}^{(0)}}=\frac{1}{|H^{1}(\Sigma,\mathbb{Z}_{n})|^{\frac{1}{2}}}\sum\limits_{h\in
H_{1}(\Sigma,\mathbb{Z}_{n})}{\mathcal{Z}}_{h}.$ (A.1)

We will show in Appendix A.1 that for $h\neq 0$,

$\displaystyle{\mathcal{Z}}_{h}=\sum\limits_{i\in{\mathcal{B}}^{\mathbb{Z}_{m}}}N_{i}^{2g-2}$
(A.2)

where $\mathbb{Z}_{m}$ is the smallest subgroup of $\mathbb{Z}_{n}$ such that
$h\in H_{1}(\Sigma,\mathbb{Z}_{m})$, and
${\mathcal{B}}^{\mathbb{Z}_{m}}\subset{\mathcal{B}}$ is the set of
$\mathbb{Z}_{m}$-fixed points. Using this result, we will show in Appendix A.2
that

$\displaystyle{\mathcal{Z}}_{{\mathcal{T}}/\mathbb{Z}_{n}^{(0)}}$
$\displaystyle=$
$\displaystyle\sum\limits_{i\in{\mathcal{B}}/\mathbb{Z}_{n}}|\text{Stab}(i)|\left(\frac{|\text{Stab}(i)|}{|\mathbb{Z}_{n}|^{\frac{1}{2}}}N_{i}\right)^{2g-2}$
(A.3)

where ${\mathcal{B}}/\mathbb{Z}_{n}$ is the set of $\mathbb{Z}_{n}$-orbits of
${\mathcal{B}}$, and for any orbit $i\in{\mathcal{B}}/\mathbb{Z}_{n}$, the
cardinality of the stabilizer of any of its representatives in ${\mathcal{B}}$
is denoted by $\\{1,2,\cdots,|\text{Stab}(i)|\\}$. Finally, in Appendix A.3 we
will discuss gauging a product of cyclic groups.

###  A.1 Computation of $\mathcal{Z}_{h}$

The main idea behind the derivation of equation (A.2) is that for any
$\mathbb{Z}_{n}$ bundle over a surface, the surface can be chopped into pairs
of pants and cylinders such that the restriction of the $\mathbb{Z}_{n}$
bundle to each pair of pants is trivial, but the restriction to the cylinders
may not be. In TQFT, this means that the only new ingredients required for the
computation of ${\mathcal{Z}}_{h}$ are the “twisted cylinders”. These enact
the $\mathbb{Z}_{n}$ action on ${\mathcal{B}}$ giving
$\eta^{i}{}_{j}=\delta^{i}{}_{z(i)}$ where $z\in\mathbb{Z}_{n}$. Now gluing
the surface back together using the twisted cylinders enforces the sum over
the diagonal basis ${\mathcal{B}}$ to collapse to the subset that is invariant
under the action of the twisted cylinders. We will now make this idea more
concrete.

Consider a standard basis $\\{A_{i},B_{i};i=1,\cdots,g\\}$ of cycles in
$H_{1}(\Sigma,\mathbb{Z}_{n})$ and write
$h=\sum\limits_{i=1}^{g}x_{i}A_{i}+y_{i}B_{i}$ for
$x_{i},y_{i}\in\mathbb{Z}_{n}$. It is convenient to chop the surface into
handles and caps as in Figure 5, and apply the action of the mapping class
group of each handle to bring $h$ to a form $\sum\limits_{i=1}^{g}d_{i}A_{i}$
where $d_{i}=\text{gcd}(x_{i},y_{i})$.212121The relevant element of
$SL(2,\mathbb{Z})$, i.e. $\begin{pmatrix}a&b\\\ c&e\end{pmatrix}$ is such that
that $c=-\frac{y}{d}$, $e=\frac{x}{d}$, and $(a,b)$ is a Bezout’s pair
satisfying $ax+by=d$. (See Figure 6). Figure 6 also shows that the handle
operator labelled by $(x_{i},y_{i})$ has a non-zero matrix entry only if the
two boundaries are labelled by the same $d_{i}$-invariant state. Gluing the
handle operators and caps back together we find

$\displaystyle{\mathcal{Z}}_{h}=\sum\limits_{i\in{\mathcal{B}}^{\mathbb{Z}_{m}}}N_{i}^{2g-2}$
(A.4)

where $\mathbb{Z}_{m}$ is the smallest subgroup containing all
$\text{gcd}(x_{i},y_{i})$. As the action of mapping class group preserves each
subgroup of $H_{1}(\Sigma,\mathbb{Z}_{n})$, $\mathbb{Z}_{m}$ is also the
smallest subgroup such that $H_{1}(\Sigma,\mathbb{Z}_{m})$ contains $h$. This
completes the derivation of equation (A.2).

Figure 5: Decomposition of $\Sigma$ into caps and handles Figure 6: A handle
operator decorated with $(x_{i},y_{i})\in\mathbb{Z}_{n}\times\mathbb{Z}_{n}$.
Under the action of modular group, it can be mapped to a handle operator with
decoration $(\text{gcd}(x_{i},y_{i}),0)$. Finally, consistent gluing requires
that $a=d_{i}(a)=b$, which implies that the handle operator is non-zero only
if this condition is satisfied.

###  A.2 Computation of $\mathcal{Z}_{{\mathcal{T}}/\mathbb{Z}_{n}^{(0)}}$

Now we will derive the result (A.3). The derivation is straightforward when
$n=p$ is prime, so we discuss this case first. We recall that $\mathbb{Z}_{p}$
has no non-trivial subgroups. Therefore, ${\mathcal{Z}}_{h}$ is the same for
all non-zero $h$, and we get

$\displaystyle{\mathcal{Z}}_{{\mathcal{T}}/\mathbb{Z}_{p}^{(0)}}$
$\displaystyle=$
$\displaystyle\frac{1}{p^{g}}\left(\sum\limits_{i\in{\mathcal{B}}}N_{i}^{2g-2}+(p^{2g}-1)\sum\limits_{i\in{\mathcal{B}}^{\mathbb{Z}_{p}}}N_{i}^{2g-2}\right)$
(A.5) $\displaystyle=$
$\displaystyle\frac{1}{p^{g}}\left(\sum\limits_{\begin{subarray}{c}i\in{\mathcal{B}}\backslash{\mathcal{B}}^{\mathbb{Z}_{p}}\end{subarray}}N_{i}^{2g-2}+p^{2g}\sum\limits_{i\in{\mathcal{B}}^{\mathbb{Z}_{p}}}N_{i}^{2g-2}\right)$
$\displaystyle=$
$\displaystyle\sum\limits_{\begin{subarray}{c}i\in{\mathcal{B}}/\mathbb{Z}_{p}\\\
|i|=p\end{subarray}}\left(\frac{N_{i}}{|\mathbb{Z}_{p}|^{\frac{1}{2}}}\right)^{2g-2}+p^{2g-1}\sum\limits_{\begin{subarray}{c}i\in{\mathcal{B}}/\mathbb{Z}_{p}\\\
|i|=1\end{subarray}}N_{i}^{2g-2}$ $\displaystyle=$
$\displaystyle\sum\limits_{i\in{\mathcal{B}}/\mathbb{Z}_{p}}|\text{Stab}(i)|\left(\frac{|\text{Stab}(i)|}{|\mathbb{Z}_{p}|^{\frac{1}{2}}}N_{i}\right)^{2g-2}$

This is precisely the result (A.3).

When $n$ is composite, $\mathbb{Z}_{n}$ has a non-trivial subgroups which are
labelled by divisors of $n$, and if $H_{1}(\Sigma,\mathbb{Z}_{m})$ is the
smallest subgroup containing $h$ then ${\mathcal{Z}}_{h}$ gets contribution
only from $\mathbb{Z}_{m}$-invariant elements of ${\mathcal{B}}$.

In order to evaluate (A.1) we need to know for each $m|n$, the number of
elements of $H_{1}(\Sigma,\mathbb{Z}_{m})$ that are not contained in any
smaller subgroup. This number is computed as follows. Let
$n=p_{1}^{a_{1}}\cdots p_{j}^{a_{j}}$ be the prime factorization of $n$, then
a divisor $m$ of $n$ has the prime factorization $m=p_{1}^{b_{1}}\cdots
p_{j}^{b_{j}}$ where $0\leq b_{i}\leq a_{i}$. In particular, we may represent
a divisor of $n$ by a tuple $(b_{1},\cdots,b_{j})$ such that $0\leq b_{i}\leq
a_{i}$, and denote the quantities depending on a divisor accordingly. For
example, let $\mathbb{Z}_{(b_{1},\cdots,b_{j})}$ be another notation for
$\mathbb{Z}_{p_{1}^{b_{1}}\cdots p_{j}^{b_{j}}}$,
${\mathcal{B}}^{(b_{1},\cdots,b_{j})}$ denote the set of elements fixed under
the action of $\mathbb{Z}_{(b_{1},\cdots,b_{j})}$, and so on.

The number of elements of $H_{1}\left(\Sigma,\mathbb{Z}_{p_{1}^{b_{1}}\cdots
p_{j}^{b_{j}}}\right)$ that are not contained in any smaller subgroup is

$\displaystyle{\mathcal{C}}_{(b_{1},\cdots,b_{j})}$ $\displaystyle=$
$\displaystyle\left(\prod\limits_{q=1}^{j}p_{q}^{b_{q}}\right)^{2g}-\sum\limits_{r=1}^{j}\left(p_{r}^{b_{r}-1}\prod\limits_{q\neq
r}p_{q}^{b_{q}}\right)^{2g}+\sum\limits_{1\leq
r<s<j}\left(p_{r}^{b_{r}-1}p_{s}^{b_{s}-1}\prod\limits_{\begin{subarray}{c}q\neq
r\\\ q\neq s\end{subarray}}p_{q}^{b_{q}}\right)^{2g}$ (A.6)
$\displaystyle-\cdots+(-1)^{j-1}\left(\prod\limits_{q=1}^{j}p_{q}^{b_{q}-1}\right)^{2g}.$

This result follows from a use of the inclusion-exclusion principle: the first
term is the number of elements in
$H_{1}(\Sigma,\mathbb{Z}_{(b_{1},\cdots,b_{j})})$; the second term is the
number of elements in a subgroup
$H_{1}(\Sigma,\mathbb{Z}_{(b_{1},\cdots,b_{r}-1,\cdots,b_{j})})$; the third
term includes again the number of elements in the pairwise intersections of
$H_{1}(\Sigma,\mathbb{Z}_{(b_{1},\cdots,b_{r}-1,\cdots,b_{j})})$ and
$H_{1}(\Sigma,\mathbb{Z}_{(b_{1},\cdots,b_{q}-1,\cdots,b_{j})})$, and so on.

Using (A.6), we obtain the following form for the sum in (A.1),

$\displaystyle\sum\limits_{h\in
H_{1}(\Sigma,\mathbb{Z}_{n})}{\mathcal{Z}}_{h}=\sum\limits_{\begin{subarray}{c}(b_{1},\cdots,b_{j})\\\
0\leq b_{j}\leq
a_{j}\end{subarray}}{\mathcal{C}}_{(b_{1},\cdots,b_{j})}\sum\limits_{i\in{\mathcal{B}}^{(b_{1},\cdots,b_{j})}}N_{i}^{2g-2}.$
(A.7)

The coefficient of $\left(\prod\limits_{q=1}^{j}p_{q}^{b_{q}}\right)^{2g}$ in
this sum gets a contribution from a tuple $(c_{1},\cdots,c_{j})$ if for all
$i\in\\{1,\cdots,j\\}$, $c_{i}=b_{i}$ or $c_{i}=b_{i}+1$. In particular, the
coefficient of $\left(\prod\limits_{q=1}^{j}p_{q}^{b_{q}}\right)^{2g}$ is

$\displaystyle\sum\limits_{i\in{\mathcal{B}}^{(b_{1},\cdots,b_{j})}}N_{i}^{2g-2}$
$\displaystyle-\sum\limits_{r=1}^{j}\sum\limits_{i\in{\mathcal{B}}^{(b_{1},\cdots,b_{r}+1,\cdots,b_{j})}}N_{i}^{2g-2}+\sum\limits_{1\leq
r<s\leq
j}\sum\limits_{i\in{\mathcal{B}}^{(b_{1},\cdots,b_{r}+1,\cdots,b_{s+1},\cdots,b_{j})}}N_{i}^{2g-2}$
(A.8)
$\displaystyle-\cdots+(-1)^{j-1}\sum\limits_{i\in{\mathcal{B}}^{(b_{1}+1,\cdots,b_{r}+1,\cdots,b_{s+1},\cdots,b_{j}+1)}}N_{i}^{2g-2}.$

This expression is equal to the sum of $N_{i}^{2g-2}$ over the elements of
${\mathcal{B}}$ that lie in a $\mathbb{Z}_{n}$ orbit of length
$\prod\limits_{q=1}^{j}p_{q}^{a_{q}-b_{q}}$.222222This can be seen as follows.
The first sum in (A.8) is the sum over elements fixed under
$\mathbb{Z}_{p_{1}^{b_{1}}\cdots p_{j}^{b_{j}}}$. Such an element lies in a
$\mathbb{Z}_{n}$ orbit of length $p_{1}^{x_{1}}\cdots p_{j}^{x_{j}}$ where
$x_{j}\leq a_{j}-b_{j}$. Using inclusion-exclusion principle, we can remove
the set of elements with $x_{j}<a_{j}-b_{j}$. This is precisely what the rest
of the terms in (A.8) accomplish. Let
$\text{Orb}_{(b_{1},\cdots,b_{j})}({\mathcal{B}})$ be the set of orbits of
${\mathcal{B}}$ of this length, then the expression (A.8) is equal to

$\displaystyle\prod\limits_{q=1}^{j}p_{q}^{a_{q}-b_{q}}\sum\limits_{i\in\text{Orb}_{(b_{1},\cdots,b_{j})}}N_{i}^{2g-2}$
(A.9)

Plugging this in equation (A.1), we get

$\displaystyle{\mathcal{Z}}_{{\mathcal{T}}/\mathbb{Z}_{n}^{(0)}}$
$\displaystyle=$
$\displaystyle\frac{1}{n^{g}}\left(\sum\limits_{\begin{subarray}{c}(b_{1},\cdots,b_{j})\\\
0\leq b_{j}\leq
a_{j}\end{subarray}}\left(\prod\limits_{q=1}^{j}p_{q}^{b_{q}}\right)^{2g}\prod\limits_{q=1}^{j}p_{q}^{a_{q}-b_{q}}\sum\limits_{i\in\text{Orb}_{(b_{1},\cdots,b_{j})}}N_{i}^{2g-2}\right)$
(A.10) $\displaystyle=$
$\displaystyle\sum\limits_{\begin{subarray}{c}(b_{1},\cdots,b_{j})\\\ 0\leq
b_{j}\leq
a_{j}\end{subarray}}\left(\prod\limits_{q=1}^{j}p_{q}^{b_{q}}\right)^{2g-1}\sum\limits_{i\in\text{Orb}_{(b_{1},\cdots,b_{j})}(B)}\left(\frac{N_{i}}{|\mathbb{Z}_{n}|^{\frac{1}{2}}}\right)^{2g-2}$
$\displaystyle=$
$\displaystyle\sum\limits_{\begin{subarray}{c}(b_{1},\cdots,b_{j})\\\ 0\leq
b_{j}\leq
a_{j}\end{subarray}}\sum\limits_{i\in\text{Orb}_{(b_{1},\cdots,b_{j})}(B)}\left(\prod\limits_{q=1}^{j}p_{q}^{b_{q}}\right)^{2g-1}\left(\frac{N_{i}}{|\mathbb{Z}_{n}|^{\frac{1}{2}}}\right)^{2g-2}$
$\displaystyle=$
$\displaystyle\sum\limits_{i\in{\mathcal{B}}/\mathbb{Z}_{n}}|\text{Stab}(i)|\left(\frac{|\text{Stab}(i)|}{|\mathbb{Z}_{n}|^{\frac{1}{2}}}N_{i}\right)^{2g-2}$

This completes the derivation of equation (A.3) for $n$ composite.

As an example of this general discussion, consider the case $n=12=2^{2}\times
3$, i.e. $p_{1}=2$, $a_{1}=2$, $p_{2}=3$, and $a_{2}=1$. 12 has six divisors,
i.e. 1, 2, 3, 4, 6, and 12. For each divisor $m$, the number of elements in
$H_{1}(\Sigma,\mathbb{Z}_{m})$ that are not contained in a smaller subgroup is
(cf. Equation (A.6))

$\displaystyle{\mathcal{C}}_{(0,0)}$ $\displaystyle=$ $\displaystyle 1$
$\displaystyle{\mathcal{C}}_{(1,0)}$ $\displaystyle=$ $\displaystyle 2^{2g}-1$
$\displaystyle{\mathcal{C}}_{(0,1)}$ $\displaystyle=$ $\displaystyle 3^{2g}-1$
$\displaystyle{\mathcal{C}}_{(2.0)}$ $\displaystyle=$ $\displaystyle
4^{2g}-2^{2g}$ $\displaystyle{\mathcal{C}}_{(1,1)}$ $\displaystyle=$
$\displaystyle 6^{2g}-3^{2g}-2^{2g}+1$ $\displaystyle{\mathcal{C}}_{(2,1)}$
$\displaystyle=$ $\displaystyle 12^{2g}-6^{2g}-4^{2g}+2^{2g}$

Plugging these numbers in (A.7), we obtain that the coefficient of $1$ is

$\displaystyle\sum\limits_{i\in
B}N_{i}^{2g-2}-\sum\limits_{i\in{\mathcal{B}}^{\mathbb{Z}_{2}}}N_{i}^{2g-2}-\sum\limits_{i\in{\mathcal{B}}^{\mathbb{Z}_{3}}}N_{i}^{2g-2}+\sum\limits_{i\in{\mathcal{B}}^{\mathbb{Z}_{6}}}N_{i}^{2g-2}.$
(A.11)

which is precisely the sum of $N_{i}^{2g-2}$ over the set of elements in
orbits of length 12. Similarly, the coefficient of $2^{2g}$ is

$\displaystyle\sum\limits_{i\in{\mathcal{B}}^{\mathbb{Z}_{2}}}N_{i}^{2g-2}-\sum\limits_{i\in{\mathcal{B}}^{\mathbb{Z}_{4}}}N_{i}^{2g-2}-\sum\limits_{i\in{\mathcal{B}}^{\mathbb{Z}_{6}}}N_{i}^{2g-2}+\sum\limits_{i\in{\mathcal{B}}^{\mathbb{Z}_{12}}}N_{i}^{2g-2}.$
(A.12)

which is the sum over elements in orbits of length 6. After making similar
computations for coefficients of $3^{2g}$, $4^{2g}$, $6^{2g}$ and $12^{2g}$ it
is straightforward to derive that the partition function of
${\mathcal{T}}/\mathbb{Z}_{12}^{(0)}$ is

$\displaystyle{\mathcal{Z}}_{{\mathcal{T}}/\mathbb{Z}_{12}^{(0)}}=\sum\limits_{i\in{\mathcal{B}}/\mathbb{Z}_{12}}|\text{Stab}(i)|\left(\frac{|\text{Stab}(i)|}{|\mathbb{Z}_{12}|^{\frac{1}{2}}}N_{i}\right)^{2g-2}$
(A.13)

###  A.3 The case of product of cyclic groups

Finally, we will discuss the case of gauging (a subgroup of) the 0-form
symmetry of a TQFT that is a product of cyclic groups. Here, we may gauge one
factor of the product at a time. At each step, the partition function can be
computed using the techniques presented in the previous two subsections. A
special case of interest that has appeared implicitly in this paper is that of
${\mathcal{T}}(Spin(4n,\mathbb{C}))$. The center of $Spin(4n,\mathbb{C})$ is
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$. As this subgroup of the 0-form symmetry
is gauged, a natural generalization of the results (A.2) and (A.3) holds.

In general, for a TQFT ${\mathcal{T}}$ with 0-form symmetry
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$, ${\mathcal{Z}}_{h}$ for $h\in
H_{1}(\Sigma,\mathbb{Z}_{2}\times\mathbb{Z}_{2})$ is given by

$\displaystyle{\mathcal{Z}}_{h}=\sum\limits_{\lambda\in
P_{k}^{\widetilde{Z}}}(\theta_{t}(f_{\lambda,t}))^{1-g}$ (A.14)

Here $\widetilde{Z}\subset\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ is the smallest
subgroup such that $H_{1}(\Sigma,\widetilde{Z})$ contains $h$. Furthermore,

$\displaystyle{\mathcal{Z}}_{{\mathcal{T}}/(\mathbb{Z}_{2}\times\mathbb{Z}_{2})^{(0)}}$
$\displaystyle=$
$\displaystyle\sum\limits_{i\in{\mathcal{B}}/(\mathbb{Z}_{2}\times\mathbb{Z}_{2})}|\text{Stab}(i)|\left(\frac{|\text{Stab}(i)|}{|\mathbb{Z}_{2}\times\mathbb{Z}_{2}|^{\frac{1}{2}}}N_{i}\right)^{2g-2}.$
(A.15)

This result follows easily by gauging the two $\mathbb{Z}_{2}$ factors one
after another, and using the result (A.3). The consistency of (A.14) with
(A.15) follows from the equation

$\displaystyle{\mathcal{Z}}_{{\mathcal{T}}/(\mathbb{Z}_{2}\times\mathbb{Z}_{2})^{(0)}}=\frac{1}{|H_{1}(\Sigma,\mathbb{Z}_{2})|^{\frac{1}{2}}}\sum\limits_{h}{\mathcal{Z}}_{h}.$
(A.16)

##  Appendix B Fusion rules of $\mathcal{T}(PSL(2,\mathbb{C})_{k})$

In this appendix we will derive the fusion rules of
${\mathcal{T}}(PSL(2,\mathbb{C})_{k})$ given in Section 4. In subsection B.1
the fusion rules of ${\mathcal{T}}(SL(2,\mathbb{C})_{k})/\mathbb{Z}_{2}^{(1)}$
are derived, while in subsection B.2 a derivation of the fusion rules of
${\mathcal{T}}(PSL(2,\mathbb{C})_{k})={\mathcal{T}}(SL(2,\mathbb{C})_{k})/(\mathbb{Z}_{2}^{(0)}\times\mathbb{Z}_{2}^{(1)})$
is given.

###  B.1 Fusion rules of
$\mathcal{T}(SL(2,\mathbb{C})_{k})/\mathbb{Z}_{2}^{(1)}$

The operators of ${\mathcal{T}}(SL(2,\mathbb{C})_{k})/\mathbb{Z}_{2}^{(1)}$
are

$\displaystyle x^{j}=\frac{1}{2}(w^{j}+w^{k-j})$ (B.1)

for $j=0,1,\cdots,\frac{k}{2}$. The fusion rules are derived as follows.

$\displaystyle x^{a}\times x^{b}$ $\displaystyle=$
$\displaystyle\frac{1}{4}(w^{a}+w^{k-a})\times(w^{b}+w^{k-b})$ (B.2)
$\displaystyle=$
$\displaystyle\frac{1}{4}\sum\limits_{c=0}^{k}\left(f^{ab}{}_{c}w^{c}+f^{a(k-b)}{}_{c}w^{c}+f^{(k-a)b}{}_{c}+f^{(k-a)(k-b)}{}_{c}w^{c}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{4}\sum\limits_{c=0}^{k}\left(f^{ab}{}_{c}w^{c}+f^{a(k-b)}{}_{k-c}w^{k-c}+f^{(k-a)b}{}_{c}+f^{(k-a)(k-b)}{}_{k-c}w^{k-c}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{4}\sum\limits_{c=0}^{k}(f^{ab}{}_{c}+f^{(k-a)b}{}_{c})(w^{c}+w^{k-c})$
$\displaystyle=$
$\displaystyle\sum\limits_{c=0}^{\frac{k}{2}-1}(f^{ab}{}_{c}+f^{(k-a)b}{}_{c})x^{c}+f^{ab}{}_{\frac{k}{2}}x^{\frac{k}{2}}$

Here in the fourth equality, we used an identity
$f^{ab}{}_{c}=f^{a(k-b)}{}_{k-c}$ which follows from the action of
$\mathbb{Z}_{2}^{(1)}$ symmetry.232323Let $\psi$ be the generator of 1-form
symmetry, i.e. $\psi\times\psi=\mathbb{1}$ and $\psi\times w^{a}=w^{k-a}$.
Then two different ways of degenerating the five-punctured sphere $f^{\psi\psi
ab}{}_{c}$ gives the required identity $f^{ab}{}_{c}=f^{a(k-b)}{}_{k-c}$.  It
follows from (B.2) that the fusion coefficients of
$\mathcal{T}(SL(2,\mathbb{C})_{k})/\mathbb{Z}_{2}^{(1)}$ are

$\displaystyle(f^{\prime})^{ab}{}_{c}$ $\displaystyle=$ $\displaystyle
f^{ab}{}_{c}+f^{(k-a)b}{}_{c}$ (B.3) $\displaystyle(f^{\prime})^{ab}{}_{\mu}$
$\displaystyle=$ $\displaystyle f^{ab}{}_{\frac{k}{2}}.$ (B.4)

###  B.2 Further gauging the $\mathbb{Z}_{2}^{(0)}$ symmetry

Now we consider gauging the $\mathbb{Z}_{2}^{(0)}$ symmetry of
$\mathcal{T}(SL(2,\mathbb{C})_{k})/\mathbb{Z}_{2}^{(1)}$. The operators of
this theory are

$\displaystyle
x^{0},x^{2},\cdots,x^{\frac{k}{2}-2},x^{{\frac{k}{2}}^{(1)}},x^{{\frac{k}{2}}^{(2)}}$
(B.5)

where $x^{{\frac{k}{2}}^{(1)}}+x^{{\frac{k}{2}}^{(2)}}$ equals the
$x^{\frac{k}{2}}$ operator of
$\mathcal{T}(SL(2,\mathbb{C})_{k})/\mathbb{Z}_{2}^{(1)}$. The fusion rules are
of six different kinds:

$\displaystyle\widetilde{f}^{ab}{}_{c},\quad\widetilde{f}^{ab}{}_{\mu},\quad\widetilde{f}^{a\mu}{}_{b},\quad\widetilde{f}^{a\mu}{}_{\nu},\quad\widetilde{f}^{\mu\nu}{}_{a},\quad\widetilde{f}^{\mu\nu}{}_{\rho}.$
(B.6)

where $a,b,c\in\\{0,2,\cdots,\frac{k}{2}-2\\}$, and
$\mu,\nu,\rho\in\\{\frac{k}{2}^{(1)}$ and $\frac{k}{2}^{{(2)}}\\}$. Rewriting
(B.2) as (when $a$ and $b$ are even, the first sum gets contribution only from
even values of $c$)

$\displaystyle x^{a}\times x^{b}$ $\displaystyle=$
$\displaystyle\sum\limits_{\begin{subarray}{c}c=0\\\ c\
\text{even}\end{subarray}}^{\frac{k}{2}-2}(f^{ab}{}_{c}+f^{(k-a)b}{}_{c})x^{c}+f^{ab}{}_{\frac{k}{2}}(x^{\frac{k}{2}^{(1)}}+x^{\frac{k}{2}^{(2)}})$
(B.7)

we obtain

$\displaystyle\widetilde{f}^{ab}{}_{c}$ $\displaystyle=$ $\displaystyle
f^{ab}{}_{c}+f^{(k-a)b}{}_{c}$ (B.8) $\displaystyle\widetilde{f}^{ab}{}_{\mu}$
$\displaystyle=$ $\displaystyle f^{ab}{}_{\frac{k}{2}}$ (B.9)

The next two fusion coefficients in (B.5) encode the fusion of $x^{a}$ with
$x^{\mu}$, but since $\sum\limits_{\mu}x^{\mu}=x^{\frac{k}{2}}$, it is useful
to sum over $\mu$ in order to find relations with the fusion rules of
${\mathcal{T}}(SL(2,\mathbb{C})_{k})$:

$\displaystyle\sum\limits_{\mu}(x^{a}\times
x^{\mu})=\frac{1}{2}(w^{a}+w^{k-a})\times w^{k/2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum\limits_{\begin{subarray}{c}b=0\\\ b\
even\end{subarray}}^{k}\left(f^{a\frac{k}{2}}{}_{b}w^{b}+f^{(k-a)\frac{k}{2}}{}_{k-b}w^{k-b}\right)$
(B.10) $\displaystyle=$
$\displaystyle\frac{1}{2}\sum\limits_{\begin{subarray}{c}b=0\\\ b\
even\end{subarray}}^{k}f^{a\frac{k}{2}}{}_{b}(w^{b}+w^{k-b})$ $\displaystyle=$
$\displaystyle\sum\limits_{\begin{subarray}{c}b=0\\\ b\
even\end{subarray}}^{\frac{k}{2}-2}f^{a\frac{k}{2}}{}_{b}(w^{b}+w^{k-b})+f^{a\frac{k}{2}}{}_{\frac{k}{2}}w^{\frac{k}{2}}$
$\displaystyle=$ $\displaystyle\sum\limits_{\begin{subarray}{c}b=0\\\ b\