Q_{n}\in\mathfrak{t}^{\vee}$. We assume that the root lattice satisfies

  1. (A1)

[$\mathsf{G}$-grading] $\Lambda_{R}\subset\Gamma^{\vee}$.

In representation theoretic terms, condition (A1) states that $R$ integrates
to a representation of $T_{\Gamma}$. Moreover, condition (A1) implies that
$\mathfrak{t}_{R}$ is the adjoint representation of the complex Lie supergroup
$G_{\Gamma}$ with bosonic subgroup $T_{\Gamma}$ and Lie superalgebra
$\mathfrak{t}_{R}$. In terms of the category $\widetilde{\mathcal{C}}_{R}$,
condition (A1) ensures that tracking weights modulo $\Gamma^{\vee}$ defines a
grading of $\widetilde{\mathcal{C}}_{R}$ by the torus
$\mathsf{G}=\mathfrak{t}^{\vee}/\Gamma^{\vee}$ which is Langlands dual to
$T_{\Gamma}$.

Let $\mathsf{Z}_{0}$ be a subgroup of $\mathfrak{t}^{\vee}$ and set
$\mathsf{Z}=\mathsf{Z}_{0}\oplus\mathbb{Z}/2\mathbb{Z}$, seen as a subgroup of
$\mathfrak{t}^{\vee}\oplus\mathbb{Z}/2\mathbb{Z}$. Consider the following
conditions on $\mathsf{Z}_{0}$:

  1. (A2)

[$\mathsf{Z}$ in degree $\bar{0}$] $\mathsf{Z}_{0}\subset\Gamma^{\vee}$.

  2. (A3)

[Existence of $\psi$] $\kappa^{\vee}(k,\lambda)\in\mathbb{Z}$ for all
$k\in\mathsf{Z}_{0}$ and $\lambda\in\Gamma^{\vee}$.

  3. (A4)

[Trivial ribbon] $-\kappa^{\vee}(k,k)+\sum_{i=1}^{n}\chi_{i}(k)\in
2\mathbb{Z}$ for all $k\in\mathsf{Z}_{0}$.

  4. (A5)

[Finiteness] The index of $\mathsf{Z}_{0}$ in $\Gamma^{\vee}$ is finite.

Conditions (A1) and (A3) imply that
$\chi_{i}(\gamma)=\kappa^{\vee}(Q_{i},\gamma)$ is an integer for each $1\leq
i\leq n$. Conditions (A2) and (A3) imply that $\kappa^{\vee}(k,k)$ is an
integer for all $k\in\mathsf{Z}_{0}$. This in turn implies that
$-\kappa^{\vee}(k,k)+\sum_{i=1}^{n}\chi_{i}(k)$ is an integer, of which
condition (A4) is a further refinement. As we will see below, conditions
(A1)-(A5) are necessary for $\mathsf{Z}$ to be the weights of a free
realization on $\widetilde{\mathcal{C}}_{R,\bar{0}}$ which is a part of a
relative pre-modular structure on $\widetilde{\mathcal{C}}_{R}$. The labeling
of each condition indicates its relevance from this perspective.

Recall the effective metric $\kappa_{\textup{\text{eff}}}$ introduced in
Section 2.4.

######  Proposition 4.1.

Let $\mathsf{Z}_{0}$ be the image of $\Gamma$ under the map
$\kappa^{\flat}:\mathfrak{t}\rightarrow\mathfrak{t}^{\vee}$. If the lattice
$\Gamma$ is even integral with respect to the effective metric
$\kappa_{\textup{\text{eff}}}$, then
$\mathsf{Z}=\mathsf{Z}_{0}\oplus\mathbb{Z}/2\mathbb{Z}$ satisfies conditions
(A2)-(A5).

###### Proof.

We verify each of the conditions. Given $\gamma\in\Gamma$, let
$\omega_{\gamma}=\kappa^{\flat}(\gamma)$.

  1. (A2)

Since the lattice $\Gamma$ is integral with respect to $\kappa$, the map
$\kappa^{\flat}:\mathfrak{t}\rightarrow\mathfrak{t}^{\vee}$ restricts to a
group homomorphism $\kappa^{\flat}:\Gamma\rightarrow\Gamma^{\vee}$. Hence,
$\omega_{\gamma}\in\Gamma^{\vee}$.

  2. (A3)

For each $\lambda\in\Gamma^{\vee}$, the definitions of $\kappa^{\vee}$ and
$\kappa^{\flat}$ give

$\kappa^{\vee}(\omega_{\gamma},\lambda)=\kappa^{\vee}(\kappa^{\flat}(\gamma),\lambda)=\lambda(\gamma).$

It follows that $\kappa^{\vee}(\omega_{\gamma},\lambda)\in\mathbb{Z}$.

  3. (A4)

Using the definition of the effective metric, we compute

$-\kappa^{\vee}(\omega_{\gamma},\omega_{\gamma})+\sum_{i=1}^{n}\chi_{i}(\omega_{\gamma})=-\kappa_{\textup{\text{eff}}}(\gamma,\gamma)+\sum_{i=1}^{n}\chi_{i}(\omega_{\gamma})(\chi_{i}(\omega_{\gamma})+1).$

Since $(\Gamma,\kappa_{\textup{\text{eff}}})$ is even integral and
$\chi_{i}(\omega_{\gamma})$ is an integer, it follows that
$-\kappa^{\vee}(\omega_{\gamma},\omega_{\gamma})+\sum_{i=1}^{n}\chi_{i}(\omega_{\gamma})\in
2\mathbb{Z}$.

  4. (A5)

This follows from non-degeneracy of $\kappa$. Indeed, the index in question is
the discriminant of $\kappa$. ∎

###  4.2. Non-zero matter

In this section we assume that $n\geq 1$, so that $R$ is not the zero
representation. The case $n=0$ is treated in Section 4.4. Continue with the
setting of Section 4.1 so that $\mathsf{G}=\mathfrak{t}^{\vee}/\Gamma^{\vee}$
and
$\mathsf{Z}_{0}=\text{{im}}(\kappa^{\flat}:\Gamma\rightarrow\Gamma^{\vee})$.
As in Proposition 4.1, assume that $(\Gamma,\kappa_{\textup{\text{eff}}})$ is
even integral. Keeping the notation of Section 3.4, let $\mathsf{X}$ be the
image of $\chi^{-1}(\mathsf{Y})$ under the quotient map
$\mathfrak{t}^{\vee}\rightarrow\mathsf{G}$.

######  Proposition 4.2.

The $\mathsf{G}$-graded category $\widetilde{\mathcal{C}}_{R}$ is generically
semisimple with small symmetric subset $\mathsf{X}$. For
$\bar{\lambda}\in\mathsf{G}\setminus\mathsf{X}$, a completely reduced
dominating set of $\widetilde{\mathcal{C}}_{R,\bar{\lambda}}$ is

$\\{V_{(\lambda,\bar{p})}\mid\lambda\mbox{ in class
}\bar{\lambda},\;\bar{p}\in\mathbb{Z}/2\mathbb{Z}\\}.$

###### Proof.

Conditions (A2), (A3) and (A5) imply that
$\chi^{-1}(\mathsf{Y})+\mathsf{Z}_{0}$ is finitely many translates of
$\chi^{-1}(\mathsf{Y})$ and so a union countably many affine hyperplanes in
$\mathfrak{t}^{\vee}$; see the proof of Proposition 3.8. It follows that
$\chi^{-1}(\mathsf{Y})+\mathsf{Z}_{0}$ is small in $\mathfrak{t}^{\vee}$ which
implies that $\mathsf{X}$ is small in $\mathsf{G}$.

Since the definition of $\mathsf{X}$ is such that any weight with image in
$\mathsf{G}\setminus\mathsf{X}$ is typical, the remainder of the proposition
can be proved in the same way as Proposition 3.15. ∎

Let $D=\operatorname{coker}(\kappa^{\flat}:\Gamma\rightarrow\Gamma^{\vee})$ be
the discriminant group of the metric $\kappa$. The group $D$ is finite abelian
of order the absolute value of the determinant of any Gram matrix of $\kappa$.
By an abuse of notation, we sometimes identify $D$ with a subset of
representatives in $\Gamma^{\vee}$ which contains $0$.

######  Proposition 4.3.

The monoidal functor

$\sigma:\mathsf{Z}\rightarrow\widetilde{\mathcal{C}}_{R},\qquad(\omega_{\gamma},\bar{p})\mapsto\sigma_{(\omega_{\gamma},\bar{p})}:=\mathbb{C}^{\mathfrak{t}}_{(\omega_{\gamma},\bar{p})}$

defines a free realization of $\mathsf{Z}$ in
$\widetilde{\mathcal{C}}_{R,\bar{0}}$ and gives $\widetilde{\mathcal{C}}_{R}$
the structure of a pre-modular $\mathsf{G}$-category relative to
$(\mathsf{Z},\mathsf{X})$.

###### Proof.

Recalling the computations from Example 3.17, we see that condition (A2)
ensures that each $\sigma_{(\omega_{\gamma},\bar{p})}$ has $\mathsf{G}$-degree
$\bar{0}$. Conditions (A1) and (A3) imply that
$\chi_{i}(\gamma)=\kappa^{\vee}(Q_{i},\gamma)\in\mathbb{Z}$, $1\leq i\leq n$,
which translates to the statement that $\sigma_{(\omega_{\gamma},\bar{p})}$ is
one dimensional and hence invertible. Finally, condition (A4) ensures that
$\theta_{\sigma_{(\omega_{\gamma},\bar{p})}}=\textup{\text{id}}_{\sigma_{(\omega_{\gamma},\bar{p})}}$.

Let $\bar{\lambda}\in\mathsf{G}\setminus\mathsf{X}$ with chosen lift
$\lambda\in\mathfrak{t}^{\vee}$. In view of the completely reduced dominating
set of Proposition 4.2, we can take
$\Theta(\bar{\lambda})=\\{V_{(\lambda+k,\bar{0})}\mid k\in D\\}$.

Finally, for $V\in\widetilde{\mathcal{C}}_{R,\bar{\lambda}}$, we compute

$c_{\sigma_{(\omega_{\gamma},\bar{p})},V}\circ
c_{V,\sigma_{(\omega_{\gamma},\bar{p})}}=q^{-4\kappa^{\vee}(\lambda,\omega_{\gamma})}\textup{\text{id}}_{V\otimes\sigma_{(\omega_{\gamma},\bar{p})}},$

where $\lambda\in\mathfrak{t}^{\vee}$ is any lift of $\bar{\lambda}$. We may
therefore take for the required bicharacter

$\psi:\mathsf{G}\times\mathsf{Z}\rightarrow\mathbb{C}^{\times},\qquad(\bar{\lambda},(\omega_{\gamma},\bar{p}))\mapsto
q^{-4\kappa^{\vee}(\omega_{\gamma},\lambda)}.$

Note that independence of $\psi$ on the choice of lift follows from condition
(A3).

In view of Propositions 3.21 and 4.2, this completes the proof. ∎

Turning to relative modularity, let $W\in\widetilde{\mathcal{C}}_{R,\bar{0}}$.
Recall that a morphism
$f\in\textup{\text{End}}_{\widetilde{\mathcal{C}}_{R}}(W)$ is called
_transparent in $\widetilde{\mathcal{C}}_{R,\bar{0}}$_ if

$\textup{\text{id}}_{U}\otimes
f=c_{W,U}\circ(f\otimes\textup{\text{id}}_{U})\circ c_{U,W}$

and

$f\otimes\textup{\text{id}}_{V}=c_{V,W}\circ(\textup{\text{id}}_{V}\otimes
f)\circ c_{W,V}$

for all $U,V\in\widetilde{\mathcal{C}}_{R,\bar{0}}$.

The following result is a variation of [DRGPM20, Lemma 2.3].

######  Lemma 4.4.

Let $W\in\widetilde{\mathcal{C}}_{R,\bar{0}}$ and
$f\in\textup{\text{End}}_{\widetilde{\mathcal{C}}_{R}}(W)$ be transparent in
$\widetilde{\mathcal{C}}_{R,\bar{0}}$. There exist an integer $m\geq 1$, one
dimensional modules
$\mathbb{C}^{\mathfrak{t}}_{(k_{i},\bar{p}_{i})}\in\widetilde{\mathcal{C}}_{R,\bar{0}}$
and morphisms
$g_{i}\in\textup{\text{Hom}}_{\widetilde{\mathcal{C}}_{R}}(W,\mathbb{C}^{\mathfrak{t}}_{(k_{i},\bar{p}_{i})})$
and
$h_{i}\in\textup{\text{Hom}}_{\widetilde{\mathcal{C}}_{R}}(\mathbb{C}^{\mathfrak{t}}_{(k_{i},\bar{p}_{i})},W)$,
where $1\leq i\leq m$, such that $f=\sum_{i=1}^{m}h_{i}\circ g_{i}$.

###### Proof.

Consider $V_{(0,\bar{0})}\in\widetilde{\mathcal{C}}_{R,\bar{0}}$ with its
highest weight vector $v_{\varnothing}$ and let $w\in W$ be a weight vector.
Because $v_{\varnothing}$ is annihilated by $E_{I}$, $I\neq\varnothing$, the
explicit form of the braiding shows that
$c_{V_{(0,\bar{0})},W}(v_{\varnothing}\otimes w)$ is proportional to $w\otimes
v_{\varnothing}$. Because $f$ is transparent,
$c_{W,V_{(0,\bar{0})}}(f(w)\otimes v_{\varnothing})$ is proportional to
$v_{\varnothing}\otimes f(w)$. Since $\\{F_{I}v_{\varnothing}\ \mid
I\subset\\{1,\dots,n\\}\\}$ is a basis of $V_{(0,\bar{0})}$, we conclude that
$E_{I}f(w)=0$ for all $I\neq\varnothing$.

Reversing the roles of $E_{i}$ and $F_{i}$ in Definition 3.4, define the Verma
module of lowest weight
$(\lambda,\bar{p})\in\mathfrak{t}^{\vee}\times\mathbb{Z}/2\mathbb{Z}$ by

$V^{-}_{(\lambda,\bar{p})}:=\widetilde{U}_{\sqrt{-1}}^{\mathfrak{t}}(\mathfrak{t}_{R})\otimes_{\widetilde{U}_{\sqrt{-1}}^{\mathfrak{t},\leq
0}(\mathfrak{t}_{R})}\mathbb{C}^{-}_{(\lambda,\bar{p})}.$

The module $V^{-}_{(\lambda,\bar{p})}$ has a homogeneous weight basis
$\\{E_{I}v^{-}_{\varnothing}\mid I\subset\\{1,\dots,n\\}\\}$, where
$v^{-}_{\varnothing}=1\otimes 1$. Applying the argument of the previous
paragraph with $V_{(0,\bar{0})}$ and its highest weight vector
$v_{\varnothing}$ replaced with $V^{-}_{(0,\bar{0})}$ and its lowest weight
vector $v^{-}_{\varnothing}$, we conclude that $F_{I}f(w)=0$ for all
$I\neq\varnothing$.

It follows from the previous two paragraphs that

$\mathcal{K}_{i}-\mathcal{K}^{-1}_{i}=(q-q^{-1})[E_{i},F_{i}]$

annihilates $f(w)$. Writing $\lambda\in\mathfrak{t}^{\vee}$ for the weight of
$f(w)$, we conclude that $q^{4\chi_{i}(\lambda)}=1$, $1\leq i\leq n$. By
Example 3.17, each homogeneous weight vector in the image of $f$ spans a one
dimensional module, necessarily in degree $\bar{0}\in\mathsf{G}$ since the
same is true of $W$. Since $f$ factors through its image, which is a direct
sum of its weight spaces, we obtain the desired one dimensional modules and
factorization of $f$. ∎

######  Remark 4.5.

As is evident from its proof, Lemma 4.4 does not require the full strength of
transparency: it suffices for $f$ to satisfy one-sided versions of
transparency only for some (anti-)Verma modules $U$ and $V$ in degree
$\bar{0}$. ∎

######  Theorem 4.6.

The category $\widetilde{\mathcal{C}}_{R}$ is a modular $\mathsf{G}$-category
relative to $(\mathsf{Z},\mathsf{X})$ with stabilization coefficients

$\Delta_{\pm}=\sum_{k\in D}q^{\mp
2\kappa^{\vee}(k,k)}\prod_{i=1}^{n}\frac{q^{2\chi_{i}(\lambda)}-q^{-2\chi_{i}(\lambda)}}{q^{2\chi_{i}(\lambda+k)}-q^{-2\chi_{i}(\lambda+k)}}$

and relative modularity parameter $\zeta=(-1)^{n}|D|$. Moreover,
$\widetilde{\mathcal{C}}_{R}$ is TQFT finite.

###### Proof.

The stabilization coefficients can be computed directly using Lemma 3.18 and
the formula for the ribbon automorphism given in the proof of Theorem 3.16.

For relative modularity, consider Definition 1.11 with $h=\bar{\gamma}$ and
$g=\bar{\lambda}$ and $V_{i}=V_{(\alpha,\bar{0})},V_{j}=V_{(\beta,\bar{0})}\in
I_{\bar{\lambda}}$. The morphism
$f_{\bar{\gamma},(\alpha,\bar{0}),(\beta,\bar{0})}\in\textup{\text{End}}_{\widetilde{\mathcal{C}}_{R}}(V_{(\alpha,\bar{0})}\otimes
V_{(\beta,\bar{0})}^{\vee})$ determined by the left hand side of diagram (2)
is transparent in $\widetilde{\mathcal{C}}_{R,\bar{0}}$; see [CGPM14, Lemma
5.9]. Lemma 4.4 therefore ensures the existence of one dimensional modules
$\mathbb{C}^{\mathfrak{t}}_{(k_{i},\bar{p}_{i})}\in\widetilde{\mathcal{C}}_{R,\bar{0}}$
such that

$f_{\bar{\gamma},(\alpha,\bar{0}),(\beta,\bar{0})}=\sum_{i=1}^{m}h_{\bar{\gamma},(\alpha,\bar{0}),(\beta,\bar{0}),i}\circ
g_{\bar{\gamma},(\alpha,\bar{0}),(\beta,\bar{0}),i}$

for some morphisms

$g_{\bar{\gamma},(\alpha,\bar{0}),(\beta,\bar{0}),i}\in\textup{\text{Hom}}_{\widetilde{\mathcal{C}}_{R}}(V_{(\alpha,\bar{0})}\otimes
V_{(\beta,\bar{0})}^{\vee},\mathbb{C}^{\mathfrak{t}}_{(k_{i},\bar{p}_{i})}),$
$h_{\bar{\gamma},(\alpha,\bar{0}),(\beta,\bar{0}),i}\in\textup{\text{Hom}}_{\widetilde{\mathcal{C}}_{R}}(\mathbb{C}^{\mathfrak{t}}_{(k_{i},\bar{p}_{i})},V_{(\alpha,\bar{0})}\otimes
V_{(\beta,\bar{0})}^{\vee}).$

The set $I_{\bar{\lambda}}$ is defined so that

$\textup{\text{Hom}}_{\widetilde{\mathcal{C}}_{R}}(V_{(\alpha,\bar{0})}\otimes
V^{\vee}_{(\beta,\bar{0})},\mathbb{C}^{\mathfrak{t}}_{(k_{i},\bar{p}_{i})})\simeq\textup{\text{Hom}}_{\widetilde{\mathcal{C}}_{R}}(V_{(\alpha,\bar{0})},V_{(\beta,\bar{0})}\otimes\mathbb{C}^{\mathfrak{t}}_{(k_{i},\bar{p}_{i})})$

is non-zero for a unique $\mathbb{C}^{\mathfrak{t}}_{(k_{i},\bar{p}_{i})}$.
Comparing weights gives $\bar{p}_{i}=0$ and $\alpha=\beta+k_{i}$. It follows
that

$f_{\bar{\gamma},(\alpha,\bar{0}),(\beta,\bar{0})}=c_{\alpha,\beta}\textup{\text{id}}_{\mathbb{C}^{\mathfrak{t}}_{(\alpha-\beta,\bar{0})}}\otimes\stackrel{{\scriptstyle\longleftarrow}}{{\operatorname{coev}}}_{V_{(\alpha,\bar{0})}}\circ\stackrel{{\scriptstyle\longrightarrow}}{{\operatorname{ev}}}_{V_{(\alpha,\bar{0})}}$
(31)

for some scalar $c_{\alpha,\beta}\in\mathbb{C}$. The modified trace of the
right hand side of equation (31) is
$q^{2\sum_{i=1}^{n}\chi_{i}(\alpha-\beta)}c_{\alpha,\beta}\mathsf{d}(V_{(\alpha,\bar{0})})$,
which vanishes if and only if $c_{\alpha,\beta}$ does. Fix a lift of
$\bar{\gamma}$ to $\gamma\in\mathfrak{t}^{\vee}$. By isotopy invariance and
the defining properties of the modified trace, the modified trace of the left
hand side of equation (31) is

$\displaystyle\mathsf{d}(V_{(\alpha,\bar{0})})\sum_{k\in
D}\mathsf{d}(V_{(\gamma+k,\bar{0})})\mathsf{tr}_{V_{(\alpha,\bar{0})}\otimes
V_{(\beta,\bar{0})}^{\vee}}(f_{(\gamma+k,\bar{0}),(\alpha,\bar{0}),(\beta,\bar{0})})$
$\displaystyle=$ $\displaystyle\mathsf{d}(V_{(\alpha,\bar{0})})\sum_{k\in
D}\mathsf{d}(V_{(\gamma+k,\bar{0})})\mathsf{tr}_{V_{(\gamma+k,\bar{0})}}(\Phi_{V^{\vee}_{(\beta,\bar{0})},V_{(\gamma+k,\bar{0})}}\circ\Phi_{V_{(\alpha,\bar{0})},V_{(\gamma+k,\bar{0})}})$
$\displaystyle=$ $\displaystyle\mathsf{d}(V_{(\alpha,\bar{0})})\sum_{\beta\in
D}\mathsf{d}(V_{(\gamma+k,\bar{0})})^{2}\langle\Phi_{V^{\vee}_{(\beta,\bar{0})},V_{(\gamma+k,\bar{0})}}\rangle\langle\Phi_{V_{(\alpha,\bar{0})},V_{(\gamma+k,\bar{0})}}\rangle$
$\displaystyle=$
$\displaystyle(-1)^{n}q^{2\sum_{i=1}^{n}\chi_{i}(\alpha-\beta)-4\kappa^{\vee}(\alpha-\beta,\gamma)}\mathsf{d}(V_{(\alpha,\bar{0})})\sum_{k\in
D}q^{-4\kappa^{\vee}(\alpha-\beta,k)},$

the final equality following from equation (29). The canonical isomorphism
$\Gamma\simeq\Gamma^{\vee\vee}$ implies that

$q^{-4\kappa^{\vee}(\alpha-\beta,-)}=e^{-2\pi\sqrt{-1}\kappa^{\vee}(\alpha-\beta,-)}:D\rightarrow\mathbb{C}^{\times}$

is the trivial character if and only if
$\alpha-\beta\in\text{{im}}\kappa^{\flat}$. Note however that the definition
of $I_{\bar{\lambda}}$ implies that $\alpha-\beta\in\text{{im}}\kappa^{\flat}$
if and only if $\alpha=\beta$. Fourier theory for the finite abelian group $D$
then gives

$\mathsf{tr}_{V_{(\alpha,\bar{0})}\otimes
V_{(\beta,\bar{0})}^{\vee}}(f_{\bar{\gamma},(\alpha,\bar{0}),(\beta,\bar{0})})=\begin{cases}(-1)^{n}\mathsf{d}(V_{(\alpha,\bar{0})})|D|&\mbox{if
}\alpha=\beta,\\\ 0&\mbox{otherwise}.\end{cases}$

It follows that $c_{\alpha,\beta}=0$ unless $\alpha=\beta$ and we conclude
that $\zeta=(-1)^{n}|D|$ is a relative modularity parameter.

Turning to TQFT finiteness (see Definition 1.13), note that
$\widetilde{\mathcal{C}}_{R}$ is locally finite abelian with enough injectives
and projectives. In particular, $\widetilde{\mathcal{C}}_{R}$ is Krull–Schmidt
and its isomorphism classes of projective indecomposable objects are in
bijection with its isomorphism classes of simple objects [Kra15, Lemma 3.5].
Since the latter are in bijection with
$\mathfrak{t}^{\vee}\times\mathbb{Z}/2\mathbb{Z}$ by Theorem 3.12, property
(F1) holds by condition (A5). Property (F2) follows from the fact that objects
of $\widetilde{\mathcal{C}}_{R}$ are finite dimensional weight modules and
property (F3) follows from the Krull–Schmidt property. ∎

Denote the relative modular category of Theorem 4.6 by
$\widetilde{\mathcal{C}}_{R,\Gamma}$ and the TQFT resulting from Theorem 1.12
by $\mathcal{Z}_{\widetilde{\mathcal{C}}_{R,\Gamma}}$. In the remainder of
this section, we argue that $\mathcal{Z}_{\widetilde{\mathcal{C}}_{R,\Gamma}}$
models the homological truncation of Chern–Simons-Rozansky–Witten theory of
the Hamiltonian $T_{c,\Gamma}$-manifold $T^{\vee}R$ at level $\kappa$ or,
equivalently, the truncation of Chern–Simons theory with gauge supergroup
$G_{c,\Gamma}$ at level $\kappa$.

####  4.2.1. Topological flavour symmetry

The grading group $\mathsf{G}$ of the category
$\widetilde{\mathcal{C}}_{R,\Gamma}$ agrees with the topological flavour
symmetry group (also known as the magnetic flavour symmetry group) of the
$T_{c,\Gamma}$-gauge theory underlying the Chern–Simons-Rozansky–Witten theory
of $T^{\vee}R$. Following the general discussion of such symmetries in
[GKSW15, §4], this can be seen as follows. The current associated to the
topological flavour symmetry is the curvature $F$ of the
$T_{c,\Gamma}$-connection $A$; it is conserved due the Bianchi identity. The
infinitesimal action of this $0$-form symmetry is implemented by surface
integrals of $\frac{1}{2\pi}F$. For example, its action on a local operator
$\mathcal{O}(x)$ is realized by integrating $\frac{1}{2\pi}F$ over a
$2$-sphere $S^{2}$ linking the point $x$ and hence measures the Chern number
of the gauge bundle sourced by $\mathcal{O}(x)$. In other words, operators
charged under this symmetry are magnetic monopoles. The weights for the
topological flavour symmetry group, that is, the allowed Chern numbers of
gauge bundles on $S^{2}$, are identified with the cocharacter lattice
$\Gamma\simeq\pi_{1}(T_{c,\Gamma})$ of $T_{c,\Gamma}$ and hence the
topological flavour symmetry group is identified with Pontryagin dual
$\hat{\Gamma}\simeq\mathfrak{t}^{\vee}/\Gamma^{\vee}\simeq\mathsf{G}$, as
claimed

Finite symmetry transformations are implemented by the topological surface
operators given by exponentiated integrals of the curvature,

$\mathcal{U}_{\bar{g}}[\Sigma]=\exp\bigg{(}\sqrt{-1}\xi\int_{\Sigma}F\bigg{)}\,,$

where $\Sigma$ is a surface in $3$-dimensional spacetime and
$\bar{g}=\xi+\Gamma^{\vee}\in\mathsf{G}$. Importantly, if $\Sigma$ has non-
empty boundary, then the surface operator is bounded by an improperly-
quantized Wilson line [ALMRP92, GKSW15]; schematically, this is due to the
relation

$\mathcal{U}_{\bar{g}}[\Sigma]=\exp\bigg{(}\sqrt{-1}\xi\int_{\Sigma}F\bigg{)}=\exp\bigg{(}\sqrt{-1}\xi\oint_{\partial\Sigma}A\bigg{)}\,.$

The way to interpret this formula is that a Wilson line labelled by the
infinitesimal weight $\xi$, that is, an object of the subcategory
$\widetilde{\mathcal{C}}_{R,\Gamma,\xi}$, is not gauge invariant on its own
unless $\xi\in\Gamma^{\vee}$, that is, $\xi$ is a weight of $T_{c,\Gamma}$. If
$\xi\notin\Gamma^{\vee}$, then the Wilson line must be attached to the
corresponding magnetic surface operator $\mathcal{U}_{\bar{g}}$, whose support
$\Sigma$ is extra data necessary to define the Wilson line. The attached
surface $\Sigma$ can be thought of as the worldsheet of the electric analogue
of a Dirac string [ALMRP92], which is visible to magnetically charged local
operators unless the Wilson line is properly quantized.

Line operators that are well-defined without the additional data of a
topological surface defect are called _genuine_ while those that require this
additional data are _non-genuine_ [KS14, §2]. A consistent network of such
topological surface defects precisely encodes (the holonomies of) a flat
$\mathsf{G}$-connection on the complement of the support of these line
operators or, equivalently, the Poincaré dual of this network realizes a
$\mathsf{G}$-valued cohomology class $\omega$. This is the physical origin of
the cohomology classes which decorate objects and morphisms of
$\mathsf{C}\mathsf{ob}^{\textnormal{ad}}_{\widetilde{\mathcal{C}}_{R,\Gamma}}$.

####  4.2.2. Screening by gauge vortex lines

To describe the full, non-perturbative, theory the effects of screening by
gauge vortex lines must be included. Namely, the physically inequivalent line
operators in the non-perturbative theory are identified with
$\mathsf{Z}$-orbits of objects in $\widetilde{\mathcal{C}}_{R,\Gamma}$. In
particular, simple line operators are labelled by their weights modulo
screening, that is, elements of $\mathfrak{t}^{\vee}/\mathsf{Z}_{0}$. We note
that all of these simple Wilson lines introduce a monodromy in the bosonic
gauge fields, a phenomenon familiar to compact Chern–Simons theories [Wit89,
§3.3]. The monodromy induced by simple line operators is controlled by the map
$\mathfrak{t}^{\vee}/\mathsf{Z}_{0}\to T_{\Gamma}$ induced by
$\kappa^{\sharp}:\mathfrak{t}^{\vee}\to\mathfrak{t}$. The simple line
operators in degree $\bar{0}$, that is, those line operators which source a
flat connection for the topological flavour symmetry with trivial holonomy,
are thus genuine and labelled by elements of the discriminant group
$D\simeq\Gamma^{\vee}/\mathsf{Z}_{0}$, as expected.

####  4.2.3. Orientations and spin structures

The boundary vertex operator algebras which generalize those of [GN23] to the
present setting have boundary monopole operators with half-integral conformal
weights/spins if and only if the lattice
$(\Gamma,\kappa_{\textup{\text{eff}}})$ is not even. In particular, if
$(\Gamma,\kappa_{\textup{\text{eff}}})$ is even, then the physical theory is
defined on oriented $3$-manifolds; other wise a spin structures is required.
Our evenness assumption in Proposition 4.1 therefore matches physics
predictions. When the lattice $(\Gamma,\kappa_{\textup{\text{eff}}})$ is odd,
we expect a spin analogue of Theorem 4.6 and the resulting TQFT
$\mathcal{Z}_{\widetilde{\mathcal{C}}_{R,\Gamma}}$, along the lines of
[BCGPM14].

####  4.2.4. Verlinde formula

Let ${\mathscr{D}}=\sqrt{(-1)^{n}|D|}$. We compute the value of
$\mathcal{Z}=\mathcal{Z}_{\widetilde{\mathcal{C}}_{R,\Gamma}}$ on trivial
circle bundles over closed surfaces with insertions and relate the result to
Euler characteristics of state spaces of $\mathcal{Z}$, yielding a Verlinde
formula for $\mathcal{Z}$.

Let ${\mathcal{S}}=(\Sigma,\\{p_{1},\ldots,p_{m}\\},\omega,\mathcal{L})$ be a
decorated connected closed surface of genus $g$. Since the Lagrangian subspace
$\mathcal{L}$ plays no role in what follows, we henceforth ignore it. For each
$\bar{\beta}\in\mathsf{G}$, consider the decorated closed $3$-manifold

${\mathcal{S}}\times S^{1}_{\bar{\beta}}=(\Sigma\times
S^{1},T=\\{p_{1},\dots,p_{m}\\}\times S^{1},\omega\oplus\bar{\beta}),$

where we use the isomorphism

$H^{1}((\Sigma\times S^{1})\setminus T;\mathsf{G})\simeq
H^{1}(\Sigma\setminus\\{p_{i}\\};\mathsf{G})\oplus\mathsf{G}$

to extend $\omega$ to $\omega\oplus\bar{\beta}$. Assuming that all colours of
${\mathcal{S}}\times S^{1}_{\bar{\beta}}$ have degree in
$\mathsf{G}\setminus\mathsf{X}$, the partition function
$\mathcal{Z}({\mathcal{S}}\times S^{1}_{\bar{\beta}})$ can be computed by the
following surgery presentation:

$\begin{array}[]{c}\raisebox{-4.0pt}{\psfig{height=129.16626pt}}\end{array}\put(-230.0,64.0){\mbox{\tiny$\Omega_{\bar{\beta}_{1}}$}}\put(-300.0,40.0){\mbox{\tiny$\Omega_{\bar{\alpha}_{1}}$}}\put(-83.0,64.0){\mbox{\tiny$\Omega_{\bar{\beta}_{g}}$}}\put(-155.0,40.0){\mbox{\tiny$\Omega_{\bar{\alpha}_{g}}$}}\put(0.0,-29.0){{\mbox{\tiny$\Omega_{\bar{\beta}}$}}}\put(-205.0,-25.0){\mbox{\tiny$V_{1}$}}\put(-140.0,-25.0){\mbox{\tiny$V_{m}$}}\;\;.$

Here $V_{i}=V_{(\mu_{i},\bar{q}_{i})}$ is the colour of the point $p_{i}$ and
$\bar{\alpha}_{j}=\omega(a_{j}),\bar{\beta}_{j}=\omega(b_{j})$ for a
symplectic basis $\\{a_{j},b_{j}\mid 1\leq j\leq g\\}$ of
$H_{1}(\Sigma;\mathbb{Z})$. By applying equation (2) first to the
$\Omega_{\bar{\alpha}_{j}}$-coloured strand and then to the
$\Omega_{\bar{\beta}_{j}}$-coloured strand, we simplify the surgery
presentation using the equalities

$\begin{array}[]{c}\raisebox{-4.0pt}{\psfig{height=86.11084pt}}\end{array}\put(-13.0,40.0){\mbox{\tiny$\Omega_{\bar{\alpha}_{j}}$}}\put(-22.0,-12.0){\mbox{\tiny$\Omega_{\bar{\beta}_{j}}$}}\put(-80.0,7.0){\mbox{\tiny$V_{(\beta+k,\bar{0})}$}}=-\frac{\zeta}{\mathsf{d}(V_{(\beta+k,\bar{0})}^{\vee})}\begin{array}[]{c}\raisebox{-4.0pt}{\psfig{height=86.11084pt}}\end{array}\put(-1.0,-12.0){\mbox{\tiny$\Omega_{\bar{\beta}_{j}}$}}\put(-5.0,-31.0){\mbox{\tiny$V_{(\beta+k,\bar{0})}$}}\put(-60.0,-25.0){\mbox{\tiny$V_{(\beta+k,\bar{0})}$}}=\frac{\zeta^{2}}{\mathsf{d}(V_{(\beta+k,\bar{0})}^{\vee})^{2}}\textup{\text{id}}_{V_{(\beta+k,\bar{0})}},$

where $V_{(\beta+k,\bar{0})}$ is any colour appearing in
$\Omega_{\bar{\beta}}$. Applying this simplification for $j=1,\dots,g$ reduces
the surgery presentation to an $\Omega_{\bar{\beta}}$-coloured unknot
encircled by $m$ pairwise unlinked unknots coloured by $V_{1},\dots,V_{m}$,
whose associated scalar we denote by
$\langle\Phi_{\\{V_{i}\\},\Omega_{\bar{\beta}}}\rangle$. Evaluating the
simplified diagram gives

$\mathcal{Z}({\mathcal{S}}\times
S^{1}_{\bar{\beta}})={\mathscr{D}}^{-2g-2}\zeta^{2g}\sum_{k\in
D}\mathsf{d}(V_{(\beta+k,\bar{0})})^{2-2g}\langle\Phi_{\\{V_{i}\\},\Omega_{\bar{\beta}}}\rangle.$

Setting $\mu=\sum_{i=1}^{m}\mu_{i}$ and $\bar{q}=\sum_{i=1}^{m}\bar{q}_{i}$,
we use Lemma 3.18 to compute

$\mathcal{Z}({\mathcal{S}}\times
S^{1}_{\bar{\beta}})=(-1)^{(g+1+m)n+\bar{q}}|D|^{g-1}q^{-4\kappa^{\vee}(\beta,\mu)+2m\sum_{i=1}^{n}\chi_{i}(\beta)+2\sum_{i=1}^{n}\chi_{i}(\mu)}\cdot\\\
\sum_{k\in
D}\prod_{i=1}^{n}(q^{2\chi_{i}(\beta+k)}-q^{-2\chi_{i}(\beta+k)})^{2g-2+m}.$
(32)

Since $\mathcal{Z}({\mathcal{S}}\times S^{1}_{\bar{\beta}})$ is holomorphic in
$\omega$, equation (32) holds whenever ${\mathcal{S}}$ is admissible.

Let $s$ be a variable and $t=(t_{1},\dots,t_{r})$ a multivariable. For
$\lambda=(\lambda_{1},\dots,\lambda_{r})\in\mathfrak{t}^{\vee}$, introduce the
notation $t^{\lambda}=t_{1}^{\lambda_{1}}\cdots t_{r}^{\lambda_{r}}$. Given
$\beta\in\mathfrak{t}^{\vee}$, we take specializing $t$ to
$q^{-4\kappa^{\vee}(-,\beta)}$ to mean setting each $t^{\lambda}$,
$\lambda\in\mathfrak{t}^{\vee}$, equal to
$q^{-4\kappa^{\vee}(\lambda,\beta)}$. Write
$\mathcal{Z}_{(k,\bar{p})}({\mathcal{S}})$ for the homogeneous summand of
$\mathcal{Z}({\mathcal{S}})$ of degree $(k,\bar{p})\in\mathsf{Z}$ and define
the generating function of graded dimensions

$\dim_{(t,s)}\mathcal{Z}({\mathcal{S}})=\sum_{(k,\bar{p})\in\mathsf{Z}}(-1)^{\bar{p}}\dim_{\mathbb{C}}\mathcal{Z}_{(k,\bar{p})}({\mathcal{S}})t^{k}s^{\bar{p}}.$

######  Theorem 4.7 (Verlinde formula).

There is an equality

$\mathcal{Z}({\mathcal{S}}\times
S^{1}_{\bar{\beta}})=\dim_{(q^{-4\kappa^{\vee}(-,\beta)},1)}\mathcal{Z}({\mathcal{S}}).$

###### Proof.

Since this can be proved in the same way as [BCGPM16, Theorem 5.9] and [GY22,
Theorem 3.5], we omit the details. ∎

More explicitly, the right hand side of the equality of Theorem 4.7 reads

$\sum_{\gamma\in\Gamma}\chi(\mathcal{Z}_{(\omega_{\gamma},\bullet)}({\mathcal{S}}))q^{-4\kappa^{\vee}(\omega_{\gamma},\beta)},$

where $\chi(\mathcal{Z}_{(\omega_{\gamma},\bullet)}({\mathcal{S}}))$ denotes
the Euler characteristic of the $\mathbb{Z}/2\mathbb{Z}$-graded subspace of
$\mathcal{Z}({\mathcal{S}})$ of $\mathsf{Z}_{0}$-degree $\omega_{\gamma}$.

######  Example 4.8.

Setting $m=0$ in equation (32) gives

$\mathcal{Z}({\mathcal{S}}\times
S^{1}_{\bar{\beta}})=(-1)^{(g+1)n}|D|^{g-1}\sum_{k\in
D}\prod_{i=1}^{n}(q^{2\chi_{i}(\beta+k)}-q^{-2\chi_{i}(\beta+k)})^{2g-2}.$

Applying Theorem 4.7, we conclude that

$\chi(\mathcal{Z}({\mathcal{S}})):=\sum_{\gamma\in\Gamma}\chi(\mathcal{Z}_{(\omega_{\gamma},\bullet)}({\mathcal{S}}))=\lim_{\beta\rightarrow
0}\mathcal{Z}({\mathcal{S}}\times S^{1}_{\bar{\beta}})\\\ =|D|^{g-1}\sum_{k\in
D}\prod_{i=1}^{n}\Big{(}2\sin\big{(}\chi_{i}(k)\pi\big{)}\Big{)}^{2g-2}.$

In particular,
$\dim_{\mathbb{C}}\mathcal{Z}({\mathcal{S}})=\chi(\mathcal{Z}({\mathcal{S}}))=|D|$
when $g=1$. ∎

######  Remark 4.9.

The Euler characteristic $\chi(\mathcal{Z}({\mathcal{S}}))$ from Example 4.8
can be reproduced physically by way of supersymmetric localization. See [CK19]
and references therein for a detailed overview of localization techniques and
[CDGG21, §5] for an implementation in a related context. In brief, this Euler
characteristic can be realized as a sum over solutions $\\{y^{*}\\}$ to the
Bethe vacuum equations of the form

$\chi(\mathcal{Z}({\mathcal{S}}))=\sum_{\\{y^{*}\\}}\mathcal{H}(y^{*})^{g-1},$

where $\mathcal{H}$ denotes the handle-gluing operator. In the present
context, the Bethe vacuum equations take the form

$\prod_{b=1}^{r}y_{b}^{\kappa_{ab}}=(-1)^{\sum_{i=1}^{n}Q_{ai}},\qquad 1\leq
a\leq r$

and the handle-gluing operator is

$\mathcal{H}(y)=|D|\prod_{i=1}^{n}\bigg{(}\big{(}1-\prod_{a=1}^{r}y_{a}^{Q_{ai}}\big{)}\big{(}1-\prod_{a=1}^{r}y_{a}^{-Q_{ai}}\big{)}\bigg{)},$

from which the above formula for $\chi(\mathcal{Z}({\mathcal{S}}))$ is a
straightforward computation. ∎

###  4.3. Compact global forms of $\mathfrak{gl}(1|1)$-Chern–Simons theory

Consider the special case of Section 4.2 corresponding to Example 2.4, so that
$r=2$ and $n=1$ with weight $Q=\left(\begin{smallmatrix}1\\\
0\end{smallmatrix}\right)$ and metric $\kappa=\left(\begin{smallmatrix}0&1\\\
1&0\end{smallmatrix}\right)$. The effective metric is
$\kappa_{\textup{\text{eff}}}=\left(\begin{smallmatrix}1&1\\\
1&0\end{smallmatrix}\right)$. We identify $\mathfrak{t}_{R}$ with
$\mathfrak{gl}(1|1)$, so that $\Lambda_{R}=\mathbb{Z}\cdot N^{\vee}$ and
$\chi_{1}(\lambda)=-\lambda_{E}$.

Let $\Gamma\subset\mathfrak{t}$ be a full rank lattice which is integral with
respect to $\kappa$ and satisfies condition (A1). As argued in [Mik15, §4.2],
there exist unique integers $s,t\in\mathbb{Z}_{>0}$ and half integer
$u\in\frac{1}{2}\mathbb{Z}/s\mathbb{Z}$ such that
$\Gamma=\mathbb{Z}\cdot\gamma_{1}\oplus\mathbb{Z}\cdot\gamma_{2}$, where

$\gamma_{1}=\frac{s}{t}E,\qquad\gamma_{2}=tN+\frac{u}{t}E.$

We furthermore assume that $(\Gamma,\kappa_{\textup{\text{eff}}})$ is even,
which translates to the condition that $t^{2}+2u$ is an even integer. In the
basis $\\{\gamma_{1},\gamma_{2}\\}$, the Gram matrix of $\kappa$ is
$B=\left(\begin{smallmatrix}0&s\\\ s&2u\end{smallmatrix}\right)$, whence
$|D|=s^{2}$. The discriminant group is

$D\simeq\mathbb{Z}/d_{1}\mathbb{Z}\oplus\mathbb{Z}/d_{2}\mathbb{Z},$

where $d_{1}=\gcd(s,2u)$ and $d_{2}=\frac{s^{2}}{d_{1}}$. We compute

$\gamma_{1}^{\vee}=-\frac{u}{ts}N^{\vee}+\frac{t}{s}E^{\vee},\qquad\gamma_{2}^{\vee}=\frac{1}{t}N^{\vee}$

and

$\omega_{\gamma_{1}}=\frac{s}{t}N^{\vee},\qquad\omega_{\gamma_{2}}=\frac{u}{t}N^{\vee}+tE^{\vee}.$

Denote by $\widetilde{\mathcal{C}}_{R,\Gamma}$ the relative modular category
constructed in Theorem 4.6. The TQFT
$\mathcal{Z}_{\widetilde{\mathcal{C}}_{R,\Gamma}}$ models Chern–Simons theory
with gauge supergroup $G_{c,\Gamma}$ at level $\kappa$, that is, a global form
of $\mathfrak{u}(1|1)$-Chern–Simons theory. Let $Y\in GL(2,\mathbb{Z})$ be a
right Smith multiplier for the Gram matrix $B$. Using the material recalled in
Appendix A with the dual basis $\\{\gamma_{1}^{\vee},\gamma_{2}^{\vee}\\}$,
the result of Example 4.8 can be written as

$\chi(\mathcal{Z}_{\widetilde{\mathcal{C}}_{R,\Gamma}}({\mathcal{S}}))=s^{2g-2}\sum_{i=0}^{d_{1}-1}\sum_{j=0}^{d_{2}-1}\Big{(}2\sin\big{(}(Y_{11}i+Y_{21}j)\frac{t}{s}\pi\big{)}\Big{)}^{2g-2}.$

######  Example 4.10.

Suppose that $(s,t,u)=(1,t,u)$ with $t^{2}+2u\in 2\mathbb{Z}$. Then $D$ is the
trivial group and

$\chi(\mathcal{Z}({\mathcal{S}}))=\begin{cases}0&\mbox{if }g\geq 2,\\\
1&\mbox{if }g=1.\end{cases}\qed$

######  Example 4.11.

Let $(s,t,u)=(s,1,\frac{s}{2})$ with $s$ odd, so that $d_{1}=d_{2}=s$ and
$D\simeq\mathbb{Z}/s\mathbb{Z}\oplus\mathbb{Z}/s\mathbb{Z}$. A right Smith
multiplier for $B=\left(\begin{smallmatrix}0&s\\\ s&s\end{smallmatrix}\right)$
is $Y=\left(\begin{smallmatrix}1&0\\\ -1&1\end{smallmatrix}\right)$ so that

$\chi(\mathcal{Z}({\mathcal{S}}))=s^{2g-2}\sum_{i,j=0}^{s-1}\Big{(}2\sin\big{(}\frac{(i-j)\pi}{s}\big{)}\Big{)}^{2g-2}=s^{2g-1}\sum_{k=0}^{s-1}\Big{(}2\sin\big{(}\frac{k\pi}{s}\big{)}\Big{)}^{2g-2}=s^{2g}{2g-2\choose
g-1},$

the final equality following from [PBM86, Eqn. (4.4.2.1)]. In physical terms,
this example is Chern–Simons theory of a pair of $U(1)$ gauge fields $A^{\pm}$
at levels $\pm s$ coupled to a B-twisted hypermultiplet $Z^{\pm}$ of
$(T_{+},T_{-})$-weights $\pm(1,-1)$. When the underlying $U(1)^{2}$-bundle is
trivial, the action for a closed $3$-manifold $M$ reads

$S(M)=\int_{M}\left(\frac{s}{4\pi}(A^{+}\wedge dA^{+}-A^{-}\wedge
dA^{-})+Z^{-}\wedge d_{A}Z^{+}\right).\qed$

######  Example 4.12.

Take $(s,t,u)=(s,1,\frac{s-1}{2})$ with $s$ even, so that $d_{1}=1$ and
$d_{2}=s^{2}$ and $D\simeq\mathbb{Z}/s^{2}\mathbb{Z}$. A right Smith
multiplier for $B=\left(\begin{smallmatrix}0&s\\\
s&s-1\end{smallmatrix}\right)$ is $Y=\left(\begin{smallmatrix}s&s-1\\\
-s+1&-s+2\end{smallmatrix}\right)$ so that

$\chi(\mathcal{Z}({\mathcal{S}}))=s^{2g-2}\sum_{j=0}^{s^{2}-1}\Big{(}2\sin\big{(}\frac{(-s+1)j\pi}{s}\big{)}\Big{)}^{2g-2}=s^{2g-1}\sum_{j=0}^{s-1}\Big{(}2\sin\big{(}\frac{j\pi}{s}\big{)}\Big{)}^{2g-2}=s^{2g}{2g-2\choose
g-1}.\qed$

######  Remark 4.13.

Another example of physical interest is $(s,t,u)=(s,1,0)$, which corresponds
to the Chern–Simons theory of a hyperbolic pair of $U(1)$ gauge fields
$A^{N}$, $A^{E}$ at level $s$ coupled to a B-twisted hypermultiplet $Z^{\pm}$
of $(N,E)$-weights $\pm(1,0)$. The action is

$S(M)=\int_{M}\left(\frac{s}{4\pi}(A^{N}\wedge dA^{E}+A^{E}\wedge
dA^{N})+Z^{-}\wedge d_{A^{N}\oplus A^{E}}Z^{+}\right).$

However, this example does not satisfy our assumptions since
$\kappa_{\textup{\text{eff}}}$ is not even integral: $t^{2}+2u=1$ is odd.
Instead, we expect this example to define a spin TQFT. ∎

###  4.4. Zero matter and toral Chern–Simons theory

Consider the degenerate setting of Section 4.1 in which $R$ is the zero
representation, that is, there is no matter. In this case
$\mathfrak{t}_{R}=\mathfrak{t}$ and there are no effective corrections, that
is, $\kappa=\kappa_{\textup{\text{eff}}}$.

Since $\widetilde{U}_{\sqrt{-1}}^{\mathfrak{t}}(\mathfrak{t})$ is concentrated
in degree $\bar{0}$, we consider its category $\widetilde{\mathcal{C}}$ of
finite dimensional ungraded weight
$\widetilde{U}_{\sqrt{-1}}^{\mathfrak{t}}(\mathfrak{t})$-modules. The category
$\widetilde{\mathcal{C}}$ is semisimple with isomorphism classes of simple
objects given by the one dimensional modules
$V_{\lambda}=\mathbb{C}^{\mathfrak{t}}_{\lambda}$ of weight
$\lambda\in\mathfrak{t}^{\vee}$ (with no condition on $\lambda$). As in
Section 4.2, grade $\widetilde{\mathcal{C}}$ by the dual torus
$\mathsf{G}=\mathfrak{t}^{\vee}/\Gamma^{\vee}$ and set
$\mathsf{Z}=\text{{im}}(\kappa^{\flat}:\Gamma\rightarrow\Gamma^{\vee})$.

######  Theorem 4.14.

The category $\widetilde{\mathcal{C}}$ is a relative modular
$\mathsf{G}$-category with respect to $(\mathsf{Z},\mathsf{X}=\varnothing)$
with with stabilization coefficients

$\Delta_{\pm}=\sum_{k\in D}q^{\mp 2\kappa^{\vee}(k,k)}$

and relative modularity parameter $\zeta=|D|$. Moreover,
$\widetilde{\mathcal{C}}$ is TQFT finite.

###### Proof.

Since $\widetilde{\mathcal{C}}$ is semisimple, we can take for the modified
trace the standard trace. Proposition 4.1 ensures that conditions (A2)-(A5)
hold. We can therefore argue as in Section 4.2 to prove that
$\widetilde{\mathcal{C}}$ is relative modular with the stated numerical
invariants. ∎

######  Remark 4.15.

There is an analogue of Theorem 4.14 for the category $\mathcal{C}$ of
ungraded weight modules over the $U_{-1}^{\mathfrak{t}}(\mathfrak{t})$; see
Remark 2.10. The results which follow can also be proved in this context. ∎

Let $\mathcal{Z}_{\widetilde{\mathcal{C}}_{\Gamma}}$ be the
$\mathsf{Z}$-graded TQFT associated to the relative modular structure of
Theorem 4.14. Since $\mathsf{X}=\varnothing$, all decorated bordisms are
admissible, that is
$\mathsf{C}\mathsf{ob}^{\textnormal{ad}}_{\widetilde{\mathcal{C}}}=\mathsf{C}\mathsf{ob}_{\widetilde{\mathcal{C}}}$.
In particular, $\mathcal{Z}_{\widetilde{\mathcal{C}}_{\Gamma}}$ produces
invariants of arbitrary decorated closed $3$-manifolds.

Working in the same topological setting as Section 4.2.4, we find for the
partition of trivial circle bundles with insertions

$\mathcal{Z}_{\widetilde{\mathcal{C}}_{\Gamma}}({\mathcal{S}}\times
S^{1}_{\bar{\beta}})=|D|^{g-1}\sum_{k\in D}q^{-2\kappa^{\vee}(\beta+k,\mu)}.$
(33)

######  Proposition 4.16.

Let ${\mathcal{S}}$ be a decorated surface of genus $g\geq 1$ with no marked
points. Then $\mathcal{Z}_{\widetilde{\mathcal{C}}_{\Gamma}}({\mathcal{S}})$
is concentrated in degree $0\in\mathsf{Z}$, where it has dimension $|D|^{g}$.

###### Proof.

Specializing equation (33) to $m=0$ gives
$\mathcal{Z}_{\widetilde{\mathcal{C}}_{\Gamma}}({\mathcal{S}}\times
S^{1}_{\bar{\beta}})=|D|^{g}$. Taking the limit $\beta\rightarrow 0$ then
gives
$\chi(\mathcal{Z}_{\widetilde{\mathcal{C}}_{\Gamma}}({\mathcal{S}}))=|D|^{g}$.
It remains to prove that
$\mathcal{Z}_{\widetilde{\mathcal{C}}_{\Gamma}}({\mathcal{S}})$ is
concentrated in degree $\bar{0}\in\mathsf{Z}$. A direct modification of the
proof of [GY22, Theorem 3.3] applies in the present setting to show that the
homogeneous summand
$\mathcal{Z}_{\widetilde{\mathcal{C}}_{\Gamma},-k}({\mathcal{S}})$ of degree
$-k\in\mathsf{Z}$ is spanned by colourings of degree $k$ of a particular
oriented trivalent spine, denoted in _loc. cit._ by $\tilde{\Gamma}$, of a
handlebody bounded by ${\mathcal{S}}$. See [GY22, Definition 3.2]. Degree $-k$
colourings are determined by solving a system of recursive equations, with
each equation governing a balancing condition associated to a node of
$\tilde{\Gamma}$, that is, the condition that the colour of the incoming edge
appears as a summand of the tensor product of the colours of the outgoing
edges. Since the category $\widetilde{\mathcal{C}}$ is pointed, the colour of
any two edges incident to a node uniquely determines the third. From this one
can verify directly that this system of equations admits a solution only if
$k=\bar{0}$. ∎

The theory $\mathcal{Z}_{\widetilde{\mathcal{C}}_{\Gamma}}$ models
$T_{c,\Gamma}$-Chern–Simons theory at level $\kappa$. Note that the latter
theory is semisimple, so that no homological truncation is needed. More
precisely, whereas standard mathematical models of toral Chern–Simons theory
incorporate only genuine Wilson line operators,
$\mathcal{Z}_{\widetilde{\mathcal{C}}_{\Gamma}}$ incorporates also non-genuine
line operators, as discussed in Section 4.2.1.

Recall that the standard Reshetikhin–Turaev approach to
$T_{c,\Gamma}$-Chern–Simons theory at level $\kappa$ is via the modular tensor
category $\mathsf{vect}_{\mathbb{C}}(D,q)$ whose underlying abelian category
is finite dimensional $D$-graded vector spaces and whose associator and
braiding are determined by the quadratic form

$q:D\rightarrow\mathbb{Q}/\mathbb{Z},\qquad d\mapsto\kappa^{\vee}(d,d),$

interpreted as an element of the abelian cohomology
$H^{3}_{\textnormal{ab}}(D;\mathbb{Q}/\mathbb{Z})$ [JS93, Sti08, KS11].
Physically, $\mathsf{vect}_{\mathbb{C}}(D,q)$ is the category of genuine
Wilson line operators. For other approaches to toral Chern–Simons theory,
again incorporating only genuine line operators, see [DW90, Man98, FHLT10].
For a generalization to spin TQFTs when the lattice $(\Gamma,\kappa)$ is odd,
see [BM05].

We explain how to recover $\mathsf{vect}_{\mathbb{C}}(D,q)$ from the relative
modular category $\widetilde{\mathcal{C}}_{\Gamma}$ and hence the
Reshetikhin–Turaev model from
$\mathcal{Z}_{\widetilde{\mathcal{C}}_{\Gamma}}$. First, note that the ribbon
subcategory $\widetilde{\mathcal{C}}_{\Gamma,\bar{0}}$ is naturally identified
with the category of finite dimensional representations of $T_{c,\Gamma}$ and
so as labels of genuine Wilson line operators in classical
$T_{c,\Gamma}$-Chern–Simons theory. Since the group $\mathsf{Z}$ acts freely
on isomorphism classes of objects of $\widetilde{\mathcal{C}}_{\Gamma}$, we
can form the orbit category
$\widetilde{\mathcal{C}}_{\Gamma,\bar{0}}/\mathsf{Z}$. Explicitly,
$\widetilde{\mathcal{C}}_{\Gamma,\bar{0}}/\mathsf{Z}$ has the same objects as
$\widetilde{\mathcal{C}}_{\Gamma,\bar{0}}$ and morphisms

$\textup{\text{Hom}}_{\widetilde{\mathcal{C}}_{\Gamma,\bar{0}}/\mathsf{Z}}(V,W)=\bigoplus_{k\in\mathsf{Z}}\textup{\text{Hom}}_{\widetilde{\mathcal{C}}_{\Gamma,\bar{0}}}(\sigma_{k}\otimes
V,W).$

Isomorphism classes of simple objects of
$\widetilde{\mathcal{C}}_{\Gamma,\bar{0}}/\mathsf{Z}$ are then in bijection
with $\Gamma^{\vee}/\mathsf{Z}\simeq D$. It follows that the category of
$D$-graded vector spaces is a skeleton of
$\widetilde{\mathcal{C}}_{\Gamma,\bar{0}}/\mathsf{Z}$. Transferring the ribbon
structure from $\widetilde{\mathcal{C}}_{\Gamma,\bar{0}}$ to this skeleton
recovers the modular tensor category $\mathsf{vect}_{\mathbb{C}}(D,q)$.
Passing to the orbit category
$\widetilde{\mathcal{C}}_{\Gamma,\bar{0}}/\mathsf{Z}$, and so identifying
classical line operators labeled representations in the same
$\mathsf{Z}$-orbit, is a non-perturbative quantum effect implemented by
monopole operators [KS11, §3.1].

A standard model for the state space of a genus $g$ surface $\Sigma_{g}$ in
$T_{c,\Gamma}$-Chern–Simons theory at level $\kappa$, obtained by geometric
quantization, is the space of holomorphic sections of a $\Theta$-line bundle
over the moduli space of flat $T_{c,\Gamma}$-bundles on $\Sigma_{g}$. This
space of sections is of dimension $|D|^{g}$, in agreement with Proposition
4.16.

##  5\. Additional relative modular structures

We continue to set $\hbar=\frac{\pi\sqrt{-1}}{2}$ and work with the category
$\widetilde{\mathcal{C}}_{R}$. We work under Assumptions A, B and C. In this
section, we consider various modifications of Section 4.1. Since many of the
proofs are similar to those of Section 4, we will at points be brief.

###  5.1. Abelian gauged $\mathcal{N}=4$ hypermultiplets

Consider again Example 2.5. Fix a basis
$\mathfrak{s}\simeq\bigoplus_{a=1}^{s}\mathbb{C}\cdot N_{a}$ and let
$\Gamma=\bigoplus_{a=1}^{s}\mathbb{Z}\cdot N_{a}$. Assumption A holds if and
only if the matrix $Q^{(\mathfrak{s})}$ has integer entries. Assumptions B and
C hold if and only if they hold with $Q$ replaced by $Q^{(\mathfrak{s})}$. The
effective metric on $\mathfrak{t}=\mathfrak{s}\oplus\mathfrak{s}^{\vee}$ is

$\kappa_{\textup{\text{eff}}}=\left(\begin{matrix}c_{2}(R)&\textup{\text{id}}_{\mathfrak{s}^{\vee}}\\\
\operatorname{can}_{\mathfrak{s}}&0\end{matrix}\right),$

where $c_{2}(R)\in\textup{\text{End}}_{\mathbb{C}}(\mathfrak{s})$ is the
quadratic Casimir of $R$; with respect to the above basis of $\mathfrak{s}$,
its entries are
$c_{2}(V)_{ab}=\sum_{i=1}^{n}Q^{(\mathfrak{s})}_{ai}Q^{(\mathfrak{s})}_{bi}$.

Since $Q^{(\mathfrak{s})}$ has integer entries, $R$ lifts to a representation
of the complex torus
$T^{(\mathfrak{s})}_{\Gamma}=(\Gamma\otimes_{\mathbb{Z}}\mathbb{C})/\Gamma$.
The above data determines a theory of $T^{(\mathfrak{s})}_{c,\Gamma}$-gauged
$\mathcal{N}=4$ hypermultiplets with charge matrix the transpose of
$Q^{(\mathfrak{s})}$ [BCDN23, §2] or, equivalently, Chern–Simons-
Rozansky–Witten theory with non-compact gauge group
$T^{(\mathfrak{s})}_{c,\Gamma}\times\mathfrak{s}^{\vee}$ at level $\kappa$ and
holomorphic symplectic target $T^{\vee}R$. A vertex operator algebra whose
(not necessarily local) modules model the category $\widehat{\mathcal{C}}_{R}$
of (not necessarily genuine) line operators in such a theory was proposed in
[BCDN23, §8]; a Kazhdan–Lusztig correspondence for these boundary vertex
operator algebras is currently being developed by Creutzig–Niu [CN24]. We note
that the associated quantum groups can be derived directly from the underlying
field theory without passing through an auxiliary vertex operator algebra
[CDN24], and should be related to the ones described here by uprolling [CR22].
As we explain below, we expect that the derived category of the full ribbon
subcategory $\widetilde{\mathcal{C}}_{R}^{\operatorname{int}}$ of
$\widetilde{\mathcal{C}}_{R}$ consisting of
$\widetilde{U}_{\sqrt{-1}}^{\mathfrak{t}}(\mathfrak{t}_{R})$-modules whose
$\mathfrak{s}$-weights lie in the integral lattice
$\Gamma^{\vee}\subset\mathfrak{s}^{\vee}$ is a full subcategory of
$\widehat{\mathcal{C}}_{R}$ after quotienting by the free realization
$\mathsf{Z}$ below. We emphasize that objects of $\widehat{\mathcal{C}}_{R}$
are not subject to any semisimplicity assumption on the action of the
subalgebra $\mathfrak{s}^{\vee}\subset\mathfrak{t}$.

Since the lattice $\Gamma\subset\mathfrak{t}$ is not full rank, the present
setting is not that of Section 4.1. In particular, the groups $\Gamma^{\vee}$
and

$\\{\lambda\in\mathfrak{t}^{\vee}\mid\lambda(\gamma)\in\mathbb{Z}\;\;\forall\gamma\in\Gamma\\}\simeq\Gamma^{\vee}\oplus\mathfrak{s}$

differ. Here, and below, we identify $\mathfrak{s}^{\vee\vee}$ with
$\mathfrak{s}$. Since $\Gamma^{\vee}\subset\Gamma^{\vee}\oplus\mathfrak{s}$,
we can grade $\widetilde{\mathcal{C}}_{R}$ by the group
$\mathsf{G}=\mathfrak{t}^{\vee}/\Gamma^{\vee}\oplus\mathfrak{s}\simeq\mathfrak{s}^{\vee}/\Gamma^{\vee}$,
which is known to be the topological flavour symmetry of abelian gauged
hypermultiplets.

######  Proposition 5.1.

Let $\mathsf{Z}_{0}$ be the image of $\Gamma$ under the map
$\kappa^{\flat}:\mathfrak{t}\rightarrow\mathfrak{t}^{\vee}$. If the integral
lattice $\Gamma$ is even with respect to $\kappa_{\textup{\text{eff}}}$, then
$\mathsf{Z}=\mathsf{Z}_{0}\oplus\mathbb{Z}/2\mathbb{Z}$ satisfies conditions
(A2)-(A4) but fails condition (A5), where in each condition the group
$\Gamma^{\vee}$ of Section 4.1 is replaced with
$\Gamma^{\vee}\oplus\mathfrak{s}$.

###### Proof.

We discuss each condition. Given $\gamma\in\Gamma$, let
$\omega_{\gamma}=\kappa^{\flat}(\gamma)$.

  1. (A2)

Since $\kappa$ is hyperbolic and $\Gamma\subset\mathfrak{s}$, we have
$\omega_{\gamma}\in\mathfrak{s}^{\vee\vee}\simeq\mathfrak{s}\subset\mathfrak{t}^{\vee}$.
Since $\mathfrak{s}\subset\Gamma^{\vee}\oplus\mathfrak{s}$, the condition
holds.

  2. (A3)

For each $\lambda\in\Gamma^{\vee}\oplus\mathfrak{s}$, we have
$\kappa^{\vee}(\omega_{\gamma},\lambda)=\lambda(\gamma)\in\mathbb{Z}$.

  3. (A4)

As in Proposition 4.1, this follows from the even integrality of
$\kappa_{\textup{\text{eff}}}$.

  4. (A5)

This condition fails since the index in question is (uncountably) infinite. ∎

In particular, we do not obtain a relative modular structure on
$\widetilde{\mathcal{C}}_{R}$ or, for the same reasons, its integral
subcategory $\widetilde{\mathcal{C}}_{R}^{\operatorname{int}}$. Theorem 3.12
implies that $\mathsf{Z}$-orbits of isomorphism classes of simple objects of
$\widetilde{\mathcal{C}}_{R,\bar{0}}^{\operatorname{int}}$ are in bijection
with $\Gamma^{\vee}\times\mathfrak{s}/\Gamma$. This agrees with the
classification of simple objects of the vertex algebraic model
$\widehat{\mathcal{C}}_{R}$ [BCDN23, Corollary 8.3.1]. The braiding and twist
of $\widetilde{\mathcal{C}}_{R}^{\operatorname{int}}$ are compatible with
those of $\widehat{\mathcal{C}}_{R}$.

######  Remark 5.2.

The category $\widetilde{\mathcal{C}}_{R}$ includes line operators sourcing
background flat connections for the topological flavour symmetry group
$\mathsf{G}$ but not the Higgs branch flavour symmetry group. While the
inclusion of the latter is a crucial ingredient in reproducing physical
expectations, it will not resolve the above lack of finiteness. See [CDGG21,
§2] for such an example (that violates Assumption B). ∎

###  5.2. Free realizations via $\ker\chi$

We consider a slight generalization of Section 4.1. Fix subgroups
$\Lambda_{0}$ and $\Lambda$ of $\mathfrak{t}^{\vee}$. Consider the following
conditions, where $k\in\Lambda_{0}$ and $\lambda\in\Lambda$:

  1. (B1)

[$\mathsf{G}$-grading] $\Lambda_{R}\subset\Lambda$.

  2. (B2)

[$\mathsf{Z}$ in degree $\bar{0}$] $\Lambda_{0}\subset\Lambda$.

  3. (B3)

[Existence of $\psi$] $\kappa^{\vee}(k,\lambda)\in\mathbb{Z}$.

  4. (B4)

[Trivial ribbon] $-\kappa^{\vee}(k,k)+\sum_{i=1}^{n}\chi_{i}(k)\in
2\mathbb{Z}$.

  5. (B5)

[Finiteness] The index of $\Lambda_{0}$ in $\Lambda$ is finite.

  6. (B6)

[Invertibility of $\mathsf{Z}$] $\chi_{i}(k)\in\mathbb{Z}$ for all $1\leq
i\leq n$.

As in Section 4.1, the above conditions are necessary for tracking weights
modulo $\Lambda$ to define a grading of $\widetilde{\mathcal{C}}_{R}$ by the
group $\mathsf{G}=\mathfrak{t}^{\vee}/\Lambda$ and for
$\mathsf{Z}=\Lambda_{0}\oplus\mathbb{Z}/2\mathbb{Z}$ to be the weights of a
free realization on $\widetilde{\mathcal{C}}_{R,\bar{0}}$ which is a part of a
relative pre-modular structure on $\widetilde{\mathcal{C}}_{R}$. In the
restricted setting of Section 4.1, where $\Lambda=\Gamma^{\vee}$ and
$\Lambda_{0}=\text{{im}}(\kappa^{\flat}:\Gamma\rightarrow\Gamma^{\vee})$, the
analogue of condition (B6) follows from conditions (A1) and (A3).

Assume that $\Lambda$ is a subgroup of
$\ker(\chi:\mathfrak{t}^{\vee}\rightarrow\mathbb{C}^{n})$ and let
$\Lambda_{0}=\Lambda$. Conditions (B2), (B3), (B5) and (B6) then hold
automatically. We assume that conditions (B1) and (B4) hold as well. Since
$\Lambda\subset\ker\chi$, there is an induced group homomorphism
$\overline{\chi}:\mathsf{G}\rightarrow\mathsf{H}$. By Lemma 1.7, the preimage
of $\mathsf{Y}$ under $\overline{\chi}$ is a small symmetric subset of
$\mathsf{G}$ which is also a subgroup; call it $\mathsf{X}$.

######  Theorem 5.3.

The monoidal functor

$\sigma:\mathsf{Z}\rightarrow\widetilde{\mathcal{C}}_{R},\qquad(k,\bar{p})\mapsto\sigma_{(k,\bar{p})}:=\mathbb{C}^{\mathfrak{t}}_{(k,\bar{p})}$

defines a free realization of $\mathsf{Z}$ on
$\widetilde{\mathcal{C}}_{R,\bar{0}}$ and gives $\widetilde{\mathcal{C}}_{R}$
the structure of a modular $\mathsf{G}$-category relative to
$(\mathsf{Z},\mathsf{X})$ with stabilization coefficients $\Delta_{\pm}=(\pm
1)^{n}$ and relative modularity parameter $\zeta=(-1)^{n}$.

###### Proof.

The definition of $\mathsf{X}$ is such that any weight with image in
$\mathsf{G}\setminus\mathsf{X}$ is typical. Generic semisimplicity of
$\widetilde{\mathcal{C}}_{R}$ can therefore be proved in the same way as
Proposition 3.15. Conditions (B2) and (B6) ensure that each
$\sigma_{(k,\bar{p})}$ has $\mathsf{G}$-degree $\bar{0}$ and is one
dimensional, respectively. Condition (B4) ensures that
$\theta_{\sigma_{(k,\bar{p})}}=\textup{\text{id}}_{\sigma_{(k,\bar{p})}}$ for
all $(k,\bar{p})\in\mathsf{Z}$. Let
$\bar{\lambda}\in\mathsf{G}\setminus\mathsf{X}$ with chosen lift
$\lambda\in\mathfrak{t}^{\vee}$; any other lift is of the form $\lambda+k$ for
a unique $k\in\Lambda$. A completely reduced dominating set of
$\mathcal{C}_{R,\bar{\lambda}}$ is

$\\{V_{(\lambda,\bar{p})}\mid\lambda\mbox{ in class
}\bar{\lambda},\;\bar{p}\in\mathbb{Z}/2\mathbb{Z}\\}$

and we may take $\Theta(\bar{\lambda})=\\{V_{(\lambda,\bar{0})}\\}$. The
required bicharacter is

$\psi:\mathsf{G}\times\mathsf{Z}\rightarrow\mathbb{C}^{\times},\qquad(\bar{\lambda},(k,\bar{p}))\mapsto
q^{-4\kappa^{\vee}(k,\lambda)}.$

This proves relative pre-modularity of $\widetilde{\mathcal{C}}_{R}$.

A direct computation using Lemma 3.18 shows that for
$\bar{\lambda}\in\mathsf{G}\setminus\mathsf{X}$ with lift
$\lambda\in\mathfrak{t}^{\vee}$, we have

$\Delta_{-}=\mathsf{d}(V_{(\lambda,\bar{0})})q^{2\kappa^{\vee}(\lambda,\lambda)-4\sum_{i=1}^{n}\chi_{i}(\lambda)}\langle\Phi_{V_{(\lambda,\bar{0})},V_{(\lambda,\bar{0})}}\rangle=(-1)^{n}$

and, similarly, $\Delta_{+}=1$.

It remains to establish the existence of a relative modularity parameter.
Consider Definition 1.11 with $h=\bar{\gamma}$ and $g=\bar{\lambda}$ with
$V_{i},V_{j}=V_{(\lambda,\bar{0})}\in I_{\bar{\lambda}}$. Applying the direct
analogue of Lemma 4.4, we can write the endomorphism of
$V_{(\lambda,\bar{0})}\otimes V_{(\lambda,\bar{0})}^{\vee}$ determined by the
left hand side of diagram (2) as

$f_{\bar{\gamma},(\lambda,\bar{0}),(\lambda,\bar{0})}=\sum_{i=1}^{m}h_{\bar{\gamma},(\lambda,\bar{0}),(\lambda,\bar{0}),i}\circ
g_{\bar{\gamma},(\lambda,\bar{0}),(\lambda,\bar{0}),i}$

for some

$g_{\bar{\gamma},(\lambda,\bar{0}),(\lambda,\bar{0}),i}\in\textup{\text{Hom}}_{\widetilde{\mathcal{C}}_{R}}(V_{(\lambda,\bar{0})}\otimes
V_{(\lambda,\bar{0})}^{\vee},\mathbb{C}^{\mathfrak{t}}_{(k_{i},\bar{p}_{i})}),$
$h_{\bar{\gamma},(\lambda,\bar{0}),(\lambda,\bar{0}),i}\in\textup{\text{Hom}}_{\widetilde{\mathcal{C}}_{R}}(\mathbb{C}^{\mathfrak{t}}_{(k_{i},\bar{p}_{i})},V_{(\lambda,\bar{0})}\otimes
V_{(\lambda,\bar{0})}^{\vee})$

and one dimensional modules
$\mathbb{C}^{\mathfrak{t}}_{(k_{i},\bar{p}_{i})}\in\widetilde{\mathcal{C}}_{R,\bar{0}}$.
Clearly

$\textup{\text{Hom}}_{\widetilde{\mathcal{C}}_{R}}(V_{(\lambda,\bar{0})}\otimes
V^{\vee}_{(\lambda,\bar{0})},\mathbb{C}^{\mathfrak{t}}_{(k_{i},\bar{p}_{i})})\simeq\textup{\text{Hom}}_{\widetilde{\mathcal{C}}_{R}}(V_{(\lambda,\bar{0})},V_{(\lambda,\bar{0})}\otimes\mathbb{C}^{\mathfrak{t}}_{(k_{i},\bar{p}_{i})})$

vanishes unless $(k_{i},\bar{p}_{i})=0$. It follows that

$f_{\bar{\gamma},(\lambda,\bar{0}),(\lambda,\bar{0})}\in\textup{\text{Hom}}_{\widetilde{\mathcal{C}}_{R}}(V_{(\lambda,\bar{0})}\otimes
V^{\vee}_{(\lambda,\bar{0})},\mathbb{C})\simeq\textup{\text{End}}_{\widetilde{\mathcal{C}}_{R}}(V_{(\lambda,\bar{0})})\simeq\mathbb{C},$

whence $f_{\bar{\gamma},(\lambda,\bar{0}),(\lambda,\bar{0})}$ is proportional
to
$\stackrel{{\scriptstyle\longleftarrow}}{{\operatorname{coev}}}_{V_{(\lambda,\bar{0})}}\circ\stackrel{{\scriptstyle\longrightarrow}}{{\operatorname{ev}}}_{V_{(\lambda,\bar{0})}}$.
By [DR22, Proposition 1.2], this proportionality constant is necessarily
$\zeta=\Delta_{+}\Delta_{-}=(-1)^{n}$.

TQFT finiteness is proved as in Theorem 4.6. ∎

####  5.2.1. Verlinde formula

Write $\widetilde{\mathcal{C}}_{R,\chi}$ for the relative modular structure of
Theorem 5.3. Considering the analogue of Section 4.2.4, we find that
$\mathcal{Z}_{\widetilde{\mathcal{C}}_{R,\chi}}({\mathcal{S}}\times
S^{1}_{\bar{\beta}})$ is equal to

$(-1)^{(g+1+m)n+\bar{q}}q^{-4\kappa^{\vee}(\beta,\mu)+2m\sum_{i=1}^{n}\chi_{i}(\beta)+2\sum_{i=1}^{n}\chi_{i}(\mu)}\prod_{i=1}^{n}(q^{2\chi_{i}(\beta)}-q^{-2\chi_{i}(\beta)})^{2g-2+m}.$

The Verlinde formula takes the form
$\mathcal{Z}_{\widetilde{\mathcal{C}}_{R,\chi}}({\mathcal{S}}\times
S^{1}_{\bar{\beta}})=\dim_{(q^{-4\kappa^{\vee}(-,\beta)},1)}\mathcal{Z}_{\widetilde{\mathcal{C}}_{R,\chi}}({\mathcal{S}})$.

######  Example 5.4.

Setting $m=0$ gives

$\mathcal{Z}_{\widetilde{\mathcal{C}}_{R,\chi}}({\mathcal{S}}\times
S^{1}_{\bar{\beta}})=(-1)^{(g+1)n+\bar{q}}\prod_{i=1}^{n}(q^{2\chi_{i}(\beta)}-q^{-2\chi_{i}(\beta)})^{-\chi({\mathcal{S}})}.$

Applying the Verlinde formula, we conclude that
$\dim_{\mathbb{C}}\mathcal{Z}_{\widetilde{\mathcal{C}}_{R,\chi}}({\mathcal{S}})=2^{n(2g-2)}$
and

$\chi(\mathcal{Z}_{\widetilde{\mathcal{C}}_{R,\chi}}({\mathcal{S}}))=\begin{cases}1&\mbox{if
}g=1,\\\ 0&\mbox{if }g\geq 2.\end{cases}\qed$

###  5.3. $\mathfrak{psl}(1|1)$-Chern–Simons theory

Consider again the setting of Example 2.4, so that
$\mathfrak{t}_{R}\simeq\mathfrak{gl}(1|1)$; see also Section 4.3. We have
$\Lambda_{R}=\mathbb{Z}\cdot N^{\vee}$. The linear map
$\chi:\mathbb{C}^{2}\rightarrow\mathbb{C}$ is represented by the matrix
$\left(\begin{matrix}0&1\end{matrix}\right)$ so that
$\Lambda:=\ker\chi=\mathbb{C}\cdot N^{\vee}$ contains $\Lambda_{R}$. This
verifies condition (B1). It follows that the restriction of $\kappa^{\vee}$ to
$\ker\chi$ is trivial. In particular, $\ker\chi$ is even integral with respect
to $\kappa^{\vee}$. Condition (B4) therefore also holds and we obtain a
relative modular category $\widetilde{\mathcal{C}}_{R,\chi}$. The grading
$\mathsf{G}$ is by $E$-weights and the free realization $\mathsf{Z}$ consists
of all one dimensional modules with vanishing $E$-weight. The relative modular
category $\widetilde{\mathcal{C}}_{R,\chi}$ recovers that constructed from the
representation theory of the unrolled quantum group
$U_{q}^{H}(\mathfrak{gl}(1|1))$ at arbitrary parameter $q$ [GY22, Theorem
2.16]. The TQFT $\mathcal{Z}_{\widetilde{\mathcal{C}}_{R,\chi}}$ models
Chern–Simons theory with gauge supergroup $\mathfrak{psl}(1|1)$ coupled to
background flat $\mathbb{C}^{\times}$-connections, as studied in [Mik15].
Alternatively, $\mathcal{Z}_{\widetilde{\mathcal{C}}_{R,\chi}}$ models the
Rozansky–Witten theory of $T^{\vee}\mathbb{C}$ with Higgs branch flavour
symmetry $\mathbb{C}^{\times}$ which acts on $T^{\vee}\mathbb{C}$ with weight
$1$ on the base and weight $-1$ on the fibre.

In particular, the results of this section allow us to effectively circumvent
Assumption B in the construction of abelian linear Gaiotto–Witten theories
with compact gauge group, given in Section 4. Indeed, if Assumption B does not
hold, then the theory splits as a product of the Gaiotto–Witten theory
associated to the quotient representation
$R/\textup{\text{Hom}}_{\mathfrak{t}}(\mathbb{C},R)$ and
$\dim_{\mathbb{C}}\textup{\text{Hom}}_{\mathfrak{t}}(\mathbb{C},R)$-many
copies of the Rozansky–Witten theory of $T^{\vee}\mathbb{C}$. The former
theory is treated in Section 4 while the latter—when considered with Higgs
branch flavour symmetry—is treated in this section.

##  Appendix A Reminders on lattices

Let $\kappa$ be a non-degenerate symmetric bilinear form on $\mathbb{R}^{r}$.
Let $\Gamma\subset\mathbb{R}^{r}$ be a full rank integral lattice with basis
$\\{\gamma_{1},\dots,\gamma_{r}\\}$. The $r\times r$ Gram matrix of
$(\Gamma,\kappa)$ is $B=(B_{ij})=(\kappa(\gamma_{i},\gamma_{j}))$ with inverse
$B^{-1}=(B^{ij})$. Integrality of the lattice is the statement that $B$ has
integer entries. If the diagonals of $B$ are even integers, then $\Gamma$ is
called even. The dual lattice $\Gamma^{\vee}$ has basis
$\\{\gamma^{\vee}_{1},\dots,\gamma^{\vee}_{r}\\}$. There is an isomorphism

$\Gamma^{\vee}\rightarrow\\{v\in\Gamma\otimes_{\mathbb{Z}}\mathbb{Q}\mid\kappa(v,\gamma)\in\mathbb{Z}\;\,\forall\gamma\in\Gamma\\},\qquad\gamma_{i}^{\vee}\mapsto\sum_{j=1}^{r}B^{ij}\gamma_{j}.$
(34)

Let $B=X\tilde{B}Y$ be the Smith normal form of $B$, so that
$\tilde{B}=\mbox{diag}(d_{1},\dots,d_{r})$ is diagonal with entries satisfying
$d_{1}|d_{2}|\cdots|d_{r}$ and the left and right Smith multipliers $X$ and
$Y$, respectively, are in $GL(r,\mathbb{Z})$. The isomorphism (34) gives

$\sum_{k=1}^{r}X^{ij}\gamma_{k}\mapsto
d_{i}\sum_{j=1}^{r}Y_{ij}\gamma_{j}^{\vee},$

where $X^{-1}=(X^{ij})$. It follows that
$\delta_{i}:=\sum_{j=1}^{r}Y_{ij}\gamma_{j}^{\vee}$ generates the $i$th factor
of the discriminant group

$D=\operatorname{coker}(\kappa^{\flat}:\Gamma\rightarrow\Gamma^{\vee})\simeq\bigoplus_{i=1}^{r}\mathbb{Z}/d_{i}\mathbb{Z}.$

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