# Non-hyperbolic 3-manifolds and 3D field theories for 2D Virasoro minimal
models

Dongmin Gang  Heesu Kang  Seongmin Kim<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Using 3D-3D correspondence, we construct 3D dual bulk field theories for
general Virasoro minimal models $M(P,Q)$. These theories correspond to Seifert
fiber spaces $S^{2}((P,P-R),(Q,S),(3,1))$ with two integers $(R,S)$ satisfying
$PS-QR=1$. In the unitary case, where $|P-Q|=1$, the bulk theory has a mass
gap and flows to a unitary topological field theory (TQFT) in the IR, which is
expected to support the chiral Virasoro minimal model at the boundary under an
appropriate boundary condition. For the non-unitary case, where $|P-Q|>1$, the
bulk theory flows to a 3D $\mathcal{N}=4$ rank-0 superconformal field theory,
whose topologically twisted theory supports the chiral minimal model at the
boundary. We also provide a concrete field theory description of the 3D bulk
theory using $T[SU(2)]$ theories. Our proposals are supported by various
consistency checks using 3D-3D relations and direct computations of various
partition functions.

††institutetext:  Department of Physics and Astronomy $\&$ Center for
Theoretical Physics,
Seoul National University, 1 Gwanak-ro, Seoul 08826, Korea

##  1 Introduction

Chiral algebras, also known as vertex algebras, are a powerful and versatile
tool in modern theoretical physics. They provide a unifying algebraic
framework for understanding diverse physical phenomena, ranging from two-
dimensional conformal field theories to the intricate structures in higher
dimensional ($D\geq 3$) topological or supersymmetric quantum field theories
Beem:2013sza ; Beem:2014kka ; Feigin:2018bkf ; Cheng:2018vpl ;
Costello:2018fnz ; Costello:2020ndc ; Cheng:2022rqr . Among chiral algebras,
rational chiral algebras have rich and rigid mathematical structures and broad
applications in physics. They allow only a finite number of irreducible
representations, whose characters form vector-valued modular forms. They are
closely related to 3D topological field theories via the so-called bulk-
boundary correspondence, which provides knot or 3-manifold invariants and
describes the universal behaviors of topological phases. The rigid
mathematical structures greatly simplify the classification program of
rational chiral algebras, and several important classes have been classified.
The most famous and successful class is the Virasoro minimal models $M(P,Q)$
belavin1984infinite , which describe universal features of critical phenomena
in various 2D systems, such as the Ising model.

In this paper, we study the 3D bulk theories related to the Virasoro minimal
models via the bulk-boundary correspondence. In the case of unitary rational
chiral algebras, the bulk theories have a mass gap, and the infrared (IR)
physics is described by unitary topological field theories. The IR topological
quantum field theories (TQFTs) share common modular tensor category structures
with the corresponding boundary rational chiral algebras witten1989quantum ;
moore1989lectures ; turaev1992modular . For non-unitary rational chiral
algebras, recent studies have shown that the bulk theories can be described by
topologically twisted theories of an exotic class of superconformal field
theories (SCFTs) called 3D ${\cal N}=4$ rank-0 SCFTs Gang:2018huc ;
Dedushenko:2018bpp ; Gang:2021hrd ; Gang:2022kpe ; Gang:2023rei ;
Ferrari:2023fez ; Gang:2023ggt ; Dedushenko:2023cvd ; Baek:2024tuo . Here,
’rank-0’ denotes the absence of Coulomb and Higgs branch operators in the
theory. This exotic property proves crucial in realizing rational chiral
algebra at the boundary Costello:2018swh ; Beem:2023dub ; Ferrari:2023fez .
Our approach begins by realizing the 3D bulk theories for minimal models
$M(P,Q)$ through the 3D-3D correspondence Terashima:2011qi ; Dimofte:2011ju ;
Gang:2018wek  with Seifert fiber spaces, as depicted below:

$\displaystyle\begin{split}&\framebox{Seifert fiber spaces}\xrightarrow{\rm
3D-3D\;}\framebox{3D theories (gapped theories or ${\cal N}=4$ rank-0
SCFTs)}\\\ &\xrightarrow{\textrm{bulk-boundary}}\framebox{2D (unitary or non-
unitary) minimal models}\end{split}$ (1)

For details of the proposal, refer to (11) and (25). Furthermore, we provide a
field-theoretic depiction (59) of these 3D theories by utilizing the
topological structures of the Seifert fiber spaces. Our construction provides
a unified framework for the bulk duals of both unitary and non-unitary minimal
models.

The rest of this paper is organized as follows. In Section 2, we introduce the
Virasoro minimal model $M(P,Q)$ and its 3D dual theory $\mathcal{T}_{(P,Q)}$,
along with the basic dictionaries of the bulk-boundary correspondence. We then
propose that the theories $\mathcal{T}_{(P,Q)}$ can be realized as 3D class R
theories associated with Seifert fiber spaces (see (25)). This proposal is
tested against various non-trivial 3D-3D relations and bulk-boundary
dictionaries, as summarized in Table 1. In Section 3, we provide an explicit
field theory description for $\mathcal{T}_{(P,Q)}$ (see (59)) along with
several non-trivial consistency checks. The Appendices contain technical
details of the supersymmetric partition function computations and the
identification of decoupled topological field theories from 1-form symmetry ’t
Hooft anomalies.

##  2 Virasoro minimal models from 3-manifolds

In this section, we begin by reviewing basic aspects of Virasoro minimal
models. Then, we introduce the bulk-boundary correspondence and the 3D-3D
correspondence, summarized in Table 1. Utilizing these correspondences as
guidelines, we propose the 3D bulk duals of minimal models as 3D class R
theories associated with 3-manifolds known as Seifert fiber spaces, as given
in (25).

###  2.1 Virasoro minimal model $M(P,Q)$

The minimal model $M(P,Q)=M(Q,P)$ is labeled by two integers, $P$ and $Q$,
subject to the following conditions:

$\displaystyle P,Q\geq 2\textrm{ and }\textrm{gcd}(P,Q)=1\;.$ (2)

The underlying chiral algebra is the Virasoro algebra, with the 2D central
charge given by:

$\displaystyle c_{2d}=1-\frac{6(P-Q)^{2}}{PQ}\;.$ (3)

The model includes critical Ising CFT $M(3,4)$, tricritical Ising CFT $M(4,5)$
and Lee-Yang CFT $M(2,5)$. The 2D RCFT can be unitary or non-unitary,
depending on $(P,Q)$:

$\displaystyle M(P,Q)\textrm{ is }\begin{cases}\textrm{unitary},&\;\textrm{if
}|P-Q|=1\\\ \textrm{non-unitary},&\;\textrm{otherwise}\;.\end{cases}$ (4)

There are $N_{(P,Q)}:=\frac{(P-1)(Q-1)}{2}$ primaries ${\cal O}_{(a,b)}$
labeled by two integers $1\leq a<P$ and $1\leq b<Q$ modulo an equivalence
relation ${\cal O}_{(a,b)}={\cal O}_{(P-a,Q-b)}$. The conformal dimensions $h$
of the primaries are given by:

$h_{(a,b)}=\frac{(Pb-Qa)^{2}-(P-Q)^{2}}{4PQ}\;,$ (5)

and the conformal characters are:

$\displaystyle\chi_{(a,b)}(q)=\frac{q^{h_{(a,b)-\frac{c_{2d}}{24}}}}{(q)_{\infty}}\sum_{n\in\mathbb{Z}}\left(q^{n^{2}PQ+n(Qa-
Pb)}-q^{(nP+a)(nQ+b)}\right)\;.$ (6)

Here $(q)_{\infty}=\prod_{k=1}^{\infty}(1-q^{k})$ as usual. Under the
S-transformation, $q:=e^{2\pi i\tau}\rightarrow\tilde{q}:=e^{2\pi
i(-\frac{1}{\tau})}$, the characters transforms as:

$\displaystyle\chi_{\alpha}(\tilde{q})=\sum_{\beta}S_{\alpha\beta}\chi_{\beta}(q)\;,$
(7)

with the modular $S$-matrix given by:

$S_{(a,b),(a^{\prime},b^{\prime})}=-\sqrt{\frac{8}{PQ}}\
(-1)^{ba^{\prime}+b^{\prime}a\ }\text{sin}(\pi\frac{P}{Q}bb^{\prime})\
\text{sin}(\pi\frac{Q}{P}aa^{\prime})\;.$ (8)

In the above, $\alpha,\beta=0,1,\ldots,N_{(P,Q)}-1$ are collective indices for
the primaries, where $\alpha=0$ corresponds to the vacuum module, i.e.,
$\alpha=0\leftrightarrow(a,b)=(1,1)$.

###  2.2 Bulk dual 3D ${\cal T}_{(P,Q)}$ theory

The bulk-boundary correspondence relates a 3D bulk (semi-simple and finite)
topological field theory ${\cal T}$ to 2D chiral rational conformal field
theories (RCFTs). The boundary chiral RCFT $\chi R$ depends on the choice of
holomorphic boundary condition $\mathbb{B}$, and the corresponding RCFT will
be denoted as $\chi R[{\cal T},\mathbb{B}]$:

$\displaystyle(\textrm{3D topological field theory }{\cal
T})\xrightarrow{\;\textrm{at boundary with $\mathbb{B}$}\;}\textrm{(2D chiral
RCFT $\chi R[\mathcal{T},\mathbb{B}]$)}\;.$ (9)

To realize the full (diagonal) RCFT ${\cal R}$, one needs to put the bulk
theory on an interval, ${\cal M}_{2}\times[0,1]$, with the holomorphic
boundary condition $\mathbb{B}$ on both boundaries. In the IR, the system
flows to the 2D RCFT ${\cal R}[{\cal T},\mathbb{B}]$ on ${\cal M}_{2}$.

For unitary case, the bulk theory is a unitary TQFT, which describes the
universal IR behavior of a (2+1)D gapped system, such as fractional quantum
Hall system. For non-unitary case, recent studies show that the bulk theories
can be described by topologically twisted theories of an exotic class of
superconformal field theories called 3D ${\cal N}=4$ rank-0 SCFTs. The bulk-
boundary correspondence for the non-unitary case can be summarized as follows:

$\displaystyle\begin{split}&\textrm{a 3D ${\cal N}=4$ rank-0 SCFT }{\cal
T}\xrightarrow{\;\textrm{a top'l twisting}\;}\textrm{non-unitary TQFT }{\cal
T}^{\rm top}\\\ &\xrightarrow{\textrm{ at boundary with $\mathbb{B}$
}}\textrm{non-unitary chiral RCFT $\chi{\cal R}[{\cal
T},\mathbb{B}]$}\;.\end{split}$ (10)

Rank-0 means there is no Coulomb and Higgs branch operators in the theory. The
exotic property turns out to be crucial to realize rational chiral algebra at
the boundary after a topological twisting. There are two possible choices of
topological twistings (‘top’=$A$ or $B$) denoted as $A$ and $B$ twistings.

Let ${\cal T}_{(P,Q)}$ be the bulk dual theory related to the chiral minimal
model $\chi M(P,Q)$ via the bulk-boundary correspondence:

$\displaystyle\begin{split}&\textbf{Def : 3D ${\cal T}_{(P,Q)}$ theory is
defined as }\\\ &\textrm{For }|P-Q|=1,\;{\cal T}_{(P,Q)}\textrm{ is a 3D
unitary TQFT with}\\\ &{\cal T}_{(P,Q)}\xrightarrow{\textrm{ at boundary with
a proper $\mathbb{B}$ }}\chi M(P,Q)\;,\\\ &\textrm{For }|P-Q|>1,\;{\cal
T}_{(P,Q)}\textrm{ is a ${\cal N}=4$ 3D rank-0 SCFT with}\\\ &{\cal
T}_{(P,Q)}\xrightarrow{\textrm{a top'l twisting}\;}\textrm{non-unitary TQFT
}{\cal T}^{\rm top}_{(P,Q)}\xrightarrow{\textrm{ at boundary with a proper
$\mathbb{B}$ }}\chi M(P,Q)\;.\end{split}$ (11)

The main goal of this paper is to construct the bulk theory ${\cal T}_{(P,Q)}$
dual to the minimal model $M(P,Q)$.

Basic dictionaries of the bulk-boundary correspondence are summarized in the
first and second column of Table 1. Refer to Gang:2021hrd ; Gang:2023rei  for
details. In the table $T_{\rm irred}[M]$ corresponds to the bulk theory ${\cal
T}$. We will realize the bulk theory as an IR fixed point of 3D ${\cal N}\geq
2$ supersymmetric gauge theories, and the bulk quantities in the table are
related to the partition functions on various symmetric backgrounds, which are
RG-invariant. When the SUSY background is ${\cal M}_{g,p}$, a degree $p$
circle bundle over a genus $g$ Riemann surface, the partition function can be
given in the following form Benini:2015noa ; Benini:2016hjo ; Closset:2016arn
; Closset:2018ghr

$\displaystyle{\cal Z}^{{\cal M}_{g,p}}=\sum_{\vec{x}_{\alpha}\in{\cal S}_{\rm
BE}}{\cal H}(\vec{x}_{\alpha})^{g-1}{\cal F}(\vec{x}_{\alpha})^{p}\;.$ (12)

Here $\vec{x}_{\alpha}\in{\cal S}_{\rm BE}$ are called Bethe-vacua, which are
ground states on a two-torus $\mathbb{T}^{2}$ when the bulk theory is a
topological field theory. ${\cal H}$ and ${\cal F}$ are called ‘handle-gluing’
and ‘fibering’ operators, respectively. Let ${\cal Z}^{S^{2}_{b}}$ denotes the
supersymmetric partition function on a squashed 3-sphere $S^{3}_{b}$
Kapustin:2009kz ; Jafferis:2010un ; Hama:2010av :

$\displaystyle
S^{3}_{b}:=\\{(z,w)\in\mathbb{C}^{2}\;:\;b^{2}|z|^{2}+b^{-2}|w|^{2}=1\\}\;.$
(13)

When the bulk theory flows to a 3D ${\cal N}=4$ rank-0 SCFT, the SUSY prtition
functions have non-trivial dependence on $(M,\nu)$. In terms of an ${\cal
N}=2$ subalgebra, the ${\cal N}=4$ theory has a flavor symmetry $U(1)_{A}$
whose charge $A$ is given by:

$\displaystyle A=J^{C}_{3}-J^{H}_{3}\;,$ (14)

where $J^{C}_{3}$ and $J^{H}_{3}$ are two Cartans of $SU(2)^{C}\times
SU(2)^{H}\simeq SO(4)$ R-symmetry of the ${\cal N}=4$ theory. They are
normalized as $J^{C/H}_{3}\in\mathbb{Z}/2$. $M$ denotes the properly rescaled
real mass of the $U(1)_{A}$ symmetry. $\nu$ parametrizes the mixing between
the $U(1)_{R}$ symmetry of the ${\cal N}=2$ subalgebra and the $U(1)_{A}$
symmetry as follows:

$\displaystyle R_{\nu}=R_{\nu=0}+\nu
A=\left(J^{C}_{3}+J^{3}_{H}\right)+\nu\left(J^{C}_{3}-J^{3}_{H}\right)\;.$
(15)

For rank-0 SCFTs, the supersymmetric partition functions in the following
limits:

$\displaystyle\begin{split}&\textrm{A-twisting limit :
}(M,\nu)\rightarrow(0,\nu=-1)\;,\\\ &\textrm{B-twisting limit :
}(M,\nu)\rightarrow(0,\nu=1)\;,\end{split}$ (16)

are known to compute the partition functions of topologically A-twisted or
B-twisted theory, respectively. In the limits, the squashed 3-sphere function
becomes independent on the squashing parameter modulo a phase factor in (82).
In the table, we define

$\displaystyle\begin{split}&\\{{\cal F}^{A},{\cal H}^{A},{\cal
Z}_{b}^{A}\\}:=\\{{\cal F},{\cal H},{\cal
Z}^{S^{3}_{b}}\\}|_{(M,\nu)\rightarrow(0,-1)}\;,\\\ &\\{{\cal F}^{B},{\cal
H}^{B},{\cal Z}_{b}^{B}\\}:=\\{{\cal F},{\cal H},{\cal
Z}^{S^{3}_{b}}\\}|_{(M,\nu)\rightarrow(0,+1)}\;.\end{split}$ (17)

On the hand, the SUSY partition functions of a 3D ${\cal N}=4$ rank-0 SCFT at
$(M=0,\nu=0)$ computes the partition functions at the superconformal point,
and we define:

$\displaystyle\begin{split}{\cal Z}_{b}^{\rm con}:={\cal
Z}^{S^{3}_{b}}|_{(M,\nu)\rightarrow(0,0)}\;.\end{split}$ (18)

When the 3D gauge theory has a mass gap and flows to a unitary TQFT in the IR,
the SUSY partition functions are independent on the $(M,\nu)$ and its squashed
3-sphere partition function is independent on the $b$ modulo a phase factor in
(82).

###  2.3 ${\cal T}_{(P,Q)}$ from non-hyperbolic 3-manifolds

Let $T_{\rm irred}[M]$ denote the 3D class R theory associated with a closed
3-manifold $M$, whose field theory description is proposed in Dimofte:2011ju ;
Gang:2018wek . The theory is believed to describe an effective 3D field theory
of 6D $A_{1}$ ${\cal N}=(2,0)$ superconformal field theory compactified on the
3-manifold $M$. The subscript ‘irred’ emphasizes that the theory only see an
irreducible component of $SL(2,\mathbb{C})$ flat connections on $M$ rather
than all flat connections Chung:2014qpa . The theory should be distinguished
from $T_{\rm full}[M]$ studied in Gadde:2013sca ; Gukov:2016gkn ;
Gukov:2017kmk ; Cheng:2018vpl ; Eckhard:2019jgg ; Chung:2019khu ;
Assel:2022row ; Chung:2023qth , which is assumed to see the all flat
connections. Unlike $T_{\rm irred}$, however, there is no known systematic
algorithm for constructing the field theory description of $T_{\rm full}[M]$
for a general 3-manifold $M$. It is even uncertain whether the theory $T_{\rm
full}[M]$ exists as a genuine 3D field theory for general $M$ Gang:2018wek .

For Seifert fiber spaces (SFSs) $M=S^{2}(\vec{p},\vec{q})$ in Figure 1,

Figure 1: A Dehn surgery representation of
$S^{2}(\vec{p},\vec{q}):=S^{2}((p_{1},q_{1}),(p_{2},q_{2}),(p_{3},q_{3}))$.

the IR phases of the 3D field theory $T_{\rm irred}[M]$ have been analyzed in
Choi:2022dju . It was empirically found that

$\displaystyle\begin{split}&T_{\rm
irred}[M=S^{2}((p_{1},q_{1}),(p_{2},q_{2}),(p_{3},q_{3}))]\\\
&\xrightarrow{\textrm{ In the IR }}\begin{cases}\textrm{an unitary
TQFT},\;\;&q_{i}=\pm 1\;(\textrm{mod }p_{i})\;\forall i=1,2,3\;,\\\ \textrm{a
rank-0 SCFT},\;\;&\textrm{otherwise}\end{cases}\end{split}$ (19)

By combining the 3D-3D correspondence for the Seifert fiber spaces and the
bulk-boundary correspondence, one can consider following correspondence:

$\displaystyle M=S^{2}(\vec{p},\vec{q})\xrightarrow{\textrm{ 3D-3D
correspondence }}T_{\rm irred}[M]\xrightarrow{\textrm{ Bulk-boundary
}}\textrm{2D chiral RCFT }\chi\mathcal{R}[M;\mathbb{B}]\;.$ (20)

Basic dictionaries for the correspondence are summarized in Table 1.

${\rm 2D\;chiral\;RCFT\;}\chi\mathcal{R}[M;\mathbb{B}]$ | $T_{\rm irred}[M]$ |  $SL(2,\mathbb{C})$ CS on $M$
---|---|---
(unitary)/(non-unitary) | (mass gap)/(rank-0 SCFT) | equation (24)
Primary ${\cal O}_{\alpha=0,\ldots,N-1}$ | Bethe-vacuum $\vec{x}_{\alpha}\in{\cal S}_{\rm BE}$ | $\rho_{\alpha}\in\chi_{\rm irred}[M]$
Conformal dimension $h_{\alpha}$ | $e^{2\pi ih_{\alpha}}={\cal F}^{\rm top}(\vec{x}_{\alpha})/{\cal F}^{\rm top}(\vec{x}_{\alpha=0})$ | $e^{2\pi ih_{\alpha}}=e^{2\pi i\left({\rm CS}[\rho_{\alpha=0}]-{\rm CS}[\rho_{\alpha}]\right)}$
$S_{0\alpha}^{2}$ | $({\cal H}^{\rm top}(\vec{x}_{\alpha}))^{-1}$ | $1/(2\textrm{Tor}[\rho_{\alpha}])$
$|S_{00}|$ | $|{\cal Z}^{\rm top}_{b}|$ | $\big{|}\sum_{\rho_{\alpha}}\frac{e^{-2\pi i{\rm CS}[\rho_{\alpha}]}}{2\textrm{Tor}[\rho_{\alpha}]}\big{|}$
$\textrm{min}_{\alpha}\\{|S_{0\alpha}|\\}$ | $e^{-F}=|{\cal Z}^{\textrm{con}}_{b=1}|$ | $\textrm{min}_{\alpha}\\{1/\sqrt{2\textrm{Tor}[\rho_{\alpha}]}\\}$
Table 1: Basic dictionaries for the correspondence among non-hyperbolic
3-manifolds $M$, 3D bulk theories $T_{\rm irred}[M]$, and 2D chiral RCFTs
$\chi\mathcal{R}[M]$ for $M=S^{2}((\vec{p},\vec{q}))$ with
$H^{1}(M,\mathbb{Z}_{2})=1$. The superscripts ‘top’=$A$ (or $B$) and ‘con’
denote the partition function in $A$ (or $B$)-twisting limit (16) and at the
superconformal point (18) of rank-0 SCFT respectively. For the mass gap case,
the superscripts can be ignored. The $S_{\alpha\beta}$ is the modular S-matrix
in (7) and (8). ${\cal H}$ and ${\cal F}$ are handle gluing and fibering
operators appearing in the twisted partition function (12). ${\cal Z}_{b}$ is
the squashed 3-sphere partition function. $\textrm{CS}[\rho]$ and
$\textrm{Tor}[\rho]$ denotes the Chern-Simons invariant and adjoint
Reidemeister torsion of an irreducible flat connection $\rho$.

Let us briefly explain the basic 3D-3D dictionaries in the table. Refer to
Gang:2018hjd ; Gang:2019uay ; Benini:2019dyp ; Cho:2020ljj ; Cui:2021lyi ;
Bonetti:2024cvq  for details. The dictionary is valid only for the 3-manifold
with trivial $H^{1}(M,\mathbb{Z}_{2})$. The $\chi_{\rm irred}(M)$ in the table
is the set of (adjoint-)irreducible $SL(2,\mathbb{C})$ characters on a
3-manifold $M$, which is defined as:

$\displaystyle\begin{split}\chi_{\rm
irred}[M]&=\\{\rho\in\textrm{Hom}\left[\pi_{1}(M)\rightarrow
SL(2,\mathbb{C})]\;:\;\textrm{$\textrm{dim}H(\rho)=0$}\right\\}/\sim\;,\\\
&\qquad\textrm{where }H(\rho):=\\{g\in
SL(2,\mathbb{C})\;:\;[g,\rho(a)]=0\;\;\forall a\in\pi_{1}(M)\\}\;.\end{split}$
(21)

The equivalence relation is defined as111In Cho:2020ljj , they consider
equivalence up to $SL(2,\mathbb{C})$ conjugation. But the trace equivalence
relation $\sim$, which is a weaker equivalence, is turned out be more relevant
in the 3D-3D correspondence Cui:2021lyi .

$\displaystyle\rho_{1}\sim\rho_{2}\textrm{ if
}\textrm{Tr}(\rho_{1}(a))=\textrm{Tr}(\rho_{2}(a))\;,\;\;\forall
a\in\pi_{1}(M)\;.$ (22)

The condition $\textrm{dim}H(\rho)=0$ corresponds to the irreducibility of the
homomorphism $\rho$. A homomorphism $\rho$ defines an $SL(2,\mathbb{C})$ flat
connection ${\cal A}_{\rho}$, $d{\cal A}_{\rho}+{\cal A}_{\rho}\wedge{\cal
A}_{\rho}=0$. The ${\rm CS}[\rho]$ and $\textrm{Tor}[\rho]$ are basic
topological invariants of the flat connection called Chern-Simons invariant
and the adjoint Reidemeister torsion, respectively. The CS invariant is
defined as

$\displaystyle{\rm CS}[\rho]:=\frac{1}{8\pi^{2}}\int\textrm{Tr}\left({\cal
A}_{\rho}d{\cal A}_{\rho}+\frac{2}{3}{\cal A}_{\rho}^{3}\right)\;\;\textrm{
(mod $1$) }.$ (23)

The adjoint torsion $\textrm{Tor}[\rho]$ appears as the 1-loop part of
perturbative expansion of $SL(2,\mathbb{C})$ CS theory around the flat
connection ${\cal A}_{\rho}$ witten1989quantum . For
$M=S^{2}(\vec{p},\vec{q})$ with trivial $H^{1}(M,\mathbb{Z}_{2})$, one can
determine the IR phase of the $T_{\rm irred}[M]$ theory in the following way
Cho:2020ljj ; Choi:2022dju

$\displaystyle\begin{split}&T_{\rm irred}[M=S^{2}(\vec{p},\vec{q})]\\\
&\xrightarrow{\;\textrm{in the IR}\;}\begin{cases}\textrm{unitary
TQFT}\;,&\textrm{if }|\sum_{\rho\in\chi}\frac{e^{-2\pi i{\rm
CS}[\rho_{\alpha}]}}{2\textrm{Tor}[\rho]}|\leq\frac{1}{\sqrt{|2\textrm{Tor}[\rho_{\alpha}]|}}\;\;\forall\;\rho_{\alpha}\in\chi\\\
\textrm{3D rank-0 SCFT}\;,&\textrm{otherwise }\end{cases}\end{split}$ (24)

Here, $\chi$ abbreviates $\chi_{\text{irred}}[M]$.

As the main result of this section, we propose that

$\displaystyle\begin{split}&{\cal T}_{(P,Q)}\simeq T_{\rm
irred}\left[S^{2}((P,P-R),(Q,S),(3,1))\right]\;.\end{split}$ (25)

The 3D theory ${\cal T}_{(P,Q)}$ is defined as the bulk dual of the Virasoro
minimal model $M(P,Q)$ as in (11). Here $(R,S)$ is chosen to satisfy

$\displaystyle\begin{pmatrix}P&Q\\\ R&S\end{pmatrix}\in SL(2,\mathbb{Z})\;.$
(26)

It fixes the $(R,S)$ modulo a shift $(R,S)\rightarrow(R,S)+\mathbb{Z}(P,Q)$,
and we will claim that the $T_{\rm
irred}\left[S^{2}((P,R-P),(Q,-S),(3,1))\right]$ is independent of the shift:

$\displaystyle\begin{split}&T_{\rm
irred}\left[S^{2}((P,P-R),(Q,S),(3,1))\right]\simeq T_{\rm
irred}\left[S^{2}((P,P-\tilde{R}),(Q,\tilde{S}),(3,1))\right]\;\\\
&\textrm{where }(\tilde{R},\tilde{S})=(R,S)+n(P,Q)\;\textrm{and for arbitrary
$n\in\mathbb{Z}$}\;.\end{split}$ (27)

Throughout this paper, we use the following two equivalence relations denoted
by $\sim$ and $\simeq$ among 3D gauge theories,

$\displaystyle\begin{split}&{\cal T}_{1}\sim{\cal T}_{2}\textrm{ if two
theories are IR equivalent up to some `topological operations'}\;,\\\ &{\cal
T}_{1}\simeq{\cal T}_{2}\textrm{ if two theories are IR equivalent up to some
`minimal topological operations'}\;.\end{split}$ (28)

Topological operations include tensoring with a unitary TQFT, gauging of
finite (generalized) symmetries, time-reversal and so on. On the other hand,
the minimal topological operations are topological operations which preserve
the absolute values of partition functions on arbitrary closed 3-manifolds.
The minimal ones include tensoring with an invertible TQFT, time-reversal and
so on. Notice that $\simeq$ is a stronger equivalence than $\sim$. The ${\cal
T}_{(P,Q)}$, like $M(P,Q)$, is invariant under the exchange of
$P\leftrightarrow Q$:

$\displaystyle\begin{split}{\cal T}_{(Q,P)}&\simeq T_{\rm
irred}\left[S^{2}((Q,Q-\tilde{R}),(P,\tilde{S}),(3,1))\right]\textrm{ with
}\begin{pmatrix}Q&P\\\ \tilde{R}&\tilde{S}\end{pmatrix}\in
SL(2,\mathbb{Z})\;.\\\ &\simeq T_{\rm
irred}\left[S^{2}((P,(\tilde{S}-P)+P),(Q,(Q-\tilde{R})),(3,1))\right]\simeq{\cal
T}_{(P,Q)}\;.\end{split}$ (29)

In the 2nd line, we use the fact that $\begin{pmatrix}P&Q\\\
-\tilde{S}+P&Q-\tilde{R}\end{pmatrix}\in SL(2,\mathbb{Z})$.

Let us check the proposal using the dictionaries in Table 1. First, note that
the 3-manifold $S^{2}((P,P-R),(Q,S),(3,1))$ has trivial
$H^{1}(M,\mathbb{Z}_{2})$222$S^{2}((\vec{p},\vec{q}))$ has trivial
$H^{1}(M,\mathbb{Z}_{2})$ if and only if
$p_{1}p_{2}p_{3}\left(\frac{q_{1}}{p_{1}}+\frac{q_{2}}{p_{2}}+\frac{q_{3}}{p_{3}}\right)\in
2\mathbb{Z}+1$. and thus one can use the dictionaries. The fundamental group
of the SFS can be presented as:

$\pi_{1}(S^{2}(\vec{p},\vec{q}))=\langle{x_{1},x_{2},x_{3},h}|{x_{i}^{p_{i}}h^{q_{i}}=1,x_{1}x_{2}x_{3}=1,\textrm{$h$
is central}}\rangle\;.$ (30)

As studied in Cui:2021lyi , irreducible characters on
$M=S^{2}(\vec{p},\vec{q})$ can be specified by the quadruple
$(n_{1},n_{2},n_{3},\lambda)$ with $n_{k=1,2,3}\in\mathbb{Z}_{\geq 0}/2$ and
$\lambda\in\\{0,\frac{1}{2}\\}$, where

$\displaystyle\textrm{Tr}(\rho(x_{k}))=2\cos(\frac{2\pi
n_{k}}{p_{k}})\;,\;\rho(h)=\exp(2\pi i\lambda)\mathbb{I}_{2}\;.$ (31)

Note that the $\rho(h)$ should be an element of center subgroup
$\mathbb{Z}_{2}$ ($\mathbb{I}_{2}$ or $-\mathbb{I}_{2}$) of $SL(2,\mathbb{C})$
in order for $\rho$ to be (adjoint)-irreducible, i.e. $\textrm{dim}H(\rho)=0$.
Otherwise, the fundamental group relation can only be met when
$\rho(x_{k})\in\\{\pm\mathbb{I}_{2}\\}$ for all $k=1,2,3$, and thus
$\textrm{dim}H(\rho)>0$.

Before going into the details of the character variety, let us first check the
proposed duality in (27) at the level of the character variety, which
corresponds to the set of Bethe-vacua in the $T_{\rm irred}[M]$ theory. One
can easily construct a natural one-to-one map between the character varieties
with different choices of $(R,S)$ as follows

$\displaystyle\begin{split}&\rho\in\chi_{\rm
irred}\left[S^{2}((P,P-R),(Q,S),(3,1))\right]\\\
&\xleftrightarrow{\quad\textrm{one-to-one}\quad}\tilde{\rho}\in\chi_{\rm
irred}\left[S^{2}((P,P-\tilde{R}),(Q,\tilde{S}),(3,1))\right]_{(\tilde{R},\tilde{S})=(R,S)+n(P,Q)}\;,\\\
&\textrm{where}\\\
&\tilde{\rho}(x_{1})=\rho(x_{1})\rho(h)^{n},\;\;\tilde{\rho}(x_{2})=\rho(x_{2})\rho(h)^{-n},\;\;\tilde{\rho}(x_{3})=\rho(x_{3}),\;\;\tilde{\rho}(h)=\rho(h)\;.\end{split}$
(32)

Basic invariants, $\textrm{CS}[\rho]$ and $\textrm{Tor}[\rho]$, of irredicible
characters are preserved under the map.

The character variety is studied in Cui:2021lyi  and they found that

$\displaystyle\begin{split}\chi_{\rm
irred}(M)=&\left\\{\left(n_{p_{1},q_{1}}(j_{1}),n_{p_{2},q_{2}}(j_{2}),n_{p_{3},q_{3}}(j_{3}),\frac{1}{2}\right)\middle|\
j_{k}\in[0,\ldots,p_{k}-2]^{e}\right\\}\\\
&\bigsqcup\bigg{\\{}\left(n_{p_{1},q_{1}}(j_{1}),n_{p_{2},q_{2}}(j_{2}),n_{p_{3},q_{3}}(j_{3}),0\right)\bigg{|}\
j_{k}\in[0,\ldots,p_{k}-2]^{o}\bigg{\\}},\end{split}$ (33)

where

$\displaystyle n_{p,q}(j)=\begin{cases}\frac{p-j-1}{2}\;,&q\textrm{ and }j\in
2\mathbb{Z}\\\ \frac{j+1}{2}\;,&\text{otherwise}\end{cases}$ (34)

Here $[0,\ldots,p]^{e/o}$ denotes the set of even/odd numbers between $0$ and
$p$. the So the irreducible characters on $S^{2}(\vec{p},\vec{q})$ are labeled
by
$\vec{j}\in\prod_{k=1}^{3}[0...p_{k}-2]^{e}\sqcup\prod_{k=1}^{3}[0,...,p_{k}-2]^{o}$.
In our case, $p_{3}=3$ and $j_{3}\in\\{0,1\\}$, so
$(j_{1},j_{2})\in\prod_{k=1}^{2}[0,...,p_{k}-2]^{e}\sqcup\prod_{k=1}^{2}[0,...,p_{k}-2]^{o}$
wholly fix the $\vec{j}$ as well as $\lambda$. In the labeling, the adjoint
Reidemeister torsion and Chern-Simons invariant (mod 1) of a
$SL(2,\mathbb{C})$ character $\rho_{\vec{j}}$ on the 3-manifold
$S^{2}((P,R-P),(Q,-S),(3,1))$ are Cui:2021lyi

$\displaystyle\begin{split}&\textrm{Tor}(\rho_{\vec{j}})=\prod_{k=1}^{3}\frac{p_{i}}{4\sin^{2}(\frac{2\pi
r_{k}n_{k}}{p_{k}})}=\frac{PQ}{16\text{sin}^{2}(\frac{2\pi
Qn_{1}}{P})\text{sin}^{2}(\frac{2\pi Pn_{2}}{Q})}\;,\\\
&\textrm{CS}(\rho_{\vec{j}})=\sum_{k=1}^{3}\frac{-c_{k}}{4p_{k}}(j_{k}+1)^{2}\;.\end{split}$
(35)

where $c_{k}$ is :

$\displaystyle c_{k}=\begin{cases}p_{k}q_{k}s_{k}-r_{k}\;,&\ q_{k}\
\text{odd}\\\ p_{k}q_{k}s_{k}-r_{k}(p_{k}-1)^{2},&\ q_{k}\
\text{even}\end{cases}$ (36)

Here $(r_{k},s_{k})\in\mathbb{Z}^{2}$ is chosen to satisfy
$p_{k}s_{k}-q_{k}r_{k}=1$, and $n_{1}=n_{(p_{1},q_{1})}(j_{1})$ and
$n_{2}:=n_{(p_{2},q_{2})}(j_{2})$ in (34).

We propose the following one-to-one map between the primaries ${\cal
O}_{(a,b)}$ of $M(P,Q)$ and the irreducible characters $\rho_{\vec{j}}$ on
$S^{2}((P,P-R),(Q,S),(3,1))$:

$\displaystyle\begin{split}&{\cal
O}_{(a,b)}\leftrightarrow\rho_{\vec{j}}\;,\textrm{ where }\\\ &\quad
i)\;\;(a,b)=\begin{cases}(j_{1}+1,j_{2}+1)\;,&j_{1},j_{2}\ \text{even}\\\
(P-j_{1}-1,j_{2}+1)\;,&j_{1},j_{2}\ \text{odd}\end{cases}\;,\\\ &\quad\textrm{
if $(P,R,S)\in(4\mathbb{Z},2\mathbb{Z}+1,2\mathbb{Z})$ or
$(Q,R,S)\in(4\mathbb{Z},2\mathbb{Z}+1,2\mathbb{Z}+1)$}\\\ &\quad
ii)\;\;(a,b)=\begin{cases}(P-j_{1}-1,j_{2}+1)\;,&j_{1},j_{2}\ \text{even}\\\
(j_{1}+1,j_{2}+1)\;,&j_{1},j_{2}\ \text{odd}\end{cases}\;,\\\
&\quad\textrm{otherwise}\;.\end{split}$ (37)

Under the map, one can check that $(2{\rm Tor}(\rho_{\alpha}))^{-1}$ (resp.
$\textrm{CS}[\rho_{\alpha=0}]-\textrm{CS}[\rho_{\alpha}]$) equals to
$S_{0\alpha}^{2}$ (resp. $h_{\alpha}$ (mod 1)) of $M(P,Q)$.

Let us give some concrete examples.

#### Example: $M(3,4)=(\textrm{Ising CFT})$ from $S^{2}((3,4),(4,-1),(3,1))$

We choose $(R,S)=(-1,-1)$. There are 3 irreducible $SL(2,\mathbb{C})$
characters $\rho_{\vec{j}}$:

$(j_{1},j_{2})$ | $(\vec{n},\lambda)$ | $\textrm{CS}[\rho]$ | $\textrm{Tor}[\rho]$ | ${\cal O}_{(a,b)}$ | $\textrm{CS}[\rho_{\alpha=0}]-\textrm{CS}[\rho]$
---|---|---|---|---|---
$(0,0)$ | $(1,\frac{1}{2},\frac{1}{2},\frac{1}{2})$ | $\frac{31}{48}$ | $2$ | ${\cal O}_{(1,1)}={\cal O}_{(2,3)}$ | $0$
$(0,2)$ | $(1,\frac{3}{2},\frac{1}{2},\frac{1}{2})$ | $\frac{7}{48}$ | $2$ | ${\cal O}_{(1,3)}={\cal O}_{(2,1)}$ | $\frac{1}{2}$
$(1,1)$ | $(1,1,1,0)$ | $\frac{7}{12}$ | $1$ | ${\cal O}_{(1,2)}={\cal O}_{(2,2)}$ | $\frac{1}{16}$
Table 2: Ising CFT from $S^{2}((3,4),(4,-1),(3,1))$. We use the characters-to-
primaries map in (37). The result is compatible with that
$S_{(1,1),(1,1)}=S_{(1,1),(2,1)}=\frac{1}{2},S_{(1,1),(2,2)}=\frac{1}{\sqrt{2}}$
and $h_{(2,1)}=\frac{1}{2},h_{(2,2)}=\frac{1}{16}$.

#### Example: $M(3,4)=(\textrm{Ising CFT})$ from $S^{2}((3,1),(4,3),(3,1))$

This time, we choose $(R,S)=(2,3)$. Due to the different choice of $(R,S)$,
the correspondence between the primaries and the irreducible characters is
different from the case when $(R,S)=(-1,-1)$.:

$(j_{1},j_{2})$ | $(\vec{n},\lambda)$ | $\textrm{CS}[\rho]$ | $\textrm{Tor}[\rho]$ | ${\cal O}_{(a,b)}$ | $\textrm{CS}[\rho_{\alpha=0}]-\textrm{CS}[\rho]$
---|---|---|---|---|---
$(0,0)$ | $(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2})$ | $\frac{7}{48}$ | $2$ | ${\cal O}_{(2,1)}={\cal O}_{(1,3)}$ | $\frac{1}{2}$
$(0,2)$ | $(\frac{1}{2},\frac{3}{2},\frac{1}{2},\frac{1}{2})$ | $\frac{31}{48}$ | $2$ | ${\cal O}_{(2,3)}={\cal O}_{(1,1)}$ | $0$
$(1,1)$ | $(1,1,1,0)$ | $\frac{7}{12}$ | $1$ | ${\cal O}_{(2,2)}={\cal O}_{(1,2)}$ | $\frac{1}{16}$
Table 3: Ising CFT from $S^{2}((3,1),(4,3),(3,1))$. We use the characters-to-
primaries map in (37). The result is compatible with that
$S_{(1,1),(1,1)}=S_{(1,1),(2,1)}=\frac{1}{2},S_{(1,1),(2,2)}=\frac{1}{\sqrt{2}}$
and $h_{(2,1)}=\frac{1}{2},h_{(2,2)}=\frac{1}{16}$.

#### Example: $M(5,2)=(\textrm{Lee-Yang CFT})$ from $S^{2}((5,3),(2,1),(3,1))$

We choose $(R,S)=(2,1)$. There are 2 irreducible $SL(2,\mathbb{C})$ characters
$\rho_{\vec{j}}$:

$(j_{1},j_{2})$ | $(\vec{n},\lambda)$ | $\textrm{CS}[\rho]$ | $\textrm{Tor}[\rho]$ | ${\cal O}_{(a,b)}$ | $\textrm{CS}[\rho_{\alpha=0}]-\textrm{CS}[\rho]$
---|---|---|---|---|---
$(0,0)$ | $(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2})$ | $\frac{53}{120}$ | $\frac{1}{4}\left(5-\sqrt{5}\right)$ | ${\cal O}_{(4,1)}={\cal O}_{(1,1)}$ | $0$
$(2,0)$ | $(\frac{3}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2})$ | $\frac{77}{120}$ | $\frac{1}{4}\left(5+\sqrt{5}\right)$ | ${\cal O}_{(2,1)}={\cal O}_{(3,1)}$ | $\frac{4}{5}$
Table 4: Lee-Yang CFT from $S^{2}((5,3),(2,1),(3,1))$. The result is
compatible with that
$S_{(1,1),(1,1)}=\sqrt{\frac{2}{5-\sqrt{5}}},S_{(1,1),(3,1)}=\sqrt{\frac{2}{5+\sqrt{5}}}$
and $h_{(3,1)}=-\frac{1}{5}$.

#### Example: $M(5,4)=(\textrm{Tricritical Ising model})$ from
$S^{2}((5,-1),(4,5),(3,1))$

We choose $(R,S)=(6,5)$. There are 6 irreducible $SL(2,\mathbb{C})$ characters
$\rho_{\vec{j}}$:

$(j_{1},j_{2})$ | $(\vec{n},\lambda)$ | $\textrm{CS}[\rho]$ | $\textrm{Tor}[\rho]$ | ${\cal O}_{(a,b)}$ | $\textrm{CS}[\rho_{\alpha=0}]-\textrm{CS}[\rho_{\alpha}]$
---|---|---|---|---|---
$(0,0)$ | $(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2})$ | $\frac{37}{240}$ | $\frac{5}{2\left(\frac{5}{8}-\frac{\sqrt{5}}{8}\right)}$ | ${\cal O}_{(4,1)}={\cal O}_{(1,3)}$ | $\frac{1}{2}$
$(2,0)$ | $(\frac{3}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2})$ | $\frac{133}{240}$ | $\frac{5}{2\left(\frac{5}{8}+\frac{\sqrt{5}}{8}\right)}$ | ${\cal O}_{(2,1)}={\cal O}_{(3,3)}$ | $\frac{1}{10}$
$(0,2)$ | $(\frac{1}{2},\frac{3}{2},\frac{1}{2},\frac{1}{2})$ | $\frac{157}{240}$ | $\frac{5}{2\left(\frac{5}{8}-\frac{\sqrt{5}}{8}\right)}$ | ${\cal O}_{(4,3)}={\cal O}_{(1,1)}$ | $0$
$(2,2)$ | $(\frac{3}{2},\frac{3}{2},\frac{1}{2},\frac{1}{2})$ | $\frac{13}{240}$ | $\frac{5}{2\left(\frac{5}{8}+\frac{\sqrt{5}}{8}\right)}$ | ${\cal O}_{(2,3)}={\cal O}_{(3,1)}$ | $\frac{3}{5}$
$(1,1)$ | $(1,1,1,0)$ | $\frac{37}{60}$ | $\frac{5}{4\left(\frac{5}{8}+\frac{\sqrt{5}}{8}\right)}$ | ${\cal O}_{(2,2)}={\cal O}_{(3,2)}$ | $\frac{3}{80}$
$(3,1)$ | $(2,1,1,0)$ | $\frac{13}{60}$ | $\frac{5}{4\left(\frac{5}{8}-\frac{\sqrt{5}}{8}\right)}$ | ${\cal O}_{(4,2)}={\cal O}_{(1,2)}$ | $\frac{7}{16}$
Table 5: Tricritical Ising model from $S^{2}((5,-1),(4,5),(3,1))$. The result
is compatible with the modular data of the tricritical Ising model, which can
be evaluated from (5) and (8)

##  3 Field theory description of ${\cal T}_{(P,Q)}$

We now present a concrete and unified field-theoretic description for the
$T_{\rm irred}[S^{2}(\vec{p},\vec{q})]$ theory. By specializing the values of
$((p_{1},q_{1}),(p_{2},q_{2}),(p_{3},q_{3}))$ to $((P,P-R),(Q,S),(3,1))$, the
theory becomes the ${\cal T}_{(P,Q)}$ theory as proposed in (25). See (59) for
the proposed ${\cal T}_{(P,Q)}$.

In principle, the theory can be constructed using the general algorithm
proposed in Dimofte:2011ju ; Gang:2018wek , which is based on a Dehn surgery
representation of the 3-manifold using a hyperbolic link and an ideal
triangulation of the link complement. As seen from various examples explored
in Choi:2022dju , however, one needs to consider different hyperbolic links
for each $T_{\rm irred}[S^{2}(\vec{p},\vec{q})]$. This makes it difficult to
succinctly describe the field theories for all $S^{2}(\vec{p},\vec{q})$ in a
unified manner.

###  3.1 Field theory description of $T_{\rm irred}[S^{2}(\vec{p},\vec{q})]$

The Seifert fiber space $S^{2}(\vec{p},\vec{q})$ depicted in Figure 1 can be
alternatively represented as follows:

$\displaystyle\begin{split}&S^{2}(\vec{p},\vec{q})=\left((\Sigma_{0,3}\times
S^{1})\bigcup_{i=1}^{3}(D_{2}\times S^{1})_{i}\right)/\sim\\\ &\textrm{ with
}p_{i}A_{i}+q_{i}B_{i}\sim[S^{1}]\subset H_{1}[\partial
D_{2},\mathbb{Z}]\;,\quad i=1,2,3\;.\end{split}$ (38)

Here, $\Sigma_{g=0,h=3}$ denotes a three-punctured sphere and $(A_{i},B_{i})$
are basis elements of $H_{1}(\partial(\Sigma_{0,3}\times S^{1}),\mathbb{Z})$m
where $A_{i}$ (resp. $B_{i}$) represents the 1-cycle circling the $i$-th
puncture (resp. the 1-cycle along the $S^{1}\in\partial(\Sigma_{0,3})\times
S^{1}$). The equivalence relation $p_{i}A_{i}+q_{i}B_{i}\simeq[S^{1}]\in
H_{1}[\partial D_{2},\mathbb{Z}]$ corresponds to the relation
$x_{i}^{p_{i}}h^{q_{i}}=1$ in the fundamental group (30) and the relation
$x_{1}x_{2}x_{3}=1$ comes from the same relation in $\pi_{1}(\Sigma_{0,3})$.

Using the geometrical representation, the field theory
$T[S^{2}(\vec{p},\vec{q})]$ can be constructed as follows, see Figure 2.

Figure 2: Generalized quiver diagrams for $T[\Sigma_{0,3}\times S^{1}]$,
$T[SU(2)]$ and $T[S^{2}(\vec{p},\vec{q})]$. The difference between $T_{\rm
full}[S^{2}(\vec{p},\vec{q})]$ and $T_{\rm irred}[S^{2}(\vec{p},\vec{q})]$
arises only from different choices of the $T[\Sigma_{0,3}\times S^{1}]$
theory, either $T_{\rm full}[\Sigma_{0,3}\times S^{1}]$ or $T_{\rm
irred}[\Sigma_{0,3}\times S^{1}]$.

First, we prepare the $T[\Sigma_{0,3}\times S^{1}]$ theory which is expected
to a ${\cal N}=4$ theory with three $SU(2)$ flavor symmetries associated with
the 3 boundary tori (punctures$\times S^{1}$). The theory is depicted by a
blue trivalent vertex with 3 legs, and the boxes attached to the legs
represent the three $SU(2)$ flavor symmetries. Gluing the solid torus is a
Dehn filling procedure, whose corresponding field-theoretic operation in 3D-3D
correspondence has been studied in literature Pei:2015jsa ; Alday:2017yxk ;
Gukov:2017kmk ; Gang:2018wek ; Assel:2022row . In the Dehn filling operation,
the $T[SU(2)]$ theory Gaiotto:2008ak  plays an important role. The $T[SU(2)]$
is a ${\cal N}=4$ theory with $SU(2)_{L}\times SU(2)_{R}$ flavor symmetry,
(see Appendix A for details) and is depicted by a red line with two boxes
representing the two $SU(2)$ symmetries. The Dehn filling operation,
$p_{i}A_{i}+q_{i}B_{i}\sim[S_{1}]\in H_{1}[\partial D_{2},\mathbb{Z}]$, at
each $i$ corresponds to coupling the $T[\Sigma_{0,3}\times S^{1}]$ theory to
$(\sharp-1)$-copies of $T[SU(2)]$ theories using the $i$-th $SU(2)$ flavor
symmetry in $T[\Sigma_{0,3}\times S^{1}]$ and $2(\sharp-1)$ $SU(2)$s in the
$T[SU(2)]^{\otimes(\sharp-1)}$, as described in 3rd quiver diagram in Figure
2. The circle denotes the ${\cal N}=3$ gauging the diagonal $SU(2)$, and the
integer $k$ next to the circle denotes the Chern-Simons (CS) level. The number
of $T[SU(2)]$s, $\sharp-1$, and the CS levels
$\vec{k}=(k^{(1)},\ldots,k^{(\sharp)})$ are related to the Dehn filling slope
$(p,q)$ as follows:

$\displaystyle\begin{split}&\frac{q}{p}=\frac{1}{k^{(1)}-\frac{1}{k^{(2)}-\frac{1}{k^{(3)}-\ldots\frac{1}{k^{(\sharp)}}}}}\;.\\\
&\textrm{or equivalently,}\\\ &g(\vec{k}):=T^{k^{(1)}}ST^{k^{(2)}}\ldots
ST^{k^{(\sharp)}}=\pm\begin{pmatrix}q&*\\\ p&*\end{pmatrix}\;.\end{split}$
(39)

Here $S$ and $T\in SL(2,\mathbb{Z})$ are chosen as

$\displaystyle S=\begin{pmatrix}0&-1\\\ 1&0\end{pmatrix},\quad
T=\begin{pmatrix}1&0\\\ 1&1\end{pmatrix}\;.$ (40)

Especially when $p/q$ is an integer, i.e., $q=1$, the Dehn filling operation
corresponds to the gauging of $SU(2)$ flavor symmetry with CS level $p$.

#### Field theory for $T_{\rm
irred}\left[S^{2}((p_{1},q_{1}),(p_{2},q_{2}),(p_{3},q_{3}))\right]$

Using the prescription above, the field theory $T_{\rm
full}[S^{2}(\vec{p},\vec{q})]$ is constructed as follows Benini:2010uu ;
Eckhard:2019jgg ; Assel:2022row

$\displaystyle\begin{split}&T_{\rm full}[\Sigma_{0,3}\times
S^{1}]=(\textrm{$S^{1}$-reduction of 4D $T_{N=2}$ theory})\\\ &=(\textrm{a
free theory with 8 half-hypermultiplets in
$\mathbf{2}\otimes\mathbf{2}\otimes\mathbf{2}$ under the
$SU(2)^{3}$})\;.\end{split}$ (41)

The $T_{\rm irred}[S^{2}(\vec{p},\vec{q})]$ can be constructed in the same way
except that the $T_{\rm full}[\Sigma_{0,3}\times S^{1}]$ should be replaced by
$T_{\rm irred}[\Sigma_{0,3}\times S^{1}]$. We propose that:

$\displaystyle T_{\rm irred}\left[\Sigma_{0,3}\times S^{1}\right]=(\textrm{a
topological field theory})\;,$ (42)

This proposal is based on the observation that $T_{\rm irred}[M]$ theories for
non-hyperbolic 3-manifolds $M$ with torus boundaries usually exhibit a mass
gap and flow to topological field theory. As we will see below, this proposal
passes several non-trivial consistency checks. Combining the above proposal
with the general Dehn surgery prescription in 3D-3D correspondence, we propose
that

$\displaystyle\begin{split}&T_{\rm
irred}\left[M=S^{2}((p_{1},q_{1}),(p_{2},q_{2}),(p_{3},q_{3}))\right]\otimes(\textrm{a
unitary TQFT})\\\ &\simeq[({\cal D}(p_{1},q_{1})\otimes{\cal
D}(p_{2},q_{2})\otimes{\cal
D}(p_{3},q_{3}))]/H^{1}(M,\mathbb{Z}_{2})\;.\end{split}$ (43)

Here $\sim$ denotes the stronger IR equivalence in (28). Here
$H^{1}(M,\mathbb{Z}_{2})$ represents the 1-form symmetry in $\prod_{i}{\cal
D}(p_{i},q_{i})$ which geometrically originates from the $\mathbb{Z}_{2}$
cohomology of the internal 3-manifold in 3D-3D correspondence Eckhard:2019jgg
; Cho:2020ljj . The theory ${\cal D}(p,q)$ is defined as follows:

$\displaystyle\begin{split}&D(\vec{k})\simeq{\cal
D}(p,q)\otimes\textrm{TFT}[\vec{k}],\textrm{ where }\\\
&D(\vec{k}):=\begin{cases}\frac{T[SU(2)]^{\otimes(\sharp-1)}}{SU(2)^{(1)}_{k^{(1)}}\otimes
SU(2)^{(2)}_{k^{(2)}}\ldots\times SU(2)^{(\sharp)}_{k^{(\sharp)}}},&\sharp\geq
2\\\ \mathcal{N}=2\;\textrm{ pure }SU(2)_{k^{(1)}}\textrm{ CS
theory},&\sharp=1\end{cases}\end{split}$ (44)

Here $/G_{k}$ denotes ${\cal N}=3$ gauging of $G$ symmetry with Chern-Simon
level $k$. The CS levels $\vec{k}=(k^{(1)},\ldots,k^{(\sharp)})$ are related
to the $(p,q)$ as in (39). The gauged $SU(2)$ symmetries are

$\displaystyle\begin{split}&SU(2)^{(1)}\;:\;SU(2)_{L}^{(1)}:=SU(2)_{L}\textrm{
of the 1st }T[SU(2)]\;,\\\ &SU(2)^{(2\leq I\leq\sharp-1)}\;:\;\textrm{diagonal
subgroup of }(SU(2)^{(I-1)}_{R}\times SU(2)^{(I)}_{L})\;,\\\
&SU(2)^{(\sharp)}\;:\;SU(2)_{R}^{(\sharp-1)}\;.\end{split}$ (45)

The field theory description can be summarized in the quiver diagram in Figure
3.

Figure 3: A quiver diagram for $D(\vec{k})$ theory.

The $D(\vec{k})$ theory has $(\mathbb{Z}_{2})^{\otimes\sharp}$ 1-form symmetry
originating from the center subgroup of $SU(2)^{\sharp}$ gauge symmetry. The
symmetry has non-trivial ‘t Hooft anomalies Gang:2018wek  which can be
characterized by the following action of the 4D anomaly theory

$\displaystyle S_{\rm anoamly}=\pi\int_{{\cal
M}_{4}}\left(\sum_{I=1}^{\sharp}k^{(I)}\frac{{\cal
P}(\omega_{2}^{(I)})}{2}+\sum_{J=1}^{\sharp-1}\omega_{2}^{(J)}\cup\omega_{2}^{(J+1)}\right)\;.$
(46)

Here $\omega_{2}^{(I)}\in H^{2}({\cal M}_{4},\mathbb{Z}_{2})$ is the two-form
background field for each $\mathbb{Z}_{2}$ 1-form symmetry from $I$-th $SU(2)$
gauge symmetry and ${\cal P}$ is the Pontryagin square operation. By matching
the anomaly with that of the decoupled topological field theory Hsin:2018vcg ,
we expect that the decoupled topological theory is given in the the following
form:

$\displaystyle\begin{split}&\textrm{TFT}[\vec{k}=(k^{(1)},k^{(2)},\ldots,k^{(\sharp)})]\\\
&=\textrm{$U(1)^{\sharp}_{\cal K}$ theory with mixed CS level }{\cal
K}_{IJ}=2\times\begin{cases}+1\textrm{ or }-1\;,&|I-J|=1\\\ 0\;,&I=J\textrm{
and }k^{(I)}\in 2\mathbb{Z}\\\ +1\textrm{ or }-1\;,&I=J\textrm{ and
}k^{(I)}\in 2\mathbb{Z}+1\\\ 0\;,&|I-J|>0\end{cases}\end{split}\;.$ (47)

Refer to Comi:2023lfm  for a similar analysis done for $T_{\rm
full}[S^{2}((k_{1},1),(k_{2},1),(k_{3},1))]$ theory. See also Appendix B for
more details of the decoupled TQFT. By removing the decoupled TQFT from
$D(\vec{k})$, we obtain the ${\cal D}(p,q)$. We expect following properties of
the ${\cal D}(p,q)$ theory

$\displaystyle\begin{split}&i)\;{\cal D}(p,q)\textrm{ does not depend on the
choice of $\vec{k}$ for given $(p,q)$. }\;,\\\ &ii)\;{\cal D}(p,q)\simeq{\cal
D}(p,q+p\mathbb{Z})\;.\end{split}$ (48)

Figure 4: Three basic dualities (modulo topological sectors) guaranteeing the
independence of $D(p,q)$ on the choices of $\vec{k}$. The 2nd and 3rd moves
follow from the $SL(2,\mathbb{Z})$ structure of the duality wall theory. The
1st move comes from the IR duality between $(T[SU(2)]/(SU(2)_{R})_{\pm
1})\sim(\textrm{empty theory only with background CS level $\mp 1$ for
}SU(2)_{L})$. See the computation in (99) for a non-trivial check of the
duality using superconformal index. The same moves were used in constructing
$T_{\rm full}[M]$ for 3-manifolds $M$ associated to plumbed graphs in
Gukov:2017kmk .

The first property follows from the basic IR dualities (modulo topological
sectors) depicted in Figure 4, which imply that

$\displaystyle D(\vec{k}_{1})\sim D(\vec{k}_{2})\quad\textrm{ if
$g(\vec{k}_{1})=g(\vec{k}_{2})\cdot T^{\pm 1}ST^{\pm 1}$}\;.$ (49)

Note that $g(\vec{k})=\pm\left(\begin{array}[]{cc}q&*\\\ p&*\\\
\end{array}\right)$ and the right multiplication of $T^{\pm 1}ST^{\pm
1}=\left(\begin{array}[]{cc}\mp 1&-1\\\ 0&\mp 1\\\ \end{array}\right)$ does
not change the slope $(p,q)$. The 2nd property follows from the move 1 in the
figure with reversed left/right orientation on each quivers, which implies
that

$\displaystyle D(\vec{k}_{1})\sim D(\vec{k}_{2})\quad\textrm{ if
$g(\vec{k}_{1})=T^{\pm 1}ST^{\pm 1}\cdot g(\vec{k}_{2})$}\;.$ (50)

The left multiplication of $T^{\pm 1}ST^{\pm 1}$ changes the $(p,q)$ to
$(p,q\pm p)$. The above IR equivalence $\sim$ modulo topological sector can
promoted into the IR equivalence $\simeq$ in (28) if the decoupled
$\textrm{TFT}(\vec{k})$s on both sides are removed.

From the two properties in (48), it is easy to see that (for $|p|\geq 2$):

$\displaystyle\begin{split}&\textrm{If $q=\pm 1\;(\textrm{mod }p)$,}\\\
&\textrm{the ${\cal D}(p,q)$ has a mass gap and flows to a unitary TQFT in the
IR}\end{split}$ (51)

since

$\displaystyle\begin{split}&{\cal D}(p,\pm 1+p\mathbb{Z})\simeq{\cal D}(p,\pm
1)\sim D(\vec{k}=(\pm p))=(\textrm{${\cal N}=3$ $SU(2)_{\pm p}$})\\\
&\simeq(\textrm{${\cal N}=2$ $SU(2)_{\pm p}$})\simeq SU(2)_{\pm
p-2\times\textrm{sign}(\pm p)}\;.\end{split}$

The pure ${\cal N}=3$ CS theory with non-zero CS level is IR equivalent to
pure ${\cal N}=2$ CS theory since the adjoint chiral mutiplet in the ${\cal
N}=3$ mutiplet has a superpotential mass term and can be integrated out. The
pure ${\cal N}=2$ Chern-Simons theory $SU(2)_{k}$ contains an auxiliary
massive gaugino, and integrating out it induces a CS level shift by
$-2\times\textrm{sign}(k)$. Furthermore, one can check that

$\displaystyle{\cal D}(2,1+2\mathbb{Z})\simeq{\cal
D}(3,1+3\mathbb{Z})\simeq(\textrm{a trivial theory}),$ (52)

The first follows from the fact that $SU(2)_{2-2}=SU(2)_{0}$ is a trivial
theory, while the 2nd follows from that $SU(2)_{3-2}=SU(2)_{1}\simeq U(1)_{2}$
and $\textrm{TFT}[\vec{k}=(3)]$ in (47) is again $U(1)_{2}$. One the other
hand, we expec that

$\displaystyle\begin{split}&\textrm{If $q\neq\pm 1\;(\textrm{mod }p)$,}\\\
&\textrm{the ${\cal D}(p,q)$ flows to an ${\cal N}=4$ rank-0 SCFT in the
IR}.\end{split}$ (53)

The non-zero CS terms lift all the Coulomb/Higgs branches of $T[SU(2)]$
theories. The UV ${\cal N}=3$ will be enhanced to ${\cal N}=4$ in the IR
thanks to the nilpotency properties of moment maps of the $T[SU(2)]$ theory
Gaiotto:2008ak ; Gang:2018huc ; Garozzo:2019ejm . Along with the proposal in
(43), (51) and (53) are compatible with the expected IR phases of
$T[S^{2}(\vec{p},\vec{q})]$ given in (19).

#### Comparison with $T_{\rm irred}[S^{2}((\vec{p},\vec{q}))]$s in
Choi:2022dju

Based on a Dehn surgery representation of $S^{2}((\vec{p},\vec{q}))$ with
hyperbolic knots, the field theories of $T_{\rm
irred}[S^{2}((\vec{p},\vec{q}))]$ for various $(\vec{p},\vec{q})$s were
analyzed in Choi:2022dju . For instance, the 3D theory $T_{\rm irred}[M]$ for
$M=(S^{3}\backslash\mathbf{5}_{2})_{p}$ (the 3-manifold obtained by a Dehn
surgery on $\mathbf{5}_{2}$ knot, a.k.a the 3-twist knot, with an integral
slop $p$) is given by

$\displaystyle\begin{split}&T_{\rm
irred}[(S^{3}\backslash\mathbf{5}_{2})_{p}]\\\
&=\begin{cases}\frac{\left(U(1)_{-\frac{1}{2}}\times SU(2)_{p-2}\textrm{
${\cal N}=2$ gauge theory coupled to a chiral in
(Adj)}_{+1}\right)}{\mathbb{Z}_{2}}\;,&p\in 2\mathbb{Z}\\\
\frac{(U(1)_{-\frac{1}{2}}\times SU(2)_{p-2}\textrm{ ${\cal N}=2$ gauge theory
coupled to a chiral in (Adj)}_{+1})\otimes U(1)_{\pm 2}}{\mathbb{Z}^{\rm
diag}_{2}}\;,&p\in 2\mathbb{Z}+1\end{cases}\end{split}$ (54)

The theory in the numerator has a $\mathbb{Z}_{2}$ 1-form symmetry originating
from the center $\mathbb{Z}_{2}$ subgroup of the $SU(2)$ gauge group. For even
$p$, the $\mathbb{Z}_{2}$ 1-form symmetry is non-anomalous (i.e., having a
trivial ‘t Hooft anomaly) and thus can be gauged. For odd $p$, however, the
1-form symmetry has a non-trivial ‘t Hooft anomaly, requiring it to be
tensored with the $U(1)_{\pm 2}$ theory, which has the same anomalous 1-form
symmetry, before gauging the non-anomalous diagonal $\mathbb{Z}_{2}$ 1-form
symmetry.

Topologically it is known that dunfield2018census

$\displaystyle(S^{3}\backslash\mathbf{5}_{2})_{p}=\begin{cases}S^{2}((2,1),(3,1),(11,-9))\;,&p=1\\\
S^{2}((2,1),(4,1),(7,-5))\;,&p=2\\\
S^{2}((3,1),(3,1),(5,-3))\;,&p=3\end{cases}$ (55)

According to our proposal in (43), we expect the following dual
description333Using $\frac{-9}{11}=\frac{1}{-1-\frac{1}{4-\frac{1}{-2}}}$,
$\frac{-5}{7}=\frac{1}{-1-\frac{1}{2-\frac{1}{-2}}}$ and
$-\frac{3}{5}=\frac{1}{-2-\frac{1}{-3}}$.

$\displaystyle T_{\rm
irred}[(S^{3}\backslash\mathbf{5}_{2})_{p}]\sim\begin{cases}D(\vec{k}=(-1,4,-2))\;,&p=1\\\
D(\vec{k}=(-1,2,-2))\;,&p=2\\\ D(\vec{k}=(-2,-3))\;,&p=3\end{cases}$ (56)

Note that ${\cal D}(p,1)$ with $p\in\mathbb{Z}$ is a (unitary) TQFT and thus
ignored in the above. Using the expression in (97), one can compute the
superconformal indices:

$\displaystyle\begin{split}&{\cal I}^{\rm
sci}_{D(\vec{k})}(\eta,\nu=0)\;\;\textrm{ with }\vec{k}=(-1,4,-2)\\\
&=1-q-\left(\eta+\frac{1}{\eta}\right)q^{3/2}-2q^{2}-\eta^{-1}q^{5/2}+(\eta^{2}-1)q^{3}+\left(\eta-\frac{1}{\eta}\right)q^{7/2}+0q^{4}+\ldots\\\
&{\cal I}^{\rm sci}_{D(\vec{k})}(\eta,\nu=0)\;\;\textrm{ with
}\vec{k}=(-1,2,-2)\\\
&=1-q-\left(\eta+\frac{1}{\eta}\right)q^{3/2}-2q^{2}-\eta^{-1}q^{5/2}+(\eta^{2}-1)q^{3}+\left(\eta-\frac{1}{\eta}\right)q^{7/2}+\eta^{2}q^{4}+\ldots\\\
&{\cal I}^{\rm sci}_{D(\vec{k})}(\eta,\nu=0)\;\;\textrm{ with
}\vec{k}=(-2,-3)\\\
&=1-q-\left(\eta+\frac{1}{\eta}\right)q^{3/2}-2q^{2}-\left(\eta+\frac{1}{\eta}\right)q^{5/2}-2q^{3}-\left(\eta+\frac{1}{\eta}\right)q^{7/2}-2q^{4}+\ldots\;.\end{split}$
(57)

These results match the superconformal indices given in Choi:2022dju  for the
$T_{\rm irred}[(S^{3}\backslash\mathbf{5}_{2})_{p}]$ theory with $p=1,2,3$
respectively. For rank-0 SCFTs, the superconformal index is defined as:

$\displaystyle{\cal I}^{\rm
sci}(\eta,\nu):=\textrm{Tr}(-1)^{R_{\nu}}q^{\frac{R_{\nu}}{2}+j_{3}}\eta^{A}\;,$
(58)

where $A$ and $R_{\nu}$ are defined in (14) and (15).

###  3.2 Field theory for ${\cal T}_{(P,Q)}$

Combining equations (25) and (43), we arrive at the final form of the field
theory description:

$\displaystyle\begin{split}&{\cal T}_{(P,Q)}\simeq T_{\rm
irred}[S^{2}((P,P-R),(Q,S),(3,1))]\\\ &\simeq\begin{cases}({\cal
D}(P,P-R)\otimes{\cal D}(Q,S)\otimes U(1)_{\pm 2},&P\textrm{ and }Q\in
2\mathbb{Z}+1\\\ \left({\cal D}(P,P-R)\otimes{\cal D}(Q,S)\otimes
U(1)_{2}\times U(1)_{\pm 2}\right)/\mathbb{Z}^{\rm
diag}_{2},&\textrm{otherwise}\end{cases}\end{split}$ (59)

The theory ${\cal D}(p,q)$ is defined in (44). This is the main result of our
paper. In the first case, we include the decoupled topological field theory,
$U(1)_{2}$ or $U(1)_{-2}$, in order that (recall that ${\cal Z}_{b}^{\rm
con}:={\cal Z}^{S^{3}_{b}}(M=0,\nu=0)$)

$\displaystyle\begin{split}&|\left({\cal Z}^{\rm con}_{b=1}\textrm{ of }{\cal
T}_{(P,Q)}\right)|=\big{|}\left({\cal Z}^{\rm con}_{b=1}\textrm{ of }{\cal
D}(P,P-R)\right)\times\left({\cal Z}^{\rm con}_{b=1}\textrm{ of }{\cal
D}(Q,S)\right)\times\frac{1}{\sqrt{2}}\big{|}\\\
&=\sqrt{\frac{8}{PQ}}\sin\left(\frac{\pi}{P}\right)\sin\left(\frac{\pi}{Q}\right)=\left(\textrm{min}_{\alpha}|S_{0\alpha}|\textrm{
of }M(P,Q)\right)\;.\end{split}$ (60)

We use the result in (125) and the fact that the $S^{3}$ partition function of
$U(1)_{\pm}$ is $\frac{1}{\sqrt{2}}$. The decoupled $U(1)_{\pm 2}$ has a
$\mathbb{Z}_{2}$ 1-form symmetry generated by an anyon, a topological line
defect, with topological spin $\pm\frac{1}{4}$ (mod $1$), which is compatible
with the fact that there is a primary operator with conformal dimension
$\pm\frac{1}{4}$ (mod $1$) in the $M(P,Q)$ with $P,Q\in 2\mathbb{Z}+1$.

When $P$ (resp. $Q$) is even, the theory ${\cal D}(P,P-R)$ (resp. ${\cal
D}(Q,S)$) has an non-anomalous $\mathbb{Z}_{2}$ 1-form symmetry generated by
an anyon with topological spin $0$ or $1/2$ Hsin:2018vcg , see (126). The
theory in the numerator has three $\mathbb{Z}_{2}$ 1-form symmetries and
$\mathbb{Z}^{\rm diag}_{2}$ denotes the gauging the diagonal one. The $\pm$ in
the $U(1)_{2}\times U(1)_{\pm 2}$ is chosen such that the $\mathbb{Z}^{\rm
diag}_{2}$ 1-form symmetry is bosonic (i.e., the symmetry generating anyon has
a topological spin $0$).

When $Q=P+1$, the $(R,S)$ can be chosen as $(-1,-1)$ and the ${\cal
T}_{(P,Q)}$ theory becomes

$\displaystyle\begin{split}&({\cal D}(P,P+1)\otimes{\cal D}(P+1,-1)\otimes
U(1)_{2}\otimes U(1)_{\pm 2})/\mathbb{Z}_{2}^{\rm diag}\;,\\\
&\simeq\frac{SU(2)_{(P-2)}\otimes SU(2)_{-(P-1)}\otimes
U(1)_{2}}{\mathbb{Z}^{\rm diag}_{2}}\;.\end{split}$ (61)

This is the coset description of unitary minimal model ${\cal T}_{(P,P+1)}$.
We use the fact that

$\displaystyle\begin{split}&\textrm{For }P\in 2\mathbb{Z}_{\geq 1},\;{\cal
D}(P+1,-1)\otimes U(1)_{\pm 2}\simeq SU(2)_{-(P-1)}\textrm{ and }\;{\cal
D}(P,P+1)\simeq SU(2)_{P-2}\;,\\\ &\textrm{For }P\in 2\mathbb{Z}_{\geq
1}+1,\;{\cal D}(P+1,-1)\simeq SU(2)_{-(P-1)}\textrm{ and }\;{\cal
D}(P,P+1)\otimes U(1)_{\mp 2}\simeq SU(2)_{(P-2)}\;.\end{split}$

The 2D chiral central charge of the theory can be computed as follows using
that $c_{2d}(SU(2)_{k})=\frac{3k}{|k|+2}$ and $c_{2d}(U(1)_{2})=1$,

$\displaystyle
c_{2d}=\frac{3(P-2)}{P}-\frac{3(P-1)}{P+1}+1=1-\frac{6P}{P(P+1)}\;,$ (62)

which matches (3).

#### Example : $(P,Q)=(3,5)$

Choosing $(R,S)=(1,2)$ and using equations (59) and (52), we find:

$\displaystyle{\cal T}_{(P,Q)}\simeq{\cal D}(3,2)\otimes{\cal D}(5,2)\otimes
U(1)_{\pm 2}\simeq{\cal D}(5,2)\otimes U(1)_{\pm 2}\;.$ (63)

From that $\frac{2}{5}=\frac{1}{3-\frac{1}{2}}$, we obtain

$\displaystyle{\cal D}(5,2)\otimes{\rm TFT}[\vec{k}]\simeq D(\vec{k})\textrm{
with }\vec{k}=(3,2)\;.$ (64)

Utilizing the explicit computation from (104), we confirm that the set
$\mathbf{HF}:={({\cal H}^{-1/2},{\cal F}/{\cal F}_{\alpha=0})}$ of the
$D((3,2))$ theory in the $A$-twisting limit can be factorized as follows:

$\displaystyle\begin{split}&\mathbf{HF}^{A}\textrm{ of }D((3,2))\\\
&=\bigg{\\{}\left(\frac{4}{\sqrt{15}}\sin(\frac{\pi}{3})\sin(\frac{2\pi}{5}),1\right),\left(\frac{4}{\sqrt{15}}\sin(\frac{\pi}{3})\sin(\frac{\pi}{5}),e^{\frac{2\pi
i}{5}}\right)\bigg{\\}}\times\bigg{\\{}\left(\frac{1}{2},1\right)^{\otimes
2},\left(\frac{1}{2},i\right),\left(\frac{1}{2},\frac{1}{i}\right)\bigg{\\}}\;.\end{split}$
(65)

The second factor can be interpreted as the contribution from the decoupled
${\rm TFT}[\vec{k}]$. By producting the first factor, which is from ${\cal
D}(5,2)$, with the $\mathbf{HF}=\\{(|S_{0\alpha}|,e^{2\pi ih_{\alpha}})\\}$ of
$U(1)_{-2}$, we obtain the set for the ${\cal T}_{(3,5)}$ theory:

$\displaystyle\begin{split}&\mathbf{HF}^{A}\textrm{ of }{\cal T}_{(3,5)}\\\
&=\bigg{\\{}\left(\frac{4}{\sqrt{15}}\sin(\frac{\pi}{3})\sin(\frac{2\pi}{5}),1\right),\left(\frac{4}{\sqrt{15}}\sin(\frac{\pi}{3})\sin(\frac{\pi}{5}),e^{\frac{2\pi
i}{5}}\right)\bigg{\\}}\times\bigg{\\{}\left(\frac{1}{\sqrt{2}},1\right),\left(\frac{1}{\sqrt{2}},\frac{1}{i}\right)\bigg{\\}}\;.\end{split}$
(66)

This set nicely matches with the set of $(|S_{0\alpha}|,e^{2\pi ih_{\alpha}})$
for $M(3,5)$, as expected from the dictionaries in Table 1.

Alternatively, using $\frac{2}{5}=\frac{1}{2-\frac{1}{-2}}$, we have:

$\displaystyle{\cal D}(5,-2)\otimes{\rm TFT}[\vec{k}]\simeq D(\vec{k})\textrm{
with }\vec{k}=(2,-2)\;.$ (67)

In this case, we find:

$\displaystyle\begin{split}&\mathbf{HF}^{A}\textrm{ of }D((2,-2))\\\
&=\bigg{\\{}\left(\frac{4}{\sqrt{15}}\sin(\frac{\pi}{3})\sin(\frac{2\pi}{5}),1\right),\left(\frac{4}{\sqrt{15}}\sin(\frac{\pi}{3})\sin(\frac{\pi}{5}),e^{\frac{2\pi
i}{5}}\right)\bigg{\\}}\times\bigg{\\{}\left(\frac{1}{2},1\right)^{\otimes
3},\left(\frac{1}{2},-1\right)\bigg{\\}}\;.\end{split}$ (68)

Again, the second factor can be interpreted as the contribution from the
decoupled ${\rm TFT}[\vec{k}]$. By omitting the topological factor, we obtain
the set for ${\cal D}(5,2)$. It is equivalent to the previous set obtained
using $D((3,2))$.

#### Example : $(P,Q)=(3,8)$

Choosing $(R,S)=(1,3)$, we have:

$\displaystyle{\cal T}_{(3,8)}\simeq\left({\cal D}(3,2)\otimes{\cal
D}(8,3)\otimes U(1)_{2}\times U(1)_{\pm 2}\right)/\mathbb{Z}^{\rm
diag}_{2}\simeq\left({\cal D}(8,3)\otimes U(1)_{2}\times U(1)_{\pm
2}\right)/\mathbb{Z}^{\rm diag}_{2}\;.$ (69)

Using $\frac{3}{8}=\frac{1}{3-\frac{1}{3}}$, we obtain:

$\displaystyle{\cal D}(8,3)\otimes{\rm TFT}[\vec{k}]\simeq D(\vec{k})\textrm{
with }\vec{k}=(3,3)\;.$ (70)

Using the explicit computation in (104), we confirm that the set
$\mathbf{HF}:=\\{({\cal H}^{-1/2},{\cal F}/{\cal F}_{\alpha=0})\\}$ of
$D((3,3))$ theory in the $A$-twisting limit can be factorized as follows:

$\displaystyle\begin{split}\mathbf{HF}^{A}&\textrm{ of }D((3,3))\\\
=\bigg{\\{}&\left(\frac{1}{2},e^{\frac{29i\pi}{16}-\frac{\pi
i}{2}}\right),\left(\frac{1}{2\sqrt{2}},e^{\frac{25i\pi}{16}-\frac{\pi
i}{2}}\right),\left(\frac{1}{2\sqrt{2}},e^{\frac{25i\pi}{16}-\frac{\pi
i}{2}}\right),\left(\frac{1}{2}\sin\left(\frac{\pi}{8}\right),i\right),\\\
&\left(\frac{1}{2}\sin\left(\frac{\pi}{8}\right),\frac{1}{i}\right),\left(\frac{1}{2}\cos\left(\frac{\pi}{8}\right),1\right),\left(\frac{1}{2}\cos\left(\frac{\pi}{8}\right),-1\right)\bigg{\\}}\times\bigg{\\{}\left(\frac{1}{\sqrt{2}},1\right),\left(\frac{1}{\sqrt{2}},i\right)\bigg{\\}}\;.\end{split}$
(71)

The 2nd factor can be regarded as the contribution from the decoupled ${\rm
TFT}[\vec{k}]$ and 1st factor is from ${\cal D}(8,-3)$. From the
$\mathbf{H}\mathbf{F}^{A}$ of ${\cal D}(8,-3)$, one can see that the ${\cal
D}(8,-3)$ theory has a non-anomalous $\mathbb{Z}_{2}$ 1-form symmetry
generated by an anyon with topological spin $\frac{1}{2}$. Hence, we have the
set for ${\cal T}_{(3,8)}$ theory:444Let ${\cal T}$ be a theory with
$\mathbb{Z}_{2}$ 1-form symmetry generated by an anyon with topological spin
$1/2$. The Bethe vacua can be divided into two sets, ${\cal S}_{\rm BE}={\cal
S}_{\rm BE}^{\rm even}\cup{\cal S}_{\rm BE}^{\rm odd}$. Two Bethe vacua in
${\cal S}_{\rm BE}^{\rm even}$, when related by the $\mathbb{Z}_{2}$ 1-form
symmetry, have the same ${\cal H}$ but their ${\cal F}$ differ by a sign. On
the other hand, two Bethe vacua (which can be identical) in ${\cal S}_{\rm
BE}^{\rm odd}$, when related by the $\mathbb{Z}_{2}$ 1-form symmetry, have the
same ${\cal H}$ and ${\cal F}$. In the theory $\tilde{{\cal T}}:=\frac{{\cal
T}\otimes U(1)_{2}\otimes U(1)_{2}}{\mathbb{Z}_{2}^{\rm diag}}$, the Bethe-
vacua set can be canonically identified with that of ${\cal T}$. The ${\cal
H}$ of Bethe-vacua in $\tilde{{\cal T}}$ are the same as in ${\cal T}$. For
Bethe vacua in ${\cal S}_{\rm BE}^{\rm even}$, their ${\cal F}$s in
$\tilde{{\cal T}}$ are equal to that in ${\cal T}$. However, for Bethe vacua
in ${\cal S}_{\rm BE}^{\rm odd}$, the ${\cal F}$s in $\tilde{{\cal T}}$ differ
by a factor of $\exp(\frac{\pi i}{2})$ compared to that in ${\cal T}$.

$\displaystyle\begin{split}\mathbf{HF}^{A}&\textrm{ of }{\cal
T}_{(3,8)}=\left({\cal D}(8,3)\otimes U(1)_{2}\otimes
U(1)_{2}\right)/\mathbb{Z}_{2}^{\rm diag}\\\
=\bigg{\\{}&\left(\frac{1}{2},e^{\frac{29i\pi}{16}}\right),\left(\frac{1}{2\sqrt{2}},e^{\frac{25i\pi}{16}}\right),\left(\frac{1}{2\sqrt{2}},e^{\frac{25i\pi}{16}}\right),\left(\frac{1}{2}\sin\left(\frac{\pi}{8}\right),i\right),\\\
&\left(\frac{1}{2}\sin\left(\frac{\pi}{8}\right),\frac{1}{i}\right),\left(\frac{1}{2}\cos\left(\frac{\pi}{8}\right),1\right),\left(\frac{1}{2}\cos\left(\frac{\pi}{8}\right),-1\right)\bigg{\\}}\;.\end{split}$
(72)

This set nicely matches the set of $(|S_{0\alpha}|,e^{2\pi ih_{\alpha}})$ of
$M(3,8)$, as expected from the dictionaries in Table 1.

###  3.3 Comparison with the ${\cal T}_{(2,2t+3)}$ by Gang-Kim-Stubbs

For the case when $(P,Q)=(2,2t+3)$, the 3D ${\cal T}_{(P,Q)}$ is (we choose
$R=1,S=t+2$)

$\displaystyle\begin{split}&{\cal T}_{(2,2t+3)}\simeq T_{\rm
irred}[S^{2}((2,1),(2t+3,t+2),(3,1))]\;,\\\ &\simeq{\cal
D}(2t+3,t+2)\otimes\frac{U(1)_{2}\otimes U(1)_{-2}}{\mathbb{Z}_{2}^{\rm
diag}}\simeq{\cal D}(2t+3,t+2)\;.\end{split}$ (73)

Here, we use the fact that both ${\cal D}(2,1)$ and $(U(1)_{2}\otimes
U(1)_{-2})/\mathbb{Z}_{2}^{\rm diag}$ are trivial theories. Using
$\frac{t+2}{2t+3}=\frac{1}{2-\frac{1}{t+2}}$, the ${\cal D}(2t+3,t+2)$ is
given as

$\displaystyle D(\vec{k})\simeq{\cal
D}(2t+3,t+2)\otimes\textrm{TFT}[\vec{k}]\textrm{ with }\vec{k}=(2,t+2)\;.$
(74)

The decoupled topological field theory is, see (116) and (118):

$\displaystyle{\rm TFT}[\vec{k}=(2,t+2)]\simeq U(1)^{2}_{{\cal K}}\textrm{
with mixed CS level }{\cal K}=\begin{cases}\begin{pmatrix}0&2\\\
2&0\end{pmatrix},&t\in 2\mathbb{Z}\\\ \begin{pmatrix}2&2\\\
2&0\end{pmatrix},&t\in 2\mathbb{Z}+1\end{cases}$ (75)

Recently, an abelian ${\cal N}=2$ gauge theory description, denoted ${\cal
T}^{GKS}_{(2,2t+3)}$, of ${\cal T}_{(2,2t+3)}$ was proposed in Gang:2023rei :

$\displaystyle{\cal T}^{GKS}_{(2,2t+3)}=\left(\frac{({\cal
T}_{\Delta})^{\otimes t}}{U(1)^{t}_{K}}+\textrm{monopole
superpotentials}\right)\;,$ (76) $\displaystyle\textrm{with mixed CS level :
}K=2\begin{pmatrix}1&1&1&\cdots&1&1\\\ 1&2&2&\cdots&2&2\\\ 1&2&3&\cdots&3&3\\\
\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\\ 1&2&3&\cdots&t-1&t-1\\\
1&2&3&\cdots&t-1&t\\\ \end{pmatrix}\;,$ (77)

Here ${\cal T}_{\Delta}$ is a free theory of chiral multiplet with background
CS level $-1/2$ for the $U(1)$ flavor symmetry Dimofte:2011ju  .

We now claim that the two descriptions for ${\cal T}_{(2,2t+3)}$ are actually
equivalent.555In Comi:2023lfm , they also found a dual description of the
${\cal T}^{GKS}_{(2,2t+3)}$ theory, which is $T_{\rm
full}[(t+1,1),(1,1),(1,1)]$. One can check the following duality:

$\displaystyle\begin{split}&{\cal
T}^{GKS}_{(2,2t+3)}\otimes\textrm{TFT}[(2,t+2)]\simeq
D(\vec{k}=(2,t+2))=\frac{T[SU(2)]}{(SU(2)_{L})_{2}\times(SU(2)_{R})_{t+2}}\;,\end{split}$
(78)

from various BPS partition function computations. For example, the
superconformal index for $D(\vec{k})$ can be computed using (97) and we find
that

$\displaystyle\begin{split}&{\cal I}^{\rm
sci}_{D(\vec{k}=(-2,-3))}(\eta,\nu=0)={\cal I}^{\rm sci}_{{\cal
T}^{GKS}_{(2,5)}}(q,\eta,\nu=0)\\\
&=1-q+\left(-\eta-\frac{1}{\eta}\right)q^{3/2}-2q^{2}+\left(-\eta-\frac{1}{\eta}\right)q^{5/2}-2q^{3}+\left(-\eta-\frac{1}{\eta}\right)q^{7/2}-2q^{4}+\ldots\\\
&{\cal I}^{\rm sci}_{D(\vec{k}=(-2,-4))}(\eta,\nu=0)={\cal I}^{\rm sci}_{{\cal
T}^{GKS}_{(2,7)}}(q,\eta,\nu=0)\\\
&=1-q+\left(-\eta-\frac{1}{\eta}\right)q^{3/2}-2q^{2}-\eta
q^{5/2}+\left(\frac{1}{\eta^{2}}-1\right)q^{3}+\left(\frac{1}{\eta}-\eta\right)q^{7/2}+\frac{q^{4}}{\eta^{2}}+\ldots\;.\end{split}$
(79)

For the round 3-sphere partition function case, we have ($\simeq$ means
equality up to an overall phase factor as defined in (82)):

$\displaystyle\begin{split}&\left({\cal Z}^{\rm con}_{b=1}\textrm{ of
}D(\vec{k}=(2,t+2))\right)\simeq\frac{1}{\sqrt{(2t+3)}}\sin\left(\frac{\pi}{2t+3}\right)\;,\\\
&\left({\cal Z}^{\rm con}_{b=1}\textrm{ of
}\textrm{TFT}[\vec{k}=(2,t+2)]\right)\simeq\frac{1}{\sqrt{|\det{\cal
K}|}}=\frac{1}{2}\;,\\\ &\left({\cal Z}^{\rm con}_{b=1}\textrm{ of }{\cal
T}^{GKS}_{(2,2t+3)})\right)\simeq\frac{2}{\sqrt{(2t+3)}}\sin\left(\frac{\pi}{2t+3}\right),\end{split}$
(80)

which again supports the proposed duality.

##  4 Discussion and Future Directions

In this paper, we provide an explicit field theory description of the 3D
theory $\mathcal{T}_{(P,Q)}$ dual to the Virasoro minimal model $M(P,Q)$.
Interestingly, the bulk theory exhibits very distinct IR phases—either gapped
or $\mathcal{N}=4$ rank-0 SCFT—depending on whether the RCFT is unitary or
not. The main results are presented in (25) and (59).

#### Boundary Condition

In this paper, we compute various partition functions on closed 3-manifolds to
test the bulk-boundary correspondence. To directly observe the boundary
rational chiral algebra, one needs to consider the theory on an open manifold
with an appropriate boundary condition Gadde:2013wq ; Gadde:2013sca ;
Yoshida:2014ssa ; Dimofte:2017tpi . Identifying the proper boundary condition
that supports the Virasoro minimal models would be an interesting direction
for future research.

#### Mirror RCFTs of Minimal Models

In 3D rank-0 SCFTs, there are two choices of topological twistings: $A$ and
$B$ twisting. Our theory $\mathcal{T}_{(P,Q)}$ is expected to support the
Virasoro minimal model at the boundary under one of these topological
twistings. The other choice of twisting generally supports a different
rational chiral algebra at the boundary. Understanding the mirror dual
rational chiral algebras of the Virasoro minimal models would be a valuable
avenue for further study.

#### Other Minimal Models

Recently, 3D dual theories for some supersymmetric $\mathcal{N}=1$ minimal
models have been proposed Baek:2024tuo . Extending our work to other classes
of minimal models, including these supersymmetric cases, would be an
intriguing direction for future research. Some progress in this direction will
be reported in BGK .

###### Acknowledgements.

We would like to thank Yuji Tachikawa for the useful discussion. The work of
DG and HK is supported in part by the National Research Foundation of Korea
grant NRF-2022R1C1C1011979. DG also acknowledges support by Creative-
Pioneering Researchers Program through Seoul National University.

##  Appendix A BPS partition functions of $D(\vec{k})$

The $T[SU(2)]$ theory, which is a basic building block of the $D(\vec{k})$
theory, is a 3D ${\cal N}=4$ SQED with $N_{f}=2$ (see Table 6).

Chiral multiplet | $U(1)_{\rm gauge}$ | $R_{\nu=0}$ | $A$ | $F$
---|---|---|---|---
$(\Phi_{1},\Phi_{2})$ | $(+1,-1)$ | $\frac{1}{2}$ | $\frac{1}{2}$ | $(+1,-1)$
$(\Phi_{3},\Phi_{4})$ | $(+1,-1)$ | $\frac{1}{2}$ | $\frac{1}{2}$ | $(-1,+1)$
$\Phi_{0}$ | $0$ | $1$ | $-1$ | $0$
Table 6: Matter contents of the $T[SU(2)]$ theory in terms of ${\cal N}=2$
chiral multiplets. $(\Phi_{1},\Phi_{2})$ and $(\Phi_{3},\Phi_{2})$ form two
${\cal N}=4$ hypermultiplets transforming as $\mathbf{2}$ under the
$SU(2)_{L}$ flavor symmetry, with their Cartan denoted by $F$, normalized as
$F\in\mathbb{Z}$. The theory has a $U(1)$ topological symmetry associated with
the $U(1)$ gauge field, which is enhanced to $SU(2)_{R}$ in the IR. ${\cal
N}=4$ vector multiplet contains a neutral chiral multiplet $\Phi_{0}$. The
theory has a superpotential ${\cal
W}\propto\Phi_{1}\Phi_{0}\Phi_{2}-\Phi_{3}\Phi_{0}\Phi_{4}$.

In terms of an ${\cal N}=2$ subalgebra, the theory possesses $SU(2)_{L}\times
SU(2)_{R}\times U(1)_{A}$ flavor symmetry that commutes with the subalgebra.
Various SUSY parition functions of the $T[SU(2)]$ theory and its variants have
been computed in various literatures. To ensure self-containment, we reproduce
these computations here.

#### Squashed 3-sphere partition function Kapustin:2009kz ; Jafferis:2010un ;
Hama:2010av

The squashed 3-sphere partition function $\mathcal{Z}^{S^{3}_{b}}$ of the
$S$-duality wall theory is

$\displaystyle\begin{split}&{\cal
Z}^{S^{3}_{b}}_{T[SU(2)]}(M_{1},M_{2};M,\nu)=\int\frac{dZ}{\sqrt{2\pi\hbar}}{\cal
I}^{\hbar}_{T[SU(2)]}(Z,M_{1},M_{2};W),\;\textrm{ where }\\\ &{\cal
I}^{\hbar}_{T[SU(2)]}:=\exp\left(\frac{Z^{2}+M_{1}^{2}+2M_{2}Z}{\hbar}\right)\psi_{\hbar}\left(-W+(i\pi+\frac{\hbar}{2})\right)\\\
&\qquad\qquad\quad\times\prod_{\epsilon_{1},\epsilon_{2}\in\\{\pm
1\\}}\psi_{\hbar}\left(\epsilon_{1}Z+\epsilon_{2}M_{1}+\frac{W}{2}+(\frac{i\pi}{2}+\frac{\hbar}{4})\right)\bigg{|}_{W:=M+(i\pi+\hbar/2)\nu}\;,\end{split}$
(81)

We define $\hbar:=2\pi ib^{2}$, where the $b$ is the squashing parameter of
the squashed 3-sphere $S^{3}_{b}$ in (13).

The partition function has following phase factor ambiguity

$\displaystyle\exp\left(i\pi(b^{2}+b^{-2})\mathbb{Q}+i\pi\mathbb{Q}\right)\;,$
(82)

which depends on the background CS levels for $R$-symmetry and flavor
symmetries, decoupled invertible TQFT, 3-manifold framing and so forth. The
partition function depends on following parameters:

$\displaystyle\begin{split}&M_{1}/M_{2}\;:\;\textrm{(rescaled) real masses for
the the Cartan $U(1)$s of $SU(2)_{L}/SU(2)_{R}$}\;,\\\
&(M,\nu)\;:\;\textrm{((rescaled) real mass, R-symmetry mixing in
\eqref{R-symmetry mixing}) for the $U(1)_{A}$}\;.\end{split}$ (83)

The rescaled real mass $M$ is $b\times(\textrm{real mass})$ and the squashed
3-sphere partition has $b\leftrightarrow b^{-1}$ symmetry when the
(unrescaled) real masses and $\nu$ are fixed. The special function
$\psi_{\hbar}$ in the integrand is called quantum dilogarithm (Q.D.L)
function, which is defined by Faddeev:1993rs

$\displaystyle\psi_{\hbar}(Z):=\begin{cases}\prod_{r=1}^{\infty}\frac{1-q^{r}e^{-Z}}{1-\widetilde{q}^{-r+1}e^{-\widetilde{Z}}}\;,\quad\textrm{if
$|q|<1$}\\\
\prod_{r=1}^{\infty}\frac{1-\widetilde{q}^{r}e^{-\widetilde{Z}}}{1-q^{-r+1}e^{-Z}}\;,\quad\textrm{if
$|q|>1$}\end{cases}$ (84)

with

$\displaystyle q=e^{2\pi ib^{2}},\quad\widetilde{q}:=e^{2\pi
ib^{-2}},\quad\widetilde{Z}=Z/b^{2}\;.$ (85)

The $\psi_{\hbar}(Z)$ computes the squashed 3-sphere partition function of the
${\cal T}_{\Delta}$ theory Dimofte:2011ju , a massless free theory of single
${\cal N}=2$ chiral multiplet with background CS level $-1/2$ for the $U(1)$
flavor symmetry. The $Z$ is $M+(i\pi+\frac{\hbar}{2}R(\Phi))$ with the
rescaled real mass $M$ for the $U(1)$ flavor symmetry and the R-charge
$R(\Phi)$ of the chiral field. The self-mirror property of the $T[SU(2)]$
theory implies that

$\displaystyle{\cal Z}^{S^{3}_{b}}_{T[SU(2)]}(M_{1},M_{2};M,\nu)\simeq{\cal
Z}^{S^{3}_{b}}_{T[SU(2)]}(M_{2},M_{1};-M,-\nu)\;.$ (86)

Here $\simeq$ means the equality modulo a phase factor of the form in (82).
Using the $T[SU(2)]$ partition function, the squashed 3-sphere partition
function of $D(\vec{k})$ in Figure 3 can be computed as follows

$\displaystyle\begin{split}&{\cal Z}^{S^{3}_{b}}_{D(\vec{k})}(M,\nu)\\\
&=\int\left(\prod_{I=1}^{\sharp}\frac{\Delta(M_{I})dM_{I}}{\sqrt{2\pi\hbar}}\exp\left(\frac{k^{(I)}M_{I}^{2}}{\hbar}\right)\right)\left(\prod_{I=1}^{\sharp-1}{\cal
Z}^{S^{3}_{b}}_{T[SU(2)]}(M_{I},M_{I+1};M,\nu)\right)\;.\end{split}$ (87)

The $\Delta(M)$ is the contribution from a $SU(2)$ vector multiplet,

$\displaystyle\Delta(M)=2\sinh(M)\sinh(2\pi iM/\hbar)\;.$ (88)

The partition function for $D(\vec{k})$ drastically simplifies when $b=1$ and
$\nu=0$. For $T[SU(2)]$ theory, the partition function becomes Nishioka:2011dq
; Gang:2021hrd

$\displaystyle{\cal
Z}^{S^{3}_{b=1}}_{T[SU(2)]}(M_{1},M_{2};M=0,\nu=0)\simeq\frac{1}{2}\frac{\sin\left(\frac{M_{1}M_{2}}{\pi}\right)}{\sinh(M_{1})\sinh(M_{2})}\;.$
(89)

For $D(\vec{k})$ theory,

$\displaystyle\begin{split}&{\cal Z}^{S^{3}_{b=1}}_{D(\vec{k})}(M=0,\nu=0)\\\
&\simeq
2\int\left(\prod_{I=1}^{\sharp}\frac{dM_{I}}{2\pi}\exp\left(\frac{k^{(I)}M_{I}^{2}}{2\pi
i}\right)\right)\sinh(M_{1})\sinh(M_{\sharp})\prod_{I=1}^{\sharp-1}\sin\left(\frac{M_{I}M_{I+1}}{\pi}\right)\;,\\\
&\simeq\frac{1}{\sqrt{2^{\sharp-2}|p|}}\sin\left(\frac{\pi}{|p|}\right)\;.\end{split}$
(90)

The integral in the middle line is simply a sum of Gaussian integrals and can
be easily evaluated to obtain the final answer.Interestingly, the final result
can be expressed as a very simple function of $p$, which is related to
$\vec{k}$ as shown in (39). In the computation, we use the following identity:

$\displaystyle\begin{split}&\det\overline{{\cal K}}(\vec{k})=|p|\;,\\\
&\textrm{ where }\overline{{\cal K}}_{IJ}:=\begin{cases}1,&|I-J|=1\\\
k^{(I)},&I=J\\\
0,&\textrm{otherwise}\end{cases}\;\quad\quad(I,J=1,\ldots,\sharp)\end{split}$
(91)

#### Superconformal index Kim:2009wb ; Imamura:2011su ; Dimofte:2011py

The generalized superconformal index ${\cal I}^{\rm sci}_{T[SU(2)]}$ for the
$T[SU(2)]$ theory is $(\tilde{\eta}^{2}:=\eta)$

$\displaystyle\begin{split}&{\cal I}^{\rm
sci}_{T[SU(2)]}(m_{1},u_{1},m_{2},u_{2};\eta,\nu)\\\
&=\sum_{m}\oint_{|u|=1}\frac{du}{2\pi
iu}u^{2(m+m_{2})}u_{2}^{2m}u_{1}^{2m_{1}}{\cal
I}_{\Delta}(0,-q^{1/2}\tilde{\eta}^{-2})\\\
&\quad\times\prod_{\epsilon_{1},\epsilon_{2}\in\\{\pm 1\\}}{\cal
I}_{\Delta}(\epsilon_{1}m+\epsilon_{2}m_{1},\tilde{\eta}u^{\epsilon_{1}}u^{\epsilon_{2}}_{1}(-q^{1/2})^{1/2})\big{|}_{\tilde{\eta}\rightarrow(-\eta
q^{1/2})^{\nu/2}}\;,\\\ &=\sum_{m}\oint_{|u|=1}\frac{du}{2\pi
iu}(-q^{1/2})^{m+m_{2}}u^{2(m+m_{2})}u_{1}^{2m_{1}}u_{2}^{2m}{\cal
I}_{\Delta}(0,-q^{1/2}\tilde{\eta}^{-2}){\cal
I}_{\Delta}(m+m_{1},-q^{1/2}\tilde{\eta}uu_{1})\\\ &\quad\quad{\cal
I}_{\Delta}(m-m_{1},-q^{1/2}\tilde{\eta}uu_{1}^{-1}){\cal
I}_{\Delta}(-m+m_{1},\tilde{\eta}u^{-1}u_{1}){\cal
I}_{\Delta}(-m-m_{1},\tilde{\eta}u^{-1}u_{1}^{-1})\big{|}_{\tilde{\eta}\rightarrow(-\eta
q^{1/2})^{\nu/2}}\;.\end{split}$ (92)

In the middle, we changed the integral variable $u$ to $u(-q^{1/2})^{1/2}$,
which corresponds to adjusting the mixing between the $U(1)$ R-symmetry and
the gauge $U(1)$ symmetry, i.e.,

$\displaystyle R_{\nu}\rightarrow\tilde{R}_{\nu}:=R_{\nu}+\frac{1}{2}G\;,$
(93)

where $G$ is the gauge charge. Since the index counts gauge-invariant
operators, it remains unaffected by the mixing. In practice, the last
expression is much easier to handle using Mathematica. The index depends on
the following parameters:

$\displaystyle\begin{split}&(m_{1},u_{1})/(m_{2},u_{2})\;:\;\textrm{(monopole
flux, fugacity) for the the Cartans of $SU(2)_{L}/SU(2)_{R}$}\;,\\\
&(\eta,\nu)\;:\;\textrm{fugacity and R-symmetry mixing parameter for the
$U(1)_{A}$ symmetry}\;.\end{split}$ (94)

Here the tetrahedron index ${\cal I}_{\Delta}(m,u)$ is defined as
Dimofte:2011py

$\displaystyle\begin{split}&{\cal
I}_{\Delta}(m,u):=\prod_{r=0}^{\infty}\frac{1-q^{r-\frac{1}{2}m+1}u^{-1}}{1-q^{r-\frac{1}{2}m}u}=\sum_{e\in\mathbb{Z}}{\cal
I}_{\Delta}^{c}(m,e)u^{e},\;\;\\\ &\textrm{where }{\cal
I}_{\Delta}^{c}(m,e)=\sum_{n=\lfloor
e\rfloor}^{\infty}\frac{(-1)^{n}q^{\frac{1}{2}n(n+1)-(n+\frac{1}{2}e)m}}{(q)_{n}(q)_{n+e}}\;.\end{split}$
(95)

It computes the generalized superconformal index of the ${\cal T}_{\Delta}$
theory with the R-charge choice $R(\Phi)=0$ where $(m,u)$ are (background
monopole flux, fugacity) for the $U(1)$ flavor symmetry. At general $R$-charge
choice, the index becomes ${\cal I}_{\Delta}(m,u(-q^{1/2})^{R(\Phi)})$. As a
consistency check for the formula, one can confirm the following self-mirror
property in $q$-expansion

$\displaystyle{\cal I}^{\rm
sci}_{T[SU(2)]}(m_{1},u_{1},m_{2},u_{2};\eta,\nu)={\cal I}^{\rm
sci}_{T[SU(2)]}(m_{2},u_{2},m_{1},u_{1};\eta^{-1},-\nu)\;.$ (96)

Using the $T[SU(2)]$ index, the superconformal index of the $D(\vec{k})$
theory can be computed as follows:

$\displaystyle\begin{split}&{\cal I}^{\rm sci}_{D(\vec{k})}(\eta,\nu)\\\
&=\sum_{m_{1},\ldots,m_{m}\in\mathbb{Z}_{\geq
0}}\oint\prod_{I=1}^{\sharp}\left(\frac{du_{I}}{2\pi
iu_{I}}\Delta(m_{I},u_{I})u_{I}^{2k^{(I)}m_{I}}\right)\left(\prod_{I=1}^{\sharp-1}{\cal
I}^{\rm
sci}_{T[SU(2)]}(m_{I},u_{I},m_{I+1},u_{I+1};\eta,\nu)\right)\;.\end{split}$
(97)

Here $\Delta(m,u)$ is the contribution from a $SU(2)$ vector multiplet

$\displaystyle\begin{split}&\Delta(m,u)=\frac{1}{\textrm{Sym}(m)}(q^{m/2}u-q^{-m/2}u^{-1})(q^{m/2}u^{-1}-q^{-m/2}u)\\\
&\textrm{with }\textrm{Sym}(m):=\begin{cases}2,\;m=0\\\ 1,\;m\neq
0\end{cases}\;.\end{split}$ (98)

One can also check the following

$\displaystyle\begin{split}&{\cal I}^{\rm sci}_{T[SU(2)]/(SU(2)_{R})_{\pm
1}}(m_{1},u_{1};\eta,\nu)\\\ &=\sum_{m_{2}\in\mathbb{Z}_{\geq
0}}\oint\frac{du_{2}}{2\pi iu_{2}}\Delta(m_{2},u_{2})u_{2}^{\pm 2m_{2}}{\cal
I}^{\rm sci}_{T[SU(2)]}(m_{1},u_{1},m_{2},u_{2};\eta,\nu)=u_{1}^{\mp
2m_{1}}\;.\end{split}$ (99)

It provides a non-trivial check for the IR duality corresponding to the first
move in Figure 4.

#### Twisted partition functions Benini:2015noa ; Benini:2016hjo ;
Closset:2016arn ; Closset:2018ghr

Now let us compute the twisted partition functions ${\cal Z}^{{\cal M}_{g,p}}$
(12) of the $D(\vec{k})$ theory. For the computation, we first consider the
integrand of the squashed 3-sphere partition function (87) in an asymptotic
limit $\hbar\rightarrow 0$ Gang:2019jut :

$\displaystyle\begin{split}&\log{\cal
I}^{\hbar}_{D(\vec{k})}\xrightarrow{\quad\hbar\rightarrow
0\quad}\frac{1}{\hbar}{\cal W}^{(\vec{k})}_{0}+{\cal
W}^{(\vec{k})}_{1}+\ldots\;,\\\ &{\cal
W}_{0}^{(\vec{k})}(\vec{Z},\vec{M};M,\nu)=\left(\sum_{I=1}^{\sharp}(\pm 2\pi
iM_{I}+k^{(I)}M_{I}^{2})\right)+\sum_{I=1}^{\sharp-1}{\cal
W}_{0}^{T[SU(2)]}(Z_{I},M_{I},M_{I+1};M,\nu)\;,\\\ &\textrm{where }{\cal
W}_{0}^{T[SU(2)]}(Z,M_{1},M_{2};M,\nu)=Z^{2}+M_{1}^{2}+2M_{2}Z+\textrm{Li}_{2}(e^{M+i\pi\nu})\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\sum_{\epsilon_{1},\epsilon_{2}\in\\{\pm
1\\}}\textrm{Li}_{2}(-e^{-\epsilon_{1}Z-\epsilon_{2}M_{1}-\frac{(M+i\pi(\nu-1))}{2}})\;,\\\
&{\cal
W}^{(\vec{k})}_{1}(\vec{Z},\vec{M};M,\nu)=\sum_{I=1}^{\sharp}\log(\sinh(M_{I}))+\sum_{I=1}^{\sharp-1}{\cal
W}_{1}^{T[SU(2)]}(Z_{I},M_{I},M_{I+1};M,\nu)\textrm{ with }\\\ &{\cal
W}_{1}^{T[SU(2)]}(Z,M_{1},M_{2})=-\frac{\nu}{2}\log(1+e^{M+i\pi\nu})\\\
&\qquad\qquad\qquad\qquad\qquad\;\;+\frac{\nu-1}{4}\sum_{\epsilon_{1},\epsilon_{2}\in\\{\pm
1\\}}\log(1+e^{-\epsilon_{1}Z-\epsilon_{2}M_{1}-\frac{(M+i\pi(\nu-1))}{2}})\;.\end{split}$
(100)

We use the following asymptotic behavior of Q.D.L,

$\displaystyle\log\psi_{\hbar}(Z)\xrightarrow{\quad\hbar\rightarrow
0\quad}\frac{\textrm{Li}_{2}(e^{-Z})}{\hbar}-\frac{1}{2}\log(1-e^{-Z})+\ldots\;.$
(101)

Then, the Bethe-vacua of the $D(\vec{k})$ theory are obtained as follows
$(I=1,\ldots,(\sharp-1)$ while $J=1,\ldots,\sharp)$

$\displaystyle\begin{split}&{\cal S}^{\rm
BE}_{D(\vec{k})}(M,\nu)=\big{\\{}\vec{z},\vec{m}\;:\;\exp(\partial_{Z_{I}}{\cal
W}_{0}^{(\vec{k})})\big{|}_{*}=\exp(\partial_{M_{J}}{\cal
W}_{0}^{(\vec{k})})\big{|}_{*}=1,\;m_{J}^{2}\neq 1\big{\\}}/\mathbf{W}\;,\\\
&\textrm{with }*:Z_{I}\rightarrow\log z_{I},\;M_{J}\rightarrow\log
m_{J}\;.\end{split}$ (102)

Here $\mathbf{W}$ denotes the Weyl subgroup $\mathbb{Z}_{2}^{\sharp}$ of the
$SU(2)^{\sharp}$ gauge symmetry, which acts on the Bethe-vacua as

$\displaystyle\mathbf{W}\;:\;m_{J}\rightarrow 1/m_{J}\;\;\textrm{for each
$J=1,\ldots,\sharp$}\;.$ (103)

Handle gluing ${\cal H}$ and fibering operator ${\cal F}$ of the $D(\vec{k})$
theory are ($\vec{X}:=(\vec{Z},\vec{M})$)

$\displaystyle\begin{split}&{\cal
H}_{D(\vec{k})}(\vec{z},\vec{m};M,\nu)=\frac{e^{i\delta}}{|\mathbf{W}|^{2}}\left(\det_{A,B}\left(\partial_{X_{A}}\partial_{X_{B}}{\cal
W}^{(\vec{k})}_{0}\right)\right)\exp\left(-2{\cal
W}_{1}^{(\vec{k})}\right)\bigg{|}_{*}\;,\\\ &{\cal
F}_{D(\vec{k})}(\vec{z},\vec{m};M,\nu)=\exp\left(-\frac{{\cal
W}^{(\vec{k})}_{0}-\vec{X}\cdot\partial_{\vec{X}}{\cal
W}^{(\vec{k})}_{0}-M\partial_{M}{\cal W}^{(\vec{k})}_{0}}{2\pi
i}\right)\bigg{|}_{*}\;.\end{split}$ (104)

Here $|\mathbf{W}|=2^{\sharp}$ and $e^{i\delta}$ is a phase factor which is
sensitive to the subtle overall phase factor in (82). We fix the phase
ambiguity by requiring that

$\displaystyle\mathcal{Z}^{{\cal M}_{g=0,p=0}}_{D(\vec{k})}(M=0,\nu=\pm
1)=\sum_{(\vec{z},\vec{m})\in{\cal S}_{\rm BE}}{\cal
H}_{D(\vec{k})}^{-1}=1\;.$ (105)

For $D(\vec{k})$ theory with $\vec{k}=(k_{1},k_{2})$, the twisted partition
functions are studied in Gang:2021hrd . There are $2\times(|k_{1}k_{2}-1|-1)$
Bethe-vacua whose handle gluing operators

$\displaystyle\\{{\cal
H}_{D(\vec{k}=(k_{1},k_{2}))}(\vec{z},\vec{m})\;:\;{(\vec{z},\vec{m})\in{\cal
S}_{\rm BE}}(M=0,\nu=\pm
1)\\}=\bigg{\\{}\frac{|k_{1}k_{2}-1|}{\sin^{2}\left(\frac{\pi
n}{k_{1}k_{2}-1}\right)}^{\otimes 2}\bigg{\\}}_{n=1}^{|k_{1}k_{2}-1|-1}\;.$
(106)

Further, one can check that the set $\mathbf{HF}:=\\{({\cal H}^{-1/2},{\cal
F})\;:\;(\vec{z},\vec{m})\in{\cal S}_{\rm BE}\\}$ of
$D(\vec{k}=(k_{1},k_{2}))$ has following factorization properties

$\displaystyle\mathbf{HF}=\widetilde{\mathbf{HF}}\times\begin{cases}\\{(\frac{1}{2},1),(\frac{1}{2},1),(\frac{1}{2},1),(\frac{1}{2},-1)\\},\quad&\textrm{$k_{1}$
and $k_{2}$ are both even}\;,\\\
\\{(\frac{1}{2},1),(\frac{1}{2},1),(\frac{1}{2},i),(\frac{1}{2},i^{-1})\\},&\textrm{one
of them is odd}\\\ \\{(\frac{1}{\sqrt{2}},1),(\frac{1}{\sqrt{2}},i)\\}\textrm{
or }\\{(\frac{1}{\sqrt{2}},1),(\frac{1}{\sqrt{2}},i^{-1})\\},&\textrm{both are
odd}\end{cases}$ (107)

We will understand the above factorization pattern by analyzing the decoupled
TQFT ${\rm TFT}[\vec{k}]$ in the next section, .

##  Appendix B Decoupled TQFT $\textrm{TFT}(\vec{k})$

From the ’t Hooft anomaly (46) of the $(\mathbb{Z}_{2})^{\sharp}$ 1-form
symmetry in $D(\vec{k})$, we expect that the $D(\vec{k})$ theory in the IR
contains a decoupled topological field theory which has the same anomaly. The
simplest choice is the $U(1)^{\sharp}_{{\cal K}}$ theory with the mixed CS
term ${\cal K}$ of the following form:

$\displaystyle\frac{{\cal K}}{2}=(\overline{{\cal K}}\textrm{ in \eqref{det of
Kbar}})\textrm{ (mod $2$)}\;.$ (108)

This implies that

$\displaystyle\begin{split}&\det({\cal K}/2)=\det(\overline{{\cal
K}})=|p|\;(\textrm{mod $2$})\;,\\\ &\Rightarrow|\det({\cal
K})|=\begin{cases}\in 2^{\sharp}\times(2\mathbb{Z}_{\geq 0}+1),&\textrm{odd
$p$}\\\ \in 2^{\sharp}\times(2\mathbb{Z}_{\geq 0}),&\textrm{even
$p$}\end{cases}\end{split}$ (109)

On the other hand, for the $U(1)^{\sharp}_{{\cal K}}$ theory $(\textrm{when
${\cal K}$ is non-degenerate})$

$\displaystyle\begin{split}&\det({\cal K})=(\textrm{the number of ground
states on $\mathbb{T}^{2}$, i.e. Bethe-vacua})\\\ &\leq
2^{\sharp}\;.\end{split}$ (110)

since all the Bethe-vacua of the decoupled TQFT are connected to each other by
an action of the $(\mathbb{Z}_{2})^{\sharp}$ 1-form symmetry. Combining (109)
and (110), the $\det{\cal K}$ is determined as follows:

$\displaystyle\det({\cal K})=\begin{cases}2^{\sharp},&\textrm{odd $p$}\\\
0,&\textrm{even $p$}\end{cases}$ (111)

####  $\sharp=2$ and $k_{1},k_{2}\in 2\mathbb{Z}$

In the case, $p$ is always odd and the possible candidates for ${\cal K}$ are
($n\in\mathbb{Z}$)

$\displaystyle{\cal K}=\begin{pmatrix}4n&\pm 2\\\ \pm 2&0\end{pmatrix}\textrm{
or }\begin{pmatrix}0&\pm 2\\\ \pm 2&4n\end{pmatrix}$ (112)

But all the ${\cal K}$s are actually equivalent to

$\displaystyle{\cal K}\simeq\begin{pmatrix}0&2\\\ 2&0\end{pmatrix}\;.$ (113)

Here ${\cal K}\simeq{\cal K}^{\prime}$ is an equivalence between
$U(1)^{\sharp}_{\cal K}$ and $U(1)^{\sharp}_{{\cal K}^{\prime}}$ up to a
following redefinition of gauge fields $\vec{A}$

$\displaystyle\vec{A}\rightarrow M\cdot\vec{A}\textrm{ with }M\in
GL(2\sharp,\mathbb{Z})\;.$ (114)

We choose $GL(2\sharp,\mathbb{Z})$ instead of $GL(2\sharp,\mathbb{R})$ to
preserve the monopole charge quantization. More explicitly, the equivalence
relation is

$\displaystyle{\cal K}\simeq{\cal K}^{\prime}\textrm{ if }{\cal
K}^{\prime}=M^{T}{\cal K}M\textrm{ with a $M\in GL(2\sharp,\mathbb{Z})$}\;.$
(115)

So, in the case, the decoupled TQFT is uniquely determined

$\displaystyle\textrm{TFT}[(k_{1},k_{2})]\simeq U(1)^{2}_{{\cal K}}\textrm{
with }{\cal K}=\begin{pmatrix}0&2\\\ 2&0\end{pmatrix}\;,$ (116)

which is the toric-code TQFT. It has the set of
$\mathbf{HF}:=\\{|S_{0\alpha}|,\exp(2\pi ih_{\alpha})\\}$ as follows 666For
the $U(1)^{\sharp}_{\cal K}$ theory with non-degenerate ${\cal K}$, there are
$\det{\cal K}$ simple objects (or Bethe-vacua), $\alpha=0,\ldots,\det{\cal
K}-1$, which are solutions of the Bethe equations, ${\cal S}_{\rm
BE}:=\\{\vec{z}\;:\prod_{I=1}^{\sharp}z_{I}^{{\cal K}_{IJ}}=1,\;\textrm{ for
}J=1,\ldots,\sharp\\}$. The $S_{0\alpha}$ of Bethe-vacuum
$\vec{z}_{\alpha}\in{\cal S}_{\rm BE}$ is $1/\sqrt{|\det{\cal K}|}$ for all
$\alpha$ and the $e^{2\pi ih_{\alpha}}$ is given by the fibering operator
${\cal F}=\exp(\frac{\sum_{I,J}{\cal K}_{IJ}Z_{I}Z_{J}}{4\pi i})$ with
$\vec{Z}=\log\vec{z}_{\alpha}$.

$\displaystyle(\mathbf{HF}\textrm{ of
}\textrm{TFT}[(k_{1},k_{2})])=\bigg{\\{}(\frac{1}{2},1)^{\otimes
3},(\frac{1}{2},-1)\bigg{\\}}\;,$ (117)

which explains the corresponding factorization property in (107).

####  $\sharp=2$ and $k_{1}\neq k_{2}\;(\textrm{mod }2)$

In the case, $p$ is always odd and there is only one consistent choice of
${\cal K}$ up to the equivalence, which is

$\displaystyle\textrm{TFT}[(k_{1},k_{2})]\simeq U(1)^{2}_{{\cal K}}\textrm{
with }{\cal K}=\begin{pmatrix}2&2\\\ 2&0\end{pmatrix}\;,$ (118)

whose $\mathbf{HF}:=\\{|S_{0\alpha}|,\exp(2\pi ih_{\alpha})\\}$ is

$\displaystyle(\mathbf{HF}\textrm{ of
}\textrm{TFT}[(k_{1},k_{2})])=\bigg{\\{}(\frac{1}{2},1)^{\otimes
2},(\frac{1}{2},i),(\frac{1}{2},\frac{1}{i})\bigg{\\}}\;,$ (119)

which explains the corresponding factorization property in (107).

####  $\sharp=2$ and $k_{1},k_{2}\in 2\mathbb{Z}+1$

In the case, $p$ is always even and the possible candidates for the ${\cal K}$
are

$\displaystyle{\cal K}\simeq 2\begin{pmatrix}a&a\\\ a&a\end{pmatrix}\simeq
2\begin{pmatrix}0&0\\\ 0&a\end{pmatrix}\textrm{ with $a\in 2\mathbb{Z}+1$}\;.$
(120)

Ignoring the 1st gauge field $A_{1}$ which does not appear in the action, the
decoupled TQFT is nothing but the $U(1)_{2a}$ theory. From the constraint in
(110), which is $|2a|\leq 2^{2}$ for our case, the possible values of $a$ are
$\pm 1$. Thus, the decoupled TQFT is $U(1)_{\pm 2}$ theory whose
$\mathbf{HF}:=\\{|S_{0\alpha}|,\exp(2\pi ih_{\alpha})\\}$ is

$\displaystyle(\mathbf{HF}\textrm{ of
}\textrm{TFT}[(k_{1},k_{2})])=\bigg{\\{}(\frac{1}{\sqrt{2}},1),(\frac{1}{\sqrt{2}},i^{\pm
1})\bigg{\\}}\;,$ (121)

which explains the corresponding factorization property in (107).

For higher values of $\sharp\geq 3$, the analysis becomes more complicated,
and we could not uniquely determine ${\cal K}$. From the factorization
properties of the set $\mathbf{HF}$ for several $D(\vec{k})$s, we observe that

$\displaystyle(\textrm{the number of Bethe-vacua of the decoupled
TQFT})=\begin{cases}2^{\sharp},&\textrm{odd }p\\\ 2^{\sharp-1},&\textrm{even
}p\end{cases}$ (122)

For odd $p$, this is compatible with (110) and (111). For abelian CS theory,
it is generally true that

$\displaystyle(\textrm{3-sphere partition function})=1/\sqrt{(\textrm{the
number of Bethe-vacua})}\;.$ (123)

Thus, from (122), we have

$\displaystyle\begin{split}&|(\mathcal{Z}^{\rm con}\textrm{ of
}\textrm{TFT}(\vec{k}))|=\begin{cases}2^{-\sharp/2},&p\in 2\mathbb{Z}+1\\\
2^{(1-\sharp)/2},&p\in 2\mathbb{Z}\end{cases}\end{split}$ (124)

Combined with (90), we obtain

$\displaystyle\begin{split}&|(\mathcal{Z}^{\rm con}\textrm{ of }{\cal
D}(p,q))|=\frac{|(\mathcal{Z}^{\rm con}\textrm{ of
}D(\vec{k}))|}{|(\mathcal{Z}^{\rm con}\textrm{ of
}\textrm{TFT}(\vec{k}))|}=\begin{cases}\frac{2}{\sqrt{|p|}}\sin\left(\frac{\pi}{|p|}\right),&p\in
2\mathbb{Z}+1\\\ \sqrt{\frac{2}{|p|}}\sin\left(\frac{\pi}{|p|}\right),&p\in
2\mathbb{Z}\end{cases}\end{split}$ (125)

Note that the partition function is independent on the choice of $\vec{k}$ as
expected from $i)$ in (48).

For even $p$, a $\mathbb{Z}_{2}$ subgroup of the UV
$(\mathbb{Z}_{2})^{\sharp}$ 1-form symmetry is absent in the decoupled
$\textrm{TFT}(\vec{k})$, which captures the anomaly of the 1-form symmetry.
This implies that

$\displaystyle\textrm{For }p\in 2\mathbb{Z},\;\;{\cal D}(p,q)\textrm{ has a
non-anomalous $\mathbb{Z}_{2}$ 1-form symmetry},$ (126)

which can be identified with the absent $\mathbb{Z}_{2}$ subgroup .

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