SISSA 05/2022/FISI

Factorization and Global Symmetries
in Holography

Francesco Benini1,2,3, Christian Copetti1,2 and Lorenzo Di Pietro2,4

1 SISSA, Via Bonomea 265, 34136 Trieste, Italy
2 INFN, Sezione di Trieste, Via Valerio 2, 34127 Trieste, Italy
3 ICTP, Strada Costiera 11, 34151 Trieste, Italy
4 Dipartimento di Fisica, Università di Trieste, Strada Costiera 11, 34151
Trieste, Italy

We consider toy models of holography arising from 3d Chern-Simons theory. In
this context a duality to an ensemble average over 2d CFTs has been recently
proposed. We put forward an alternative approach in which, rather than summing
over bulk geometries, one gauges a one-form global symmetry of the bulk
theory. This accomplishes two tasks: it ensures that the bulk theory has no
global symmetries, as expected for a theory of quantum gravity, and it makes
the partition function on spacetimes with boundaries coincide with that of a
modular-invariant 2d CFT on the boundary. In particular, on wormhole
geometries one finds a factorized answer for the partition function. In the
case of non-Abelian Chern-Simons theories, the relevant one-form symmetry is
non-invertible, and its “gauging” corresponds to the condensation of a
Lagrangian anyon.

###### Contents

  1. 1 Introduction and summary
  2. 2 The free compact scalar
    1. 2.1 2d CFT
    2. 2.2 $U(1)_{k}\times U(1)_{-k}$ Chern-Simons theory
    3. 2.3 Gauging a $\mathbb{Z}_{k}$ one-form symmetry
    4. 2.4 Partition functions and factorization
    5. 2.5 Correlators
    6. 2.6 Comments on symmetries
  3. 3 Abelian generalizations
    1. 3.1 Multi-component compact scalar
    2. 3.2 More general Lagrangian subgroups
    3. 3.3 Non-rational theories
  4. 4 Non-Abelian case
    1. 4.1 Review of anyon condensation in modular tensor categories
    2. 4.2 Partition functions and factorization
    3. 4.3 Correlators
    4. 4.4 Examples
  5. 5 Discussion
  6. A Higher-genus partition functions
  7. B Boundary symmetry and contact term
  8. C Gauging formulas with boundary
  9. D Factorization for higher genus

##  1 Introduction and summary

According to the holographic principle [1, 2] and the AdS/CFT correspondence
[3, 4, 5], a theory of quantum gravity in asymptotically anti-de-Sitter (AdS)
spacetime is dual to a conformal field theory (CFT) placed on the conformal
boundary of that spacetime. Various examples of low-dimensional gravities,
however, appeared in the literature in the last few years, which are dual not
to a conventional unitary quantum system, but rather to an average (with
respect to some measure) over a family of quantum systems. A smooking gun for
that is the lack of factorization of the partition function in the presence of
disconnected boundaries. If the spacetime on which the dual boundary theory
lives is of the form $X=X_{1}\sqcup\ldots\sqcup X_{n}$ (a disconnected sum),
the lack of factorization amounts to

$\mathcal{Z}_{\text{grav}}[X]\neq\mathcal{Z}_{\text{grav}}[X_{1}]\cdots\mathcal{Z}_{\text{grav}}[X_{n}]\;,$
(1.1)

where $\mathcal{Z}_{\text{grav}}[X]$ is the gravitational partition function
with fixed boundary conditions $X$. This can be associated to the presence of
non-trivial (Euclidean) spacetime wormholes connecting different boundaries.

One example [6, 7, 8, 9] is provided by two-dimensional Jackiw-Teitelboim (JT)
dilaton gravity [10, 11], dual to the Sachdev-Ye-Kitaev (SYK) model [12, 13],
namely a family of quantum mechanical systems with a large number of fermions,
averaged over the couplings among those fermions. A complete calculation of
the path integral on the gravity side was possible, confirming the duality to
an ensemble average [14]. A different example [15, 16] is provided by a three-
dimensional Euclidean gravity defined as a sum over hyperbolic geometries, in
which the small fluctuations around each geometry are described by an Abelian
Chern-Simons (CS) theory. Its dual is the family of free conformal field
theories of $D$ compact bosons, averaged over their Narain moduli space [17].
Variations of this example have been considered, e.g., in [18, 19, 20, 21, 22,
23]. The non-factorized contribution due to off-shell wormholes in 3d gravity
was studied in [24].

In view of these examples, it is natural to wonder whether there exist two
types of holographic correspondences — the standard one and the averaged one —
and thus of gravitational theories. Since an average over unitary theories is
not a unitary quantum system, at least not in the traditional sense, one asks
whether the corresponding gravitational theories suffer from some pathology
that prevents them from being UV complete.111In the case of JT gravity, it has
recently been noticed [25] that a certain non-local deformation can resolve
the lack of factorization and lead to well-defined quantum mechanical systems
with discrete spectrum. Various approaches to this question have been taken,
e.g., in [26, 27, 28, 29, 30, 31, 32].

Within the class of solvable 3d gravitational theories defined in terms of CS
theories with compact gauge group, we propose an answer. We notice that those
theories possess a global symmetry, generated by the topological line
operators in the theory [33]. Abelian CS theories have a conventional 1-form
symmetry, while in the non-Abelian case the full 1-form symmetry is non-
invertible [34].222Besides, in both cases there usually are 0-form symmetries
as well. This is in tension with the expectation that quantum gravity should
not have any global symmetry [35, 36, 37, 38, 39]. Thus, we propose a
connection between the presence of global symmetries in the gravitational
theory, and averaging in the boundary theory.333A connection between averaging
in the boundary theory and the presence of global symmetries in the bulk
theory (with a focus on 2d gravity) has also been proposed in [40] from a
different point of view.

Indeed, a simple way to remove the global symmetry is to gauge a suitable non-
anomalous subgroup thereof. In the Abelian case this is a standard gauging of
a discrete 1-form symmetry, that may be implemented by coupling the theory to
dynamical gauge fields, while in the non-Abelian case one needs to resort to
anyon condensation [41, 42] — a sort of analog of gauging for non-invertible
symmetries. In both cases, after gauging, the bulk Chern-Simons theory
acquires the following pleasant properties:

  1. 1)

The theory becomes trivial in the bulk, in the sense that the Euclidean
partition function on any (oriented) closed 3-manifold equals 1. After all,
this is what we expect from a holographic theory: the degrees of freedom only
live at the boundary.

  2. 2)

Given a (possibly disconnected) 2d boundary and a boundary condition, the
Euclidean partition function on any 3-manifold with that boundary gives the
same result (this follows from point 1). Chern-Simons theory is a generally-
covariant theory [43, 44], in the sense that its partition function on closed
3-manifolds does not depend on a choice of metric — but it does depend on the
topology. After gauging, the theory becomes independent from the bulk topology
as well.

  3. 3)

The partition function of the gravitational theory is defined as the one on an
arbitrary 3-manifold with the given boundary conditions (not as a sum over all
possible geometries, or some subset thereof), because of point 2).
Factorization in the case of disconnected boundaries immediately follows.

  4. 4)

The partition function with boundary conditions equals the partition function
of a single and well-defined boundary CFT. The details of such a CFT are
encoded in the bulk gauge group and in the specific chosen gauging of the
1-form symmetry.

  5. 5)

What we described extends to correlation functions of the boundary theory, and
hinges on the fact that all lines are transparent in the bulk (because of
point 1).

The properties 2) and 3) have already been observed in a much more complicated
example, namely string theory on AdS${}_{3}\times S^{3}\times T^{4}$ in the
tensionless limit, after performing the full non-perturbative path integral
[45, 46]. A property analogous to 3) was also observed in [47] for the SYK
model reduced to one dimension [48]. The above observations clarify the
relationship between averaging and global symmetries, at least in the class of
models under consideration. The bulk theory has a (possibly non-invertible)
global 1-form symmetry if and only if it has non-trivial topological line
operators. In that case, the partition function with fixed boundary conditions
depends on the topology of the chosen bulk 3-manifold, and it is natural to
define the gravitational theory as a sum over those topologies. This leads to
a lack of factorization, i.e., to averaging over a family of boundary CFTs.
This can be rephrased in the language of Coleman’s $\alpha$-states [49] (see
also [50, 51]) with each $\alpha$-state being the dual to a specific CFT. In
our class of models, each CFT is built from the 2d left- and right-moving
chiral algebras dual [44] to the bulk CS theory before gauging.444This was
conjectured in [18, 21] from the point of view of the boundary CFT.

After gauging, we obtain a bulk theory which is holographically dual — in the
standard sense — to a specific 2d rational conformal field theory (RCFT) at a
finite value of the central charge, defined in terms of its holomorphic and
anti-holomorphic halves and a prescription for gluing them. The relation
between 3d topological quantum field theories (TQFTs) and the holomorphic half
of 2d RCFTs has been known for a long time [44, 52, 53, 54, 55, 56]. Besides,
the formulation of modular-invariant parings between the holomorphic and anti-
holomorphic sectors of a CFT in the language of TQFT, in particular in terms
of Frobenius algebra objects (that we review in Section 4.1), has also been
understood [57]. The latter has been more physically interpreted in terms of
topological interfaces and boundary conditions in TQFT [58, 59] and in terms
of anyon condensation [60].555In the context of 2d gravity, it has been
understood that $\alpha$-eigenstates can be related to certain types of brane
insertions [61, 62, 63].

In this paper we elaborate on the interpretation in terms of gauging, showing
that it makes more transparent the construction of generally-covariant and
topology-independent theories — i.e., of gravitational theories — in terms of
CS theory.666Ref. [64] described a different approach to promote the
correspondence between Chern-Simons theory and the holomorphic half of a RCFT,
to a holographic duality to a full CFT. As we already mentioned, exploiting
the categorical formulation (in terms of Frobenius algebras) of the notion of
gauging of (possibly higher-form) symmetries and its non-invertible analog,
anyon condensation, is very useful in order to treat non-Abelian Chern-Simons
theories. See [65, 66, 67, 68, 69, 70, 71] for interesting recent works that
explore the gauging of non-invertible symmetries.

The paper is organized as follows. In Section 2 we describe the main ideas in
the simplest example: a free compact scalar at rational radius. This serves as
a gentle introduction to most of the relevant concepts. It is shown explicitly
how to recover a well-defined dual RCFT by gauging a maximal (Lagrangian)
subgroup of the one-form symmetry in the bulk, and how this gauging factorizes
wormhole contributions automatically. In Section 3 we discuss the extension to
more general Abelian theories with $U(1)^{D}$ chiral algebras. In Section 4 we
discuss the extension to the non-Abelian case, for which we implement the
machinery of anyon condensation. We describe how the relevant concepts on
discrete gauging naturally extend to this case, by introducing the
condensation of commuting Lagrangian anyons [42]. The properties of such
objects are then used explicitly to show factorization of wormhole geometries
and the projection to a well-defined RCFT. We conclude in Section 5 with
implications and possible extensions of our results. Technical parts are
collected in appendices.

##  2 The free compact scalar

The simplest example in which we can exhibit the main physical ideas of this
paper is that of the 2d CFT of a free compact real scalar field. We will first
review a few facts about such a CFT, and then move on to its holographic bulk
description.

###  2.1 2d CFT

We consider a free massless compact real boson $\varphi$, with action777We
follow the notation of [72] with $g=1/4\pi$.

$S=\frac{1}{8\pi}\int
d^{2}\sigma\,\partial_{\mu}\varphi\,\partial^{\mu}\varphi\;.$ (2.1)

The theory enjoys a $U(1)$ symmetry that shifts $\varphi$, which is enhanced
to $\mathfrak{u}(1)\times\mathfrak{u}(1)$ current algebra by the holomorphic
currents $\partial\varphi$ and $\bar{\partial}\varphi$. We identify
$\varphi\cong\varphi+2\pi R$ (where $R$ is dimensionless). The Euclidean torus
partition function can be written as

$Z(R,\tau)=\frac{\Theta(R,\tau)}{|\eta(\tau)|^{2}}\;,$ (2.2)

where $\tau=\tau_{1}+i\tau_{2}$ is the modular parameter of the torus, one
defines $q=e^{2\pi i\tau}$ as usual, $\eta(\tau)$ is the Dedekind eta
function, and

$\Theta(R,\tau)=\sum_{n,w\,\in\,\mathbb{Z}}\exp\biggl{[}-2\pi\tau_{2}\biggl{(}\frac{n^{2}}{R^{2}}+\frac{w^{2}R^{2}}{4}\biggr{)}+2\pi
i\tau_{1}nw\biggr{]}=\sum_{n,w\,\in\,\mathbb{Z}}q^{\frac{1}{2}\left(\frac{n}{R}+\frac{wR}{2}\right)^{2}}\bar{q}^{\frac{1}{2}\left(\frac{n}{R}-\frac{wR}{2}\right)^{2}}$
(2.3)

is the Siegel-Narain theta function. The integers $n,w$ are the electric
charge (or momentum) and the magnetic charge (or vorticity), respectively,
under the $U(1)$ symmetry. From the partition function one reads off the
dimension $\Delta$ and spin $s$ of the primary operators
$e^{in\varphi/R}e^{iwR\tilde{\varphi}/2}$ (where $\tilde{\varphi}$ is the dual
field such that
$\partial_{\mu}\tilde{\varphi}=\epsilon_{\mu\nu}\partial^{\nu}\varphi$) of the
current algebra:

$\Delta_{n,w}=\frac{n^{2}}{R^{2}}+\frac{w^{2}R^{2}}{4}\;,\qquad s_{n,w}=nw\;.$
(2.4)

Electric/magnetic T-duality acts as $R\leftrightarrow 2/R$. The left- and
right-moving dimensions of primary operators are

$h_{n,w}=\frac{1}{2}\biggl{(}\frac{n}{R}+\frac{wR}{2}\biggr{)}^{2}\;,\qquad\qquad\bar{h}_{n,w}=\frac{1}{2}\biggl{(}\frac{n}{R}-\frac{wR}{2}\biggr{)}^{2}\;.$
(2.5)

The theory has an infinite number of primaries, whose dimensions, for generic
$R$, are real numbers. However when $R^{2}\in\mathbb{Q}$, the theory is a
rational CFT: the dimensions are rational numbers, the chiral algebra is
enhanced, and the primaries organize into a finite number of modules under the
extended chiral algebra. The left-moving part of the extended chiral algebra
is given by the fields with $\bar{h}_{n,w}=0$, in other words one has to solve
$n=\frac{R^{2}}{2}w$ for $n,w\in\mathbb{Z}$. Therefore one sets

$\frac{R^{2}}{2}=\frac{p^{\prime}}{p}\qquad\text{with
}p^{\prime},p\in\mathbb{N}\text{ coprime}\;.$ (2.6)

The solutions to $\bar{h}_{n,w}=0$ are $n=p^{\prime}\ell$, $w=p\ell$ with
$\ell\in\mathbb{Z}$, and yield the spectrum
$h=\frac{2p^{\prime}p}{2}\ell^{2}$. This corresponds to the chiral algebra
(see, e.g., [73]):

$\mathfrak{u}(1)_{k}\qquad\text{with}\qquad k=2p^{\prime}p\;.$ (2.7)

Since $k\in 2\mathbb{N}$, this is a bosonic chiral algebra (all chiral fields
have integer dimension).

The characters of $\mathfrak{u}(1)_{k}$ are

$K_{\lambda}^{(k)}(\tau)\equiv\operatorname{Tr}q^{L_{0}-\frac{c}{24}}=\frac{1}{\eta(\tau)}\sum_{n\in\mathbb{Z}}q^{\frac{(kn+\lambda)^{2}}{2k}}$
(2.8)

for $\lambda=0,\ldots,k-1$ (defined modulo $k$).888The characters for
$\lambda$ and $-\lambda$ (corresponding to charge conjugate representations)
are identical, however they could be distinguished by considering the refined
characters with a fugacity for the $U(1)$ symmetry. For the sake of
simplicity, we will not do that here. The modular transformations of the
characters can be written as

$K_{\lambda}^{(k)}(\gamma\cdot\tau)=\sum_{\mu=0}^{k-1}M_{\lambda\mu}^{(\gamma)}\;K_{\mu}^{(k)}(\tau)\;,$
(2.9)

where $\gamma\in SL(2,\mathbb{Z})$ and $\gamma\cdot\tau$ is its action on the
modular parameter $\tau$. One finds

$\displaystyle T:\tau$ $\displaystyle\mapsto\tau+1\;,\qquad\qquad$
$\displaystyle T_{\lambda\mu}$ $\displaystyle=e^{2\pi
i\left[\frac{\lambda^{2}}{2k}-\frac{1}{24}\right]}\,\delta_{\lambda\mu}\;,$
(2.10) $\displaystyle S:\tau$
$\displaystyle\mapsto-\frac{1}{\tau}\;,\qquad\qquad$ $\displaystyle
S_{\lambda\mu}$ $\displaystyle=\frac{1}{\sqrt{k}}\,e^{-2\pi
i\frac{\lambda\mu}{k}}\;.$

They satisfy $S^{2}=C$ (where $C:\lambda\mapsto-\lambda$ is charge
conjugation), $SS^{\dagger}=S^{4}=\mathbbm{1}$, and $(ST)^{3}=C$.

The torus partition function can be written in terms of characters, by
correctly combining left- and right-moving fields into physical fields. In
order to construct the pairing of primaries, one finds integers $r_{0},s_{0}$
such that

$pr_{0}-p^{\prime}s_{0}=1$ (2.11)

and then defines

$\omega=pr_{0}+p^{\prime}s_{0}\mod k\;.$ (2.12)

Since $k\in 2\mathbb{N}$, it follows that $\omega^{2}=1\text{ mod }2k$.
Therefore the map

$\omega:\lambda\mapsto\omega\lambda\qquad\qquad\text{for
}\lambda\in\mathbb{Z}_{k}$ (2.13)

provides an involution on the set of integrable representations of the chiral
algebra. This involution is an outer automorphism of the algebra that
preserves the chiral dimension $h(\lambda)=\frac{\lambda^{2}}{2k}$ modulo 1,
and therefore the $S$-matrix. With some algebra, one shows that999Given the
$SL(2,\mathbb{Z})$ matrix $S=\big{(}\begin{smallmatrix}p&p^{\prime}\\\
s_{0}&r_{0}\end{smallmatrix}\big{)}$, one performs the redefinition
$\big{(}\begin{smallmatrix}N\\\
-\bar{\ell}\end{smallmatrix}\big{)}=S\big{(}\begin{smallmatrix}n\\\
w\end{smallmatrix}\big{)}$. The inverse transformation is
$S^{-1}=\big{(}\begin{smallmatrix}r_{0}&-p^{\prime}\\\
-s_{0}&p\end{smallmatrix}\big{)}$. Then (2.2) takes the form
$Z=\frac{1}{|\eta|^{2}}\sum_{N,\bar{\ell}\in\mathbb{Z}}q^{\frac{1}{2k}\left(\rule{0.0pt}{4.20004pt}N\right)^{2}}\bar{q}^{\frac{1}{2k}\left(\rule{0.0pt}{4.20004pt}k\bar{\ell}+\omega
N\right)^{2}}$. Now rewrite $\sum_{N\in\mathbb{Z}}$ as
$\sum_{\lambda=0}^{k-1}\sum_{\ell\in\mathbb{Z}}$ setting $N=k\ell+\lambda$.
Shifting $\bar{\ell}\to\bar{\ell}-\omega\ell$, one obtains (2.14).

$Z(R,\tau)=\sum_{\lambda=0}^{k-1}K_{\lambda}^{(k)}(\tau)\;\overline{K_{\omega\lambda}^{(k)}(\tau)}\;.$
(2.14)

In particular, the pairing of representations $(\lambda,\omega\lambda)$
guarantees that all physical fields have integer spin, because the left and
right dimensions agree modulo 1. Notice that for $R^{2}\in 2\mathbb{N}$ one
gets $k=R^{2}$ and $\omega=1$, which yields a diagonal partition function. In
the dual case $R^{2}=\frac{4}{k}$ with $k\in 2\mathbb{N}$, one gets
$k=4/R^{2}$ and $\omega=-1$: in this case the partition function is non-
diagonal, and the pairing of representations is through charge conjugation
$\lambda\leftrightarrow k-\lambda$.

The partition function (2.14) is modular invariant. This is guaranteed by the
expressions of $T,S$ in (2.10), as long as multiplication by $\omega$ is a
group automorphism of $\mathbb{Z}_{k}$ that preserves the quadratic function

$q(\lambda)=\frac{\lambda^{2}}{2k}\mod 1$ (2.15)

(the chiral dimension mod 1). This is equivalent to $\omega^{2}=1\text{ mod
}2k$. In that case, the symmetric bilinear form
$q(\lambda+\mu)-q(\lambda)-q(\mu)=\frac{\lambda\mu}{k}$ mod 1 is preserved as
well.

The equations (2.6), (2.7), (2.11) and (2.12) provide a map from
$R^{2}\in\mathbb{Q}$ to the pair $(k,\omega)$. The inverse map (assuming that
$\omega$, which is an element of the multiplicative group $\mathbb{Z}_{k}^{*}$
of integers mod $k$, preserves (2.15)) can be constructed as follows. If we
restrict to momentum modes (i.e., to modes with equal left and right charge),
then the minimal dimension is $h=\bar{h}=\frac{1}{2R^{2}}$. Momentum modes are
solutions to $\lambda+kn=\omega\lambda+k\bar{n}$ for $n,\bar{n}\in\mathbb{Z}$,
and given a solution for $\lambda$ there always is a solution in which $n=0$.
We conclude that $\frac{1}{2R^{2}}$ is equal to the minimal value of
$\frac{\lambda^{2}}{2k}$ over those solutions, or in other words,

$R^{2}=\frac{k}{\lambda^{2}}\;,$ (2.16)

where $\lambda>0$ is the minimal positive integer solution to

$\lambda(\omega-1)=0\mod k\;.$ (2.17)

In particular notice that, for a given chiral algebra determined by $k$, there
is a family of RCFTs — one for each choice of $\omega$ — which might be
constructed from it.

#### Higher genus.

The complex structure of a Riemann surface $\Sigma$ of genus $g\geq 1$ is
described by a period matrix $\Omega$:101010Given a canonical basis
$\\{A_{i},B_{j}\\}$ of 1-cycles on $\Sigma$ with intersection numbers
$(A_{i},A_{j})=(B_{i},B_{j})=0$, $(A_{i},B_{j})=\delta_{ij}$, and a basis of
holomorphic 1-forms $\omega_{i}$ such that
$\oint_{A_{i}}\omega_{j}=\delta_{ij}$, one defines
$\Omega_{ij}=\oint_{B_{i}}\omega_{j}$. a symmetric $g\times g$ matrix, that we
split into real and imaginary part as $\Omega_{ij}=x_{ij}+i\,y_{ij}$, with
positive-definite imaginary part, $\operatorname{\mathbb{I}m}\Omega=y>0$. The
Euclidean partition function on $\Sigma$ is

$Z_{\Sigma}(R,\Omega)=\frac{\Theta(R,\Omega)}{\Phi}\;,$ (2.18)

where

$\Theta(R,\Omega)=\sum_{n,w\,\in\,\mathbb{Z}^{g}}\exp\biggl{[}-2\pi\,y_{ij}\biggl{(}\frac{n^{i}n^{j}}{R^{2}}+\frac{w^{i}w^{j}R^{2}}{4}\biggr{)}+2\pi
i\,x_{ij}n^{i}w^{j}\biggr{]}$ (2.19)

is the Siegel-Narain theta function. The denominator $\Phi$ can be written as
[74]

$\Phi=\bigl{\lvert}\det\nolimits^{\prime}_{\Gamma}\bar{\partial}_{0}\bigr{\rvert}\;,$
(2.20)

where $\bar{\partial}_{0}$ is the Dolbeault operator mapping functions to
$(0,1)$-forms, with zero-modes removed, while $\Gamma$ indicates a certain
regularization (we provide more details in Appendix A). For $g>1$, $\Phi$
suffers from the conformal anomaly and holomorphic factorization fails.
However both $Z_{\Sigma}(R,\Omega)$,
$\bigl{(}\det\operatorname{\mathbb{I}m}\Omega\bigr{)}^{1/2}\,\Theta(R,\Omega)$,
and $\bigl{(}\det\operatorname{\mathbb{I}m}\Omega\bigr{)}^{1/2}\,\Phi$ are
modular invariant under the $Sp(2g,\mathbb{Z})$ action of the mapping class
group.

The theta function can be rewritten as

$\Theta=\sum_{n,w\,\in\,\mathbb{Z}^{g}}\exp\biggl{[}\pi
i\biggl{(}\frac{n}{R}+\frac{wR}{2}\biggr{)}^{\\!\\!{\sf{T}}}\Omega\biggl{(}\frac{n}{R}+\frac{wR}{2}\biggr{)}-\pi
i\biggl{(}\frac{n}{R}-\frac{wR}{2}\biggr{)}^{\\!\\!{\sf{T}}}\Omega^{*}\biggl{(}\frac{n}{R}-\frac{wR}{2}\biggr{)}\biggr{]}\;.$
(2.21)

In the rational case $R^{2}\in\mathbb{Q}$, with $(k,\omega)$ defined as
before, following the same steps as in footnote 9, one obtains

$Z_{\Sigma}=\\!\\!\sum_{\lambda\in(\mathbb{Z}_{k})^{g}}\frac{1}{\Phi}\sum_{\ell\in\mathbb{Z}^{g}}\exp\biggl{[}\frac{2\pi
i}{2k}\,\bigl{(}k\ell+\lambda\bigr{)}^{\\!{\sf{T}}}\Omega\,\bigl{(}k\ell+\lambda\bigr{)}\biggr{]}\sum_{\bar{\ell}\in\mathbb{Z}^{g}}\exp\biggl{[}-\frac{2\pi
i}{2k}\,\bigl{(}k\bar{\ell}+\omega\lambda\bigr{)}^{\\!{\sf{T}}}\Omega^{*}\bigl{(}k\bar{\ell}+\omega\lambda\bigr{)}\biggr{]}\;.$
(2.22)

This is a sum over higher-genus conformal blocks labelled by $\lambda$.

###  2.2 $\boldsymbol{U(1)_{k}\times U(1)_{-k}}$ Chern-Simons theory

Two-dimensional current algebra is intimately related to three-dimensional
Chern-Simons theory [44]. In particular [56], the quantization of Chern-Simons
theory with gauge group $G$ at level $k$ on $D_{2}\times\mathbb{R}$ (where
$D_{2}$ is a two-dimensional spatial ball, or disk, while $\mathbb{R}$ is
time) with holomorphic boundary conditions yields, when $G$ is a connected and
simply-connected simple Lie group, the chiral WZW model $G_{k}$ [75], i.e.,
the Hilbert space on $D_{2}$ is the Kac-Moody current algebra. Adding a Wilson
line in an integral representation $\lambda$ along $\mathbb{R}$ through the
disk, yields a Hilbert space which is the representation $\lambda$ of the
current algebra [56]. If $G$ is not simply-connected, one obtains an extended
chiral algebra [55]. In particular, $U(1)_{k}$ Cherns-Simons theory (with
$k\in 2\mathbb{N}$) yields the $\mathfrak{u}(1)_{k}$ chiral algebra and its
representations $\lambda\in\mathbb{Z}_{k}$.

Indeed, $U(1)_{k}$ CS theory is an Abelian topological quantum field theory
(TQFT) with $k$ line operators, labelled by $\lambda\in\mathbb{Z}_{k}$. Under
fusion, the lines reproduce the group structure of $\mathbb{Z}_{k}$, which is
the one-form symmetry of the theory (in the terminology of [33]). The lines
are the unitary defect operators that implement $\mathbb{Z}_{k}$ one-form
symmetry transformations.

Euclidean path integrals of $U(1)_{k}$ CS theory on solid tori and higher-
genus handlebodies, with holomorphic (conformal) boundary conditions on the
boundary Riemann surface $\Sigma$, and with Wilson line insertions along the
non-contractible cycles, have been addressed with a variety of approaches, see
for instance [56, 76, 77, 78, 79]. The boundary conditions fix the anti-
holomorphic part of the pull-back of the connection to the boundary,
introducing a dependence on the complex structure of $\Sigma$ [56]:

$A_{\bar{z}}d\bar{z}=\partial_{\bar{z}}\chi\,d\bar{z}+i\pi
u(\operatorname{\mathbb{I}m}\Omega)^{-1}\bar{\omega}\;.$ (2.23)

Here $z$ is a local complex coordinate on $\Sigma$, $\omega_{I}$ for
$I=1,\dots,g$ are a basis of holomorphic differentials that define $\Omega$,
$\chi$ is a periodic function on $\Sigma$, and the vector $u_{I}$ of complex
numbers fixes the harmonic part of the differential $A_{\bar{z}}d\bar{z}$.
Then, with the insertion of Wilson lines parametrized by
$\lambda\in(\mathbb{Z}_{k})^{g}$ along the non-contractible cycles of the
handlebody, the Euclidean path integral reads [79]:111111We adopt here a
slightly different regularization than in [79], more natural for the
$U(1)_{k}\times U(1)_{-k}$ theory we discuss below, leading to a non-
holomorphic factor in front of the Euclidean path integral.

$Z[A_{\bar{z}},\lambda,\Omega]=\frac{(\mathrm{det}^{\prime}\Delta_{0})^{\frac{3}{4}}}{(\mathrm{det}^{\prime}\Delta_{1})^{\frac{1}{4}}}\,e^{\frac{k\pi}{2}u(\operatorname{\mathbb{I}m}\Omega)^{-1}u}\,e^{\frac{k}{2\pi}\int_{\Sigma}d^{2}x\,\partial_{z}\chi\,\partial_{\bar{z}}\chi}\;\theta\biggl{[}\begin{matrix}\lambda/k\\\
0\end{matrix}\biggr{]}(ku,k\Omega)\;.$ (2.24)

Here

$\theta\bigl{[}\begin{smallmatrix}a\\\
b\end{smallmatrix}\bigr{]}(u,\Omega)=\sum_{n\in\mathbb{Z}^{g}}e^{\pi
i(n+a)^{\sf{T}}\Omega(n+a)+2\pi i(n+a)^{\sf{T}}(u+b)}\qquad\text{with}\quad
u\in\mathbb{C}^{g}\;,\quad a,b,\in\mathbb{R}^{g}\;.$ (2.25)

On the other hand, $\Delta_{0}$ and $\Delta_{1}$ are Laplacian operators
acting on scalars and one-forms, respectively. As long as we are dealing with
only half of the theory, here $U(1)_{k}$, the definition of the determinants
in (2.24) requires some care, reflecting the obstruction to holomorphic
factorization in the boundary theory for $g>1$. However, we are eventually
interested in the $U(1)_{k}\times U(1)_{-k}$ CS theory with
holomorphic/antiholomorphic boundary conditions for the two factors. As
discussed in [16], in that case the determinants combine to give

$\frac{(\mathrm{det}^{\prime}\Delta_{0})^{\frac{3}{2}}}{(\mathrm{det}^{\prime}\Delta_{1})^{\frac{1}{2}}}=\frac{1}{\Phi}\leavevmode\nobreak\
.$ (2.26)

The $\Phi$ appearing here is equal to the one defined in (2.20), in particular
it gives $|\eta(\tau)|^{-2}$ in the case of $g=1$, see Appendix A for more
details. For the sake of simplicity, we simply take holomorphic boundary
conditions $A_{\bar{z}}=0$ for $U(1)_{k}$, and similarly antiholomorphic
boundary conditions for $U(1)_{-k}$. With the insertion of Wilson lines
$\lambda,\mu\in(\mathbb{Z}_{k})^{g}$ for the two group factors, respectively,
we obtain the Euclidean path-integral on a genus-$g$ handlebody:121212The
ambiguity due to the $2d$ Euler counterterm is fixed by setting to 1 the
result for $g=0$, after which the normalization of the $g>1$ partition
functions is also fixed.

$Z=\begin{cases}1&g=0\;,\\\\[3.00003pt]
K^{(k)}_{\lambda}(\tau)\,\overline{K^{(k)}_{\mu}(\tau)}&g=1\;,\\\\[5.0pt]
\dfrac{1}{\Phi}\,\biggl{(}\sum\limits_{\ell\in\mathbb{Z}^{g}}e^{\frac{\pi
i}{k}(k\ell+\lambda)^{\sf{T}}\Omega(k\ell+\lambda)}\biggr{)}\,\biggl{(}\sum\limits_{\bar{\ell}\in\mathbb{Z}^{g}}e^{\frac{\pi
i}{k}(k\bar{\ell}+\mu)^{\sf{T}}\Omega(k\bar{\ell}+\mu)}\biggr{)}^{\\!*}&g\geq
1\;.\end{cases}$ (2.27)

In the case $g=1$ of a solid torus, we have expressed the path integral in
terms of $\mathfrak{u}(1)_{k}$ characters (2.8).

The expressions above are not modular invariant, so a further prescription is
needed to get a candidate physical RCFT dual. One possibility is to sum the
expressions above over their modular images. This imitates the sum over
handlebodies [80] for three-dimensional gravity. Such a setup has been
studied, e.g., in [18, 21, 81] with the conclusion that it gives rise to an
_ensemble average_ over different RCFTs with the same chiral algebras (i.e.,
over different choices of $\omega$ in the language of Section 2.1). In the
remainder of this section, instead, we will introduce an alternative procedure
which selects a member of the ensemble and gives rise to a single dual RCFT.

###  2.3 Gauging a $\boldsymbol{\mathbb{Z}_{k}}$ one-form symmetry

The $U(1)_{k}\times U(1)_{-k}$ CS theory has 1-form global symmetry
$\mathbb{Z}_{k}\times\mathbb{Z}_{k}$. If we are going to regard this theory,
being generally covariant in the language of [43, 44], as a sort of three-
dimensional theory of gravity, along the lines of [16], we encounter a tension
with the expectation that theories of gravity should not have global
symmetries (see, e.g., [35, 36, 37, 38]). We might hope that, if we remove the
global symmetry, the behavior of the theory as a unitary quantum system
improves.

A simple way to remove a global symmetry is to gauge it, that is, to couple it
to dynamical gauge fields. Therefore, we would like to gauge the 1-form
symmetry group, or a subgroup thereof. From the point of view of the spectrum
of lines, this gauging was understood in [55]. The 1-form symmetry subgroup we
gauge is given by a subset $\mathcal{A}$ of the simple lines. Such a subgroup
is gaugeable, i.e., its ’t Hooft anomaly vanishes [33], if and only if the
lines in $\mathcal{A}$ are mutually transparent — i.e., they have trivial
mutual braiding — and moreover they have integer spin.131313The condition on
the spin of the lines guarantees that, if we start from a bosonic theory
(i.e., a theory that does not depend on a spin structure), after gauging the
theory is still bosonic. It is possible to gauge $\mathcal{A}$ even when some
lines in it have semi-integer spin, however at the expense of making the
theory fermionic (i.e., dependent on a choice of spin structure). The 1-form
symmetry group is necessarily Abelian [33], and for Abelian lines the braiding
is completely characterized by their spins:

$B(\lambda,\mu)=e^{2\pi
i\left[\rule{0.0pt}{4.20004pt}h(\lambda+\mu)-h(\lambda)-h(\mu)\right]}\;.$
(2.28)

Here $h(\lambda)$ mod 1 is the spin of the line $\lambda$, which is equal to
the chiral dimension mod 1 of any field in the integrable representation
$\lambda$ of the associated 2d chiral algebra. Therefore, it is enough to
require that all lines in the Abelian subgroup $\mathcal{A}$ have integer
spin.

The algorithm in [55] tells us what the spectrum of lines after gauging is. In
an Abelian theory, if we gauge a 1-form subgroup of order $p$, we reduce the
number of lines by a factor of $p^{2}$. Thus, if we start from
$\mathbb{Z}_{k}\times\mathbb{Z}_{k}$ and we want to get rid of all lines but
the identity, then we should gauge a subgroup of order $k$. This is a
_Lagrangian subgroup_ , i.e., a subgroup of lines with integer spin and whose
order squares to the order of the group. Let us assume that we gauge a
$\mathbb{Z}_{k}$ subgroup generated by the line $(1,\omega)$ for some
$\omega\in\mathbb{Z}_{k}$.141414This is not the most general case. We discuss
some generalizations in Section 3.2. The condition that the generator has
integer spin boils down to

$\omega^{2}=1\mod 2k\;.$ (2.29)

We have already encountered this condition in Section 2.1. It implies that
$\omega,k$ are coprime, and that such an $\omega\in\mathbb{Z}_{k}^{*}$ defines
an involutive group automorphism of $\mathbb{Z}_{k}$ which pairs the lines of
$U(1)_{k}$ with those of $U(1)_{-k}$. We will indicate the subgroup generated
by $(1,\omega)$ as $\mathbb{Z}_{k}^{(\omega)}$. All its lines have integer
spin, and thus it can be gauged. This leads us to study

$\mathcal{T}[k,\omega]\,\equiv\,\frac{U(1)_{k}\times
U(1)_{-k}}{\mathbb{Z}_{k}^{(\omega)}}\qquad\text{Chern-Simons theory.}$ (2.30)

In particular, we are interested in its Euclidean partition and correlation
functions.

Gauging of the 1-form symmetry can be described in two equivalent ways. The
conventional point of view is that we couple the symmetry to an external
background 2-form gauge field, and then make the latter dynamical. For a
discrete (Abelian) symmetry $\mathcal{A}$, the second step is particularly
simple because it reduces to a discrete sum over 1-form bundles. On an
oriented closed three-manifold $M$, this is a sum over the singular cohomology
group $H^{2}(M;\mathcal{A})$. The other point of view is that flat bundles can
be engineered by inserting networks of symmetry defects, which here are the
lines in $\mathcal{A}$. For Abelian lines, the networks can be broken into
simple lines wrapping the non-contractible cycles of $M$. Therefore the sum
over 1-form bundles can be expressed as a sum over the insertion of lines
along the homology cycles of $M$, namely over $H_{1}(M;\mathcal{A})\cong
H^{2}(M;\mathcal{A})$.

We fix the normalization. Let $G=\widehat{\mathcal{A}}$ be the Pontriagyn dual
to $\mathcal{A}$, namely the Abelian group of linear functions from
$\mathcal{A}$ to $\mathbb{R}/\mathbb{Z}$, which is isomorphic to
$\mathcal{A}$. For an oriented closed 3-manifold $M$, gauging of a discrete
Abelian 0-form symmetry $G$ gives the partition function

$Z^{\text{0-form
gauged}}=\frac{1}{\bigl{\lvert}H^{0}(M;G)\bigr{\rvert}}\sum_{\alpha\in
H^{1}(M;G)}Z[\alpha]\;,$ (2.31)

where $Z[\alpha]$ is the partition function of the original theory coupled to
the bundle $\alpha$. This could be equivalently written as a sum over the
insertion of codimension-1 symmetry defects $\alpha\in H_{2}(M;G)$. The
normalization is standard [82]: for each bundle, we divide by the number of
automorphisms of that bundle, which, for $G$ Abelian, does not depend on the
bundle and is the number of global gauge transformations, i.e.,
transformations that are constant on each connected component of $M$.
Similarly, gauging of a discrete (Abelian) 1-form symmetry $\mathcal{A}$ gives
the partition function

$Z^{\text{1-form
gauged}}=\frac{\bigl{\lvert}H^{0}(M;\mathcal{A})\bigr{\rvert}}{\bigl{\lvert}H^{1}(M;\mathcal{A})\bigr{\rvert}}\sum_{\lambda\in
H^{2}(M;\mathcal{A})}Z[\lambda]\;.$ (2.32)

This could be written as a sum over the insertion of line symmetry defects
$\lambda\in H_{1}(M;\mathcal{A})$. We have divided by the number of global
1-form gauge transformations, but we have removed overcounting by 0-form gauge
transformations of 1-form gauge transformations.

The normalizations in (2.31) and (2.32) are compatible. When we gauge a
discrete Abelian 0-form symmetry $\widehat{\mathcal{A}}$, the new theory
acquires a 1-form symmetry $\mathcal{A}$, and gauging the latter we get back
the original theory (as in [83]). The simple identity

$\frac{\bigl{\lvert}H^{0}(M;\mathcal{A})\bigr{\rvert}}{\bigl{\lvert}H^{1}(M;\mathcal{A})\bigr{\rvert}}\sum_{\beta\in
H^{2}(M;\mathcal{A})}e^{2\pi
i\int_{M}\gamma\cup\beta}\,\frac{1}{\bigl{\lvert}H^{0}(M;\widehat{\mathcal{A}})\bigr{\rvert}}\sum_{\alpha\in
H^{1}(M;\widehat{\mathcal{A}})}e^{2\pi
i\int_{M}\alpha\cup\beta}\,Z[\alpha]=Z[\gamma]\;,$ (2.33)

where $\gamma\in H^{1}(M;\widehat{\mathcal{A}})$, expresses this fact. Here
$\cup$ is the cup product, the bilinear form
$\widehat{\mathcal{A}}\times\mathcal{A}\to\mathbb{R}/\mathbb{Z}$ is the
natural one, and we used that $|H^{1}|=|H^{2}|$.

For holographic applications, we need a generalization of the gauging formula
(2.32) to oriented three-manifolds $M$ with boundary, with Dirichelet boundary
conditions $b$. The formula is the following:151515We are grateful to Pavel
Putrov for explaining this formula to us. See also Section 5 of [84].

$Z^{\text{1-form gauged}}_{b}=\left\lvert\frac{H^{0}(M,\partial
M\smallsetminus P)}{H^{1}(M,\partial M\smallsetminus
P)}\right\rvert\sum_{\begin{subarray}{l}a\in H^{2}(M,\partial M\smallsetminus
P)\\\ i^{*}(a)=b\end{subarray}}Z[a]\;.$ (2.34)

Here $\partial M$ is the two-dimensional boundary of $M$, $P$ is a set made of
one point in each connected component of $\partial M$, $H^{*}(M,B)$ is the
singular cohomology of $M$ relative to $B$,161616See, e.g., [85] for the
definitions of relative singular cohomology and homology. $i$ is the inclusion
map $\partial M\stackrel{{\scriptstyle i}}{{\hookrightarrow}}M$, $b\in
H^{2}(\partial M)$ is the boundary condition, while $Z[a]$ is the partition
function of the original theory with background $a$. All cohomology groups
take values in $\mathcal{A}$, that we have kept implicit. We provide a
derivation of this formula in Appendix C. It is convenient to use homology —
as opposed to cohomology — classes, since the former have a direct
interpretation as line insertions. Using Poincaré duality, we obtain:

$Z^{\text{1-form
gauged}}_{b}=\left\lvert\frac{H_{3}(M)}{H_{2}(M)}\right\rvert\sum_{\begin{subarray}{l}a\in
H_{1}(M,P)\\\ \partial a=b\end{subarray}}Z[a]\;,$ (2.35)

where $H_{*}(M,P)$ is singular homology of $M$ relative to $P$. Here $b\in
H_{0}(P)$, and the map
$H_{1}(M,P)\stackrel{{\scriptstyle\partial}}{{\to}}H_{0}(P)$ is in the long
exact sequence for the pair $(M,P)$. Notice that $H_{1}(M,P)$ is larger than
$H_{1}(M)$ because it includes relative 1-cycles going from one connected
component of the boundary to another, however these extra 1-cycles are
precisely fixed by the boundary conditions.

The theory $\mathcal{T}$ in (2.30) is completely trivial on closed three-
manifolds. As we explained, it only contains a single transparent line. As a
consequence, the Hilbert space is one-dimensional on any closed spatial two-
manifold and thus $\mathcal{T}$ is an invertible TQFT. Besides, the partition
function $Z_{\mathcal{T}}[M]=1$ on any (oriented) closed three-manifold $M$,
irrespective of its topology [86].171717In general, the invertible TQFT could
be a multiple of the $(E_{8})_{1}$ TQFT with $c_{-}=8$. In our construction,
as long as the left and right part before gauging are related by orientation
reversal, the invertible TQFT is completely trivial. In the general case, it
can be made trivial by stacking with $(E_{8})_{1}$. We interpret this as the
fact that $\mathcal{T}$ is not just a generally-covariant theory: rather, it
is independent both of a local metric _and_ of the global topology, and
therefore it behaves as a full-fledged theory of gravity without the need of
extra sums over topologies. The total Hilbert space on closed manifolds is
referred to as the “baby universes” Hilbert space [49, 50, 51], and in
$\mathcal{T}$ it is one-dimensional. As we discuss in the following sections,
things become more interesting in the presence of boundaries.

###  2.4 Partition functions and factorization

We proceed to compute the Euclidean partition function of the gauged Chern-
Simons theory $\mathcal{T}[k,\omega]\equiv\bigl{[}U(1)_{k}\times
U(1)_{-k}\bigr{]}/\mathbb{Z}^{(\omega)}_{k}$ on general (oriented) three-
manifolds with boundary, and compare with 2d CFT. We focus on _factorization_
: we show that the partition function of $\mathcal{T}$ on a three-manifold
whose boundary is made of one or more connected components, is equal to the
product of partition functions on handlebodies, each one having as boundary
one of the connected components. In particular, the partition function of
$\mathcal{T}$ on any oriented three-manifold with a given (connected or
disconnected) boundary is the same, and it only depends on the boundary.

We will always take holomorphic and antiholomorphic boundary conditions for
the gauge fields of the two factors $U(1)_{k}\times U(1)_{-k}$, respectively.
We also define the so-called _algebra object_

$\mathcal{A}=\bigoplus_{\lambda\in\mathbb{Z}_{k}}(\lambda,\omega\lambda)\;,$
(2.36)

as the set of lines which generate the group to be gauged. Note that
$\mathcal{A}$ is a line itself. To gauge the 1-form symmetry
$\mathbb{Z}_{k}^{(\omega)}$ we use (2.35) with trivial boundary conditions,
$b=0$ (we discuss more general boundary conditions in Section 4.3). The sum
over line insertions can be viewed as the insertion of the single line
$\mathcal{A}$ on each generator of $H_{1}(M)$. With some abuse of notation we
indicate as $\mathcal{A}$ both the 1-form symmetry group
$\mathbb{Z}_{k}^{(\omega)}$ to be gauged, and the set of lines in CS theory
that implement it.

The line $\mathcal{A}$ has the following properties:

  1. 1)

$\mathcal{A}$ is transparent with respect to itself (this is the anomaly-free
condition).

  2. 2)

Wrapping $\mathcal{A}$ around a generic line $(\mu,\bar{\mu})$ is equal to
$k\,\delta_{\omega\mu,\bar{\mu}}$ times that line, i.e., it is proportional to
a projector onto $\mathcal{A}$.

  3. 3)

Parallel fusion of $\mathcal{A}$ with a line $(\lambda,\omega\lambda)$ in
$\mathcal{A}$ gives back $\mathcal{A}$.

These properties are represented in Figure 1. We reviewed in (2.27) the
partition function of $U(1)_{k}\times U(1)_{-k}$ CS theory on handlebodies
with line insertions along the non-contractible cycles. Here we will first
discuss a few simple examples of factorization, and argue for the general case
at the end. We will sometimes use latin letters $a,b,c,\dots$ to denote the
composite lines $(\mu,\bar{\mu})$ of the full theory. The orientation reversal
of $\lambda$ will be denoted as $\check{\lambda}$.

$\mathcal{A}$$\mathcal{A}$$=$$\mathcal{A}$$\mathcal{A}$

$(\mu,\bar{\mu})$$\mathcal{A}$$=\;k\,\delta_{\omega\mu,\bar{\mu}}$$(\mu,\bar{\mu})$

$\mathcal{A}$$(\lambda,\omega\lambda)$$=$$\mathcal{A}$$\mathcal{A}$$\mathcal{A}$$(\lambda,\omega\lambda)$$(\lambda,\omega\lambda)$

Figure 1: Basic properties of the Abelian Lagrangian algebra object
$\mathcal{A}$.

#### Genus $\boldsymbol{g=0}$.

We introduce the notation ${\sf{S}}\Sigma_{g}$ for a genus $g$ handlebody.
Notice that ${\sf{S}}\Sigma_{0}={\sf{S}}S^{2}=D_{3}$ is a solid ball, while
${\sf{S}}\Sigma_{1}={\sf{S}}T^{2}=D_{2}\times S^{1}$ is a solid torus. We also
introduce the geometries $X_{n}=S^{3}\smallsetminus n\mathring{D}_{3}$: their
boundaries $\partial X_{n}$ consist of $n$ disconnected $S^{2}$’s. Note that
$X_{1}=D_{3}$ is a solid ball, while $X_{2}=S^{2}\times I$ is a spherical
cylinder, or “Euclidean wormhole”, where $I=[0,1]$ is a closed interval. The
homology groups are

$H_{3}(X_{n};\mathcal{A})=0\;,\qquad
H_{2}(X_{n};\mathcal{A})=\mathcal{A}^{n-1}\;,\qquad
H_{1}(X_{n},P;\mathcal{A})\big{|}_{b=0}=0\;.$ (2.37)

Note that $H_{1}(X_{n},P;\mathcal{A})=\mathcal{A}^{n-1}$, but the boundary
condition $b=0$ reduces it to zero. This implies that on this class of
geometries the gauging is trivial, and it can only affect the normalization.

The partition function of $\mathcal{T}$ on a solid ball $X_{1}=D_{3}$,
according to (2.27), is

$Z_{\mathcal{T}}[D_{3}]=Z_{\text{CS}}[D_{3}]=1\;,$ (2.38)

in agreement with the CFT result.

Next, consider the cylinder $X_{2}=S^{2}\times I$. In order to compute
$Z_{\text{CS}}[S^{2}\times I]$ we exploit the completeness relation. The
Hilbert space on $S^{2}$ is one-dimensional and the unique state is produced
by the path integral on $D_{3}$, therefore we can split the cylinder
$S^{2}\times I$ in two disks, up to a normalization factor $c_{0}$. In order
to compute $c_{0}$, we split the cylinder twice: $Z_{\text{CS}}[S^{2}\times
I]=c_{0}\,Z_{\text{CS}}[D_{3}]\,Z_{\text{CS}}[D_{3}]=c_{0}^{2}\,Z_{\text{CS}}[D_{3}]\,Z_{\text{CS}}[S^{3}]\,Z_{\text{CS}}[D_{3}]$.
This implies

$c_{0}=Z_{\text{CS}}[S^{3}]^{-1}=S_{00}^{-1}=\mathcal{D}\;,$ (2.39)

where $\mathcal{D}=\sqrt{\sum_{a\in\mathcal{C}}d_{a}^{2}}=k$ is the total
quantum dimension of the TQFT. For $\mathcal{A}$ Lagrangian,
$|\mathcal{A}|=c_{0}$. Since $Z_{\mathcal{T}}[S^{2}\times
I]=\frac{1}{|\mathcal{A}|}Z_{\text{CS}}[S^{2}\times I]$, we conclude that

$Z_{\mathcal{T}}[S^{2}\times
I]=Z_{\mathcal{T}}[D_{3}]\,Z_{\mathcal{T}}[D_{3}]=1\;.$ (2.40)

The partition function of $\mathcal{T}$ exhibits factorization, as expected in
the boundary CFT.

The argument easily generalizes to all geometries $X_{n}$ by applying the
completeness relation $n$ times, once around each of the $S^{2}$ boundaries.
One obtains

$Z_{\mathcal{T}}[X_{n}]=\frac{1}{|\mathcal{A}|^{n-1}}Z_{\text{CS}}[X_{n}]=\frac{c_{0}^{n}}{|\mathcal{A}|^{n-1}}Z_{\text{CS}}[D_{3}]^{n}Z_{\text{CS}}[S^{3}]=Z_{\mathcal{T}}[D_{3}]^{n}=1\;,$
(2.41)

again exhibiting factorization.

$\mathcal{A}_{1}$$\mathcal{A}_{2}$$\mathcal{A}_{3}$$\mathcal{A}_{n-1}$$\mathcal{A}^{\perp}$
Figure 2: Line configuration for the gauging on $Y_{n}$. We draw
$S^{2}\smallsetminus n\mathring{D}_{2}$, while the transverse $S^{1}$ is
implicit. The $n-1$ lines denoted as $\mathcal{A}$ wrap one of the boundary
circles, while $\mathcal{A}^{\perp}$ (drawn as a broken line) wraps the
transverse $S^{1}$.

#### Genus $\boldsymbol{g=1}$.

We introduce the geometries $Y_{n}=(S^{2}\smallsetminus
n\mathring{D}_{2})\times S^{1}$: their boundaries $\partial Y_{n}$ consist of
$n$ disconnected $T^{2}$’s. Note that $Y_{1}=D_{2}\times S^{1}={\sf{S}}T^{2}$
is a solid torus, while $Y_{2}=T^{2}\times I$ is a toroidal cylinder.181818In
the context of a different approach to the factorization problem [87], Chern-
Simons theory on this wormhole geometry was also studied recently in [88]. The
homology groups are

$H_{3}(Y_{n};\mathcal{A})=0\;,\qquad
H_{2}(Y_{n};\mathcal{A})=\mathcal{A}^{n-1}\;,\qquad
H_{1}(Y_{n},P;\mathcal{A})\big{|}_{b=0}=\mathcal{A}^{n}\;.$ (2.42)

We can represent $Y_{n}$ with line insertions along $H_{1}$ as in Figure 2. In
the case $n=1$ of the solid torus, $H_{1}$ is generated by the non-
contractible cycle and the partition function is

$Z_{\mathcal{T}}[{\sf{S}}T^{2}]=Z_{\text{CS}}[{\sf{S}}T^{2};\mathcal{A}]=\sum_{a\in\mathcal{A}}Z_{\text{CS}}[{\sf{S}}T^{2};a]=Z(R,\tau)\;,$
(2.43)

where $Z(R,\tau)$ is the modular-invariant CFT torus partition function
(2.14). In the middle we have a sum over the insertion of lines
$a\equiv(\lambda,\omega\lambda)\in\mathcal{A}$ along the non-contractible
cycle, which, according to (2.27), produce the characters of the
representations $a$ in RCFT.

In order to discuss factorization, we need once again the completeness
relation in CS theory, which allows us to perform surgery around a boundary
component. The Hilbert space on $T^{2}$ has dimension equal to the total
number of lines, and a basis of states — that we indicate as
$|{\sf{S}}T^{2};a\rangle$ — is produced by the path integral on a solid torus
with line insertions $a$. The inner product between these states is obtained
by taking two solid tori with opposite orientation, each with a line
insertion, and gluing them along their boundaries. We obtain $S^{2}\times
S^{1}$ with two lines wrapping $S^{1}$ at different points on $S^{2}$. The
resulting partition function is $Z_{\text{CS}}[S^{2}\times
S^{1};\check{a},b]=\langle{\sf{S}}T^{2};a|{\sf{S}}T^{2};b\rangle=\delta_{ab}$,
expressing the orthonormality of states. The completeness relation reads

$\mathbbm{1}_{T^{2}}=\sum\nolimits_{a}|{\sf{S}}T^{2};a\rangle\,\langle{\sf{S}}T^{2};a|\;,$
(2.44)

where the sum is over all simple lines in the theory. In terms of the
partition function of CS theory, it gives $Z_{\text{CS}}[T^{2}\times
I]=\sum_{a}Z_{\text{CS}}[{\sf{S}}T^{2};a]\,Z_{\text{CS}}[{\sf{S}}T^{2};\check{a}]$.

Hence, in the case $n=2$ of the toroidal cylinder (or wormhole), using the
properties of the line $\mathcal{A}$ spelled after (2.36) and the completeness
relation, we obtain:

$\displaystyle Z_{\mathcal{T}}[T^{2}\times I]$
$\displaystyle=\frac{1}{|\mathcal{A}|}\,Z_{\text{CS}}[T^{2}\times
I;\mathcal{A},\mathcal{A}^{\perp}]=\frac{1}{|\mathcal{A}|}\sum_{a}Z_{\text{CS}}[{\sf{S}}T^{2};\mathcal{A},(\mathcal{A}),a]\,Z_{\text{CS}}[{\sf{S}}T^{2};\check{a}]$
(2.45)
$\displaystyle=\sum_{a\in\mathcal{A}}Z_{\text{CS}}[{\sf{S}}T^{2};\mathcal{A},a]\,Z_{\text{CS}}[{\sf{S}}T^{2};\check{a}]=\sum_{a\in\mathcal{A}}Z_{\text{CS}}[{\sf{S}}T^{2};\mathcal{A}]\,Z_{\text{CS}}[{\sf{S}}T^{2};\check{a}]$
$\displaystyle=Z_{\text{CS}}[{\sf{S}}T^{2};\mathcal{A}]\,Z_{\text{CS}}[{\sf{S}}T^{2};\mathcal{A}]=Z_{\mathcal{T}}[{\sf{S}}T^{2}]\,Z_{\mathcal{T}}[{\sf{S}}T^{2}]\;.$

Here $\mathcal{A}^{\perp}$ is inserted perpendicularly to $\mathcal{A}$. After
performing surgery, a cycle becomes contractible in the bulk, and the line
inserted around that cycle (which wraps $a$) is denoted by $(\mathcal{A})$.
The main steps are graphically represented in Figure 3.

$\displaystyle\frac{1}{|\mathcal{A}|}$$\mathcal{A}$$\mathcal{A}$$\displaystyle=\;\frac{1}{|\mathcal{A}|}\sum_{a}$$a$$\mathcal{A}$$\mathcal{A}$$\check{a}$$=$$\mathcal{A}$$\mathcal{A}$
Figure 3: Surgery performed on $Y_{2}=T^{2}\times I$ (the transverse $S^{1}$
is kept implicit) in order to reduce it to ${\sf{S}}T^{2}\times{\sf{S}}T^{2}$
and show factorization of the partition function of theory $\mathcal{T}$.

The same procedure can be applied inductively to $Y_{n}$, as is clear from
Figure 2. Indeed, applying the completeness relation to the neighborhood of
one boundary $T^{2}$, one detaches a copy of ${\sf{S}}T^{2}$ with
$\mathcal{A}$ inserted, produces a factor of $|\mathcal{A}|$, and is left with
$Y_{n-1}$ with its insertions of $\mathcal{A}$. Finally:

$Z_{\mathcal{T}}[Y_{n}]=\Bigl{(}Z_{\mathcal{T}}[{\sf{S}}T^{2}]\Bigr{)}^{n}\;.$
(2.46)

#### Genus $\boldsymbol{g>1}$.

We will only consider two geometries: the handlebody ${\sf{S}}\Sigma_{g}$, and
the cylinder $\Sigma_{g}\times I$. For the handlebody, the homology groups are
$H_{3}({\sf{S}}\Sigma_{g};\mathcal{A})=H_{2}({\sf{S}}\Sigma_{g};\mathcal{A})=0$
and $H_{1}({\sf{S}}\Sigma_{g},P;\mathcal{A})\big{|}_{b=0}=\mathcal{A}^{g}$,
generated by the non-contractible cycles. Therefore the partition function of
the gauged theory is

$Z_{\mathcal{T}}[{\sf{S}}\Sigma_{g}]=\sum_{a_{1},\dots,a_{g}\in\mathcal{A}}Z_{\text{CS}}[{\sf{S}}\Sigma_{g};a_{1},\dots,a_{g}]\;.$
(2.47)

According to (2.27), the sum on the RHS is over conformal blocks and thus this
reproduces the modular invariant partition function (2.22) of the compact
boson theory.

The completeness relation says that

$Z_{\text{CS}}[\Sigma_{g}\times
I]=c_{g}\sum_{a_{1},\dots,a_{g}}Z_{\text{CS}}[{\sf{S}}\Sigma_{g};a_{1},\dots,a_{g}]\,Z_{\text{CS}}[{\sf{S}}\Sigma_{g};\check{a}_{1},\dots,\check{a}_{g}]\;,$
(2.48)

where the lines $a_{i}$ are inserted along the non-contractible cycles, and
the coefficient $c_{g}$ is determined by the normalization of states:
$c_{g}^{-1}=\langle{\sf{S}}\Sigma_{g};a_{1}\ldots
a_{g}|{\sf{S}}\Sigma_{g};a_{1}\ldots a_{g}\rangle$. In order to compute
$c_{g}$, one uses the following argument. Represent
$S^{3}={\sf{S}}\Sigma_{g}\cup{\sf{S}}\Sigma_{g}$ as the gluing of two
handlebodies:191919Cut $S^{3}$ along an $S^{2}$ so as to divide it in two
balls, $S^{3}=D_{3}\cup D_{3}$. This is the case $g=0$. Now modify one $D_{3}$
by removing from its interior a solid handle attached to its boundary $S^{2}$,
and add that handle to the other $D_{3}$. This gives
$S^{3}={\sf{S}}T^{2}\cup{\sf{S}}T^{2}$, which is the case $g=1$. By repeating
the removal/addition of handles, one obtains
$S^{3}={\sf{S}}\Sigma_{g}\cup{\sf{S}}\Sigma_{g}$ for any $g$. the $g$ non-
contractible cycles of the first handlebody form $g$ disconnected Hopf links
with the cycles of the second handlebody. Since all states have the same
normalization $c_{g}^{-1}$, we find

$\langle{\sf{S}}\Sigma_{g};a_{1},\ldots,a_{g}|S_{1}\cdots
S_{g}|{\sf{S}}\Sigma_{g};b_{1},\ldots,b_{g}\rangle=\langle a_{1}\cdot
b_{1},\ldots,a_{g}\cdot b_{g}\rangle_{S^{3}}\\\ =e^{-2\pi
i\sum_{j}(a_{j},b_{j})}\,\langle 1\rangle_{S^{3}}=c_{g}^{-1}\,S_{a_{1}\ldots
a_{g},b_{1}\ldots b_{g}}\;.$ (2.49)

Here $a_{i}\cdot b_{i}$ indicates a Hopf link, $(a,b)$ is the product that for
$\mathfrak{u}(1)_{k}$ is $(\mu,\nu)=\frac{\mu\nu}{k}$, and in the last
expression we have rewritten the first expression in terms of the matrix
elements of $S\equiv S_{1}\cdots S_{g}$, where each of the matrices $S_{j}$
performs an $S$-transformation on one of the handles of $\Sigma_{g}$. Equating
the last two terms and using $\langle 1\rangle_{S^{3}}=\mathcal{D}^{-1}$ and
the expression (2.10) for the $S$-matrix gives $c_{g}=\mathcal{D}^{1-g}$. Thus
the completeness relation reads

$\mathbbm{1}_{\Sigma_{g}}=\mathcal{D}^{1-g}\sum_{a_{1},\ldots,a_{g}}|{\sf{S}}\Sigma_{g};a_{1}\ldots
a_{g}\rangle\,\langle{\sf{S}}\Sigma_{g};a_{1}\ldots a_{g}|\;.$ (2.50)

Consider now the cylinder $\Sigma_{g}\times I$, with homology groups
$H_{3}[\Sigma_{g}\times I;\mathcal{A}]=0$, $H_{2}[\Sigma_{g}\times
I;\mathcal{A}]=\mathcal{A}$, and $H_{1}[\Sigma_{g}\times
I,P;\mathcal{A}]\big{|}_{b=0}=\mathcal{A}^{2g}$. The partition function of
$\mathcal{T}$ is

$Z_{\mathcal{T}}[\Sigma_{g}\times
I]=\frac{1}{|\mathcal{A}|}\,Z_{\text{CS}}\bigl{[}\Sigma_{g}\times
I;\\{\mathcal{A}\\}_{i},\\{\mathcal{A}^{\perp}\\}_{j}\bigr{]}\;.$ (2.51)

Using the completeness relation we find:

$\displaystyle Z_{\mathcal{T}}[\Sigma_{g}\times I]$
$\displaystyle=|\mathcal{A}|^{-g}\sum_{a_{1}\dots
a_{g}}Z_{\text{CS}}\bigl{[}{\sf{S}}\Sigma_{g};\\{\mathcal{A}\\}_{i},\\{(\mathcal{A})\\}_{j},\\{a_{i}\\}\bigr{]}\;Z_{\text{CS}}[{\sf{S}}\Sigma_{g};\\{\check{a}_{i}\\}]$
(2.52) $\displaystyle=\sum_{a_{1}\dots
a_{g}\in\mathcal{A}}Z_{\text{CS}}\bigl{[}{\sf{S}}\Sigma_{g};\\{\mathcal{A}\\}_{i}\bigr{]}\;Z_{\text{CS}}[{\sf{S}}\Sigma_{g},\\{\check{a}_{i}\\}]=\Bigl{(}Z_{\mathcal{T}}[{\sf{S}}\Sigma_{g}]\Bigr{)}^{2}\;.$

Once again, the partition function factorizes as expected in the boundary CFT.

The CFT partition functions are modular invariant, which means that the
partition function of $\mathcal{T}$ on a handlebody does not depend on which
particular handlebody (distinguished by the set of boundary 1-cycles that are
contractible) is attached to the boundary. This is a manifestation of the fact
that the path integral of $\mathcal{T}$ is completely independent of the bulk
geometry, since this theory is trivial in the bulk.

#### Bulk independence and factorization.

Let us finally discuss the general case, after having analyzed several
explicit examples. We can prove factorization of the partition function from
the fact that the gauged theory $\mathcal{T}$ is trivial in the bulk.

$\qquad\longleftrightarrow\qquad$

Figure 4: Surgery around a boundary component for a trivial bulk TQFT.

Let $M$ be an oriented three-manifold with boundary
$B=\bigsqcup_{i}\Sigma_{(i)}$, where each $\Sigma_{(i)}$ is a Riemann surface
of genus $g_{i}$. In order to compute the partition function of $\mathcal{T}$
on $M$, we inductively use surgery around each of the boundary components
$\Sigma_{(i)}$ [44] (see Figure 4). Since $Z_{\mathcal{T}}=1$ on any closed
3-manifold, $Z_{\mathcal{T}}[M]=Z_{\mathcal{T}}[M\sqcup S^{3}]$. We divide
$S^{3}$ in two handlebodies with the same genus $g_{i}$ as $\Sigma_{(i)}$.
Then we use that the Hilbert space of $\mathcal{T}$ on any Riemann surface is
one-dimensional, and that in a one-dimensional Hilbert space we can swap
$\langle\chi_{1}|\chi_{2}\rangle\langle\chi_{3}|\chi_{4}\rangle=\langle\chi_{1}|\chi_{4}\rangle\langle\chi_{3}|\chi_{2}\rangle$.
We end up with the disconnected sum of a handlebody with boundary
$\Sigma_{(i)}$, and a manifold $M^{\prime}$ whose boundary is
$B\smallsetminus\Sigma_{(i)}$. Repeating the procedure for all boundary
components, we end up with a disconnected sum of handlebodies with boundaries
$\Sigma_{(i)}$, and a closed manifold. The partition function of $\mathcal{T}$
on the latter is 1. Hence

$Z_{\mathcal{T}}[M]=\prod\nolimits_{i}Z_{\mathcal{T}}[{\sf{S}}\Sigma_{(i)}]\;.$
(2.53)

This shows that the partition function is completely independent of the choice
of $M$ with given boundary condition $B$, and that, therefore, it factorizes
as expected in the CFT.

###  2.5 Correlators

Besides partition functions, one can also reproduce the correlation functions
of local operators in the compact boson RCFT from the bulk $U(1)_{k}\times
U(1)_{-k}$ description with gauged $\mathbb{Z}_{k}^{(\omega)}$ 1-form
symmetry. In the bulk we follow a two-step procedure: first, we consider
correlation functions of bulk lines with endpoints at the boundary, in the
ungauged $U(1)_{k}\times U(1)_{-k}$ theory. Then, we gauge
$\mathbb{Z}_{k}^{(\omega)}$ as in the previous section. After gauging, only a
subset of the correlators is left, which is in bijection with the physical
correlators of the boundary CFT. (We provide a slightly different and more
general perspective in Section 4.3.)

#### Sphere two-point function.

We start by considering the two-point correlation function on the sphere. In
the $U(1)_{k}\times U(1)_{-k}$ theory, the two-point function is given by a
$D_{3}$ path integral with a Wilson line anchored to two points at the
boundary. We denote Wilson lines as $(\lambda,\bar{\mu})$ as before. We pick
the same orientation for the two lines, so that one endpoint $z_{1}$ has
charge $(\lambda,\bar{\mu})$ while the other endpoint $z_{2}$ has charge
$(-\lambda,-\bar{\mu})$. For the purpose of obtaining the most general two-
point function after gauging, it is sufficient to consider the case in which
the line is unknotted in the bulk (by the same bulk-independence argument as
in the previous section, any knotted line would yield the same result after
gauging). The resulting two-point correlator in the ungauged theory is [44,
89]202020Here we choose $-\frac{k}{2}<\lambda,\bar{\mu}\leq\frac{k}{2}$,
representing correlators of primary operators of chiral algebra, because with
that choice $h=\frac{\lambda^{2}}{2k}$ is the dimension of the primary. Shifts
of $\lambda$ by $k$ represent chiral-algebra descendants.

$\bigl{\langle}\lambda_{z_{1}\to z_{2}}\,\bar{\mu}_{z_{1}\to
z_{2}}\bigr{\rangle}_{D_{3}}=\left(\frac{z_{1}-z_{2}}{\ell}\right)^{-\frac{\lambda^{2}}{k}}\left(\frac{\bar{z}_{1}-\bar{z}_{2}}{\ell}\right)^{-\frac{\bar{\mu}^{2}}{k}}\;.$
(2.54)

The normalization depends on the arbitrary length scale $\ell$, and it
reflects the arbitrariness in the normalization of vertex operators on the
boundary.212121The length scale $\ell$ can be thought of as that appearing in
the propagator of the chiral boson:
$\bigl{\langle}\phi(z_{1})\phi(z_{2})\bigr{\rangle}\propto\log(\frac{z_{1}-z_{2}}{\ell})$.
Such a correlation function is not a singled-valued function of the endpoints,
and its monodromy is related to the spin of the bulk line.

Next, we gauge $\mathbb{Z}_{k}^{(\omega)}$. This consists in inserting the
object $\mathcal{A}$ along all generators of $H_{1}$, see eqn. (2.35). The
insertion of the line $(\lambda,\bar{\mu})$ inside the ball adds an element to
$H_{1}$, generated by the 1-cycle linking the line:

$\mathcal{A}$$z_{1}$$z_{2}$$\left(\lambda\,,\bar{\mu}\right)$ . (2.55)

The effect of $\mathcal{A}$ is to project onto physical fields:

$\bigl{\langle}\lambda_{z_{1}\to z_{2}}\,\bar{\mu}_{z_{1}\to
z_{2}}\bigr{\rangle}_{\mathbb{Z}^{(\omega)}_{k}\text{
gauged}}=k\,\delta_{[\omega\lambda]_{k},\bar{\mu}}\left(\frac{z_{1}-z_{2}}{\ell}\right)^{-\frac{\lambda^{2}}{k}}\left(\frac{\bar{z}_{1}-\bar{z}_{2}}{\ell}\right)^{-\frac{[\omega\lambda]_{k}^{2}}{k}}\,,$
(2.56)

where $[x]_{k}$ denotes the integer $-\frac{k}{2}<[x]_{k}\leq\frac{k}{2}$
which equals $x$ mod $k$. The only non-vanishing two-point functions coincide,
up to normalization, with those of the primary operators of the compact boson
RCFT, which have
$(h,\bar{h})=\Bigl{(}\frac{\lambda^{2}}{2k},\frac{[\omega\lambda]_{k}^{2}}{2k}\Bigr{)}$.
The constraint $\omega^{2}=1$ mod $2k$ ensures that all correlation functions
are single-valued.

#### Sphere $\boldsymbol{n}$-point function.

It is straightforward to extend the above procedure to the case of $n$-point
function of vertex operators with $n=2m$ an even integer, and assuming that
the operators come in $m$ pairs of opposite charge. The starting point is now
the correlator of $m$ lines of the $U(1)_{k}\times U(1)_{-k}$ theory. We
denote the endpoints of the $i$-th line as $z_{1i}$ and $z_{2i}$, where $i$
runs from $1$ to $m$. We take unknotted and unlinked lines. We have [89]:

$\biggl{\langle}\prod_{i=1}^{m}\lambda^{(i)}_{z_{1i}\to
z_{2i}}\,\bar{\mu}^{(i)}_{z_{1i}\to
z_{2i}}\biggr{\rangle}=\prod_{i=1}^{m}\left(\frac{z_{1i}-z_{2i}}{\ell}\right)^{-\frac{\lambda^{(i)\,2}}{k}}\left(\frac{\bar{z}_{1i}-\bar{z}_{2i}}{\ell}\right)^{-\frac{\bar{\mu}^{(i)\,2}}{k}}\\\
{}\times\prod_{i<j}^{m}\left[\frac{(z_{1i}-z_{1j})(z_{2i}-z_{2j})}{(z_{1i}-z_{2j})(z_{1j}-z_{2i})}\right]^{\frac{\lambda^{(i)}\lambda^{(j)}}{k}}\left[\frac{(\bar{z}_{1i}-\bar{z}_{1j})(\bar{z}_{2i}-\bar{z}_{2j})}{(\bar{z}_{1i}-\bar{z}_{2j})(\bar{z}_{1j}-\bar{z}_{2i})}\right]^{\frac{\bar{\mu}^{(i)}\bar{\mu}^{(j)}}{k}}\;.$
(2.57)

Gauging $\mathbb{Z}^{(\omega)}_{k}$ entails inserting $\mathcal{A}$ along the
$m$ 1-cycles of the complement of the lines inside the ball, i.e., the cycles
that link one of the $m$ lines. Each insertion of $\mathcal{A}$ projects the
line that it links to the subspace of charges
$\bar{\mu}^{(i)}=\bigl{[}\omega\lambda^{(i)}\bigr{]}_{k}$ that are allowed for
RCFT primaries. As a result:

$\biggl{\langle}\prod_{i=1}^{m}\lambda^{(i)}_{z_{1i}\to
z_{2i}}\bar{\mu}^{(i)}_{z_{1i}\to
z_{2i}}\biggr{\rangle}_{\mathbb{Z}^{(\omega)}_{k}\text{
gauged}}=k^{m}\,\prod_{i=1}^{m}\delta_{[\omega\lambda^{(i)}]_{k},\bar{\mu}^{(i)}}\left(\frac{z_{1i}-z_{2i}}{\ell}\right)^{-\frac{\lambda^{(i)\,2}}{k}}\left(\frac{\bar{z}_{1i}-\bar{z}_{2i}}{\ell}\right)^{-\frac{[\omega\lambda^{(i)}]_{k}^{2}}{k}}\\\
{}\times\prod_{i<j}^{m}\left[\frac{(z_{1i}-z_{1j})(z_{2i}-z_{2j})}{(z_{1i}-z_{2j})(z_{1j}-z_{2i})}\right]^{\frac{\lambda^{(i)}\lambda^{(j)}}{k}}\left[\frac{(\bar{z}_{1i}-\bar{z}_{1j})(\bar{z}_{2i}-\bar{z}_{2j})}{(\bar{z}_{1i}-\bar{z}_{2j})(\bar{z}_{1j}-\bar{z}_{2i})}\right]^{\frac{[\omega\lambda^{(i)}]_{k}[\omega\lambda^{(j)}]_{k}}{k}}\;.$
(2.58)

A few comments are in order:

  * •

The $k^{m}$ prefactor may be adsorbed consistently by an appropriate
normalization of the two-point function.222222This is achieved by rescaling an
operator with scaling dimensions $(h,\bar{h})$ by the factor
$\sqrt{k}\,\ell^{h+\bar{h}}$. After that, all correlators coincide with those
of the physical vertex operators in the boundary RCFT.

  * •

As for the two-point function, these correlation functions are single-valued
after gauging. Furthermore, since the surviving lines are transparent, the
result does not depend on the bulk knotting pattern.

More general charge-conserving $n$-point correlation functions can be obtained
starting from the $2m$-point functions above and performing OPEs. They can be
obtained considering configurations of anchored lines that fuse in the bulk
before gauging.

#### Correlators on wormhole geometries.

One may ask whether, after gauging, correlators on “wormhole” geometries
factorize as it happens for partition functions. The topical example is a
four-point function on $T^{2}\times I$ (or, more generally, on
$\Sigma_{g}\times I$) with two operator insertions on each end of the
interval, connected by two Wilson lines threading the bulk. We denote doubled
lines $(\lambda\,,\bar{\mu})$ by $a,b$:

$a$$b$$=$$a$$b$ .

Before gauging, this does not factorize (see also [90]). Based on the previous
discussion, after gauging $\mathbb{Z}^{(\omega)}_{k}$ one can explain
factorization in two ways:

  1. 1.

Lines stretching in the bulk can now “recombine” with the defect network for
the gauging, as on the right in Figure 1. Resolving the four-valent junctions
and using crossing leads to a factorized correlator.

  2. 2.

One can use the fact that the bulk Hilbert space is one-dimensional. One
performs surgery on the interval and glues a representative of the (only)
state on the two-punctured torus (in the figure we keep the transverse $S^{1}$
implicit):232323In this case the bulk theory is _gauged_ , but we do not draw
the network for simplicity.

$\displaystyle\mathbbm{1}_{\mathcal{H}\left(T^{2};a,b\right)}\;=\;\delta_{\check{a},b}\,\delta_{a\in\mathcal{A}}$$a$$\check{a}$
.

This leads to the factorized answer:

$a$$b$$\displaystyle=\;\delta_{\check{a},b}\,\delta_{a\in\mathcal{A}}$$a$$\check{a}$
.

While we have presented the arguments for a fixed configuration, the
generalization to more complicated geometries and operator insertions seems
quite straightforward.

###  2.6 Comments on symmetries

#### Dual symmetry.

In our discussion, two bulk theories play a prominent role: the Chern-Simons
theory $\mathcal{U}$ and its gauging, i.e., the two following gauge theories:

$\mathcal{U}=U(1)_{k}\times
U(1)_{-k}\;,\qquad\qquad\mathcal{T}=\frac{U(1)_{k}\times
U(1)_{-k}}{\mathbb{Z}_{k}^{(\omega)}}\;.$ (2.59)

Theory $\mathcal{T}$ is obtained from $\mathcal{U}$ by gauging a discrete
1-form symmetry, therefore it should feature a dual global $\mathbb{Z}_{k}$
0-form symmetry [83]. On the other hand, $\mathcal{T}$ is trivial and thus the
0-form symmetry does not act on anything. If we gauge it, we go back to
$\mathcal{U}$: this shows that $\mathcal{U}$ should be equivalent to a pure
$\mathbb{Z}_{k}$ gauge theory. Restricting for simplicity to the diagonal case
$\omega=1$, this is easy to see with a simple field redefinition in the
Lagrangian of $\mathcal{U}$:

$\mathcal{U}:\qquad\mathcal{L}=\frac{k}{4\pi}\,ada-\frac{k}{4\pi}\,\tilde{a}d\tilde{a}=\frac{k}{4\pi}\,bdb+\frac{k}{2\pi}\,bd\tilde{b}\;,$
(2.60)

where we defined $a=b+\tilde{b}$, $\tilde{a}=\tilde{b}$, and lowercase fields
are dynamical. We see that $\mathcal{U}$ is $(\mathbb{Z}_{k})_{k}$, which is
parity invariant (because CS level $\ell$ is equivalent to $\ell+2k$). On the
other hand, theory $\mathcal{T}$ can be written as

$\mathcal{T}^{\prime}:\qquad\mathcal{L}=\frac{k}{4\pi}\,BdB+\frac{k}{2\pi}\,Bd\tilde{b}$
(2.61)

where $B$ is a background field, and $\tilde{b}$ is a Lagrange multiplier
restricting $B$ to $\mathbb{Z}_{k}$. We see that, if one activates a non-
trivial background $B$ for the 0-form symmetry, $\mathcal{T}$ behaves as an
invertible TQFT. We have used a prime here because, although
$\mathcal{T}=\mathcal{T}^{\prime}$ as theories, the two descriptions are
different.

#### Non-diagonal case.

The $\mathbb{Z}_{2}$ automorphism of $U(1)_{k}$ CS theory used to define
$\mathbb{Z}_{k}^{(\omega)}$ can be given a Lagrangian description as
follows.242424This symmetry has also been studied in [91]. The technique used
here is similar to the one in [92]. We first integrate in two auxiliary gauge
fields $a_{1}$, $a_{2}$ and consider the Lagrangian

$\mathcal{L}=\frac{k}{4\pi}\,ada+\frac{1}{2\pi}\,a_{1}da_{2}\;.$ (2.62)

The fields $a_{1,2}$ are Lagrange multipliers that simply set each other to
zero [86]. As in Section 2.1, let $k=2pp^{\prime}$ with $p,p^{\prime}$ coprime
integers, find $r_{0},s_{0}$ such that $pr_{0}-p^{\prime}s_{0}=1$, and define
$\omega=pr_{0}+p^{\prime}s_{0}$ so that $\omega^{2}=1+2r_{0}s_{0}k$. Then
perform the $SL(3,\mathbb{Z})$ field redefinition

$\begin{pmatrix}a\\\ a_{1}\\\
a_{2}\end{pmatrix}\to\begin{pmatrix}\omega&1&-r_{0}s_{0}\\\
ks_{0}&p&s_{0}(1-pr_{0})\\\
-kr_{0}&-p^{\prime}&r_{0}(1+p^{\prime}s_{0})\end{pmatrix}\begin{pmatrix}a\\\
a_{1}\\\ a_{2}\end{pmatrix}\;.$ (2.63)

The matrix has unit determinant. The transformation leaves the action
invariant, but it acts on the lines as

$e^{i\,q\int a}\to e^{i\,\omega q\int a}\;,$ (2.64)

where we used that $a_{1,2}$ are set to zero by the equations of motion.

By performing this transformation on $\tilde{a}$ in (2.60) before changing
variables to $b,\tilde{b}$, we obtain the same Lagrangian for $\mathcal{U}$ on
the RHS of (2.60), but with a redefinition $b=a-\omega\tilde{a}$ (up to
auxiliary gauge fields that can be set to zero). This shows that $b$ gauges
precisely the $\mathbb{Z}_{k}$ 0-form symmetry (parametrized by $\omega$) that
does not act on physical operators.

#### Global symmetry on the boundary.

Let us now discuss the global symmetries present in the boundary theory. In
the standard holographic dictionary, a gauge symmetry in the bulk corresponds
to a global symmetry on the boundary. In our setup, we have 0-form $U(1)$
gauge symmetries in the bulk associated to the two Chern-Simons gauge fields,
and they induce $U(1)$ global symmetries of both chiralities on the boundary.
An important role is played by the boundary action

$S_{\partial}=\frac{k}{4\pi}\int_{\partial}d^{2}x\,\sqrt{g}\,g^{z\bar{z}}\,\bigl{(}a_{z}a_{\bar{z}}-\tilde{a}_{z}\tilde{a}_{\bar{z}}\bigr{)}$
(2.65)

that is needed to impose holomorphic/antiholomorphic boundary conditions, as
we review in Appendix B, where we also discuss the relevance of a certain
boundary contact term.

In addition, since we gauge a discrete $\mathbb{Z}_{k}$ 1-form symmetry in the
bulk, one might naively expect the existence of this global symmetry on the
boundary. In the holographic dictionary the boundary charge operators are
obtained by letting the bulk charge operators, over whose insertions we are
summing to gauge the symmetry, end on the boundary. Therefore, for the
$\mathbb{Z}_{k}$ 1-form symmetry, one might expect topological local operators
on the boundary arising from the endpoint of bulk lines generating the
$\mathbb{Z}_{k}$ subgroup:

$(\lambda,\omega\lambda)$

.

On the other hand, we have already seen that actually these endpoints give
rise to the physical primary operators of the compact boson CFT, which are not
topological. We conclude that this symmetry is explicitly broken by the
holomorphic boundary conditions we have imposed. Indeed, the boundary action
(2.65) gives rise to a boundary Sugawara stress tensor that assigns a non-zero
energy to the endpoints of lines, corresponding to the conformal dimension of
the primaries. More directly, one can check that the boundary term (2.65)
would not be invariant under $\mathbb{Z}_{k}$ 1-form gauge transformations of
$a$ that are non-vanishing at the boundary.252525This is also related to the
fact that when gauging $\mathbb{Z}_{k}^{(\omega)}$ in Section 2.3, we impose
Dirichelet boundary conditions to the $\mathbb{Z}_{k}$ 2-form gauge fields at
the boundary.

Finally, let us comment on the implications of the dual $\mathbb{Z}_{k}$
0-form symmetry on the boundary. We can let a dual symmetry charge operator
end on a line on the boundary, defining a topological defect line. In the
bulk, a line bounded by a $\mathbb{Z}_{k}$ surface can be thought of as a line
of the initial theory $\mathcal{U}$, that is not gauge invariant under the
$\mathbb{Z}_{k}$ one-form symmetry and therefore can be defined only as a
semi-local operator with a surface attached to it. Such semi-local operators
are defined modulo fusion with $\mathcal{A}$. For convenience let us label a
basis of these lines by $L_{i}$. By the the braiding rules of the initial
theory $\mathcal{U}$, these lines will act on operators $\phi_{a}$ at the
boundary (which are the endpoints of $a\in\mathcal{A}$) by
$L_{i}\left[\phi_{a}\right]=\frac{S_{ia}}{S_{0a}}\phi_{a}$. This is the action
of Verlinde lines in RCFTs on primary operators [93, 94, 95]. Thus we conclude
that this procedure gives rise to the set of Verlinde lines of the physical
RCFT.

##  3 Abelian generalizations

The simple example of a 2d compact scalar, dual to the $U(1)_{k}\times
U(1)_{-k}/\mathbb{Z}_{k}^{(\omega)}$ Chern-Simons theory, can be generalized
in many ways. In this section we consider a few Abelian generalizations,
including a multi-component compact scalar and a non-rational case.

###  3.1 Multi-component compact scalar

Consider $D$ real free compact bosons $X^{j}\cong X^{j}+2\pi$, with Euclidean
action:262626To make contact with (2.1), set $D=1$ and then $G_{ii}=R^{2}$
here and $\varphi=RX$ there.

$S=\frac{1}{8\pi}\int
d^{2}\sigma\biggl{(}G_{ij}\,\delta^{\alpha\beta}\,\partial_{\alpha}X^{i}\,\partial_{\beta}X^{j}+iB_{ij}\,\varepsilon^{\alpha\beta}\,\partial_{\alpha}X^{i}\,\partial_{\beta}X^{j}\biggr{)}\;.$
(3.1)

The target space is a torus $T^{D}$ with metric $G_{ij}$ and B-field $B_{ij}$.
There exists a moduli space $\mathcal{M}_{D}$ worth of CFTs obtained by
varying the metric and the B-field, modulo field redefinitions and
dualities,272727The set of equivalences was clearly reviewed in [96]. It
includes unimodular transformations of the fields $X^{j}$, as well as gauge
transformations that shifts the components of $B_{ij}$ by even integers. and
it is known as the Narain moduli space [17]:

$\mathcal{M}_{D}=O(D,D;\mathbb{Z})\backslash O(D,D;\mathbb{R})/O(D)\times
O(D)\;.$ (3.2)

Moreover, the theory has $\mathfrak{u}(1)^{D}\times\mathfrak{u}(1)^{D}$
current algebra generated by the currents $\partial X^{i}$ and
$\bar{\partial}X^{i}$, respectively.

Given a CFT identified by $m\in\mathcal{M}_{D}$, the torus partition function
is

$Z(m,\tau)=\frac{\Theta(m,\tau)}{|\eta(\tau)|^{2D}}\;,$ (3.3)

where

$\Theta(m,\tau)=\sum_{\vec{n},\vec{w}\,\in\,\mathbb{Z}^{D}}\exp\biggl{[}-2\pi\tau_{2}\Bigl{(}G^{ij}v_{i}v_{j}+\tfrac{1}{4}G_{ij}w^{i}w^{j}\Big{)}+2\pi
i\tau_{1}n_{i}w^{i}\biggr{]}=\sum_{\vec{n},\vec{w}\,\in\,\mathbb{Z}^{D}}q^{h_{\vec{n},\vec{w}}}\;\bar{q}^{\bar{h}_{\vec{n},\vec{w}}}$
(3.4)

is the Siegel-Narain theta function, while

$v_{i}=n_{i}+\tfrac{1}{2}B_{ij}w^{j}$ (3.5)

is the velocity (in the presence of B-field). In the rightmost expression we
used the left and right dimensions of primary operators,

$h_{\vec{n},\vec{w}}=\frac{1}{2}\,\left\lvert\vec{n}+\frac{G+B}{2}\vec{w}\right\rvert^{2}_{G^{-1}}\;,\qquad\qquad\bar{h}_{\vec{n},\vec{w}}=\frac{1}{2}\,\left\lvert\vec{n}-\frac{G-B}{2}\vec{w}\right\rvert^{2}_{G^{-1}}\;.$
(3.6)

In order to avoid clutter, we introduced the notation
$|\vec{x}|^{2}_{M}=x^{i}M_{ij}x^{j}$.

#### Chiral algebras.

When $G,B\in\mathbb{Q}^{D\times D}$, the theory is a RCFT. In order to
determine the chiral algebras, we proceed as follows. First define the matrix

$M=\frac{G+B}{2}\qquad\Rightarrow\qquad M^{\sf{T}}=\frac{G-B}{2}\;,\qquad
G=M+M^{\sf{T}}\;,\qquad B=M-M^{\sf{T}}\;.$ (3.7)

The left-moving chiral algebra is given by all operators with
$\bar{h}_{\vec{n},\vec{w}}=0$, which are solutions to the equation

$\vec{n}=M^{\sf{T}}\vec{w}\qquad\qquad\text{with}\qquad\vec{n},\vec{w}\in\mathbb{Z}^{D}\;.$
(3.8)

This equation gives rise to two lattices. One is the lattice $\Lambda_{L}$ of
vectors $\vec{w}$ such that $M^{\sf{T}}\vec{w}\in\mathbb{Z}^{D}$. We package a
set of generators of $\Lambda_{L}$ into an integer matrix $P_{L}$ (defined up
to multiplication by unimodular matrices from the right) as its columns. In
other words, $\Lambda_{L}=P_{L}\,\mathbb{Z}^{D}$. The other lattice is
$M^{\sf{T}}\Lambda_{L}$ of image values of $\vec{n}$, and a set of generators
is given by the columns of the integer matrix
$\widetilde{P}_{L}=M^{\sf{T}}P_{L}$. We can rewrite this relation as

$M^{\sf{T}}=\widetilde{P}_{L}\,P_{L}^{-1}\;,$ (3.9)

which could be regarded as the matrix version of writing a fraction in the
irreducible form.282828Special representatives for $P_{L},\widetilde{P}_{L}$
can be found by performing Smith decomposition of $M^{\sf{T}}$, namely writing
$M^{\sf{T}}=UD_{\mathbb{Q}}V$ where $U,V$ are integer unimodular matrices,
while $D_{\mathbb{Q}}$ is a diagonal rational matrix. We write
$D_{\mathbb{Q}}=\text{Num}\cdot\text{Den}^{-1}$ where Num, Den are the two
diagonal integer matrices of numerators and denominators of the entries of
$D_{\mathbb{Q}}$ written in the irreducible form. Then
$P_{L}=V^{-1}\text{Den}$ and $\widetilde{P}_{L}=U\,\text{Num}$. We also find
$P_{R}=U^{-1{\sf{T}}}\,\text{Den}$ and
$\widetilde{P}_{R}=V^{\sf{T}}\,\text{Num}$ for the two other matrices defined
below.  The chiral operators are labelled by
$\vec{n}=M^{\sf{T}}P_{L}\vec{\ell}$, $\vec{w}=P_{L}\vec{\ell}$ with
$\vec{\ell}\in\mathbb{Z}^{D}$ and their left-moving dimensions are
$h=\frac{1}{2}\vec{\ell}^{\,\,{\sf{T}}}P_{L}^{\sf{T}}GP_{L}\vec{\ell}$. We
recognize the chiral algebra

$\mathfrak{u}(1)^{D}_{K_{L}}\qquad\qquad\text{with}\qquad
K_{L}=P_{L}^{\sf{T}}GP_{L}=\widetilde{P}_{L}^{\sf{T}}P_{L}+P_{L}^{\sf{T}}\widetilde{P}_{L}\;.$
(3.10)

These relations should be compared with (2.6) and (2.7) in the case $D=1$. We
see that $K_{L}$ is a positive symmetric even integer matrix, namely
$K_{L}>0$, $(K_{L})_{ij}\in\mathbb{Z}$ and $(K_{L})_{ii}\in 2\mathbb{Z}$
(corresponding to a bosonic Chern-Simons theory).

The chiral algebra $\mathfrak{u}(1)^{D}_{K_{L}}$ has integrable
representations labelled by the discriminant group
$\mathcal{D}_{L}=\mathbb{Z}^{D}/K_{L}\mathbb{Z}^{D}$, where
$K_{L}\mathbb{Z}^{D}$ is the integer lattice generated by the columns of
$K_{L}$. The order of the discriminant is $|\mathcal{D}_{L}|=\det K_{L}$, the
dimensions of chiral primary operators are

$h=\frac{1}{2}\,|\vec{\lambda}|^{2}_{K_{L}^{-1}}$ (3.11)

where $\vec{\lambda}$ is the left-moving charge of the primary operator, and
the characters are

$K_{\vec{\lambda}}^{(K_{L})}(\tau)=\frac{1}{\eta(\tau)^{D}}\sum_{\vec{\ell}\,\in\,\mathbb{Z}^{D}}q^{\frac{1}{2}\left|\vec{\lambda}+K_{L}\vec{\ell}\,\right|^{2}_{K_{L}^{-1}}}$
(3.12)

for $\vec{\lambda}\in\mathcal{D}_{L}$. Indeed, comparing (3.11) with (3.6), we
identify the left-moving charge of a generic operator as
$\vec{q}_{L}=P_{L}^{\sf{T}}\vec{n}+P_{L}^{\sf{T}}M\vec{w}$. This can be
written as $\vec{\lambda}+K_{L}\vec{\ell}$, where the integer vector
$\vec{\ell}$ parametrizes the chiral algebra descendants. The left-moving
dimension $h$ mod 1 provides a quadratic function
$h:\mathcal{D}_{L}\to\mathbb{R}/\mathbb{Z}$ that can be used to construct the
bilinear form
$K_{L}^{-1}(\cdot,\cdot):\mathcal{D}_{L}\times\mathcal{D}_{L}\to\mathbb{R}/\mathbb{Z}$.

The right-moving chiral algebra is obtained in a similar way. Its operators
are solutions to $h_{\vec{n},\vec{w}}=0$, namely to $\vec{n}=-M\vec{w}$ with
$\vec{n},\vec{w}\in\mathbb{Z}^{D}$. One defines the lattice $\Lambda_{R}$ of
integer vectors $\vec{w}$ such that $M\vec{w}\in\mathbb{Z}^{D}$, and packages
a set of generators into the integer matrix $P_{R}$ as its columns. The
lattice $M\Lambda_{R}$ of image values of $\vec{n}$ is generated by the
columns of the integer matrix $\widetilde{P}_{R}=MP_{R}$. The chiral algebra
is then $\mathfrak{u}(1)^{D}_{K_{R}}$ with

$M=\widetilde{P}_{R}\,P_{R}^{-1}\;,\qquad\qquad
K_{R}=P_{R}^{\sf{T}}GP_{R}=\widetilde{P}_{R}^{\sf{T}}P_{R}+P_{R}^{\sf{T}}\widetilde{P}_{R}\;.$
(3.13)

Notice that $\det K_{L}=\det K_{R}$.

#### Left-right pairing.

Let us understand how left- and right-moving representations of chiral algebra
are paired into physical fields. The operator $(\vec{n},\vec{w})$ has left-
and right-moving charges
$\vec{q}_{L}=P_{L}^{\sf{T}}\vec{n}+P_{L}^{\sf{T}}M\vec{w}$ and
$\vec{q}_{R}=P_{R}^{\sf{T}}\vec{n}-P_{R}^{\sf{T}}M^{\sf{T}}\vec{w}$,
respectively. If we mod out the lattice $\mathbb{Z}^{2D}$ of vectors
$\big{(}\begin{smallmatrix}\vec{n}\\\ \vec{w}\end{smallmatrix}\big{)}$ by the
holomorphic and anti-holomorphic fields corresponding to the sublattice
$J\,\mathbb{Z}^{2D}$ where

$J=\begin{pmatrix}M^{\sf{T}}P_{L}&-MP_{R}\\\ P_{L}&P_{R}\end{pmatrix}\;,$
(3.14)

then we are left with the group $\mathbb{Z}^{2D}/J\,\mathbb{Z}^{2D}$ of order
$\det J=\det K_{L}=\det K_{R}$. These are the integrable representations,
paired by a group isomorphism. We say that two integer matrices $P,Q$ are
coprime if $\mathbb{Z}^{D}$ is the only integer lattice such that both
$P\mathbb{Z}^{D}$ and $Q\mathbb{Z}^{D}$ are sublattices thereof (this
condition is invariant under multiplication of $P,Q$ by integer unimodular
matrices from the right). In our case, the matrices $P_{L}^{\sf{T}}$ and
$\widetilde{P}_{L}^{\sf{T}}$ are coprime.292929Suppose that they are not. Then
there exists an integer lattice $L\mathbb{Z}^{D}$ (with $\det L>1$) such that
$P_{L}^{\sf{T}}\mathbb{Z}^{D}$ and $\widetilde{P}_{L}^{\sf{T}}\mathbb{Z}^{D}$
are sublattices thereof. Hence one can write $P_{L}^{\sf{T}}=LR^{\sf{T}}$,
$\widetilde{P}_{L}^{\sf{T}}=L\widetilde{R}^{\sf{T}}$ in terms of integer
matrices $R$, $\widetilde{R}$ providing a decomposition
$M^{\sf{T}}=\widetilde{R}R^{-1}$. However $R\,\mathbb{Z}^{D}$, which is finer
than $P_{L}\mathbb{Z}^{D}$, is mapped by $M^{\sf{T}}$ into $\mathbb{Z}^{D}$,
in contradiction with the hypothesis that $P_{L}\mathbb{Z}^{D}$ is the
totality of vectors $\vec{w}$ with that property. It follows that there exist
integer matrices $N_{1,2}$ such that

$S=\begin{pmatrix}P_{L}^{\sf{T}}&P_{L}^{\sf{T}}M\\\
N_{1}&N_{2}\end{pmatrix}\in SL(2D,\mathbb{Z})\;.$ (3.15)

Indeed the columns of $P_{L}^{{\sf{T}}}$,
$\widetilde{P}_{L}^{\sf{T}}=P_{L}^{\sf{T}}M$ generate $\mathbb{Z}^{D}$, and
thus there exists a linear integer and invertible change of coordinates $S$
from $(\vec{n},\vec{w})$ to $(\vec{q}_{L},\vec{\ell}\,)$, for some integer
coordinate $\vec{\ell}$ that parametrizes the chiral algebra descendants on
the anti-holomorphic side. Let the inverse integer matrix be

$S^{-1}=\begin{pmatrix}R_{0}&MN_{3}\\\ -S_{0}&-N_{3}\end{pmatrix}\;,$ (3.16)

so that $\big{(}\begin{smallmatrix}\vec{n}\\\
\vec{w}\end{smallmatrix}\big{)}=S^{-1}\big{(}\begin{smallmatrix}\vec{q}_{L}\\\
\vec{\ell}\end{smallmatrix}\big{)}$. In particular303030In the representation
of footnote 28, $R_{0},S_{0}$ are easily determined. Set
$R_{0}=V^{\sf{T}}R_{0}^{\text{red}}$ and
$S_{0}=U^{-1{\sf{T}}}S_{0}^{\text{red}}$. The reduced matrices satisfy:
$\text{Den}\,R_{0}^{\text{red}}-\text{Num}\,S_{0}^{\text{red}}=\mathbbm{1}$,
which is a diagonal equation, hence $R_{0}^{\text{red}},S_{0}^{\text{red}}$
are diagonal. Then $N_{1}=S_{0}^{\text{red}}V^{-1{\sf{T}}}$,
$N_{2}=R_{0}^{\text{red}}U^{\sf{T}}$, and $N_{3}=-P_{R}$.
$P_{L}^{\sf{T}}R_{0}-\widetilde{P}_{L}^{\sf{T}}S_{0}=\mathbbm{1}$.
Substituting into the right-moving charge $\vec{q}_{R}$, we determine313131One
finds
$\vec{q}_{R}=P_{R}^{\sf{T}}(R_{0}+M^{\sf{T}}S_{0})\vec{q}_{L}+P_{R}^{\sf{T}}GN_{3}\vec{\ell}$.
Since $\vec{\ell}$ parametrizes the anti-holomorphic sector,
$N_{3}=P_{R}\check{U}$ for some unimodular integer matrix $\check{U}$. Note
that different solutions to the problem in (3.15)-(3.16) are related by
$N_{1}\to N_{1}-WP_{L}^{\sf{T}}$, $N_{2}\to
N_{2}-W\widetilde{P}_{L}^{\sf{T}}$, $R_{0}\to R_{0}+MN_{3}W$, $S_{0}\to
S_{0}+N_{3}W$ for integer matrices $W$. The matrix $\omega$ shifts by
$K_{R}(\check{U}W)$, showing that $\omega$ is well-defined as a map to
$\mathcal{D}_{R}$. By some algebra one shows that $\omega
K_{L}=-K_{R}\check{U}(N_{1}\widetilde{P}_{L}+N_{2}P_{L})$ so that $\omega$ is
a map from $\mathcal{D}_{L}$ to $\mathcal{D}_{R}$. that
$\vec{q}_{R}=\omega\,\vec{q}_{L}$ mod $K_{R}\mathbb{Z}^{D}$, where

$\omega=P_{R}^{\sf{T}}R_{0}+\widetilde{P}_{R}^{\sf{T}}S_{0}\;.$ (3.17)

This integer matrix gives the group isomorphism
$\omega:\mathcal{D}_{L}\to\mathcal{D}_{R}$. With some algebra, one can show
that
$\omega^{\sf{T}}K_{R}^{-1}\omega=K_{L}^{-1}+S_{0}^{\sf{T}}R_{0}+R_{0}^{\sf{T}}S_{0}$.
This implies that

$\bar{h}\bigl{(}\omega\vec{\lambda}\bigr{)}=h\bigl{(}\vec{\lambda}\bigr{)}\pmod{1}\qquad\text{for
all }\vec{\lambda}\in\mathcal{D}_{L}\;,$ (3.18)

namely, that $\omega$ maps the quadratic function $h$ on $\mathcal{D}_{L}$ to
the quadratic function $\bar{h}$ on $\mathcal{D}_{R}$. As proven in [97], the
Abelian Chern-Simons theories constructed with the matrices $K_{L}$ and
$K_{R}$ are equivalent, either at the classical level (by a field
redefinition, if $K_{R}=\check{V}^{\sf{T}}K_{L}\check{V}$ for some integer
unimodular matrix $\check{V}$) or at the quantum level.

Eventually, the torus partition function can be written as

$Z(m,\tau)=\sum_{\vec{\lambda}\,\in\,\mathcal{D}_{L}}\,K_{\vec{\lambda}}^{(K_{L})}(\tau)\;\overline{K_{\omega\vec{\lambda}}^{(K_{R})}(\tau)}\;.$
(3.19)

In the following, in order to avoid clutter, we will assume that the left and
right matrices defining the chiral algebras are equal,

$K_{L}=K_{R}\equiv K\;,$ (3.20)

so that $\omega$ defines an automorphism of
$\mathcal{D}_{0}=\mathbb{Z}^{D}/K\,\mathbb{Z}^{D}$.

#### Chern-Simons bulk dual.

In the three-dimentional bulk we consider the Chern-Simons theory
$U(1)^{D}_{K}\times U(1)^{D}_{-K}$ with action

$S_{\text{CS}}=\frac{K_{ij}}{4\pi}\int\bigl{(}A_{L}^{i}dA_{L}^{j}-A_{R}^{i}dA_{R}^{j}\bigr{)}\;,$
(3.21)

where $i=1,\ldots,D$. Here $K$ is the positive symmetric even integer matrix
constructed above, so that the resulting Chern-Simons theory is bosonic and
has vanishing chiral central charge.323232More generally, one could relax the
condition that $K$ is positive, or take a more general (symmetric even
integer) Chern-Simons matrix that does not have the block-diagonal form
$\big{(}\begin{smallmatrix}K&0\\\ 0&-K\end{smallmatrix}\big{)}$ as we did.
These cases have been studied in [64]. On the left side, $U(1)^{D}_{K}$, the
lines correspond to the elements of $\mathcal{D}_{0}$, and the $S$ and $T$
matrices — that can be read off from the characters (3.12) — are

$\displaystyle S_{\vec{\lambda},\vec{\mu}}$
$\displaystyle=\frac{1}{|K|^{1/2}}\,e^{-2\pi
i\,\vec{\lambda}^{\,{\sf{T}}}K^{-1}\vec{\mu}}$ (3.22) $\displaystyle
T_{\vec{\lambda},\vec{\mu}}$ $\displaystyle=e^{-2\pi
iD/24}\,\theta_{\vec{\lambda}}\,\delta_{\vec{\lambda},\vec{\mu}}\qquad\text{with}\qquad\theta_{\vec{\lambda}}=e^{\pi
i\,|\vec{\lambda}|^{2}_{K^{-1}}}\;,$

where $|K|=\lvert\det{K}\rvert$ is the order of $\mathcal{D}_{0}$. The phase
of $\theta_{\vec{\lambda}}$ is the chiral dimension $h$ in (3.11) mod 1, i.e.,
the spin, and it provides a quadratic refinement of the braiding matrix,
$B(\vec{\lambda},\vec{\mu})=S\makebox[0.0pt]{\;\;$\rule{0.0pt}{8.00003pt}^{*}$}_{\vec{\lambda},\vec{\mu}}S_{0,0}/(S_{0,\vec{\lambda}}S_{0,\vec{\mu}})=\theta_{\vec{\lambda}+\vec{\mu}}/\theta_{\vec{\lambda}}\,\theta_{\vec{\mu}}$.
In the presence of a boundary, we impose holomorphic boundary conditions for
$U(1)^{D}_{K}$ and antiholomorphic for $U(1)^{D}_{-K}$.

The lines of an Abelian Chern-Simons theory form a group under fusion, the
1-form symmetry group, which in this case is
$\mathcal{D}_{0}\times\mathcal{D}_{0}$. We want to gauge a non-anomalous
subgroup $\mathcal{A}$. The anomaly cancelation condition is that each line in
$\mathcal{A}$ has integer spin [55, 33], implying that each line in
$\mathcal{A}$ has trivial braiding with all other lines. Now the total number
of lines is $|K|^{2}$, therefore if $\mathcal{A}$ has order $|K|$ then after
gauging the theory has only one line — the identity — and is trivial in the
bulk (without gravitational anomaly). A particularly simple way to satisfy
both conditions is to find a group isomorphism
$\omega:\mathcal{D}_{0}\to\mathcal{D}_{0}$ that preserves the spin of the
lines, eqn. (3.18).333333In the presence of generic left and right sectors
$K_{L}$, $K_{R}$, the condition is that
$\omega:\mathcal{D}_{L}\to\mathcal{D}_{R}$ satisfies
$\omega^{\sf{T}}K_{R}^{-1}\omega=K_{L}^{-1}+N$ for some symmetric even integer
matrix $N$. Then

$\mathcal{A}=\bigoplus_{\vec{\lambda}\,\in\,\mathcal{D}_{0}}(\vec{\lambda},\omega\vec{\lambda})\;,$
(3.23)

where $(\vec{\lambda},\omega\vec{\lambda})$ is a line of $U(1)^{D}_{K}\times
U(1)^{D}_{-K}$. The gauging procedure and the factorization of the partition
functions follow exactly the same steps as in the previous section, with $k$
replaced by the matrix $K$. After gauging, this theory is holographically dual
to the multi-component compact boson RCFT defined by $(K,\omega)$. In
particular, the partition function on a solid torus is $Z(m,\tau)$ in (3.19).
The triviality of the bulk theory implies factorization and independence of
the bulk geometry.

###  3.2 More general Lagrangian subgroups

As repeated above, the lines of an Abelian Chern-Simons theory form the 1-form
symmetry group of the theory. A subgroup $\mathcal{A}$ is non-anomalous if
each line has integer spin. A non-anomalous subgroup $\mathcal{A}$ is called
_Lagrangian_ if it has maximal order, namely if $|\mathcal{A}|$ is equal to
the square root of the total number of lines. Gauging a Lagrangian subgroup
generates a theory that is trivial in the bulk.

We are interested in theories of the form
$\mathcal{C}_{L}\times\overline{\mathcal{C}}_{R}$, where both
$\mathcal{C}_{L}$ and $\mathcal{C}_{R}$ are Abelian Chern-Simons theories with
positive-definite matrix. On general grounds, Lagrangian subgroups are in
correspondence with topological boundary conditions [58, 59, 67], and we can
use the folding trick to map them to topological interfaces between
$\mathcal{C}_{L}$ and $\mathcal{C}_{R}$. One possibility is that the interface
is invertible: it then represents an isomorphism between the 1-form symmetry
groups of $\mathcal{C}_{L}$ and $\mathcal{C}_{R}$ that preserves the spin.
This is precisely the isomorphism $\omega$ that we described before, and the
Lagrangian subgroup is (3.23).

However, there are more general possibilities. For instance, $\mathcal{C}_{L}$
and $\mathcal{C}_{R}$ could be non-isomorphic. Or, $\mathcal{C}_{L}$ could
have a Lagrangian subgroup $\mathcal{A}_{L}$, $\mathcal{C}_{R}$ could have a
Lagrangian subgroup $\mathcal{A}_{R}$, so that
$\mathcal{A}_{L}\otimes\mathcal{A}_{R}$ is a Lagrangian subgroup of
$\mathcal{C}_{L}\times\overline{\mathcal{C}}_{R}$. This case would produce a
partition function which is the product of a left and a right part, each
separately modular invariant. In the general case, Lagrangian subgroups are
still in correspondence with topological interfaces between $\mathcal{C}_{L}$
and $\mathcal{C}_{R}$, but these might be non-invertible. Two necessary
conditions for the existence of Lagrangian subgroups are that the total number
of lines is a perfect square, and that the signature of the total Chern-Simons
matrix is a multiple of 24.343434The signature of the total CS matrix can just
be a multiple of 8, if we allow stacking by the bosonic invertible theory
$(E_{8})_{1}$. Necessary and sufficient conditions, easily computable, for the
existence of Lagrangian subgroups have recently been found in [67].
Unfortunately, they do not tell us which nor how many inequivalent Lagrangian
subgroups there exist.

We should mention an interesting subtlety: a model whose maximally extended
chiral algebra is $\mathcal{C}$, can also be described in terms of a sub-
algebra $\widetilde{\mathcal{C}}\subset\mathcal{C}$. Let us give a simple
example. The theory $U(1)_{2}\times U(1)_{-2}$ has a unique Lagrangian
subgroup $\mathcal{A}=(0,0)\oplus(1,1)$, which gives rise to the compact boson
CFT at self-dual radius $R^{2}=2$. Also the theory $U(1)_{8}\times U(1)_{-2}$
has a unique Lagrangian subgroup
$\mathcal{A}=(0,0)\oplus(2,1)\oplus(4,0)\oplus(6,1)$, isomorphic to
$\mathbb{Z}_{4}$. Gauging of $\mathcal{A}$ produces the modular-invariant
torus partition function353535One uses the identity
$\sum_{i=0}^{p-1}K^{(kp^{2})}_{p(\lambda+ki)}(\tau)=K^{(k)}_{\lambda}(\tau)$,
which extends to the characters refined by $U(1)$.

$Z=\Bigl{(}K^{(8)}_{0}(\tau)+K^{(8)}_{4}(\tau)\Bigr{)}\overline{K^{(2)}_{0}(\tau)}+\Bigl{(}K^{(8)}_{2}(\tau)+K^{(8)}_{6}(\tau)\Bigr{)}\overline{K^{(2)}_{1}(\tau)}=\left\lvert
K^{(2)}_{0}(\tau)\right\rvert^{2}+\left\lvert
K^{(2)}_{1}(\tau)\right\rvert^{2}$ (3.24)

which describes, once again, the compact boson at $R^{2}=2$. The two models
describe the very same RCFT, but in the latter one uses the chiral algebra
$\mathfrak{u}(1)_{8}$ which is a subalgebra of $\mathfrak{u}(1)_{2}$. The
theory $U(1)_{8}\times U(1)_{-8}$ has three Lagrangian subgroups:
$\mathbb{Z}_{8}^{(1)}$ and $\mathbb{Z}_{8}^{(-1)}$ that we already described,
as well as
$\mathcal{A}=(0,0)\oplus(0,4)\oplus(4,0)\oplus(4,4)\oplus(2,2)\oplus(2,6)\oplus(6,2)\oplus(6,6)$
which is isomorphic to $\mathbb{Z}_{2}\times\mathbb{Z}_{4}$. Gauging the first
two gives the compact boson CFT at $R^{2}=8$ and $R^{2}=1/2$, respectively,
that are dual to each other. Gauging the latter gives, once again, the compact
boson CFT at $R^{2}=2$.

###  3.3 Non-rational theories

We conclude this section with a non-rational example. The starting point is a
non-compact Abelian theory: Chern-Simons with gauge group $\mathbb{R}$ (which
can be easily generalized to $\mathbb{R}^{D}$). We do not have full control
over all details (in particular the normalization of states should be treated
carefully), but we present some interesting ideas. When the gauge group is
$\mathbb{R}$, the level is not physical (it can be rescaled by a field
redefinition) and the action reads

$S_{\mathbb{R}}=\frac{1}{4\pi}\int A\,dA\;.$ (3.25)

This theory has a continuous family of Wilson lines $W_{s}=e^{is\\!\int\\!A}$
labeled by $s\in\mathbb{R}$. With this normalization, the solid torus
partition function gives the $\mathfrak{u}(1)$ characters

$\chi^{\mathbb{R}}_{s}(\tau)=\frac{1}{\eta(\tau)}\,q^{\frac{s^{2}}{2}}\;.$
(3.26)

The modular matrices are

$T_{s,t}=e^{2\pi
i\left(\frac{s^{2}}{2}-\frac{1}{24}\right)}\,\delta(s-t)\;,\qquad
S_{s,t}=e^{-2\pi ist}\;,\qquad C_{s,t}=\delta(s+t)\;.$ (3.27)

We can now stack the theory $\mathbb{R}$ with its parity reversal, that we
indicate as $\overline{\mathbb{R}}$, and look for maximal non-anomalous
subgroups of the 1-form symmetry to be gauged. The definition of a Lagrangian
subgroup should be modified because the theory has an infinite number of
lines. We define a Lagrangian (Abelian) subgroup by:

  1. 1)

The lines in $\mathcal{A}$ must form a group.

  2. 2)

The lines in $\mathcal{A}$ must have integer spin, and thus be mutually
transparent.

  3. 3)

Every other line in the theory must link non-trivially with at least one line
in $\mathcal{A}$.

Condition 3) is the same as asking that $\mathcal{A}$ be maximal, in the sense
that no other line can be added to $\mathcal{A}$ while preserving 1) and 2).
Each line in $\mathcal{A}$ will be of the form $L_{s}\otimes\overline{L}_{t}$.
Asking the lines in $\mathcal{A}$ to have integer spin implies

$(s+t)(s-t)=0\mod 2\;.$ (3.28)

This automatically implies that the lines in $\mathcal{A}$ are mutually
transparent. There are two types of solutions.

One possibility to solve (3.28) is to set $s=\pm t$. If we include in
$\mathcal{A}$ lines with both signs, their composition does not close inside
$\mathcal{A}$. Therefore, without loss of generality, let us consider the
diagonal case $s=t$ (the antidiagonal case $s=-t$ leads to the same result).
The Lagrangian condition implies that all diagonal lines
$L_{s}\otimes\overline{L}_{s}$ appear in $\mathcal{A}$:

$\mathcal{A}_{\infty}=\int ds\,L_{s}\otimes\overline{L}_{s}\;.$ (3.29)

This is a Lagrangian subgroup, whose gauging gives the CFT of a non-compact
free boson (with zero-mode removed). In particular the torus partition
function is

$Z_{T^{2}}=\int
ds\,\left\lvert\chi_{s}^{\mathbb{R}}(\tau)\right\rvert^{2}=\frac{1}{\sqrt{2\tau_{2}}\,|\eta(\tau)|^{2}}\;.$
(3.30)

The integral in (3.29) could have in principle included a non-trivial measure
factor $\mu(s)$. A more refined argument363636In Section 4.1 we describe a
more general formalism, based on modular tensor categories. In (4.18),
$Z_{s}^{\mathcal{A}}$ plays the role of $\mu(s)$. Since the $F$-matrix is
$[F^{stu}_{s+t+u}]_{t+u,s+t}=1$, the associativity condition (4.23) implies
$\mu(s+t)=\mu(t+u)$ for all $s,t,u\in\mathbb{R}$, from which it follows that
$\mu(s)$ must be constant. shows that the measure must be constant, and we set
it to one.

Another possibility to solve (3.28) is to set

$s+t=\frac{2n}{R}\;,\qquad s-t=wR\;,\qquad\text{with }n,w\in\mathbb{Z}$ (3.31)

for some real positive number $R$. The Lagrangian condition forces us to
include all possible values of $n,w$. Thus:

$\mathcal{A}_{R}=\bigoplus_{n,w\in\mathbb{Z}}L_{\frac{n}{R}+\frac{wR}{2}}\otimes\overline{L}_{\frac{n}{R}-\frac{wR}{2}}\;.$
(3.32)

Gauging of this Lagrangian subgroup gives the compact boson CFT at generic
radius $R$, and the torus partition function is indeed (2.2).

The treatment above can be extended to the multi-component case
($\mathbb{R}^{D}$ Chern-Simons theory) in a straightforward way. As expected,
one finds that all the free CFTs of $D$ compact or non-compact (with zero-
modes removed) scalars are described by Lagrangian algebras.

##  4 Non-Abelian case

Let us now move to the case of non-Abelian Chern-Simons theories. They
describe interacting boundary RCFTs, for instance Wess-Zumino-Witten (WZW)
models.373737Coset models can also be described in this way [55], but we will
not consider them here. In the bulk we take $G_{k}\times G_{-k}$ Chern-Simons
theory, where $G$ is a compact Lie group and $k$ is its level. As before, we
impose holomorphic and anti-holomorphic boundary conditions for the two
factors, respectively, in order to obtain a
$\mathfrak{g}_{k}\oplus\overline{\mathfrak{g}}_{k}$ chiral algebra [56].

In the non-Abelian case, the fusion of lines does not give rise to a group
structure but rather to a full-fledged fusion algebra. There might be a subset
of the simple lines, called Abelian, with the property that their fusion with
any other simple line gives a single simple line (as opposed to a composite
line). Under fusion, the set of Abelian lines has the structure of an Abelian
group, the 1-form symmetry of the theory. A non-anomalous subgroup of it could
be gauged, however this is not sufficient to render the theory
trivial.383838For instance, the $SU(2)_{k}$ CS theory has $k+1$ lines
$L_{\lambda}$ labeled by the Dynkin label $\lambda\in\\{0,1,\dots,k\\}$. The
1-form symmetry group is $\mathbb{Z}_{2}=\\{L_{0},L_{k}\\}$. In
$SU(2)_{k}\times SU(2)_{-k}$, the diagonal $\mathbb{Z}_{2}$ given by
$(L_{0},L_{0})$ and $(L_{k},L_{k})$ is non-anomalous and can be gauged,
however the resulting theory is a non-trivial TQFT. The reason is that most
lines are not Abelian. The fact that they are topological, though, implies
that they constitute a sort of symmetry. Indeed, the full set of lines form
the so-called non-invertible 1-form symmetry of the theory, which is a modular
tensor category (MTC).

There exists an analogous concept to the gauging of a symmetry group that
applies to non-invertible symmetries, called anyon condensation [41, 42]. This
is an algebraic procedure that, given an initial MTC and a piece of data
called an “algebra object” $\mathcal{A}$, produces a new MTC corresponding to
the TQFT after gauging. The algebra object must satisfy certain consistency
conditions, that are the analog of picking a non-anomalous subgroup. If the
algebra object has maximal dimension, it is called a Lagrangian algebra and
its condensation produces a trivial theory. It turns out that in $G_{k}\times
G_{-k}$ Chern-Simons theory there always are one or more Lagrangian algebra
objects, whose condensation produces the holographic duals to many RCFTs.

###  4.1 Review of anyon condensation in modular tensor categories

To set our notation, let us review the main data of a modular tensor category
and the procedure of anyon condensation. For more detailed expositions, we
refer, e.g., to [52, 54, 57, 98, 99] for the former and to [42] for the
latter.

#### Modular tensor categories (MTC).

To a Chern-Simons theory (or to a chiral algebra) one associates a category
$\mathcal{C}$ whose objects are the Wilson lines (or the integrable
representations of chiral algebra), and whose morphisms are maps between lines
(or intertwiners between representations). We indicate the objects as $L$.

In a tensor category one can take tensor products of objects and morphisms,
and there is a unique trivial line $L_{0}$ (sometimes indicated also as
$L_{\mathbbm{1}}$) providing the neutral element. The tensor product is
associative. A MTC is semisimple: simple objects $L_{a}$ are the ones whose
only endomorphisms are multiples of the identity, and a category is semisimple
if every object is the direct sum of finitely many simple objects. We label
the simple objects by $a,b,c,\dots$ (comparing with previous sections, they
are the lines $(\lambda,\bar{\mu})$ discussed before). Morphisms from
$L_{a}\otimes L_{b}$ to $L_{c}$ form a vector space
$V_{ab}^{c}=\operatorname{Hom}(L_{a}\otimes L_{b},L_{c})$ over $\mathbb{C}$,
and we label a basis of it by the index $\mu_{ab}^{c}$. The number of
independent morphisms is the fusion coefficient $N_{ab}^{c}=\dim V_{ab}^{c}$,
which is required to be symmetric in $ab$.

For each object $L$ there is a dual object $\check{L}$, associated to the
conjugate representation of chiral algebra, such that the product of $L$ and
$\check{L}$ contains the trivial line: $L\otimes\check{L}=L_{0}+\dots$ (notice
that $\check{L}_{0}=L_{0}$). Dual objects allow us to define dual morphisms
from $L_{c}$ to $L_{a}\otimes L_{b}$. Morphisms and dual morphisms are
associated to trivalent junctions of lines:

$c$$b$$a$$\mu_{ab}^{c}$ $c$$b$$a$$b$$a$$\mu^{ab}_{c}$ . (4.1)

The dimensions of the morphism vector spaces satisfy
$N_{ab}^{c}=N_{\check{a}c}^{b}$ as well as the other relations that follow
from symmetry. Morphisms admit orthogonality and completeness relations:

$a$$b$$\mu$$\mu^{\prime}$$c$$d\quad=\quad\delta_{ab}\,\delta_{\mu\mu^{\prime}}$$a$
$a$$b$$\quad=\quad\displaystyle\sum_{c}\sum_{\mu\in
V_{ab}^{c}}$$a$$b$$a$$b$$c$$\mu$$\mu$ . (4.2)

On the left, $\delta_{\mu\mu^{\prime}}$ can be non-vanishing only if
$N_{cd}^{a}>0$. Besides, one can define a trace and associate to each object
$L$ a number called the quantum dimension and denoted by $\dim L$:

$L$$=\operatorname{Tr}\bigl{(}\operatorname{id}_{L}\bigr{)}=\dim
L=\dim\check{L}\;.$ (4.3)

For simple lines we introduce the notation $d_{a}=\dim L_{a}$. Such quantum
dimensions are completely fixed by $N_{ab}^{c}$. The total dimension of the
category is $\dim\mathcal{C}\equiv\mathcal{D}=\sqrt{\sum_{a}d_{a}^{2}}$.

The MTC is characterized by braiding and fusion matrices, that define linear
relations between isomorphic spaces of morphisms. The brading matrix $R$ is
defined as

$c$$a$$b$$\mu$$=\quad\displaystyle\sum_{\nu\in
V_{ab}^{c}}\;[R^{ab}_{c}]_{\mu\nu}$$c$$b$$a$$\nu$ (4.4)

while the fusion matrix $F$ is defined as393939One can also define the inverse
$G$ of the fusion matrix $F$, satisfying
$\sum\nolimits_{f,\alpha\beta}\;[F^{abc}_{d}]_{(e;\mu\nu)(f;\alpha\beta)}\,[G^{abc}_{d}]_{(f;\alpha\beta)(g;\rho\sigma)}=\delta_{eg}\delta_{\mu\rho}\delta_{\nu\sigma}\;.$
(4.5) One can show that $G$ is expressed in terms of $F$ as
$[G^{abc}_{d}]_{(e;\mu\nu)(f;\rho\sigma)}=\sum\nolimits_{\alpha\beta\gamma\delta}\;[R^{ab}_{e}]^{-1}_{\mu\alpha}[R^{ec}_{d}]^{-1}_{\nu\beta}[F^{cba}_{d}]_{(e;\alpha\beta)(f;\gamma\delta)}[R^{bc}_{f}]_{\gamma\rho}[R^{af}_{d}]_{\delta\sigma}\;.$
(4.6)

$d$$a$$c$$e$$b$$\nu$$\mu$$\displaystyle=\quad\sum_{f}\sum_{\begin{subarray}{c}\rho\in
V_{ab}^{f}\\\ \sigma\in
V_{fc}^{d}\end{subarray}}\;[F^{abc}_{d}]_{(e;\mu\nu)(f;\rho\sigma)}$$d$$a$$f$$c$$b$$\sigma$$\rho$
(4.7)

and it encodes associativity of the fusion product $a\otimes b\otimes c$.
These matrices satisfy several consistency conditions [52]. One set is given
by the pentagon equations

$\sum_{\lambda}[F^{abf}_{e}]_{(g;\beta\gamma)(\ell;\sigma\lambda)}[F^{\ell
cd}_{e}]_{(f;\alpha\lambda)(k;\psi\rho)}=\sum_{h}\sum_{\delta\mu\nu}[F^{abc}_{k}]_{(h;\delta\nu)(\ell;\sigma\psi)}[F^{ahd}_{e}]_{(g;\mu\gamma)(k;\nu\rho)}[F^{bcd}_{g}]_{(f;\alpha\beta)(h;\delta\mu)}\;.$
(4.8)

The indices of morphisms run over the corresponding spaces (when they are not
vanishing), in particular $\lambda\in V_{\ell f}^{e}$, $\delta\in V_{bc}^{h}$,
$\mu\in V_{hd}^{g}$, and $\nu\in V_{ah}^{k}$. Another set is given by the
hexagon equations:

$\sum_{\gamma\delta}[R^{ac}_{e}]_{\alpha\gamma}[F^{bac}_{d}]_{(e;\gamma\beta)(f;\delta\lambda)}[R^{ab}_{f}]_{\delta\mu}=\sum_{g}\sum_{\nu\rho\sigma}[F^{bca}_{d}]_{(e;\alpha\beta)(g;\nu\rho)}[R^{ag}_{d}]_{\rho\sigma}[F^{abc}_{d}]_{(g;\nu\sigma)(f;\mu\lambda)}\;.$
(4.9)

Here $\gamma\in V_{ac}^{e}$, $\delta\in V_{ba}^{f}$, $\nu\in V_{bc}^{g}$,
$\rho\in V_{ga}^{d}$, and $\sigma\in V_{ag}^{d}$.

The matrices $R$ and $F$ are “gauge dependent”, in the sense that they depend
on a choice of basis in the vector spaces of morphisms. Two important gauge-
invariant quantities are the topological spin $\theta_{a}$ and the
$S$-matrix.404040Another one is the Frobenius-Schur indicator
$\varkappa_{a}=d_{a}[F^{aaa}_{a}]_{00}=\pm 1$ for lines such that
$\check{a}=a$. The former is defined by

$\displaystyle\raisebox{-25.6073pt}{ \leavevmode\hbox to138.11pt{\vbox
to51.21pt{\pgfpicture\makeatletter\hbox{\hskip 12.1519pt\lower-28.24806pt\hbox
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}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}{{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{-22.76228pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{{}{}}}{{}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-8.8189pt}{-24.91505pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$a$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{}{{}{}{}{{}}{{{{}{}{}{}}}{{}{}{}{}}}}{} {} {} {}
{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{4.71431pt}{3.8216pt}{8.5359pt}{8.5359pt}{8.5359pt}\pgfsys@curveto{13.25021pt}{8.5359pt}{17.07181pt}{4.71431pt}{17.07181pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{}{{}{}{}{{}}{{{{}{}{}{}}}{{}{}{}{}}}}{} {} {} {}
{}{}{}\pgfsys@moveto{17.07182pt}{0.0pt}\pgfsys@curveto{17.07182pt}{-4.71431pt}{13.25023pt}{-8.5359pt}{8.53592pt}{-8.5359pt}\pgfsys@curveto{6.52907pt}{-8.5359pt}{4.58643pt}{-7.82877pt}{3.04912pt}{-6.5388pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{8.5359pt}\pgfsys@lineto{0.0pt}{22.76228pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$=$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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{}{}{}\pgfsys@moveto{62.59598pt}{0.0pt}\pgfsys@lineto{62.59598pt}{22.76228pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{}{{}{}{}{{}}{{{{}{}{}{}}}{{}{}{}{}}}}{} {} {} {}
{}{}{}\pgfsys@moveto{62.59598pt}{0.0pt}\pgfsys@curveto{62.59598pt}{-4.71431pt}{58.77438pt}{-8.5359pt}{54.06007pt}{-8.5359pt}\pgfsys@curveto{49.34576pt}{-8.5359pt}{45.52417pt}{-4.71431pt}{45.52417pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{}{{}{}{}{{}}{{{{}{}{}{}}}{{}{}{}{}}}}{} {} {} {}
{}{}{}\pgfsys@moveto{45.52415pt}{0.0pt}\pgfsys@curveto{45.52415pt}{4.71431pt}{49.34575pt}{8.5359pt}{54.06006pt}{8.5359pt}\pgfsys@curveto{56.06691pt}{8.5359pt}{58.00955pt}{7.82877pt}{59.54686pt}{6.5388pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{{}}{}\pgfsys@moveto{62.59598pt}{-8.5359pt}\pgfsys@lineto{62.59598pt}{-22.76228pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{{}{}}}{{}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{53.77708pt}{-24.91505pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$a$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{77.51035pt}{-2.86945pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$=\;\;\theta_{a}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}{{}}
{{}{}}{}{}\pgfsys@moveto{113.81104pt}{-22.76228pt}\pgfsys@lineto{113.81104pt}{22.7623pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{117.34404pt}{-24.91505pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$a$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\qquad\qquad\text{or}\qquad\qquad\theta_{a}=\theta_{\check{a}}=\frac{1}{d_{a}}\;\;\raisebox{-19.91684pt}{
\leavevmode\hbox to81.19pt{\vbox
to37.57pt{\pgfpicture\makeatletter\hbox{\hskip 40.59354pt\lower-19.7072pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{}}{{}}{{}}
{}{}{}\pgfsys@moveto{-11.38092pt}{-11.38092pt}\pgfsys@lineto{11.38092pt}{11.38092pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ }}{ }
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-10.69319pt}{-16.37419pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$a$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{{}{}}{}{}\pgfsys@moveto{-11.38092pt}{11.38092pt}\pgfsys@lineto{-3.33344pt}{3.33344pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{{}{}}{}{}\pgfsys@moveto{11.38092pt}{-11.38092pt}\pgfsys@lineto{3.33344pt}{-3.33344pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{}{{}{}{}{{}}{{{{}{}{}{}}} {{{}{}{}{}}} {{}{}{}{}}}}{} {} {} {}
{}{}{}\pgfsys@moveto{-11.38092pt}{11.38092pt}\pgfsys@curveto{-17.65883pt}{17.65883pt}{-27.83772pt}{17.65883pt}{-34.11563pt}{11.38092pt}\pgfsys@curveto{-40.39354pt}{5.10301pt}{-40.39354pt}{-5.07588pt}{-34.11563pt}{-11.35379pt}\pgfsys@curveto{-27.83772pt}{-17.6317pt}{-17.65883pt}{-17.6317pt}{-11.38092pt}{-11.35379pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{}{{}{}{}{{}}{{{{}{}{}{}}}{{{}{}{}{}}}{{}{}{}{}}}}{} {} {} {}
{}{}{}\pgfsys@moveto{11.38092pt}{11.38092pt}\pgfsys@curveto{17.65883pt}{17.65883pt}{27.83772pt}{17.65883pt}{34.11563pt}{11.38092pt}\pgfsys@curveto{40.39354pt}{5.10301pt}{40.39354pt}{-5.07588pt}{34.11563pt}{-11.35379pt}\pgfsys@curveto{27.83772pt}{-17.6317pt}{17.65883pt}{-17.6317pt}{11.38092pt}{-11.35379pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\;,$ (4.10)

it is a phase, $|\theta_{a}|=1$, and it satisfies

$\theta_{a}=\frac{d_{a}\,[F^{a\check{a}a}_{a}]_{00}}{[R^{a\check{a}}_{0}]}=\sum_{c,\mu}\frac{d_{c}}{d_{a}}\,[R^{aa}_{c}]_{\mu\mu}\;,\qquad\qquad\sum_{\nu}[R^{ab}_{c}]_{\mu\nu}[R^{ba}_{c}]_{\nu\rho}=\frac{\theta_{c}}{\theta_{a}\theta_{b}}\,\delta_{\mu\rho}\;.$
(4.11)

The absolute value of the first relation provides an alternative formula for
$d_{a}$. The $S$-matrix is defined by

$S_{ab}=\frac{1}{\mathcal{D}}\;\raisebox{-19.91684pt}{ \leavevmode\hbox
to80.07pt{\vbox to45.62pt{\pgfpicture\makeatletter\hbox{\hskip
44.56107pt\lower-22.81052pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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{}{{}}{}{}{{}{}{}{{}}{{{{}{}{}{}}}{{{}{}{}{}}}{{{}{}{}{}}}{{}{}{}{}}}}{} {} {}
{} {}{}{} {}
{{{}}{{}}}{{{{{}}{{}}}{}}{{{}}{{}}}}{{}}{{}}{}\pgfsys@moveto{2.27626pt}{12.51927pt}\pgfsys@curveto{9.34665pt}{4.09315pt}{8.24771pt}{-8.46974pt}{-0.1784pt}{-15.54013pt}\pgfsys@curveto{-8.60452pt}{-22.61052pt}{-21.16742pt}{-21.51158pt}{-28.23781pt}{-13.08546pt}\pgfsys@curveto{-35.3082pt}{-4.65935pt}{-34.20926pt}{7.90355pt}{-25.78314pt}{14.97394pt}\pgfsys@curveto{-19.3692pt}{20.35582pt}{-10.27325pt}{21.15175pt}{-3.02235pt}{16.96544pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-41.22806pt}{-2.08821pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$a$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\;=\frac{1}{\mathcal{D}}\sum_{c}N^{c}_{\check{a}b}\,d_{c}\,\frac{\theta_{c}}{\theta_{a}\theta_{b}}$
(4.12)

which implies

$b$$a$$\displaystyle=\;\;\frac{S_{ab}}{S_{0b}}$$b$ . (4.13)

In particular $S_{00}=1/\mathcal{D}$ and $S_{0a}=d_{a}/\mathcal{D}$. One can
prove that $S_{ab}=S_{ba}=S_{\check{a}b}^{*}$ and that $S$ is a unitary
matrix. The Verlinde formula [93] is

$N_{ab}^{c}=\sum_{x\in\mathcal{C}}\frac{S_{ax}S_{bx}S^{*}_{cx}}{S_{0x}}\;.$
(4.14)

Together with the matrix $T_{ab}=e^{-2\pi ic_{-}/24}\theta_{a}\delta_{ab}$ and
the charge-conjugation matrix $C_{ab}=\delta_{a\check{b}}$, the $S$-matrix
obeys the modular relations

$(ST)^{3}=S^{2}=C\;,\qquad\qquad C^{2}=\mathbbm{1}\;.$ (4.15)

The chiral central charge $c_{-}$, defined modulo 8, is given by
$\frac{1}{\mathcal{D}}\sum_{a}\theta_{a}d_{a}^{2}=e^{2\pi ic_{-}/8}$.

$b_{12}$$\mu_{1}$$a_{1}$$\nu_{2}$$c_{2}$$b_{23}$$\mu_{2}$$a_{2}$$b_{g-1,g}$$\nu_{g}$$a_{g}$

Figure 5: Left: a basis of line insertions in a genus-$g$ handlebody that
generate the Hilbert space of states on the boundary Riemann surface
$\Sigma_{g}$. Right: example of insertions for $g=2$.

The Hilbert space $\mathcal{H}_{g}$ of the TQFT on a Riemann surface
$\Sigma_{g}$ of genus $g$ is generated by a handlebody ${\sf{S}}\Sigma_{g}$
with line insertions. A basis for the Hilbert space is provided by the
insertion of simple lines $a_{1},\dots,a_{g},c_{2},\dots,c_{g-1}$ along the
non-contractible cycles (similarly to the Abelian case), together with $g-1$
lines $b_{12},b_{23},\dots,b_{g-1,g}$ such that $b_{i,i+1}$ connects the
$i$-th to the $(i+1)$-th non-contractible cycle and at each junction the
homomorphism space is non-trivial. See Figure 5. The total number of states is
thus

$\dim\mathcal{H}_{g}=\sum_{\begin{subarray}{c}a_{1},\dots,a_{g}\\\
c_{2},\dots,c_{g-1}\end{subarray}\,\in\,\mathcal{C}}\;\;\sum_{b_{12},\dots,b_{g-1,g}\,\in\,\mathcal{C}}N_{a_{1}\check{a}_{1}}^{b_{12}}N^{b_{12}}_{a_{2}c_{2}}N_{a_{2}c_{2}}^{b_{23}}\dots
N_{a_{g-1}c_{g-1}}^{b_{g-1,g}}N^{b_{g-1,g}}_{a_{g}\check{a}_{g}}\;.$ (4.16)

Using Verlinde’s formula, this reduces to

$\dim\mathcal{H}_{g}=\mathcal{D}^{2g-2}\sum\nolimits_{a\in\mathcal{C}}d_{a}^{2-2g}\;.$
(4.17)

#### Algebra objects and anyon condensation.

In the case of MTCs whose lines are not all Abelian, gauging a one-form
symmetry is not enough to make the theory trivial in the bulk (and thus
achieve a good holographic dual to a boundary CFT). Instead, one should invoke
a procedure called _anyon condesation_ which is the analog of gauging in the
case of non-invertible symmetries. We keep following the notation of [57], and
refer to [42] for details on condensation. The procedure can be summarized as
follows.

First, one packages a set of lines (possibly with degeneracies) into a
composite line $\mathcal{A}$:

$\mathcal{A}=\sum_{a\in\mathcal{C}}Z_{a}^{\mathcal{A}}\,L_{a}\qquad\qquad
Z_{a}^{\mathcal{A}}\in\mathbb{Z}_{\geq 0}\;.$ (4.18)

What substitutes the group structure is a morphism
$m:\mathcal{A}\otimes\mathcal{A}\to\mathcal{A}$, which we denote by a red dot.
The trivial line is required to be inside $\mathcal{A}$, and this is
represented by a unit morphism $\eta:L_{0}\to\mathcal{A}$:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathcal{A}$$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathcal{A}$$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathcal{A}$$m$$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathcal{A}$$\eta$
. (4.19)

In components, $m$ is represented by a tensor
$m^{c\gamma;\mu}_{a\alpha,b\beta}$ as follows:

$c$$b$$a$$\gamma$$\beta$$\alpha$$\displaystyle=\quad\sum_{\mu\in
V_{ab}^{c}}m_{a\alpha,b\beta}^{c\gamma;\mu}$$c$$a$$b$$\mu$ . (4.20)

Here $\alpha=1,\dots,Z_{a}^{\mathcal{A}}$ labels a choice of basis vectors in
$\operatorname{Hom}(L_{a},\mathcal{A})$ (and similarly for $\beta,\gamma$),
and the blue cup is the projector from $\mathcal{A}$ to the corresponding
vector. The projectors satisfy orthogonality and completeness relations:

$a$$b$$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathcal{A}$$\alpha$$\beta$$=\;\;\delta_{ab}\,\delta_{\alpha\beta}$$a$$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathcal{A}$$\displaystyle=\;\;\;\sum_{a,\alpha}$$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathcal{A}$$a$$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathcal{A}$$\alpha$$\alpha$
. (4.21)

The morphism $m$ is required to be associative and compatible with the unit
morphism:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathcal{A}$$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathcal{A}$$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathcal{A}$$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathcal{A}$$=$$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathcal{A}$$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathcal{A}$$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathcal{A}$$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathcal{A}$$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathcal{A}$$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathcal{A}$$=$$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathcal{A}$$=$$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathcal{A}$$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathcal{A}$
. (4.22)

The pictures corresponds to the equations

$\displaystyle\sum_{\varphi=1}^{Z_{f}^{\mathcal{A}}}\,m_{a\alpha,b\beta}^{f\varphi;\rho}\;m_{f\varphi,c\gamma}^{d\delta;\sigma}=\sum_{e\in\mathcal{C}}\sum_{\varepsilon=1}^{Z_{e}^{\mathcal{A}}}\sum_{\mu\nu}\,m_{a\alpha,e\varepsilon}^{d\delta;\nu}\;m_{b\beta,c\gamma}^{e\varepsilon;\mu}\;[F^{abc}_{d}]_{(e;\mu\nu)(f;\rho\sigma)}$
(4.23)
$\displaystyle\sum_{\gamma=1}^{Z_{0}^{\mathcal{A}}}\sum_{\mu}\,\eta_{\gamma}\,m_{0\gamma,b\beta}^{a\alpha;\mu}=\sum_{\gamma=1}^{Z_{0}^{\mathcal{A}}}\sum_{\mu}\,\eta_{\gamma}\,m_{b\beta,0\gamma}^{a\alpha;\mu}=\delta_{ab}\delta_{\alpha\beta}\;,$
(4.24)

where $\eta_{\gamma}$ are the components of the unit morphism. These
properties define a so-called _algebra object_ $\mathcal{A}$. One requires it
to be _haploid_ , or _connected_ : there is a unique copy of the trivial line
inside $\mathcal{A}$, i.e.,
$\dim\operatorname{Hom}(L_{0},\mathcal{A})=Z_{0}^{\mathcal{A}}=1$.

Besides $m$, one also defines an associative co-morphism
$\Delta:\mathcal{A}\to\mathcal{A}\otimes\mathcal{A}$ and a co-unit
$\epsilon:\mathcal{A}\to L_{0}$, making $\mathcal{A}$ into a co-algebra as
well. One requires $\mathcal{A}$ to be _special_ : in a normalization of
$\Delta$ such that $m\circ\Delta=\operatorname{id}_{\mathcal{A}}$, the
condition is

$\mathcal{A}$$\mathcal{A}$$=$$\mathcal{A}$$\mathcal{A}$$\displaystyle=\;\dim\mathcal{A}=\sum_{a\in\mathcal{C}}Z_{a}^{\mathcal{A}}\,d_{a}\;.$
(4.25)

One requires the Frobenius condition, that allows one to relate morphisms and
co-morphisms by crossing:414141In components, the relations read:
$\sum_{e,\varepsilon}\sum_{\mu\nu}\Delta^{c\gamma,e\varepsilon}_{a\alpha;\mu}\,m^{d\delta;\nu}_{e\varepsilon,b\beta}\,[F^{ceb}_{f}]_{(d;\nu\sigma)(a;\mu\rho)}=\sum_{\phi}m^{f\phi;\rho}_{a\alpha,b\beta}\,\Delta^{c\gamma,d\delta}_{f\phi;\sigma}=\sum_{e,\varepsilon}\sum_{\mu\nu}\Delta^{e\varepsilon,d\delta}_{b\beta;\mu}\,m^{c\gamma;\nu}_{a\alpha,e\varepsilon}\,[G^{aed}_{f}]_{(c;\nu\sigma)(b;\mu\rho)}\;.$
(4.26)

$\mathcal{A}$$\mathcal{A}$$\mathcal{A}$$\mathcal{A}$$=$$\mathcal{A}$$\mathcal{A}$$\mathcal{A}$$\mathcal{A}$$=$$\mathcal{A}$$\mathcal{A}$$\mathcal{A}$$\mathcal{A}$
(4.27)

These properties define a special Frobenius algebra object $\mathcal{A}$.

As opposed to the Abelian case, the line $\mathcal{A}$ is in general not
transparent with respect to itself. Instead, anomaly cancellation corresponds
to the condition

$\mathcal{A}$$\mathcal{A}$$\mathcal{A}$$=$$\mathcal{A}$$\mathcal{A}$$\mathcal{A}$
(4.28)

which in components reads

$m_{a\alpha,b\beta}^{c\gamma;\nu}=\sum_{\mu\in
V_{ba}^{c}}\,m_{b\beta,a\alpha}^{c\gamma;\mu}\;[R^{ab}_{c}]_{\mu\nu}\;.$
(4.29)

An algebra with this property is called _commutative_. Therefore, a gaugeable
(or condensable) algebra $\mathcal{A}$ — the non-invertible analog of a non-
anomalous 1-form symmetry — is a connected commutative special Frobenius
algebra, or equivalently a connected commutative separable algebra, in
$\mathcal{C}$ [42].

The partition function of the “gauged theory” after anyon condensation on an
orientable 3-manifold $M$ is defined as the partition function of the original
theory with the insertion of a fine mesh of lines $\mathcal{A}$ along the dual
edges of a triangulation of $M$, with morphisms $m$ or $\Delta$ at the
vertices (see for instance Figure 7).424242The dual edges of a triangulation
have 4-valent vertices, each of which should be resolved into two 3-valent
vertices. Associativity, commutativity, and the Frobenius property guarantee
that the result is independent from the resolution, as well as from the chosen
triangulation. Note that, since $\mathcal{A}$ is not transparent with respect
to itself, generic meshes not dual to a triangulation would lead to different
(and inconsistent) results. Only in the case that $\mathcal{A}$ is Abelian
(i.e., that all lines $L_{a}$ appearing in the decomposition (4.18) are
Abelian and $Z_{a}^{\mathcal{A}}\in\\{0,1\\}$), a fine mesh can be reduced to
a copy of $H_{1}(M;\mathcal{A})$ — and the normalization factors appearing in
(2.32) are reproduced. We give more details on the Abelian case below. When
$M$ has boundaries with Dirichelet boundary conditions, there are dual edges
of the triangulation that terminate at the boundary (inside the faces that
triangulate it), and one should fix projection maps for all of them. To
compute partition functions, one imposes trivial boundary conditions, as in
Sections 2.3 and 2.4: one terminates each boundary dual edge with a unit/co-
unit morphism $\eta$ (4.19) and includes a boundary term
$(\dim\mathcal{A})^{-\chi(\text{boundary})/2}$ in order to normalize to 1 the
$S^{2}$ partition function of the boundary theory. More general boundary
conditions come into play when computing correlation functions of local
operators placed at the boundary (as discussed in Section 4.3). In this case,
one considers a triangulation such that each insertion point is inside a
boundary face (and no face contains more than one point). The dual edges of
the triangulation can end at those points, and for each of them one chooses a
projector map $\alpha$, as in (4.20), from $\mathcal{A}$ to the representation
$L_{a}$ of chiral algebra (with $Z^{\mathcal{A}}_{a}>0$) that corresponds to
the inserted 2d primary operator. When $Z^{\mathcal{A}}_{a}>1$ there are
multiple choices for the projector, as there exist multiple operators in the
same representation $L_{a}$ in the spectrum of the RCFT.

The lines of the gauged theory are local $\mathcal{A}$-modules $\mathcal{M}$,
and their category $\mathcal{C}_{\mathcal{A}}^{\text{loc}}$ inherits a MTC
structure from $\mathcal{C}$. A (left) $\mathcal{A}$-module is a pair
$(\mathcal{M},\rho_{\mathcal{M}})$ of an object $\mathcal{M}\in\mathcal{C}$
and an action $\rho_{\mathcal{M}}:\mathcal{A}\otimes\mathcal{M}\to\mathcal{M}$
(below on the left), satisfying the compatibility condition in the middle of
the following picture:

$\mathcal{M}$$\mathcal{M}$$\mathcal{A}$$\rho_{\mathcal{M}}$$\mathcal{M}$$\mathcal{M}$$\mathcal{A}$$\mathcal{A}$$=$$\mathcal{M}$$\mathcal{M}$$\mathcal{A}$$\mathcal{A}$$\mathcal{A}$$\mathcal{M}$$\mathcal{M}$$=$$\mathcal{M}$$\mathcal{M}$$\mathcal{A}$
(4.30)

as well as the condition