$\rho_{\mathcal{M}}\circ(\eta\otimes\operatorname{id}_{\mathcal{M}})=\operatorname{id}_{\mathcal{M}}$.
The module is local if it has trivial braiding with $\mathcal{A}$, as on the
right in the picture above. This allows one to define a canonical right action
in terms of the left action,

$\mathcal{M}$$\mathcal{M}$$\mathcal{A}$$\stackrel{{\scriptstyle\text{def}}}{{=}}$$\mathcal{M}$$\mathcal{M}$$\mathcal{A}$$=$$\mathcal{A}$$\mathcal{M}$$\mathcal{M}$
(4.31)

and, by virtue of the commutativity of $\mathcal{A}$, the two actions commute.
This makes $\mathcal{M}$ into an $\mathcal{A}$-bimodule as well. Notice that
$(\mathcal{A},m)$ is always a local $\mathcal{A}$-module, which corresponds to
the trivial line of $\mathcal{C}_{\mathcal{A}}^{\text{loc}}$ and of the gauged
theory. The total dimension of the category
$\mathcal{C}_{\mathcal{A}}^{\text{loc}}$ is
$\dim\mathcal{C}_{\mathcal{A}}^{\text{loc}}=\mathcal{D}/\dim\mathcal{A}$ [100,
101]. Hence, the gauged theory is trivial in the bulk if and only if

$\dim\mathcal{A}=\mathcal{D}\;.$ (4.32)

This condition defines a Lagrangian algebra object (or anyon) $\mathcal{A}$.

One can prove that a condensable anyon has trivial spin,434343One considers a
loop as in (4.10), but where the anyon running in the loop is $\mathcal{A}$.
On the one hand, $\mathcal{A}$ can be decomposed into simple lines. On the
other hand, the properties of a special Frobenius algebra together with
commutativity allow one to untwist the loop. This leads to the equation
$\sum_{a}Z^{\mathcal{A}}_{a}d_{a}\theta_{a}=\sum_{a}Z^{\mathcal{A}}_{a}d_{a}$.
in other words

$\theta_{a}=1\qquad\forall\,a\text{ such that }Z^{\mathcal{A}}_{a}>0\;.$
(4.33)

Moreover, a Lagrangian condensable anyon is modular invariant [57, 67]:

$\sum_{b\in\mathcal{C}}S_{ab}\,Z_{b}^{\mathcal{A}}=Z_{a}^{\mathcal{A}}\;.$
(4.34)

#### Gauss law.

Given a commuting special Frobenius algebra $\mathcal{A}$, the effect of its
condensation is encoded in a projector444444This projector was defined in [57]
for a generic Frobenius algebra object, but it simplifies in the commuting
case. $P_{\mathcal{A}}(L)\in\operatorname{Hom}(\mathcal{A}\otimes
L,\mathcal{A}\otimes L)$. In the following, we will only apply it to simple
lines $L_{b}$ and write $P_{\mathcal{A}}(b)$. The projector is defined as

$P_{\mathcal{A}}(b)\;\;\equiv$$\mathcal{A}$$b$$=$$\mathcal{A}$$b$ (4.35)

and is easily seen to satisfy $P_{\mathcal{A}}(b)\circ
P_{\mathcal{A}}(b)=P_{\mathcal{A}}(b)$ [57]. In the case that $\mathcal{A}$ is
Lagrangian, using modular invariance of the vector $Z_{a}^{\mathcal{A}}$, one
computes the trace of $P_{\mathcal{A}}(b)$:

$\operatorname{Tr}P_{\mathcal{A}}(b)\;\;=$$\mathcal{A}$$b$$=$$b$$\mathcal{A}$$=\;\;\mathcal{D}\,Z_{b}^{\mathcal{A}}\;.$
(4.36)

This shows that $P_{\mathcal{A}}$ only has support on the lines
$L_{b}\in\mathcal{A}$.

For Lagrangian $\mathcal{A}$, the projector can equivalently be represented as

$\displaystyle
P_{\mathcal{A}}(b)\,=\,\frac{\dim\mathcal{A}}{d_{b}}\;\sum_{\beta=1}^{Z_{b}^{\mathcal{A}}}$$\mathcal{A}$$\mathcal{A}$$\mathcal{A}$$b$$b$$\beta$$\beta$
. (4.37)

To prove it, one uses the orthogonality between $\mathcal{A}$-modules [57],
depicted in Figure 6 on the left. Specializing to the case that the two
modules are $(\mathcal{A},m)$, composing with projection maps $\delta$ and
$\gamma$ above and below, respectively, and summing over them, one obtains the
relation in Figure 6 on the right. Let us call $\widehat{P}_{\mathcal{A}}(b)$
the operator in (4.37) and relax the assumption that $\mathcal{A}$ is
Lagrangian. The relation in Figure 6 implies that
$\widehat{P}_{\mathcal{A}}(b)$ is a projector,
$\widehat{P}_{\mathcal{A}}(b)\circ\widehat{P}_{\mathcal{A}}(b)=\widehat{P}_{\mathcal{A}}(b)$.
One also easily shows that placing the operator $\widehat{P}_{\mathcal{A}}(b)$
along a 1-cycle is equivalent to placing the line
$Z_{b}^{\mathcal{A}}\mathcal{A}$, that
$\operatorname{Tr}\widehat{P}_{\mathcal{A}}(b)=Z_{b}^{\mathcal{A}}\dim\mathcal{A}$,
as well as that
$\mathcal{P}_{\mathcal{A}}(b)\circ\widehat{\mathcal{P}}_{\mathcal{A}}(b)=\widehat{\mathcal{P}}_{\mathcal{A}}(b)\circ\mathcal{P}_{\mathcal{A}}(b)=\widehat{\mathcal{P}}_{\mathcal{A}}(b)$.
In the case that $\mathcal{A}$ is Lagrangian, they imply that
$\widehat{P}_{\mathcal{A}}(b)=P_{\mathcal{A}}(b)$ [102].

$c$$b$$d$$\gamma$$\beta$$\alpha$$\delta$$\mathcal{M}_{\kappa^{\prime}}$$\mathcal{A}$$\mathcal{M}_{\kappa}$$\displaystyle=\;\;\frac{d_{b}}{\dim\mathcal{M}_{\kappa}}\;\delta_{\kappa\kappa^{\prime}}\,\delta_{\alpha\beta}\,\delta^{(d,\delta)}_{(c,\gamma)}$$c$

$\mathcal{A}$$b$$\mathcal{A}$$\alpha$$\beta$$\mathcal{A}$$\displaystyle=\;\;\frac{d_{b}}{\dim\mathcal{A}}\;\delta_{\alpha\beta}$$\mathcal{A}$

Figure 6: Left: orthogonality of $\mathcal{A}$-modules. The index $\kappa$
labels different modules. Right: this relation is obtained from the left by
setting $\mathcal{M}_{\kappa}=\mathcal{M}_{\kappa^{\prime}}=\mathcal{A}$ and
$\rho_{\mathcal{M}}=m$, composing with the projections $\gamma$ below and
$\delta$ above, and then summing over $c,\gamma,d,\delta$.

#### The Abelian case.

A line $L_{a}$ is said to be Abelian when $d_{a}=1$. Abelian lines have unique
fusion with all other lines, namely $\sum_{c}N_{ab}^{c}=1$ for all $b$. The
set of Abelian lines forms a finite Abelian group under fusion: the 1-form
symmetry group of the bulk theory. An algebra object $\mathcal{A}$ is said to
be invertible when it is made of Abelian lines with no degeneracies, namely
when $Z_{a}^{\mathcal{A}}\in\\{0,1\\}$ and
$d_{a}Z_{a}^{\mathcal{A}}=Z_{a}^{\mathcal{A}}$. In this case the components
$L_{i}$ of $\mathcal{A}$ form an Abelian subgroup of the total 1-form symmetry
group, that we also denote as $\mathcal{A}$. The morphism components reduce to
$m_{ij}^{k}$, which can be non-zero only when $i\otimes j=k$. We can thus
regard them as a normalized group-cohomology 2-cochain $m\in
C^{2}(\mathcal{A};\mathbb{C}^{*})$ labelled by
$ij\in\mathcal{A}\otimes\mathcal{A}$, with $m_{0a}^{a}=m_{a0}^{a}$. If we
regard $[F^{ijk}_{\ell}]_{mn}$ as a 3-cochain labelled by $ijk$ (because
$m=j\otimes k$, $n=i\otimes j$ and $\ell=i\otimes j\otimes k$), then the
pentagon equation (4.8) reduces to $dF=1$ (using multiplicative notation),
while the associativity condition (4.23) reduces to

$F=(dm)^{-1}\;,$ (4.38)

namely $F$ must be trivial in $H^{3}(\mathcal{A};\mathbb{C}^{*})$. If we
regard $[R^{ij}_{k}]$ as a 2-cochain labelled by $ij$, then the commutativity
condition (4.28) reduces to $R^{ij}_{k}=m_{ij}^{k}/m_{ji}^{k}$ and one easily
checks that the anomaly cancelation condition $\theta_{i}=1$ is satisfied.

One can prove the following relation for invertible special Frobenius algebras
$\mathcal{A}$:

$\displaystyle=\;\;\;\frac{1}{\dim\mathcal{A}}$ . (4.39)

The dotted notation means that the line should close, however there could be
other lines passing in the middle, or the line could close through a
topologically non-trivial 1-cycle. The relation holds because, in the
invertible case, only the identity can run along the horizontal channel.
Applying the relation to $P_{\mathcal{A}}(b)$ we immediately reproduce the
construction we used in the Abelian case, in Section 2.4. In particular,
$P_{\mathcal{A}}(b)=Z_{b}^{\mathcal{A}}\,\operatorname{id}_{\mathcal{A}}\otimes\operatorname{id}_{b}$.

###  4.2 Partition functions and factorization

There is a correspondence between RCFTs with a given chiral algebra, and
condensable Lagrangian anyons in the MTC
$\mathcal{C}=\mathcal{C}_{L}\times\mathcal{C}_{R}$, where $\mathcal{C}_{L}$
and $\overline{\mathcal{C}}_{R}$ (the overline stands for orientation
reversal) correspond to the left and right part of the chiral algebra (that in
general could be different), respectively [53, 57, 102, 59]. On the one hand,
given a (bosonic) RCFT, its torus partition function takes the form

$Z_{T^{2}}^{\mathcal{A}}=\sum_{\mu\in\mathcal{C}_{L},\;\bar{\nu}\in\mathcal{C}_{R}}Z^{\mathcal{A}}_{\mu\bar{\nu}}\;\chi_{\mu}(\tau)\,\chi_{\bar{\nu}}(\bar{\tau})\;,$
(4.40)

where $Z_{\mu\bar{\nu}}^{\mathcal{A}}$ are non-negative integers while $\chi$
are characters. One can prove that

$\mathcal{A}\,\equiv\,\sum_{\mu\in\mathcal{C}_{L},\;\bar{\nu}\in\mathcal{C}_{R}}Z^{\mathcal{A}}_{\mu\bar{\nu}}\;L_{\mu}\otimes
L_{\bar{\nu}}$ (4.41)

is a condensable Lagrangian anyon, with a suitable morphism $m$. With respect
to our previous notation, simple lines are labeled by $a=(\mu,\bar{\nu})$. For
instance, modular invariance of the torus partition function,
$\sum_{a\in\mathcal{C}}S_{ab}Z_{b}^{\mathcal{A}}=Z_{a}^{\mathcal{A}}$, and
uniqueness of the vacuum, $Z_{0}^{\mathcal{A}}=1$ where the index $0=(0,0)$,
imply that $\mathcal{A}$ is Lagrangian (recall that
$d_{a}=\mathcal{D}S_{0a}$):

$\dim\mathcal{A}=\sum_{a\in\mathcal{C}}Z_{a}^{\mathcal{A}}d_{a}=\mathcal{D}\;.$
(4.42)

Commutativity, instead, follows from the fact that physical conformal
primaries $\mathcal{O}_{a}$, with $a\subset\mathcal{A}$, of the RCFT have
trivial braiding.

On the contrary, consider a bulk Chern-Simons theory $G_{k}\times G_{-k}$
(that can be generalized in various ways) with a given condensable Lagrangian
line $\mathcal{A}$ as in (4.41). We want to determine the partition functions
of the theory $\mathcal{T}$ obtained after condensation of $\mathcal{A}$, on
oriented three-manifolds with boundaries and holomorphic/anti-holomorphic
boundary conditions for $G_{k}$ and $G_{-k}$, respectively. Let us start from
handlebodies.

Figure 7: Left: triangulation of the ball ${\sf{S}}S^{2}=D_{3}$ by a single
tetrahedron, and dual network. Right: triangulation of the solid torus
${\sf{S}}T^{2}$ by three tetrahedra. We identify the right face with the left
face, and the two faces in the front with the two in the back, while the two
upper faces triangulate the boundary $T^{2}$. The dual network has been drawn
only after simplification, and it wraps the non-contractible cycle.

The case of ${\sf{S}}S^{2}=D_{3}$ is simple: this geometry can be triangulated
by a single tetrahedron, and its faces triangulate the boundary $S^{2}$
(Figure 7 on the left). The dual network reduces to the segment in (4.25) on
the right, equal to $\dim\mathcal{A}=\mathcal{D}$. The Chern-Simons partition
function is $Z_{\text{CS}}[D_{3}]=1$, therefore, in order to normalize to 1
the $S^{2}$ partition function of the boundary theory, we include a boundary
counterterm454545The choice of counterterm is related to the choice of
normalizations in (4.25). For instance, in a normalization in which the bubble
is equal to $\dim\mathcal{A}$ while the segment is equal to 1, the boundary
counterterm would be trivial, but there would be a bulk counterterm to ensure
independence from the triangulation.
$(\dim\mathcal{A})^{-\chi(\text{boundary})/2}$. This gives

$Z_{\mathcal{T}}[D_{3}]=Z_{\text{CS}}[D_{3}]=1\;.$ (4.43)

The solid torus ${\sf{S}}T^{2}$ can be triangulated by three tetrahedra (with
suitable gluing of the faces, see Figure 7 on the right), and two faces
triangulate the boundary $T^{2}$. The dual network reduces to a loop of
$\mathcal{A}$ along the non-contractible cycle. The partition function of
Chern-Simons theory on the solid torus with holomorphic boundary conditions
and a simple line insertion reproduces the corresponding character of chiral
algebra (or extensions thereof, if $G$ is not simply connected) [56, 77, 79].
This has been shown very explicitly in [79]. The partition function of the
gauged theory, thus, reproduces the torus partition function (4.40).

Triangulation of a handlebody ${\sf{S}}\Sigma_{g}$ produces (after
simplifications) a dual network as in Figure 5 on right, in which the lines
are $\mathcal{A}$ and the (co)morphisms are $m$ or $\Delta$. Once again, the
partition function of the three-dimensional gauged theory $\mathcal{T}$, with
holomorphic boundary conditions, is equal to the partition function of the
RCFT on $\Sigma_{g}$. Using the identity on the right of (4.21) on each
segment of the line $\mathcal{A}$, and the definition (4.20) of the components
of $m$ (and the similar definition for $\Delta$), the network gets rewritten
in terms of the insertions in Figure 5 on the left, with: $(i)$ for each
segment of $\mathcal{A}$, a sum over representations $a\in\mathcal{A}$ and a
sum over projectors $\alpha:\mathcal{A}\to L_{a}$ at the two ends; $(ii)$ for
each junction between lines with labels $a,b,c$, a sum over morphisms $\mu\in
V_{ab}^{c}$ (or co-morphisms $\mu\in V^{ab}_{c}$, depending on orientation).
Each term in these sums contains a coefficient
$m_{a\alpha,b\beta}^{c\gamma;\mu}$ for each morphism and
$\Delta^{a\alpha,b\beta}_{c\gamma;\mu}$ for each co-morphism.464646Each
segment connecting the loops in the network of Figure 5 will have a morphism
at one junction, and a co-morphism at the other junction. The choice of
normalization (4.25) for the bubble fixes the normalization of the components
of $\Delta$ in terms of those of $m$. Since we have $g-1$ insertions of
$\Delta$, we see that this normalization enters with a power of $g-1$, and is
related to the Euler counterterm $(\dim\mathcal{A})^{1-g}$. The partition
function of $G_{k}\times G_{-k}$ Chern-Simons theory with the insertions of
Figure 5 (left) gives the (product of left and right) conformal blocks of the
chiral algebra on $\Sigma_{g}$ [44]. The conformal blocks are labeled by the
primaries $a,\dots$ and the morphisms $\mu,\dots$ in Figure 5, while do not
depend on the projector labels $\alpha,\dots$ which instead reflect the
multiplicities of chiral algebra representations in the operator content of
the RCFT (if one works with the maximally extended chiral algebra, then there
are no multiplicities [53]). What we described from the condensation of
$\mathcal{A}$ are precisely the ingredients of the partition function of the
RCFT on $\Sigma_{g}$ [52, 53, 54] (in the Abelian case one recovers the
construction of higher-genus partition functions of, e.g., [74] that we
reviewed in Section 2.1). Indeed, when the bulk geometry is a handlebody, it
is always possible to pull the condensation network of $\mathcal{A}$ to the
boundary: the lines become Verlinde lines [93] and one reduces to the pair-of-
pants decomposition of 2d partition functions.

The partition functions of $\mathcal{T}$ on general oriented three-dimensional
manifolds $M$ can be computed using the fact that $\mathcal{T}$ is completely
trivial in the bulk — the Hilbert space on any closed two-dimensional spatial
manifold is one-dimensional, and the partition function on any closed three-
manifold is $1$ — and then applying the very same formal argument that led to
(2.53) and depicted in Figure 4. As in the Abelian case, it implies that:

  * •

The partition function only depends on the boundary conditions, and not on the
particular 3d geometry that is used in the bulk. In particular, such a
background independence implies that no sum over geometries should be
performed.

  * •

When the boundary consists of a connected Riemann surface $\Sigma_{g}$, the
partition function is the one computed by the corresponding handlebody. When
the boundary consists of multiple components, the partition function
factorizes as the product of partition functions on handlebodies, as in
(2.53).

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Factorization of the $T^{2}\times I$ partition function of the gauged theory
$\mathcal{T}$. Left: the $T^{2}\times I$ geometry (the upper and lower faces
are indentified in all pictures) with the network of algebra object
$\mathcal{A}$ inserted. Middle: two solid tori, obtained by inserting the
completeness of states at some point along $I$. We recognise the projector
$\mathcal{P}_{\mathcal{A}}(b)$. Right: The ${\sf{S}}T^{2}\times{\sf{S}}T^{2}$
partition function of $\mathcal{T}$. This picture is the non-Abelian analog of
Figure 3.

Alternatively, one can exhibit factorization by explicitly constructing a
triangulation of the multi-boundary geometry and computing the expectation
value of the dual network of $\mathcal{A}$. Let us show how this works for the
simple case of the $T^{2}\times I$ Euclidean wormhole, confirming that it
factorizes into the product of two partition functions on solid tori, as
expected from the point of view of the boundary theory. The manifold
$T^{2}\times I$ is an annulus times a circle, and we represent it on the LHS
of Figure 8. In the same figure we indicate the network of $\mathcal{A}$
produced by a triangulation [57]. We cut $T^{2}\times I$ along a $T^{2}$ at
some point of the interval $I$, and insert a complete basis of states on
$T^{2}$ represented by solid tori with the insertion of lines $L_{b}$. This
produces the expression in the middle of Figure 8, in which the projector
$\mathcal{P}_{\mathcal{A}}(b)$ of (4.35) appears. Using the expression (4.37)
for $\mathcal{P}_{\mathcal{A}}(b)$, and its reduction to
$Z^{\mathcal{A}}_{b}\mathcal{A}$ when placed on a cycle, we finally obtain the
RHS of Figure 8, which is the partition function of $\mathcal{T}$ on
${\sf{S}}T^{2}\times{\sf{S}}T^{2}$. A similar argument may be extended to the
more general case $\Sigma_{g}\times I$ (see Appendix D).

###  4.3 Correlators

$=$

Figure 9: Triangulation of the ball ${\sf{S}}S^{2}=D_{3}$ with three local
operator insertions on the boundary. It is triangulated by a single
tetrahedron, with three of the boundary faces containing an insertion point.
The dual triangulation is attached to those faces via a projector from
$\mathcal{A}$ to the chiral algebra representation of the corresponding
operator. The network that computes the three-point function can be simplified
as shown on the right.

Let us now discuss how to obtain correlation functions of the RCFT using the
dual description in terms of the topological theory $\mathcal{T}$ after
condensation of $\mathcal{A}$. The physical primary operators are labeled by
the representations $a$ of the (left and right) chiral algebra with
$Z^{\mathcal{A}}_{a}>0$, and in addition by the projectors $\alpha$ from
$\mathcal{A}$ to $L_{a}$, that run over the $Z^{\mathcal{A}}_{a}$ copies of
the representation in the physical spectrum of the RCFT. In order to obtain
correlations functions, as explained in Section 4.1, we take a fine enough
triangulation of the bulk three-manifold $M$ such that each boundary local
operator is inside one boundary face, and no face contains more than one
operator. One of the edges of the dual triangulation will terminate on the
local operator $(a,\alpha)$ at the boundary, and the prescription is to place
the projector $\alpha$ at that point (while edges landing at boundary points
where no operator is inserted, terminate with the unit projector $\eta$). This
is depicted in Figure 9 for the case of a three-point function on $S^{2}$. The
properties of $\mathcal{A}$ ensure that the correlators obtained this way
satisfy modular invariance in the extended sense of a surface with punctures,
that includes crossing symmetry. This construction of physical correlators,
when the bulk geometry is a handlebody, can be mapped to that of [52, 53, 54]
by pulling the network to the boundary.

###  4.4 Examples

We conclude this section by presenting some concrete examples of algebra
objects in $U(1)_{k}$ and $SU(2)_{k}$, and by algebraically checking the
factorization of the $T^{2}\times I$ partition function after condensation. We
exploit a formula given in [57] for the topological partition function
$Z_{ab}^{\mathcal{A}}$ with insertion of a dual network of the algebra object
$\mathcal{A}$ in the topology $T^{2}\times I$, in the case that $N_{ab}^{c}$
and $Z_{a}^{\mathcal{A}}$ take values in $\\{0,1\\}$:474747With the notation
$a,b,c,\ldots\in\mathcal{A}$ we indicate all simple anyons $L_{a}$ such that
$Z^{\mathcal{A}}_{a}>0$.

$Z^{\mathcal{A}}_{ab}=\sum_{d,e,f\in\mathcal{A}}m_{de}^{f}\,\Delta^{de}_{f}\sum_{g\in\mathcal{C}}\frac{\theta_{g}}{\theta_{b}}\,[F^{deb}_{\hat{a}}]_{gf}\,[G^{deb}_{\hat{a}}]_{fg}\;.$
(4.44)

This quantity is the transition amplitude from the state $|a\rangle$ in
$\mathcal{H}(T^{2})$ to the state $|b\rangle$, and the physical partition
function of the gauged theory is then

$Z_{\mathcal{T}}[T^{2}\times
I]=\sum_{(\mu,\bar{\nu}),(\rho,\bar{\sigma})\in\mathcal{C}}Z_{(\mu,\bar{\nu})(\rho,\bar{\sigma})}^{\mathcal{A}}\,\chi_{\mu}(\tau_{1})\chi_{\bar{\nu}}(\bar{\tau}_{1})\,\chi_{\rho}(\tau_{2})\chi_{\bar{\sigma}}(\bar{\tau}_{2})\;,$
(4.45)

where $\tau_{1}$, $\tau_{2}$ are the modular parameters of the two torus
boundaries. Here we have specialized the formula of [57] to the case of a
commuting algebra $\mathcal{A}$. We also introduced the inverse matrix $G$, as
in (4.6), and the co-morphism $\Delta$, which can be expressed as

$[G^{abc}_{d}]_{ef}=\frac{R^{bc}_{f}\,R^{af}_{d}}{R^{ab}_{e}\,R^{ec}_{d}}\,[F^{cba}_{d}]_{ef}\;,\qquad\qquad\Delta^{ab}_{c}=\frac{1}{\dim\mathcal{A}}\,\frac{m_{\check{a}c}^{b}\,[F^{a\check{a}c}_{c}]_{b0}}{m_{\check{a}a}^{0}\,[F^{a\check{a}a}_{a}]_{00}}\;.$
(4.46)

Since here we consider the non-chiral case that the total category is
$\mathcal{C}=\mathcal{C}_{L}\times\overline{\mathcal{C}}_{L}$, using the
double-index notation we have484848The matrices of
$\overline{\mathcal{C}}_{L}$ are obtained from those of $\mathcal{C}_{L}$ with
a parity transformation. This leads to
$\bigl{[}\,\overline{R}^{ab}_{c}\bigr{]}_{\mu\nu}=[R^{ab}_{c}]^{-1}_{\nu\mu}\;,\qquad\qquad\bigl{[}\,\overline{F}^{abc}_{d}\,\bigr{]}_{(e;\mu\nu)(f;\rho\sigma)}=[G^{abc}_{d}]_{(f;\rho\sigma)(e;\mu\nu)}\;.$
(4.47) Exploiting the fact that $R^{ab}_{c}$, $F^{abc}_{d}$ and $G^{abc}_{d}$
are unitary matrices (and specializing to the case that the fusion
coefficients $N\in\\{0,1\\}$), one obtains the formulas in the main text.

$R^{(\mu,\bar{\mu})\,(\nu,\bar{\nu})}_{(\rho,\bar{\rho})}=R^{\mu\nu}_{\rho}\,\bigl{(}R^{\bar{\mu}\bar{\nu}}_{\bar{\rho}}\bigl{)}^{*}\;,\qquad\qquad\bigl{[}F^{(\mu,\bar{\mu})\,(\nu,\bar{\nu})\,(\rho,\bar{\rho})}_{(\sigma,\bar{\sigma})}\bigr{]}_{(\lambda,\bar{\lambda})\,(\kappa,\bar{\kappa})}=[F^{\mu\nu\rho}_{\sigma}]_{\lambda\kappa}\,[F^{\bar{\mu}\bar{\nu}\bar{\rho}}_{\bar{\sigma}}]^{*}_{\bar{\lambda}\bar{\kappa}}\;,$
(4.48)

and so on. Factorization is the statement that

$Z_{ab}^{\mathcal{A}}=Z^{\mathcal{A}}_{a}\,Z^{\mathcal{A}}_{b}\;.$ (4.49)

This immediately follows from Figure 8, i.e., it is a special case of the
Gauss law action.

#### The case of $\boldsymbol{U(1)_{k}}$.

We consider $k\in 2\mathbb{N}$. There are $k$ simple lines labelled by
$\mu\in\mathbb{Z}_{k}$, forming the Abelian group $\mathbb{Z}_{k}$ under
fusion, $L_{\mu}\otimes L_{\nu}=L_{\mu+\nu}$, thus
$N_{\mu\nu}^{\rho}=\delta_{\mu+\nu=\rho\text{ mod }k}$. The conjugate line is
$\check{\mu}=k-\mu$. The theory is Abelian and the quantum dimensions of all
simple lines are $d_{\mu}=1$. The topological spins, $R$-matrices and
$F$-matrices are

$\theta_{\mu}=e^{i\pi\frac{\mu^{2}}{k}}\;,\qquad
R^{\mu\nu}_{\rho}=(-1)^{\mu\left\lfloor\\!\frac{\nu}{k}\\!\right\rfloor+\,\nu\left\lfloor\\!\frac{\mu}{k}\\!\right\rfloor}\,e^{i\pi\frac{\mu\nu}{k}}\;,\qquad[F^{\mu\nu\rho}_{\sigma}]_{\lambda\kappa}=(-1)^{\mu\,\left(\left\lfloor\\!\frac{\nu+\rho}{k}\\!\right\rfloor-\left\lfloor\\!\frac{\nu}{k}\\!\right\rfloor-\left\lfloor\\!\frac{\rho}{k}\\!\right\rfloor\right)}\;,$
(4.50)

where $\lfloor x\rfloor\in\mathbb{Z}$ is the floor function. Recall that
$R^{\mu\nu}_{\rho}$ is defined only when $N_{\mu\nu}^{\rho}>0$, and similarly
$[F^{\mu\nu\rho}_{\sigma}]_{\lambda\kappa}$ is defined only when
$N_{\nu\rho}^{\lambda}N_{\mu\lambda}^{\sigma}N_{\mu\nu}^{\kappa}N_{\kappa\rho}^{\sigma}>0$.
The total quantum dimension is $\mathcal{D}=\sqrt{k}$ and the chiral central
charge is $c_{-}=1$. The $S$-matrix is $S_{ab}=e^{-2\pi i\frac{ab}{k}}$. The
two Frobenius-Schur indicators are $\varkappa_{0}=1$,
$\varkappa_{k/2}=(-1)^{k/2}$.

The modular invariant associated to the integer $\omega$ is described by the
Lagrangian condensable anyon

$\mathcal{A}=\bigoplus_{\mu\in\mathbb{Z}_{k}}L_{\mu}\otimes
L_{\omega\mu}\;,\qquad\qquad m_{ab}^{c}\equiv
m_{(\mu,\omega\mu)(\nu,\omega\nu)}^{(\rho,\omega\rho)}=(-1)^{\mu\left\lfloor\\!\frac{\nu}{k}\\!\right\rfloor+\,\omega\mu\left\lfloor\\!\frac{\omega\nu}{k}\\!\right\rfloor}\;,$
(4.51)

where $m_{ab}^{c}$ is defined only when $N_{\mu\nu}^{\rho}>0$, and the unit
morphism is $\eta=1$. In particular
$Z^{\mathcal{A}}_{(\mu,\bar{\mu})}=\delta_{\bar{\mu}=\omega\mu\text{ mod }k}$.
One can check that the factorization property (4.49) is indeed satisfied.

#### The case of $\boldsymbol{SU(2)_{k}}$.

Next, we consider the WZW model $SU(2)_{k}$. To this end, we recall some basic
facts about that the $SU(2)_{k}$ TQFT. There are $k+1$ simple lines labeled by
the $SU(2)$ Dynkin label $\mu=0,1,\dots,k$. The fusion algebra is

$L_{\mu}\otimes L_{\nu}=\sum_{\begin{subarray}{c}\rho=|\mu-\nu|\\\
\rho=\mu-\nu\text{ mod
}2\end{subarray}}^{\min(\mu+\nu,\,2k-\mu-\nu)}L_{\rho}\;,$ (4.52)

from which the fusion coefficients $N_{\mu\nu}^{\rho}$ follow. Each line is
self-conjugate: $\check{\mu}=\mu$. The quantum dimensions, topological spins,
$R$-matrices and $F$-matrices are

$\displaystyle d_{\mu}$
$\displaystyle=[\mu+1]\;,\qquad\qquad\theta_{\mu}=e^{2\pi
i\Delta_{\mu}}\;,\qquad\qquad\Delta_{\mu}=\frac{\mu(\mu+2)}{4(k+2)}\;,$ (4.53)
$\displaystyle R^{\mu\nu}_{\rho}$
$\displaystyle=(-1)^{\frac{\mu+\nu-\rho}{2}}\,e^{i\pi\left(\rule{0.0pt}{4.20004pt}\Delta_{\rho}-\Delta_{\mu}-\Delta_{\nu}\right)}\,N^{\rho}_{\mu\nu}\;,$
$\displaystyle[F^{\mu\nu\rho}_{\sigma}]_{\lambda\kappa}$
$\displaystyle=\biggl{\\{}\begin{matrix}\rho/2&\nu/2&\lambda/2\\\
\mu/2&\sigma/2&\kappa/2\end{matrix}\biggr{\\}}\,N_{\mu\lambda}^{\sigma}N^{\lambda}_{\nu\rho}N^{\kappa}_{\mu\nu}N^{\sigma}_{\kappa\rho}\;,$

where we introduced the $q$-number and $q$-factorial:

$[n]=\frac{\sin\left(\frac{\pi
n}{k+2}\right)}{\sin\left(\frac{\pi}{k+2}\right)}\;,\qquad\qquad[n]!=\prod_{m=1}^{n}[m]\;,\qquad\qquad[0]!=[1]!=1\;.$
(4.54)

Then the curly bracket symbol is defined by

$\displaystyle\biggl{\\{}\begin{matrix}a&b&e\\\ c&d&f\end{matrix}\biggr{\\}}$
$\displaystyle=(-1)^{a+b-c-d-2e}\sqrt{[2e+1]\,[2f+1]}\,\Delta(a,b,e)\,\Delta(a,d,f)\,\Delta(c,b,f)\,\Delta(c,d,e)$
$\displaystyle\quad\times\sum_{z=z_{\text{min}}}^{z_{\text{max}}}(-1)^{z}\,[z+1]!\,\Bigl{(}[z{-}a{-}b{-}e]!\,[z{-}a{-}d{-}f]!\,[z{-}c{-}b{-}f]!\,[z{-}c{-}d{-}e]!$
$\displaystyle\hskip
99.58464pt[a{+}b{+}c{+}d{-}z]!\,[a{+}c{+}e{+}f{-}z]!\,[b{+}d{+}e{+}f{-}z]!\Bigr{)}^{-1}\;,$
(4.55)

where the integer $z$ is summed in the interval

$\displaystyle z_{\text{max}}$
$\displaystyle=\min\bigl{(}a+b+c+d,\,a+c+e+f,\,b+d+e+f\bigr{)}$ (4.56)
$\displaystyle z_{\text{min}}$
$\displaystyle=\max\bigl{(}a+b+e,\,a+d+f,\,c+b+f,\,c+d+e\bigr{)}\;.$

We used the function

$\Delta(a,b,c)=\sqrt{\frac{[-a+b+c]!\,[a-b+c]!\,[a+b-c]!}{[a+b+c+1]!}}\;.$
(4.57)

The total quantum dimension and the chiral central charge are

$\mathcal{D}=\sqrt{\frac{k+2}{2}}\,\frac{1}{\sin\bigl{(}\frac{\pi}{k+2}\bigr{)}}\;,\qquad\qquad
c_{-}=\frac{3k}{k+2}\;.$ (4.58)

The $S$-matrix is

$S_{\mu\nu}=\frac{1}{\mathcal{D}}\,\bigl{[}(\mu+1)(\nu+1)\bigr{]}\;.$ (4.59)

It is real, symmetric, and it squares to the identity. The FS indicators are
$\varkappa_{\mu}=(-1)^{\mu}$.

The modular invariants of $SU(2)_{k}$ follow an ADE classification [103]. The
simplest one, that exists for every $k\geq 1$, is the diagonal $A_{k+1}$
modular invariant:

$Z^{\mathcal{A}}_{(\mu,\bar{\mu})}=\delta_{\mu\bar{\mu}}\;,\qquad\qquad
m^{(\rho,\rho)}_{(\mu,\mu)\,(\nu,\nu)}=N^{\rho}_{\mu\nu}\;,$ (4.60)

with unit morphism $\eta=1$. One can explicitly verify that the factorization
(4.49) holds.

The other permutation modular invariant (i.e., in which the left and right
modes are paired through a permutation) exists for $k=2\text{ mod }4$ with
$k\geq 6$, and is called the $D_{\frac{k}{2}+2}$ invariant. The permutation
$\pi$ acts as

$\pi(\mu)=\begin{cases}\mu&\text{for $\mu$ even,}\\\ k-\mu&\text{for $\mu$
odd,}\end{cases}$ (4.61)

and is an isomorphism of the $SU(2)_{k}$ TQFT. The algebra object is given by

$Z^{\mathcal{A}}_{(\mu,\bar{\mu})}=\delta_{\mu_{\pi}\bar{\mu}}\;,\qquad\qquad
m^{(\rho,\rho_{\pi})}_{(\mu,\mu_{\pi})\,(\nu,\nu_{\pi})}=N_{\mu\nu}^{\rho}\times\begin{cases}1&\text{for
}m^{\text{even}}_{\text{even,even}}\;,\\\ i^{\,\rho+\mu+1}&\text{for
}m^{\text{even}}_{\text{odd,odd}}\;,\\\ i^{\,\nu}&\text{for
}m^{\text{odd}}_{\text{odd,even}}\;,\\\ i^{\,\mu+\nu-\rho}&\text{for
}m^{\text{odd}}_{\text{even,odd}}\;,\end{cases}$ (4.62)

where we used the notation $\mu_{\pi}\equiv\pi(\mu)$, and with unit morphism
$\eta=1$.

In general, a permutation invariant is defined by an automorphism $\pi$ such
that

$R^{\mu\nu}_{\rho}=R^{\mu_{\pi}\nu_{\pi}}_{\rho_{\pi}}\frac{\Pi_{\mu\nu}^{\rho}}{\Pi_{\nu\mu}^{\rho}}\;,\qquad\qquad[F^{\mu\nu\rho}_{\sigma}]_{\lambda\kappa}=[F^{\mu_{\pi}\nu_{\pi}\rho_{\pi}}_{\sigma_{\pi}}]_{\lambda_{\pi}\kappa_{\pi}}\frac{\Pi_{\nu\rho}^{\kappa}\Pi_{\mu\kappa}^{\sigma}}{\Pi_{\mu\nu}^{\lambda}\Pi_{\lambda\rho}^{\sigma}}\;,$
(4.63)

where $\Pi_{\mu\nu}^{\rho}$ is a gauge transformation. Using the fact that $F$
and $R$ are unitary, it is simple to show that the choice

$m_{(\mu,\bar{\mu})\,(\nu,\bar{\nu})}^{(\rho,\bar{\rho})}=\delta_{\mu\bar{\mu}_{\pi}}\,\delta_{\nu\bar{\nu}_{\pi}}\,\delta^{\rho\bar{\rho}_{\pi}}\left(\Pi_{\mu\nu}^{\rho}\right)^{-1}N_{\mu\,\nu}^{\rho}$
(4.64)

satisfies both associativity and commutativity. We can then perform a gauge
transformation on the right movers by $\Pi^{-1}$. This sets
$\widetilde{m}_{(\mu,\bar{\mu})\,(\nu,\bar{\nu})}^{(\rho,\bar{\rho})}=\delta_{\mu\bar{\mu}_{\pi}}\,\delta_{\nu\bar{\nu}_{\pi}}\,\delta^{\rho\bar{\rho}_{\pi}}N_{\mu\,\nu}^{\rho}$,
and the computation of the partition function reduces to the one of the
diagonal invariant, which exhibits factorization.

The case of non-permutation invariants, such as the $D_{\frac{k}{2}+2}$
invariant for $k=0\text{ mod }4$, is a bit more complicated. Writing the
theory as $SU(2)_{4l}$, the D-type invariant is described by the Lagrangian
anyon

$\mathcal{A}=\Biggl{[}\,\bigoplus_{\begin{subarray}{c}\lambda=0\\\
\lambda\text{
even}\end{subarray}}^{k/2-2}\bigl{(}L_{\lambda}+L_{k-\lambda}\bigr{)}\otimes\bigl{(}L_{\lambda}+L_{k-\lambda}\bigr{)}\,\Biggr{]}+2L_{k/2}\otimes
L_{k/2}\;.$ (4.65)

Note in particular that $Z^{\mathcal{A}}_{\frac{k}{2},\frac{k}{2}}=2$. Let us
give some details on the simplest case, $SU(2)_{4}$:

$\mathcal{A}=L_{0}\otimes L_{0}+L_{0}\otimes L_{4}+L_{4}\otimes
L_{0}+L_{4}\otimes L_{4}+2L_{2}\otimes L_{2}\;.$ (4.66)

Denoting $e=(0,0)$, $q=(0,4)$, $\tilde{q}=(4,0)$, $b=(4,4)$, and $n=(2,2)$
with $\alpha=1,2$, the non-vanishing components of the algebra morphism
are:494949There is a continuous family of solutions that depends on the chosen
basis in the space of projectors $\mathcal{A}\to n$. This is not a physical
parameter, and we set it to a nice value to simplify the expressions.

$\displaystyle m_{a\alpha,e}^{a\alpha}$
$\displaystyle=m_{a\alpha,a\alpha}^{e}=m_{e,a\alpha}^{a\alpha}=1$ (4.67)
$\displaystyle m_{q,\tilde{q}}^{b}$
$\displaystyle=m_{q,b}^{\tilde{q}}=m_{\tilde{q},q}^{b}=m_{\tilde{q},b}^{q}=m_{b,q}^{\tilde{q}}=m_{b,\tilde{q}}^{q}=1$
$\displaystyle m_{q,n2}^{n1}$
$\displaystyle=-m_{q,n1}^{n2}=m_{\tilde{q},n1}^{n2}=-m_{\tilde{q},n2}^{n1}=m_{n1,q}^{n2}=-m_{n2,q}^{n1}=m_{n2,\tilde{q}}^{n1}=-m_{n1,\tilde{q}}^{n2}=i$
$\displaystyle m_{n1,n2}^{q}$
$\displaystyle=-m_{n2,n1}^{q}=m_{n2,n1}^{\tilde{q}}=-m_{n1,n2}^{\tilde{q}}=-i$
$\displaystyle m_{b,n1}^{n1}$
$\displaystyle=m_{b,n2}^{n2}=m_{n1,b}^{n1}=m_{n2,b}^{n2}=m_{n1,n1}^{b}=m_{n2,n2}^{b}=-1$
$\displaystyle m_{n1,n1}^{n1}$
$\displaystyle=-m_{n1,n2}^{n2}=-m_{n2,n1}^{n2}=-m_{n2,n2}^{n1}=1\;.$

With this choice and the version of formula (4.44) valid when
$Z_{a}^{\mathcal{A}}$ can be bigger than 1, one can check that factorization
holds.

##  5 Discussion

In this paper we have shown how the known relation [52, 57, 59] between
modular invariants in 2d RCFT, Lagrangian commutative special Frobenius
algebras, and topological boundary conditions in 3d TQFTs, can be recast in
the following form. A three-dimensional theory $\mathcal{T}$, constructed by
starting with a Chern-Simons theory and then gauging a maximal one-form
symmetry (or, in the non-Abelian case, performing anyon condensation of a
maximal non-invertible one-form symmetry), behaves non-perturbatively as a
theory of gravity and indeed is holographically dual, in the standard sense,
to a well-defined (rational) conformal field theory. The bulk theory
$\mathcal{T}$ has a number of interesting properties. First, it is background
independent: the full non-perturbative partition function is computed by
choosing any three-dimensional manifold with the correct boundary conditions,
with no need to sum over geometries (in a sense, the theory automatically sums
over all needed geometries). A similar behavior has been observed in examples
of full-fledged string theories in AdS3 [45, 46]. Second, $\mathcal{T}$ is a
unitary theory of gravity: its partition functions with fixed boundary
conditions are equal to the partition functions of a unitary 2d boundary CFT.
In particular, the partition functions with disconnected boundaries factorize.
Third, $\mathcal{T}$ does not have any bulk observable, rather, it is
completely blind to the bulk geometry, which follows from the fact that it is
trivial in the bulk. This is what one would expect from a holographic theory,
once the full non-perturbative path integral has been done. Fourth, the bulk
theory does not have any global symmetry: bulk symmetries are precisely
removed by gauging the one-form symmetry.

Our initial motivation was to establish a link between the presence of
(possibly non-invertible) global symmetries in the bulk, and the fact that the
dual boundary system is an ensemble of theories (as signalled by the failure
of factorization of the partition functions). At least in the class of models
we have considered here — 3d bulk theories constructed out of Chern-Simons or
other topological field theories — it is easy to convince oneself that bulk
(possibly non-invertible) global symmetries imply averaging. Symmetries imply
that there are local or non-local topological operators [33], therefore the
bulk theory has non-trivial Hilbert spaces, hence there is a dependence on 3d
topologies and factorization fails. It would be interesting to extend this
argument to more general theories of gravity, in order to provide yet another
argument that quantum gravities cannot have global symmetries.

A given Chern-Simons theory, in general, admits multiple maximal gaugings —
described by multiple Lagrangian condensable anyons $\mathcal{A}$. Each
gauging produces the dual to a precise 2d CFT, while the partition functions
of the original CS theory, summed over 3d geometries in order to ensure
modular invariance, display ensemble averaging. Therefore, it is tempting to
interpret the gauging of $\mathcal{A}$ as a concrete realization of the
projectors to the so-called $\alpha$-states of the bulk system (using the
language of [49]).

In this paper we have only considered very simple topological theories, dual
to RCFTs. It would be fascinating if similar ideas could be extended to non-
compact Chern-Simons theories (or other non-semisimple TQFTs), possibly
related to three-dimensional Einstein gravity. We considered in Section 3.3
the straighforward example of $\mathbb{R}$ CS theory, but it is clear that
such an example is too simplistic to be representative of the general case.

Let us conclude with a few remarks and comments.

#### Semiclassical limits.

One might want to compare the results emerging from the formalism used here to
some sort of semiclassical gravity. To do this we need to discuss possible
semiclassical limits. In a semiclassical limit the central charge
$c=c_{L}+c_{R}$ should go to infinity. In WZW models for the current algebra
$\widehat{\mathfrak{g}}_{k}$, the central charge is
$c_{L}=k\dim(\mathfrak{g})/(k+\check{c}_{\mathfrak{g}})$, where
$\check{c}_{\mathfrak{g}}$ is the dual Coxeter number. One possibility is to
take $\text{dim}(\mathfrak{g})$ large, which is somewhat akin to the standard
large $N$ limit. Another possibility is to take
$k\to-\check{c}_{\mathfrak{g}}$. At least for
$\widehat{\mathfrak{su}}(2)_{k}$, it might be possible to interpret this limit
through analytic continuation to $\widehat{\mathfrak{sl}}(2,\mathbb{R})_{k}$,
for which the central charge formula is
$c_{L}=k\dim(\mathfrak{g})/(k-\check{c}_{\mathfrak{g}})$ (see [104] where
ideas along these lines have been applied to 3d de Sitter gravity).

Let us also notice that the dimensions of chiral vertex operators are

$h_{\lambda}=\frac{\mathcal{Q}_{\lambda}}{2(k+\check{c}_{\mathfrak{g}})}\;,$
(5.1)

where $\mathcal{Q}_{\lambda}$ the quadratic Casimir of the representation
$\lambda$. This implies that, in the second type of limit, a gap of order
$\Delta_{\text{gap}}\sim c$ opens up between the identity module and the first
integral representation of chiral algebra. This behavior resembles a
semiclassical gravity.

#### Dissecting Poincaré sums.

Our construction offers a suggestive way to think of Poincaré sums of RCFT
characters (i.e., sums over their modular images). Consider the Poincaré sum

$Z^{\text{avg}}_{\lambda\bar{\mu}}(\tau,\bar{\tau})=\sum_{\gamma\in\Gamma}\,\chi_{\lambda}(\gamma\cdot\tau)\,\overline{\chi_{\bar{\mu}}(\gamma\cdot\tau)}\;,$
(5.2)

for a “seed” character $\chi_{\lambda}\bar{\chi}_{\bar{\mu}}$. Here $\Gamma$
is the quotient of the modular group that acts non-trivially on the seed
character. For RCFTs this is a finite group, so the sum is finite. Such sums
were considered, e.g., in [18, 21] and could be thought of as an avatar of
sums over gravitational geometries. Since $Z^{\text{avg}}_{\lambda\bar{\mu}}$
is a modular invariant function, constructed as a linear combination of
character bilinears $\chi_{\rho}\bar{\chi}_{\bar{\sigma}}$, it can be expanded
in a complete basis of modular invariant functions

$Z^{\mathcal{A}}(\tau,\bar{\tau})=\sum_{\rho,\bar{\sigma}}Z^{\mathcal{A}}_{\rho\bar{\sigma}}\,\chi_{\rho}(\tau)\,\overline{\chi_{\bar{\sigma}}(\tau)}\;,$
(5.3)

labelled by the index $\mathcal{A}$ and forming a finite-dimensional vector
space. The expansion reads

$Z^{\text{avg}}_{\lambda\bar{\mu}}(\tau,\bar{\tau})=\sum_{\mathcal{A}}p_{\lambda\bar{\mu}}^{\mathcal{A}}\,Z^{\mathcal{A}}(\tau,\bar{\tau})\;,$
(5.4)

in terms of coefficients $p^{\mathcal{A}}_{\lambda\bar{\mu}}$. In general the
basis elements $Z^{\mathcal{A}}$ cannot be all chosen to correspond to
physical modular invariants of the corresponding chiral algebra: some of them
might involve coefficients $Z^{\mathcal{A}}_{\rho\bar{\sigma}}$ that are not
non-negative integers. However [18, 21] found by inspection that, in some
cases, the $Z^{\mathcal{A}}$’s involved are all physical. We will assume that
we are in such a favorable situation.

The expansion (5.4) looks like an ensemble average over theories, in a basis
of Coleman’s $\alpha$-states [49]. In our formalism, we can give each of them
a bulk interpretation as the (unique) state resulting from the condensation of
the bulk Lagrangian anyon $\mathcal{A}$. Thus, our prescription could be
viewed as defining a projection to a fixed $\alpha$-state.

We can be more explicit and compute the “probabilities”
$p_{\lambda\bar{\mu}}^{\mathcal{A}}$ by inverting (5.4). One first defines the
positive-definite scalar product

$M_{\mathcal{A}\mathcal{A}^{\prime}}=\langle\mathcal{A}|\mathcal{A}^{\prime}\rangle=\sum_{\rho,\bar{\sigma}}\,Z^{\mathcal{A}}_{\rho\bar{\sigma}}\,Z^{\mathcal{A}^{\prime}}_{\rho\bar{\sigma}}\;.$
(5.5)

Then, using modular invariance of the elements $Z^{\mathcal{A}}$, one finds

$p^{\mathcal{A}}_{\lambda\bar{\mu}}=|\Gamma|\,\sum_{\mathcal{A}^{\prime}}\,[M^{-1}]_{\mathcal{A}\mathcal{A}^{\prime}}\,Z^{\mathcal{A}^{\prime}}_{\lambda\bar{\mu}}\;.$
(5.6)

Unfortunately, the “probabilities” $p^{\mathcal{A}}$ are not guaranteed to be
non-negative. This is an important indicator for the possibility of
interpreting the sum as a proper ensemble average.

The $\alpha$-states associated to two distinct Lagrangian anyons
$\mathcal{A}$, $\mathcal{A}^{\prime}$ fail, in general, to be orthogonal using
the product (5.5), because both anyons contain the vacuum line. Since
different $\alpha$-states should be orthogonal as they represent different
superselection sectors, this seems to hint that a different scalar product
should be used.

The procedure we outlined allows us to “dissect” Poincaré sums with one
boundary in terms of bulk data. The case with multiple boundaries is in
principle also tractable. One could start from a sum over higher-genus
handlebodies and take a pinching limit in which the complex structure $\Omega$
becomes block diagonal, as in [105]. The computation of overlaps in the non-
Abelian case is tedious, although in principle doable using surgery
techniques.

#### Non-compact theories.

It would be very interesting to understand whether the formulation of modular
invariants in terms of Lagrangian condensable anyons can be extended to the
case of Chern-Simons theories with non-compact gauge group, or to non-
semisimple TQFTs. A natural objective would be towards Einstein gravity in
AdS3 in its Chern-Simons formulation.

## Acknowledgements

We are grateful to Andreas Blommaert, Luca Iliesiu, Ying-Hsuan Lin, Pavel
Putrov, and Rajath K. Radhakrishnan for useful discussions and correspondence.
We also thank the organizers and participants of the GGI workshop “Topological
properties of gauge theories”, where part of these results were announced. The
authors are partially supported by the INFN “Iniziativa Specifica ST&FI”. F.B.
and C.C. are supported by the ERC-COG grant NP-QFT No. 864583 “Non-
perturbative dynamics of quantum fields: from new deconfined phases of matter
to quantum black holes”, and by the MIUR-PRIN contract 2015 MP2CX4. F.B. is
also supported by the MIUR-SIR grant RBSI1471GJ. L.D. also acknowledges
support by the program “Rita Levi Montalcini” for young researchers.

##  Appendix A Higher-genus partition functions

Here we review the computation of the non-holomorphic function $\Phi$
appearing in the higher-genus partition functions (2.18) and (2.27). From the
point of view of the 2d CFT, $\Phi$ can be written as (see Section 3 of [74]
and references therein):

$\Phi=\bigl{\lvert}\det\nolimits^{\prime}_{\Gamma}\bar{\partial}_{0}\bigr{\rvert}\;,$
(A.1)

where $\bar{\partial}_{0}$ is the Dolbeault operator mapping functions to
$(0,1)$-forms, and the symbol $\det\nolimits^{\prime}$ denotes that the zero-
modes are removed. The subscript $\Gamma$ reminds us that this regularized
determinant depends on a choice of Lagrangian (primitive) sublattice
$\Gamma\subset H_{1}(\Sigma,\mathbb{Z})$, i.e., on a choice of contractible
cycles in the corresponding three-dimensional handlebody. On the other hand,
using two-dimensional holomorphic factorization [106, 107, 108]

$\det\nolimits^{\prime}\Delta_{0}=\bigl{(}\det\operatorname{\mathbb{I}m}\Omega\bigr{)}\,\cdot\,\bigl{\lvert}\det\nolimits^{\prime}_{\Gamma}\bar{\partial}_{0}\bigr{\rvert}^{2}$
(A.2)

where $\Delta_{0}$ is the scalar Laplacian on $\Sigma$ and the choice of
$\Omega$ is related to $\Gamma$ (see [16], sections 3.1 and 4.2), one gets the
alternative expression

$\Phi=\biggl{(}\frac{\det^{\prime}\Delta_{0}}{\det\operatorname{\mathbb{I}m}\Omega}\biggr{)}^{1/2}\;.$
(A.3)

Notice that both
$\bigl{\lvert}\det^{\prime}_{\Gamma}\bar{\partial}_{0}\bigr{\rvert}$ and
$(\det^{\prime}\Delta_{0})^{1/2}$ suffer from the conformal anomaly for $g\neq
1$. The Laplacian determinant $\det^{\prime}\Delta_{0}$ is modular invariant.

On the other hand, following the discussion in Section 4 of [16], we can make
contact with the partition function of 3d CS theory by using Zograf’s three-
dimensional holomorphic factorization formula [109, 110]:

$\Phi=\biggl{(}\frac{\det^{\prime}\Delta_{0}}{\det\operatorname{\mathbb{I}m}\Omega}\biggr{)}^{1/2}=e^{-S_{\text{L}}/24\pi}\;\left\lvert\,\prod_{\gamma\in\mathcal{P}}\prod_{n=1}^{\infty}\bigl{(}1-q_{\gamma}^{n}\bigr{)}\right\rvert^{2}\;.$
(A.4)

Here $\mathsf{G}$ is a Schottky group such that $\mathbb{H}^{3}/\mathsf{G}$ is
a handlebody whose boundary is $\Sigma$ and whose contractible cycles are
indicated by $\Gamma$, while $\mathcal{P}$ is the set of primitive subgroups
of $\mathsf{G}$ and $\gamma$ are their generators. For each generator
$\gamma$, we write $\gamma\in\mathsf{G}\subset PSL(2,\mathbb{C})$ as a
$2\times 2$ matrix, then use $2\cos\pi\tau_{\gamma}=\operatorname{Tr}\gamma$
to define $\tau_{\gamma}$ (the real part of $\tau_{\gamma}$ is only defined
mod 1, while we choose the sign of $\tau_{\gamma}$ in such a way that
$\operatorname{\mathbb{I}m}\tau_{\gamma}>0$), and finally define
$q_{\gamma}=e^{2\pi i\tau_{\gamma}}$. The function $S_{\text{L}}$ is a
Liouville action, defined in [111], which reproduces the conformal anomaly and
obstructs holomorphic factorization [112].

It was proven in [113] that $S_{\text{L}}$ is equal to the holographically
regularized volume of the handlebody $\mathbb{H}^{3}/\mathsf{G}$:

$S_{\text{L}}=-4\,\mathrm{Vol}\bigl{(}\mathbb{H}^{3}/\mathsf{G}\bigr{)}+\text{counterterms}\;.$
(A.5)

For instance, for $g=1$ one finds
$\mathrm{Vol}(\mathbb{H}^{3}/\sf{G})=-\pi^{2}\tau_{2}$ while $\mathcal{P}$ has
a unique element, so that
$\Phi=e^{-\pi\tau_{2}/6}\bigl{\lvert}\prod_{n}(1-q^{n})\bigr{\rvert}^{2}=\bigl{\lvert}\eta(\tau)\bigr{\rvert}^{2}$.505050Note
that in this case of $g=1$, the prefactor $e^{-S_{\text{L}}/24\pi}$ in eqn.
(A.4) gives a factorized contribution
$e^{-\pi\tau_{2}/6}=q^{\frac{1}{24}}\,\bar{q}^{\frac{1}{24}}$. We can also
compare with eqn. (2.3) of [79], taking into account that they include a
factorized contribution in their definition of $\widetilde{F}$, see their
footnote 3. Finally, one can use heat kernel methods to express the ratio of
determinants in (2.26) [114, 16], arising from the perturbative partition
function of $U(1)_{k}\times U(1)_{-k}$ Chern-Simons theory on a handlebody
with conformal boundary conditions, in terms of the quantities discussed
above:

$Z_{\text{CS}}^{\text{pert}}\,\equiv\,\frac{(\det^{\prime}\Delta_{0})^{3/2}}{(\det^{\prime}\Delta_{1})^{1/2}}=\exp\biggl{[}-\frac{\mathrm{Vol}(\mathbb{H}^{3}/\sf{G})}{6\pi}\biggr{]}\;\left\lvert\prod_{\gamma\in\mathcal{P}}\prod_{n=1}^{\infty}\frac{1}{1-q_{\gamma}^{n}}\right\rvert^{2}=\frac{1}{\Phi}\;.$
(A.6)

This partition function does not include the sum over disconnected flat
connections.

##  Appendix B Boundary symmetry and contact term

Consider the Abelian Chern-Simons theory with action

$S_{\text{CS}}=i\frac{k}{4\pi}\int_{M_{3}}ada\;.$ (B.1)

We study it on a manifold $M_{3}$ with a boundary $\partial M_{3}=M_{2}$. We
pick a complex structure on $M_{2}$ and we write the boundary term

$S_{\partial}=\frac{k}{4\pi}\int_{M_{2}}d^{2}x\,\sqrt{g}\,g^{z\bar{z}}\,a_{z}a_{\bar{z}}\;,$
(B.2)

using a Hermitean metric. The Hermitean metric and the volume form which are
compatible with the complex structure are related by
$\epsilon^{z\bar{z}}=ig^{z\bar{z}}$. Using this, we get the following
variation of the total action:

$\delta\bigl{(}S_{\text{CS}}+S_{\partial}\bigr{)}=i\frac{k}{2\pi}\int_{M_{3}}\delta
a\,da+\frac{k}{2\pi}\int_{M_{2}}d^{2}x\,\sqrt{g}\,g^{z\bar{z}}\,a_{z}\,\delta
a_{\bar{z}}\;.$ (B.3)

Therefore the equation of motion is $da=0$ and the boundary condition that
makes the action stationary is $a_{\bar{z}}\big{|}_{M_{2}}=A_{\bar{z}}$, where
the uppercase letter denotes a fixed $c$-number connection.

Evaluating the EOM on the boundary we get

$0=da\big{|}_{M_{2}}=\left(\partial_{z}a_{\bar{z}}-\partial_{\bar{z}}a_{z}\right)\bigr{|}_{M_{2}}\,dz\wedge
d\bar{z}=\left(\partial_{z}A_{\bar{z}}-\partial_{\bar{z}}a_{z}\bigr{|}_{M_{2}}\right)dz\wedge
d\bar{z}\;.$ (B.4)

We see that $a_{z}\big{|}_{M_{2}}$ is a holomorphic current when the boundary
value $A_{\bar{z}}$ is set to zero, and otherwise it satisfies the “anomalous”
conservation equation

$\partial_{\bar{z}}a_{z}\big{|}_{M_{2}}=\partial_{z}A_{\bar{z}}\;.$ (B.5)

Defining the partition function

$\displaystyle Z[A_{\bar{z}}]$
$\displaystyle=\int_{a_{\bar{z}}|_{M_{2}}=A_{\bar{z}}}\,\mathcal{D}a\,\exp\bigl{[}-S_{\text{CS}}-S_{\partial}\bigr{]}$
(B.6)
$\displaystyle=\int_{a_{\bar{z}}|_{M_{2}}=A_{\bar{z}}}\,\mathcal{D}a\,\exp\biggl{[}-i\frac{k}{4\pi}\int_{M_{3}}ada-\frac{k}{4\pi}\int_{M_{2}}d^{2}x\,\sqrt{g}\,g^{z\bar{z}}\,A_{\bar{z}}a_{z}\biggr{]}\;,$

a natural normalization of the holomorphic current on $M_{2}$ is to take

$\langle
j_{z}\rangle_{A_{\bar{z}}}=-\frac{g_{z\bar{z}}}{\sqrt{g}}\frac{\delta}{\delta
A_{\bar{z}}}\log Z[A_{\bar{z}}]\;,$ (B.7)

from which we get

$j_{z}=\frac{k}{4\pi}\,a_{z}\big{|}_{M_{2}}\;.$ (B.8)

There are several related unpleasant features in these formulas:

  * (i)

The current is not invariant under gauge transformations $a\to a+d\lambda$,
namely

$j_{z}\to j_{z}+\frac{k}{4\pi}\,\partial_{z}\Lambda\;,$ (B.9)

where here and in the following $\Lambda(z,\bar{z})$ denotes the restriction
of the gauge parameter $\lambda$ to $M_{2}$.

  * (ii)

The current does not satisfy the standard anomaly equation:

$\partial_{\bar{z}}j_{z}=\frac{k}{4\pi}\,\partial_{z}A_{\bar{z}}\neq\frac{k}{4\pi}\,F_{z\bar{z}}\;,$
(B.10)

which is a trivial consequence of the fact that the holomorphic component
$A_{z}$ does not enter in our formulas.

  * (iii)

The partition function does not transform in the expected way under a
background gauge transformation. To compute how the partition function
transforms under a gauge transformation $a\to a+d\lambda\equiv a^{\prime}$, we
perform a change of variables in the path integral setting
$a=a^{\prime}-d\lambda$. We obtain

$\displaystyle Z[A_{\bar{z}}]$
$\displaystyle=\int_{a^{\prime}_{\bar{z}}|_{M_{2}}=A_{\bar{z}}+\partial_{\bar{z}}\Lambda}\,\mathcal{D}a^{\prime}\,\exp\biggl{[}-i\frac{k}{4\pi}\int_{M_{3}}a^{\prime}da^{\prime}-\frac{k}{4\pi}\int_{M_{2}}d^{2}x\,\sqrt{g}\,g^{z\bar{z}}\bigl{(}A_{\bar{z}}+\partial_{\bar{z}}\Lambda\bigr{)}a^{\prime}_{z}$
$\displaystyle\qquad\quad-\frac{k}{2\pi}\int_{M_{2}}d^{2}x\,\sqrt{g}\,g^{z\bar{z}}\,\Lambda\,\partial_{z}A_{\bar{z}}+\frac{k}{4\pi}\int_{M_{2}}d^{2}x\,\sqrt{g}\,g^{z\bar{z}}\,\partial_{z}\Lambda\,\partial_{\bar{z}}\Lambda\biggr{]}\;.$
(B.11)

Collecting the $c$-numbers on the second line as a factor in front of the path
integral, we recognize the remaining path integral to be precisely the
partition function with source $A_{\bar{z}}+\partial_{\bar{z}}\Lambda$, so we
obtain

$Z[A_{\bar{z}}+\partial_{\bar{z}}\Lambda]=\exp\biggl{[}\frac{k}{2\pi}\int_{M_{2}}d^{2}x\sqrt{g}g^{z\bar{z}}\Lambda\,\partial_{z}A_{\bar{z}}-\frac{k}{4\pi}\int_{M_{2}}d^{2}x\sqrt{g}g^{z\bar{z}}\partial_{z}\Lambda\,\partial_{\bar{z}}\Lambda\biggr{]}Z[A_{\bar{z}}]\;.$
(B.12)

All of these unpleasant features can be fixed simply by adding a boundary
contact term:

$S_{\text{ct}}=-\frac{k}{4\pi}\int_{M_{2}}d^{2}x\,\sqrt{g}\,g^{z\bar{z}}\,A_{z}A_{\bar{z}}\;,$
(B.13)

and defining accordingly a new partition function

$\widetilde{Z}[A_{z},A_{\bar{z}}]=\exp\biggl{[}\frac{k}{4\pi}\int_{M_{2}}d^{2}x\,\sqrt{g}\,g^{z\bar{z}}\,A_{z}A_{\bar{z}}\biggr{]}\,Z[A_{\bar{z}}]\;.$
(B.14)

With the same definition as before,

$\langle\tilde{\jmath}_{z}\rangle_{A_{z},A_{\bar{z}}}=-\frac{g_{z\bar{z}}}{\sqrt{g}}\frac{\delta}{\delta
A_{\bar{z}}}\,\log\widetilde{Z}[A_{z},A_{\bar{z}}]\;,$ (B.15)

we now get

$\tilde{\jmath}_{z}=\frac{k}{4\pi}\,\bigl{(}a_{z}\big{|}_{M_{2}}-A_{z}\bigr{)}\;,$
(B.16)

which has the advantage of being gauge-invariant, at least if we accompany the
gauge transformation of the dynamical field by a background gauge
transformation with the same parameter. At the same time, with this
modification the current now satisfies the standard anomaly equation:

$\partial_{\bar{z}}\tilde{\jmath}_{z}=\frac{k}{4\pi}\,F_{z\bar{z}}\;.$ (B.17)

Moreover, using (B.12) we can easily check that the new partition function has
a standard anomalous transformation under background gauge transformations:

$\displaystyle\widetilde{Z}\bigl{[}A_{z}+\partial_{z}\Lambda,\,A_{\bar{z}}+\partial_{\bar{z}}\Lambda\bigr{]}$
$\displaystyle=\exp\biggl{[}\frac{k}{4\pi}\int_{M_{2}}d^{2}x\,\sqrt{g}\,g^{z\bar{z}}\,\Lambda\,F_{z\bar{z}}\biggr{]}\,\widetilde{Z}[A_{z},A_{\bar{z}}]$
(B.18)
$\displaystyle=\exp\biggl{[}-i\frac{k}{4\pi}\int_{M_{2}}\Lambda\,F\biggr{]}\,\widetilde{Z}[A_{z},A_{\bar{z}}]\;.$

##  Appendix C Gauging formulas with boundary

In this appendix we derive formula (2.34) for the partition function of a 3d
theory in which a discrete (Abelian) 1-form symmetry is gauged, on a manifold
with boundary.515151We are grateful to Pavel Putrov for explaining the context
of this appendix to us.

Let $M$ be a closed (oriented) 3-manifold. Gauging of a discrete 1-form
symmetry $\mathcal{A}$, as in (2.32), leads to the partition function

$Z^{\text{gauged}}[M]=\frac{|H^{0}(M)|}{|H^{1}(M)|}\sum_{a\in
H^{2}(M)}Z[M;a]\;.$ (C.1)

Here and in the following we make explicit the manifolds on which the
Euclidean path integrals $Z$ are computed. On the other hand, all cohomology
groups are with coefficients in the Abelian group $\mathcal{A}$, which we then
keep implicit.

Now, let $M=M_{+}\cup M_{-}$ such that $\partial M_{+}=\partial M_{-}=\Sigma$
and $M_{+}\cap M_{-}=\Sigma$. Here $\Sigma$ is a 2-dimensional closed
manifold, not necessarily connected. We would like to define the partition
functions $Z[M_{\pm}]$ on $M_{\pm}$ with Dirichelet boundary conditions on
$\Sigma$, in such a way that $Z[M]$ factorizes into them, with a sum over the
boundary conditions. We would like to use the Mayer-Vietoris sequence (see,
e.g., [85]):

$\ldots\,\stackrel{{\scriptstyle\Phi}}{{\longrightarrow}}\,H^{n-1}(\Sigma)\,\stackrel{{\scriptstyle\delta}}{{\longrightarrow}}\,H^{n}(M)\,\stackrel{{\scriptstyle\Psi}}{{\longrightarrow}}\,H^{n}(M_{+})\oplus
H^{n}(M_{-})\,\stackrel{{\scriptstyle\Phi}}{{\longrightarrow}}\,H^{n}(\Sigma)\,\stackrel{{\scriptstyle\delta}}{{\longrightarrow}}\,\ldots$
(C.2)

to express $H^{2}(M)$ in terms of $H^{2}(M_{+})\oplus H^{2}(M_{-})$. The map
$\Phi$ restricts cocycles in $M_{+}$ and in $M_{-}$ to $\Sigma$ and then takes
the difference. Therefore the kernel of $\Phi$ inside $H^{2}(M_{+})\oplus
H^{2}(M_{-})$ consists of a sum over 1-form bundles on $M_{\pm}$ with equal
boundary conditions (summed over such fixed and equal boundary conditions).
However this is not isomorphic to $H^{2}(M)$ because $\Psi$ has a kernel,
equal to $\delta\bigl{(}H^{1}(\Sigma)\bigr{)}$.

To solve the problem, we consider cohomology relative to a subspace of
$\Sigma$, such that $H^{1}$ of $\Sigma$ becomes trivial but $H^{2}$ of
$\Sigma$ remains the same. This can be achieved using $\Sigma\smallsetminus
P$, where $P$ consists of one point in each connected component of $\Sigma$.
Heuristically, the relative cohomology group
$H^{2}(\Sigma,\Sigma\smallsetminus P)$ focuses the 2-cocycles to a point in
each connected component, and this indeed captures the full cohomology. Thus,
we apply the Mayer-Vietoris sequence to $(M,\Sigma\smallsetminus
P)=(M_{+},\Sigma\smallsetminus P)\cup(M_{-},\Sigma\smallsetminus P)$.

Let us first apply the long exact sequence to the pair
$(\Sigma,\Sigma\smallsetminus P)$, which involves the local cohomology groups
$H^{n}(\Sigma,\Sigma\smallsetminus P)$:

$\displaystyle 0$
$\displaystyle\,\longrightarrow\,\underbrace{H^{0}(\Sigma,\Sigma\smallsetminus
P)}_{0}\,\longrightarrow\,H^{0}(\Sigma)\,\stackrel{{\scriptstyle\approx}}{{\longrightarrow}}\,H^{0}(\Sigma\smallsetminus
P)\,\longrightarrow\,$ (C.3)
$\displaystyle\,\longrightarrow\,\underbrace{H^{1}(\Sigma,\Sigma\smallsetminus
P)}_{0}\,\longrightarrow\,H^{1}(\Sigma)\,\stackrel{{\scriptstyle\approx}}{{\longrightarrow}}\,H^{1}(\Sigma\smallsetminus
P)\,\stackrel{{\scriptstyle 0}}{{\longrightarrow}}\,$
$\displaystyle\,\stackrel{{\scriptstyle
0}}{{\longrightarrow}}\,H^{2}(\Sigma,\Sigma\smallsetminus
P)\,\stackrel{{\scriptstyle\approx}}{{\longrightarrow}}\,H^{2}(\Sigma)\,\longrightarrow\,\underbrace{H^{2}(\Sigma\smallsetminus
P)}_{0}\,\longrightarrow\,0\;.$

Here $\approx$ means isomorphism. For a connected 2-manifold $N_{2}$, $x\in
N_{2}$ a point, and $D_{2}$ a disk around $x$, Poincaré duality implies
$H^{n}(N_{2}\smallsetminus\\{x\\})\approx
H^{n}(N_{2}\smallsetminus\mathring{D}_{2})\approx
H_{2-n}(N_{2}\smallsetminus\mathring{D}_{2},\partial D_{2})$. The Mayer-
Vietoris sequence then gives:

$\displaystyle 0$
$\displaystyle\,\longrightarrow\,H^{0}(M,\Sigma\smallsetminus
P)\,\stackrel{{\scriptstyle\approx}}{{\longrightarrow}}\,H^{0}(M_{+},\Sigma\smallsetminus
P)\oplus H^{0}(M_{-},\Sigma\smallsetminus
P)\,\longrightarrow\,\underbrace{H^{0}(\Sigma,\Sigma\smallsetminus
P)}_{0}\,\longrightarrow\,$ (C.4)
$\displaystyle\,\longrightarrow\,H^{1}(M,\Sigma\smallsetminus
P)\,\stackrel{{\scriptstyle\approx}}{{\longrightarrow}}\,H^{1}(M_{+},\Sigma\smallsetminus
P)\oplus H^{1}(M_{-},\Sigma\smallsetminus
P)\,\longrightarrow\,\underbrace{H^{1}(\Sigma,\Sigma\smallsetminus
P)}_{0}\,\longrightarrow\,$
$\displaystyle\,\longrightarrow\,H^{2}(M,\Sigma\smallsetminus
P)\,\longrightarrow\,H^{2}(M_{+},\Sigma\smallsetminus P)\oplus
H^{2}(M_{-},\Sigma\smallsetminus
P)\,\stackrel{{\scriptstyle\Phi}}{{\longrightarrow}}\,\underbrace{H^{2}(\Sigma,\Sigma\smallsetminus
P)}_{H^{2}(\Sigma)}\,\longrightarrow\,\ldots$

which implies

$H^{2}(M,\Sigma\smallsetminus
P)\approx\operatorname{Ker}\Phi\,\subset\,H^{2}(M_{+},\Sigma\smallsetminus
P)\oplus H^{2}(M_{-},\Sigma\smallsetminus P)\;.$ (C.5)

The inclusion $\Sigma\stackrel{{\scriptstyle i}}{{\hookrightarrow}}M_{+}$
induces the homomorphism $H^{2}(M_{+},\Sigma\smallsetminus
P)\stackrel{{\scriptstyle i^{*}}}{{\to}}H^{2}(\Sigma,\Sigma\smallsetminus
P)\approx H^{2}(\Sigma)$ therefore

$H^{2}(M_{+},\Sigma\smallsetminus P)=\bigoplus_{b\in
H^{2}(\Sigma)}H^{2}(M_{+},\Sigma\smallsetminus
P)\Big{|}_{\operatorname{Im}i^{*}=b}$ (C.6)

where the groups on the RHS have fixed boundary conditions, and the sum is
over those boundary conditions. The restriction to $\operatorname{Ker}\Phi$
corresponds to imposing that the boundary conditions are the same on $M_{+}$
and $M_{-}$.

On the other hand, we would like to obtain an expression for the group
$H^{2}(M)$, as opposed to the relative cohomology group
$H^{2}(M,\Sigma\smallsetminus P)$. The long exact sequence for the pair
$(M,\Sigma\smallsetminus P)$ is:

$\displaystyle 0$
$\displaystyle\,\longrightarrow\,H^{0}(M,\Sigma\smallsetminus
P)\,\longrightarrow\,H^{0}(M)\,\longrightarrow\,\underbrace{H^{0}(\Sigma\smallsetminus
P)}_{H^{0}(\Sigma)}\,\longrightarrow\,$ (C.7)
$\displaystyle\,\longrightarrow\,H^{1}(M,\Sigma\smallsetminus
P)\,\longrightarrow\,H^{1}(M)\,\longrightarrow\,\underbrace{H^{1}(\Sigma\smallsetminus
P)}_{H^{1}(\Sigma)}\,\stackrel{{\scriptstyle\gamma}}{{\longrightarrow}}\,$
$\displaystyle\,\stackrel{{\scriptstyle\gamma}}{{\longrightarrow}}\,H^{2}(M,\Sigma\smallsetminus
P)\,\longrightarrow\,H^{2}(M)\,\longrightarrow\,\underbrace{H^{2}(\Sigma\smallsetminus
P)}_{0}\,\longrightarrow\,\ldots$

and similar sequences can be written for $M_{\pm}$. In particular

$H^{2}(M)\,\approx\,H^{2}(M,\Sigma\smallsetminus
P)\,/\,\operatorname{Im}\Bigl{(}H^{1}(\Sigma\smallsetminus
P)\,\stackrel{{\scriptstyle\gamma}}{{\longrightarrow}}\,H^{2}(M,\Sigma\smallsetminus
P)\Bigr{)}\;.$ (C.8)

Notice that $H^{2}(M,\Sigma\smallsetminus P)$ is given by 2-cocycles that
vanish on $\Sigma\smallsetminus P$ modulo coboundaries of 1-cochains that
vanish on $\Sigma\smallsetminus P$; while the former represent the whole
cohomology $H^{2}(\Sigma)$ — see (C.3) — the latter do not represent any
element of $H^{1}(\Sigma)$, and as a result $H^{2}(M,\Sigma\smallsetminus P)$
is larger than $H^{2}(M)$. However $Z[M;a]$ only depends on $a\in H^{2}(M)$,
therefore if we vary $a$ by an element in the kernel of the map
$H^{2}(M,\Sigma\smallsetminus P)\to H^{2}(M)$ (which is equal to
$\operatorname{Im}\gamma$) then $Z[M;a]$ remains invariant. This shows that if
we parametrize bundles using $H^{2}(M,\Sigma\smallsetminus P)$ we only suffer
from an overcounting problem, which can be taken care of by dividing by
$|\operatorname{Im}\gamma|$. From (C.7) we obtain

$\displaystyle|\operatorname{Im}\gamma|$
$\displaystyle=\frac{|H^{1}(\Sigma)|\;|H^{1}(M,\Sigma\smallsetminus
P)|\;|H^{0}(M)|}{|H^{1}(M)|\;|H^{0}(\Sigma)|\;|H^{0}(M,\Sigma\smallsetminus
P)|}$ (C.9)
$\displaystyle=\frac{|H^{1}(\Sigma)|\;|H^{0}(M)|}{|H^{0}(\Sigma)|\;|H^{1}(M)|}\prod_{i=+,-}\frac{|H^{1}(M_{i},\Sigma\smallsetminus
P)|}{|H^{0}(M_{i},\Sigma\smallsetminus P)|}\;,$

where in the second equality we used (C.4).

We conclude that the partition function (C.1) takes the form:

$\displaystyle Z^{\text{gauged}}[M]$
$\displaystyle=\left\lvert\frac{H^{0}(M)}{H^{1}(M)}\right\rvert\,\frac{1}{|\operatorname{Im}\gamma|}\sum_{a\in
H^{2}(M,\Sigma\smallsetminus P)}Z[M;a]$ (C.10)
$\displaystyle=\left\lvert\frac{H^{0}(M)}{H^{1}(M)}\right\rvert\,\frac{1}{|\operatorname{Im}\gamma|}\sum_{b\in
H^{2}(\Sigma)}\;\sum_{\begin{subarray}{l}a_{+}\in
H^{2}(M_{+},\Sigma\smallsetminus P)\\\
i^{*}(a_{+})=b\end{subarray}}\;\sum_{\begin{subarray}{l}a_{-}\in
H^{2}(M_{-},\Sigma\smallsetminus P)\\\
i^{*}(a_{-})=b\end{subarray}}Z[M_{+};a_{+}]\,Z[M_{-};a_{-}]\;.$

This prompts the following definition of the gauged partition function with
Dirichelet boundary conditions $b\in H^{2}(\Sigma)$, which coincides with
(2.34):

$Z^{\text{gauged}}_{b}[M_{+}]=\left\lvert\frac{H^{0}(M_{+},\Sigma\smallsetminus
P)}{H^{1}(M_{+},\Sigma\smallsetminus
P)}\right\rvert\sum_{\begin{subarray}{l}a\in H^{2}(M_{+},\Sigma\smallsetminus
P)\\\ i^{*}(a)=b\end{subarray}}Z[M_{+};a]\;,$ (C.11)

and similarly for $M_{-}$. It yields the gluing formula

$Z^{\text{gauged}}[M]=\left\lvert\frac{H^{0}(\Sigma)}{H^{1}(\Sigma)}\right\rvert\sum_{b\in
H^{2}(\Sigma)}Z^{\text{gauged}}_{b}[M_{+}]\,Z^{\text{gauged}}_{b}[M_{-}]\;.$
(C.12)

The sum over boundary conditions corresponds to gauging a 1-form symmetry on
the two-dimensional closed manifold $\Sigma$, and the normalization factor
agrees with (2.32).

If $M$ is orientable, we can express (C.11) and (2.34) in terms of homology,
as opposed to cohomology, groups. Using Poincaré duality and with a few steps,
we can prove

$H^{n}(M_{+},\partial M_{+}\smallsetminus P)\,\approx\,H_{d-n}(M_{+},P)\;.$
(C.13)

In particular with $d=3$ we have $H^{0}(M_{+},\Sigma\smallsetminus P)\approx
H_{3}(M_{+})$ and $H^{1}(M_{+},\Sigma\smallsetminus P)\approx H_{2}(M_{+})$.
The group $H_{1}(M_{+},P)$ 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. This leads to the alternative formula (2.35).

To finish, let us prove (C.13). We let $M_{+}$ be an orientable compact
manifold of dimension $d$ with boundary, and $P\subset\partial M_{+}$ be a set
of points consisting of one point in each connected component of $\partial
M_{+}$. Moreover, let $D\subset\partial M_{+}$ be a set of closed disks
$D_{d-1}$, one disk around each point of $P$. The long exact sequence for the
triple $(M_{+},\partial M_{+}\smallsetminus P,\partial
M_{+}\smallsetminus\mathring{D})$ is

$\ldots\,\longrightarrow\,H^{n}(M_{+},\partial M_{+}\smallsetminus
P)\,\longrightarrow\,H^{n}(M_{+},\partial
M_{+}\smallsetminus\mathring{D})\,\longrightarrow\,H^{n}(\partial
M_{+}\smallsetminus P,\partial
M_{+}\smallsetminus\mathring{D})\,\longrightarrow\,\ldots$ (C.14)

Using excision and homotopy invariance, for each $i$-th connected component of
$\partial M_{+}$ we get

$H^{n}(\partial M_{+}\smallsetminus P,\partial
M_{+}\smallsetminus\mathring{D})\Big{|}_{i}\approx
H^{n}(\mathbb{R}^{d-1}\smallsetminus\\{0\\},\mathbb{R}^{d-1}\smallsetminus\mathring{D})\approx
H^{n}(\mathbb{R}^{d-1}\smallsetminus\mathring{D},\mathbb{R}^{d-1}\smallsetminus\mathring{D})=0\;,$
(C.15)

where, by abuse of notation, $D$ is also a $(d-1)$-dimensional disk in
$\mathbb{R}^{d-1}$ around the origin. This implies $H^{n}(M_{+},\partial
M_{+}\smallsetminus P)\approx H^{n}(M_{+},\partial
M_{+}\smallsetminus\mathring{D})$. To the spaces on the RHS we can apply
Poincaré duality, $\approx H_{d-n}(M_{+},D)$, and by homotopy invariance
$\approx H_{d-n}(M_{+},P)$.

##  Appendix D Factorization for higher genus

Here we discuss factorization in the case of two boundaries with higher genus.
We explicitly perform the calculation that proves factorization on
$\Sigma_{2}\times I$, while for the case of $g>2$ we explain all the relevant
ingredients but do not perform the full calculation.

The defect network for gauging on $\Sigma_{2}\times I$ can be reduced to a
network lying on $\Sigma_{2}$ at a certain point in $I$. We represent that
$\Sigma_{2}$ by gluing the following building blocks:

$\mathcal{A}$$1$$2$$\mathcal{A}$$2$$1$$3$$\mathcal{A}$$3$$4$$5$$\mathcal{A}$$5$$4$
(D.1)

with the understanding that we glue along matching numbers. We will henceforth
omit $\mathcal{A}$ labels for ease of notation. As a convention, the lines are
oriented upwards, which automatically defines for us which intersections are
morphisms or co-morphisms. Such choice is carried throughout the computations.

We perform surgery along a $\Sigma_{2}$ slice by inserting a complete set of
states on $\Sigma_{2}$:

$\mathbbm{1}=\sum_{\begin{subarray}{c}a,b,c\,\in\,\mathcal{A}\\\
\mu\,\in\,\operatorname{Hom}(a\otimes\check{a},b),\,\,\nu\,\in\,\operatorname{Hom}(c\otimes\check{c},b)\end{subarray}}c_{abc}^{\mu\nu}\;|a,b,c;\mu,\nu\rangle\langle
a,b,c;\mu,\nu|\;,$ (D.2)

where $c_{abc}^{\mu\nu}=\langle a,b,c;\mu,\nu|a,b,c;\mu,\nu\rangle^{-1}$. The
state $|a,b,c;\mu,\nu\rangle$ is defined by the following path integral on
${\sf{S}}\Sigma_{2}$ (gluings are as in the previous figure, and brown disks
indicate that the surfaces are filled):

$|a,b,c;\mu,\nu\rangle\;=\;$$a$$\check{a}$$a$$b$$\mu$$\nu$$b$$c$$\check{c}$$c$
. (D.3)

We can also represent this more economically as follows:

$=$$a$$b$$c$$\mu$$\nu$$\times$$\times$ (D.4)

where crosses indicate non contractible cycles. A computation leads
to525252The gist of the computation is as follows: when one glues together the
two ${\sf{S}}\Sigma_{2}$, the contractible disks combine (locally) into an
$S^{2}$. Thus the glued geometry locally looks like $S^{2}\times\mathbb{R}$,
with the $\mathbb{R}$ direction winding around in a non-trivial way carrying
the lines $a,\hat{a}$, $b,\hat{b}$, and $c,\hat{c}$. Since the Hilbert space
on a bi-punctured $S^{2}$ is one dimensional if the punctures are conjugate
and empty otherwise, we use this to perform surgery locally. We cut
transversally to the local $\mathbb{R}$ direction, and on each side of the cut
we glue a $D_{3}$ containing a line that connects the two conjugate punctures,
for instance $a,\hat{a}$. When we do that, we need to divide by the norm of
the state, which is $d/\mathcal{D}$, where $d$ is the dimension of the line.
We cut in this way each of the lines $a,b,c$ (with their conjugates).
Eventually, we are left with a simple configuration of lines in $S^{3}$.

$c_{abc}^{\mu\nu}=\frac{d_{a}\,d_{c}}{\mathcal{D}\,d_{b}}\;.$ (D.5)

On one side of the surgery we find the diagram

$\displaystyle{}[{\sf{S}}\Sigma_{2}]^{+}$
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