# Non-unitary TQFTs from 3D $\mathcal{N}=4$ rank 0 SCFTs

Dongmin Gang c Sungjoon Kim d Kimyeong Lee e Myungbo Shim f Masahito Yamazaki
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>Department of Physics and
Astronomy $\&$ Center for Theoretical Physics,
Seoul National University, 1 Gwanak-ro, Seoul 08826, KoreaAsia Pacific Center
for Theoretical Physics (APCTP), Pohang 37673, KoreaDepartment of Physics,
Pohang University of Science and Technology (POSTECH),
Pohang 37673, Republic of KoreaKorea Institute for Advanced Study, 85 Hoegiro,
Dongdaemun-Gu, Seoul 02455, KoreaDepartment of Physics and Research Institute
of Basic Science,
Kyung Hee University, Seoul 02447, KoreaKavli Institute for the Physics and
Mathematics of the Universe (WPI),
University of Tokyo, Kashiwa, Chiba 277-8583, Japan

###### Abstract

We propose a novel procedure of assigning a pair of non-unitary topological
quantum field theories (TQFTs), TFT${}_{\pm}[{\cal T}_{\rm rank\;0}]$, to a
(2+1)D interacting ${\cal N}=4$ superconformal field theory (SCFT) ${\cal
T}_{\rm rank\;0}$ of rank 0, i.e. having no Coulomb and Higgs branches. The
topological theories arise from particular degenerate limits of the SCFT.
Modular data of the non-unitary TQFTs are extracted from the supersymmetric
partition functions in the degenerate limits. As a non-trivial dictionary, we
propose that
$F=\max_{\alpha}\left(-\log|S^{(+)}_{0\alpha}|\right)=\max_{\alpha}\left(-\log|S^{(-)}_{0\alpha}|\right)$,
where $F$ is the round three-sphere free energy of ${\cal T}_{\rm rank\;0}$
and $S^{(\pm)}_{0\alpha}$ is the first column in the modular S-matrix of TFT±.
From the dictionary, we derive the lower bound on $F$,
$F\geq-\log\left(\sqrt{\frac{5-\sqrt{5}}{10}}\right)\simeq 0.642965$, which
holds for any rank 0 SCFT. The bound is saturated by the minimal ${\cal N}=4$
SCFT proposed by Gang-Yamazaki, whose associated topological theories are both
the Lee-Yang TQFT. We explicitly work out the (rank 0 SCFT)/(non-unitary
TQFTs) correspondence for infinitely many examples.

##  1 Introduction

Supersymmetric quantum field theories with 8 supercharges (8 $Q$s) provide a
fertile ground for many interesting research topics connecting various areas
in theoretical and mathematical physics. For example, Seiberg-Witten’s
approach Seiberg:1994rs  to 4-dimensional (4D) ${\cal N}=2$ supersymmetric
gauge theories provides analytic understanding of non-perturbative phenomena,
such as confinement, in strongly coupled gauge theories. 4D ${\cal N}=2$
superconformal field theories can be geometrically constructed using wrapped
M5-branes in M-theory, and the 4D-2D correspondence connects physics of 4D
supersymmetric field theories with mathematical structures on 2D Riemann
surfaces in an unexpected way Gaiotto:2009we ; Alday:2009aq ; Gaiotto:2009hg .
Interestingly, there exist non-trivial superconformal field theories with 8
supercharges in higher dimensional space-time, 5D and 6D, as predicted by
String/M-theory Witten:1995zh ; Seiberg:1996bd ; Intriligator:1997pq . More
recently, it is found that 2D chiral algebras (resp. 1D topological quantum
mechanics) appear as protected subsectors of 4D ${\cal N}=2$ (resp. 3D ${\cal
N}=4$) superconformal field theories Beem:2013sza ; Chester:2014fya ;
Chester:2014mea ; Beem:2016cbd ; Dedushenko:2016jxl ; Dedushenko:2017avn ;
Panerai:2020boq .

Extended SUSY gauge theories have rich structures in their vacuum moduli
space, and one natural invariant is the rank, i.e. the complex dimension of
the Coulomb branch. There have been numerous efforts in classifying SCFTs with
8 $Q$s for a given low rank in various space-time dimensions Chang:2019dzt ;
Argyres:2020nrr ; Heckman:2013pva ; Heckman:2015bfa ; Bhardwaj:2015xxa ;
Jefferson:2017ahm ; Jefferson:2018irk ; Bhardwaj:2018yhy ; Bhardwaj:2018vuu ;
Apruzzi:2019opn ; Bhardwaj:2020gyu . For 3D ${\cal N}=4$ SCFTs, however, the
rank is in general not a duality-invariant concept since the Coulomb and Higgs
branches are exchanged under the 3D mirror symmetry Intriligator:1996ex . For
this reason, we hereby modify the definition of rank as the maximum of the
dimension of Coulomb branch and that of the Higgs branch. Another peculiar
fact about 3D ${\cal N}=4$ theories is that there exist non-trivial
interacting SCFTs of rank 0, as studied by two of the authors of this present
paper in Gang:2018huc . This is in contrast with the case of $D\geq 4$, where
it is often implicitly assumed that there is no non-trivial interacting rank 0
SCFTs with 8 $Q$s, so that the classification program starts with rank 1.
(Recently, 4D/5D rank 0 SCFTs were found through a geometrical engineering but
it is yet unclear if they are interacting SCFTs Closset:2020scj .) Note that
most of the classification schemes in previous studies do not work for rank 0
cases since the existence of Coulomb or Higgs branch operators is an crucial
assumption in the analysis.

In this paper, we initiate the classification of rank 0 3D ${\cal N}=4$ SCFTs
by establishing the following correspondence:

$\displaystyle\begin{split}&\textrm{3D ${\cal N}=4$ superconformal field
theories of rank 0 }\\\ &\longleftrightarrow\textrm{A pair of 3D non-unitary
topological quantum field theories (TQFTs)}\;.\end{split}$ (1)

The non-unitary TQFTs emerge at particular choices of non-superconformal
R-symmetry, $\nu=\pm 1$ in (5), of rank 0 ${\cal N}=4$ SCFTs. In the limits,
due to huge Bose/Fermi cancellations the unrefined superconformal index gets
contributions only from Coulomb-branch or Higgs-branch operators and their
descendants. For rank 0 theories, the index becomes trivial (i.e. $1$) since
there are no non-trivial Coulomb/Higgs branch operators. Other partition
functions on various rigid supersymmetric Euclidean backgrounds also
drastically simplify in the degenerate limits for rank 0 theories. Our
correspondence says that the simplified partition functions are actually equal
to the partition functions of corresponding non-unitary TQFTs on the same 3D
spacetime. (See (13) for a precise statement.) Concrete dictionaries of the
correspondence are given in Table 1. In the degenerate limits, as seen in the
superconformal index case, contributions from local operators become
unimportant for rank 0 theories and only non-local loop operators become
relevant physical observables. Similarly, loop operators are the only physical
observables in general TQFTs. Using the correspondence, one can map the
problem of classifying rank 0 SCFTs to the classification of non-unitary
TQFTs, which is much easier to handle. Mathematically, TQFTs are described by
modular tensor categories (MTCs) and classification of MTCs has been studied
intensively in the literature turaev1992modular ; rowell2006quantum ;
rowell2009classification ; wen2016theory ; bruillard2016rank ; Cho:2020ljj .
The most basic quantity characterizing a 3D CFT is the round three-sphere free
energy, usually denoted as $F$. The $F$ always monotonically decreases under
the renormalization group (RG) flow and thus is regarded as a proper measure
of the degrees of freedom of 3D CFT Jafferis:2011zi ; Liu:2012eea ;
Casini:2012ei . In one of the most interesting and surprising dictionaries of
the correspondence, the $F$ of a rank 0 CFT is related to the modular S-matrix
of a non-unitary TQFT in a very simple way as given in Table 2. Combining the
dictionary and universal algebraic properties of the S-matrix, we derive
following lower bound on $F$

$\displaystyle F\geq-\log\left(\sqrt{\frac{5-\sqrt{5}}{10}}\right),$ (2)

which should hold for any rank 0 SCFTs. Interestingly, the lower bound is
saturated by the minimal theory studied in Gang:2018huc .

The correspondence is similar in spirit with the (4D ${\cal N}=2$ SCFT)/(2D
chiral algebra) correspondence mentioned above. In both correspondences, non-
unitary algebraic structures, chiral algebras on the one hand and modular
tensor category on the other, appear as protected subsectors of unitary
superconformal field theories. But there are several crucial differences.
First, two theories in our correspondence are defined on the same 3D space-
time while the 2D chiral algebra lives in the 2D subspace of 4D space-time of
the SCFT. Secondly, basic physical objects are BPS local operators in the (4D
SCFTs)/(2D chiral algebra) story while BPS non-local loop operators are basic
objects in our correspondence. That non-local loop operators play crucial
roles can be a great advantage of our classification approach over the
conventional conformal bootstrap approaches, since the latter are based on
correlation functions of local operators. We note that the 3D non-unitary
TQFTs are sensitive to the global structure of the 3D rank 0 SCFTs and two
theories in the correspondence share the same one-form symmetry as well as
their ’t Hooft anomalies. In 3D, the quantity $F$ (unlike the stress-energy
tensor central charge $C_{T}$) is sensitive to the global structure of CFT and
the conformal bootstrap approach never give a constraint on $F$ but only on
$C_{T}$, which is not a proper measure of the degrees of freedom in a strict
sense Nishioka:2013gza .

The remaining part of paper is organized as follows. In section 2, we present
the precise statement of the correspondence with several concrete
dictionaries. As an application of the correspondence, we derive interesting
lower bounds on $F$ for rank 0 SCFTs. In section 3, we explicitly work out the
correspondence in detail with several classes of infinitely many rank 0 SCFTs.
The results are summarized in Table 2. In Appendix A, we give brief reviews on
supersymmetric partition functions of 3D ${\cal N}\geq 3$ gauge theories and
modular data of 3D TQFTs which are basic ingredients of the dictionaries. In
other Appendices, we collect technical details and supplementary materials.

##  2 (Rank 0 SCFT)/(Non-unitary TQFTs) correspondence

In this section, we establish a correspondence between

$\displaystyle\begin{split}&{\cal T}_{\rm rank\;0}\;:\;\textrm{a 3D
$\mathcal{N}=4$ interacting SCFT with empty Coulomb and Higgs branches}\\\
&\xrightarrow{\qquad}{\rm TFT}_{\pm}[{\cal T}_{\rm rank\;0}]\;:\;\textrm{a
pair of 3D non-unitary TQFTs}\;.\end{split}$ (3)

The basic dictionaries for the correspondence are summarized in Table 1.

###  2.1 Non-unitary TQFTs from ${\cal N}=4$ SCFTs of rank 0

##### 3D Rank 0 ${\cal N}=4$ SCFTs

In this paper, 3D ${\cal N}=4$ rank 0 SCFT is defined as

$\displaystyle(\textrm{Rank 0 theory}):=(\textrm{Theory with no Coulomb and
Higgs branches})\;.$ (4)

3D $\mathcal{N}=4$ SCFTs have $SO(4)\simeq SU(2)_{L}\times SU(2)_{R}$
R-symmetry. The Coulomb (Higgs) branch is parametrized by chiral primary
operators charged under the $SU(2)_{R}$ ($SU(2)_{L}$) symmetry while neutral
under the $SU(2)_{L}$ ($SU(2)_{R}$) symmetry. In our definition, the rank 0
theory can have mixed branches parametrized by chiral primaries charged under
both of $SU(2)_{L}$ and $SU(2)_{R}$. Rank 0 SCFTs in Section 3 with ${\cal
N}=5$ supersymmetry actually have the mixed branches. Rank 0 theories cannot
have a continuous flavor symmetry commuting with the $SO(4)$ R-symmetry, since
a flavor current multiplet contains Higgs- or Coulomb- branch operators. By
the same reasoning, the rank 0 theories cannot have SUSY more than ${\cal
N}=5$.

##### Axial $U(1)$ symmetry and R-symmetry mixing

Let $R_{\nu}$ and $A$ be the two Cartan generators of the $SO(4)$ R-symmetry:

$\displaystyle\begin{split}&R_{\nu=0}:=R+R^{\prime}\;,\quad
A:=R-R^{\prime}\;,\\\ &R_{\nu}:=R_{\nu=0}+\nu A\;.\end{split}$ (5)

Here $R$ and $R^{\prime}$ are the Cartans of $SU(2)_{L}$ and $SU(2)_{R}$
respectively. In our convention, they are normalized as
$R,R^{\prime}\in\frac{1}{2}\mathbb{Z}$. In terms of an $\mathcal{N}=2$
subaglebra, $R_{\nu}$ is the R-charge while $A$ is the charge of a $U(1)$
flavor symmetry (commuting with the ${\cal N}=2$ supersymmetry) called the
axial $U(1)$ symmetry. The mixing between the $U(1)$ R-symmetry and the axial
flavor symmetry is parametrized by $\nu$. The IR superconformal R-symmetry
corresponds to $\nu=0$.

##### Supersymmetric partition functions

Generally, the partition function ${\cal Z}^{\mathbb{B}}_{{\cal
T}}(b^{2},m,\nu;s)$ of a 3d ${\cal N}=4$ SCFT ${\cal T}$ on a rigid
supersymmetric background $\mathbb{B}$ depends on the followings:

$\displaystyle\begin{split}{\cal M}&:\;\textrm{topology of $\mathbb{B}$}\;,\\\
b^{2}\;(\textrm{or }q)&:\;\textrm{squashing (or $\Omega$-deformation)
parameter }\;,\\\ m\;(\textrm{or }\eta=e^{m})\;&:\;\textrm{real mass (or
fugacity variable) for axial $U(1)$ symmetry}\;,\\\
\nu\;&:\;\textrm{R-symmetry mixing parameter in \eqref{R-symmetry
mixing}}\;,\\\ s\in H^{1}({\cal M},\mathbb{Z}_{2})\;&:\;\textrm{SUSY-
compatible spin-structure}\;.\end{split}$ (6)

We consider supersymmetric backgrounds $\mathbb{B}$ whose topologies are given
as ${\cal M}_{g,p}$, a degree $p$ bundle over a genus $g$ Riemann surface
$\Sigma_{g}$:

(11)

We refer readers to Appendix A.1 for a brief review on localizations on
supersymmetric backgrounds. We can turn on the $\Omega$-deformation parameter
(sometimes called squashing parameter) only for $g=0$. For even $p$, one can
consider two supersymmetric backgrounds depending on the choice of the spin
structure along the fiber $[S^{1}]\in H_{1}({\cal M},\mathbb{Z}_{2})$. We
denote $s=+1$ ($s=-1$) for the periodic (anti-periodic) boundary condition for
fermionic fields along the $S^{1}$. For $p=0$, the partition function can be
regarded as a version of the BPS index ${\cal I}(s)$ and its spin structure
dependence can be interpreted as

$\displaystyle{\cal I}(s)=\begin{cases}\textrm{Tr}_{{\cal H}_{\rm
BPS}}(-1)^{R_{\nu}},\quad\textrm{for }s=-1\;,\\\ \textrm{Tr}_{{\cal H}_{\rm
BPS}}(-1)^{2j_{3}},\quad\textrm{for }s=1\;.\end{cases}$ (12)

As BPS indices, they can be defined without overall phase factor ambiguities.
For $p\neq 0$, on the other hand, local counterterms affect the phase factor
of the partition function Closset:2012vg  and it is non-trivial to keep track
of the local counterterms. Throughout this paper, we are for the most part
interested in the absolute value of partition functions. For $g=1$ and $p=0$,
the supersymmetric partition function is independent of all the parameters and
is simply an integer number called the Witten index.

##### Emergence of non-unitary TQFT in the limits $\nu\rightarrow\pm 1$

As main result of the paper, we propose that for any rank 0 ${\cal N}=4$ SCFT
${\cal T}_{\rm rank\;0}$ we can associate a pair of non-unitary TQFTs denoted
by $\textrm{TFT}_{\pm}[{\cal T}_{\rm rank\;0}]$, satisfying the following
relation

$\displaystyle\begin{split}\textbf{Main proposal : }&{\cal
Z}^{\mathbb{B}}_{{\cal T}_{\rm
rank\;0}}\left(b^{2},m\;(\textrm{or}\;\eta),\nu;s\right)\xrightarrow{\quad
m\rightarrow 0\;(\textrm{or}\;\eta\rightarrow 1),\;\nu\rightarrow\pm
1\quad}{\cal Z}^{{\cal M}_{g,p}}_{{\rm TFT}_{\pm}[{\cal T}_{\rm
rank\;0}]}(s)\;.\end{split}$ (13)

The partition function ${\cal Z}^{{\cal M}_{g,p}}_{{\rm TFT}}$ of the
topological theory depends only on the topological structure ${\cal M}_{g,p}$
of $\mathbb{B}$ and (possibly) a choice of a spin-structure $s$ on it.111The
overall phase factor of the partition function depends also on the choice of
the framing. As with supersymmetric partition functions, there is no canonical
choice of the framing for non-zero $p$ and we mostly focus on the absolute
values of the partition functions.  We claim that in the degenerate limits i)
the rigid supersymmetry partition function becomes independent on the
squashing parameter $b^{2}$ (or $\Omega$-deformation parameter $q$) and ii) it
becomes the partition function of a non-unitary topological quantum field
theory.

We call a topological quantum field theory a non-spin (or bosonic) TQFT when
its partition function is independent on the choice of the spin structure, and
a spin (or fermionic) TQFT otherwise. We propose that

$\displaystyle\begin{split}&\textrm{ The non-unitary $\textrm{TFT}_{\pm}[{\cal
T}_{\rm rank\;0}]$ is a spin (fermionic) TQFT}\;,\\\ &\qquad\textrm{if
$R_{\nu=\pm 1}+2j_{2}\in 2\mathbb{Z}+1$ for some BPS local operators \;. }\\\
&\qquad\textrm{or equivalently,}\;\\\ &\qquad\textrm{if ${\cal I}^{\rm
sci}(q,\eta,\nu=\pm 1;s)$ contains $q^{\frac{1}{2}(\textrm{odd integer})}$
terms\;.}\end{split}$ (14)

Here ${\cal I}^{\rm sci}$ is the superconformal index defined in (163) and the
index at $\nu=\pm 1$ is in general a power series in $q^{1/2}$ since
$R_{\nu=\pm 1}+2j_{3}\in\mathbb{Z}$. From (12), we expect that the
supersymmetric partition on ${\cal M}_{g,p=0}$ depends on the spin-structure
$s$ if the above condition is met. This is because that the condition
$R_{\nu=\pm 1}+2j_{3}\in 2\mathbb{Z}+1$ implies that $(-1)^{R_{\nu=\pm
1}}\neq(-1)^{2j_{3}}$ for some BPS local operators which acts on the Hilbert-
space ${\cal H}_{\rm BPS}$ non-trivially. The above condition gives sufficient
but not necessary condition for $\textrm{TFT}_{\pm}$ to be fermionic. To see
this, consider a rank 0 SCFT ${\cal T}_{\rm rank\;0}$ not satisfying the above
condition whose associated non-unitary topological field theories,
$\textrm{TFT}_{\pm}[{\cal T}_{\rm rank\;0}]$, are bosonic. Then, ${\cal
T}_{\rm rank\;0}\otimes{\cal T}_{\rm spin\;top}$ with an unitary fermionic
topological field theory ${\cal T}_{\rm spin\;top}$ is still a rank 0 SCFT not
satisfying the above condition since the decoupled topological sector does not
contribute to the superconformal index. But its associated non-unitary TQFTs
$\textrm{TFT}_{\pm}[{\cal T}_{\rm rank\;0}\otimes{\cal T}_{\rm
spin\;top}]=\textrm{TFT}_{\pm}[{\cal T}_{\rm rank\;0}]\otimes{\cal T}_{\rm
spin\;top}$ are fermionic due to the decoupled unitary spin TQFT ${\cal
T}_{\rm spin\;top}$.

The proposal in (13) can be easily proven for the case when ${\cal
Z}^{\mathbb{B}}$ is the superconformal index (163). In the degenerate limit,
$\nu\rightarrow 1$ and $\eta=1$, the index becomes

$\displaystyle\mathcal{I}^{\rm
sci}(q,\eta,\nu;s)\xrightarrow{\;\;\eta=1,\nu\rightarrow
1\;\;}\begin{cases}\textrm{Tr}_{\mathcal{H}_{\rm
rad}(S^{2})}(-1)^{2j_{3}}q^{R+j_{3}}\;\;,\quad s=+1\;,\\\
\textrm{Tr}_{\mathcal{H}_{\rm rad}(S^{2})}(-1)^{R}q^{R+j_{3}}\;\;,\quad
s=-1\;.\end{cases}$ (15)

All unitary multiplets of 3D ${\cal N}=4$ superconformal algebra are listed in
Cordova:2016emh . From the classification, it is not difficult to check that
the index above gets contributions only from operators in a short multiplet
denoted by $B_{1}[0]^{(2R,2R^{\prime})}_{\Delta}$ with $R=0$ in
Cordova:2016emh . The bottom state inside the multiplet corresponds to a
Coulomb branch operator parametrizing the Coulomb branch. From the
superconformal index in the degenerate limit, one can actually compute the
Hilbert-series of the Coulomb branch Razamat:2014pta . Since we are
considering a rank 0 theory ${\cal T}_{\rm rank\;0}$ with empty Coulomb
branch, the index gets contributions only from the identity operator and
becomes simply ($q$-independent) $1$. Similarly, one can also confirm that the
index becomes $1$ in the other degenerate limit, $\nu\rightarrow-1$ and
$\eta=1$, since there is no Higgs branch. In summary, from the superconformal
multiplet analysis, we conclude that

$\displaystyle\mathcal{I}_{{\cal T}_{\rm rank\;0}}^{\rm
sci}(q,\eta,\nu;s)\xrightarrow{\;\;\eta=1,\nu\rightarrow\pm
1\;\;}1,\;\;\textrm{for any rank 0 theory }{\cal T}_{\rm rank\;0}\;.$ (16)

This proves the proposal in (13) for the case when ${\cal Z}^{\mathbb{B}}$=
(superconformal index) since ${\cal Z}^{S^{2}\times S^{1}}=1$ for all
topological theories.

We currently do not know the full proof of the proposal (13) for other
supersymmetric partition functions ${\cal Z}^{\mathbb{B}}$. As noticed in
Cho:2020ljj , however, the triviality of superconformal index for a non-
conformal choice of R-symmetry is a strong evidence for the appearance of non-
unitary TQFT, while the triviality of the index at the superconformal
R-symmetry implies an emergence of a unitary TQFT at the infra-red fixed
point. We explicitly test the proposal with infinitely many rank 0 SCFTs and
various supersymmetric partition functions in section 3.

###  2.2 Dictionaries

In Table 1, we summarize basic dictionaries of the correspondence. The
dictionary in the first line is the most basic one and other dictionaries
(except for the last one for $F$) follow from it. On the one hand, partition
functions (with insertion of loop operators along the fiber $S^{1}$) of a
topological field theory on the geometries ${\cal M}_{g,p}$ are determined by
the modular data, i.e. $S$ and $T$ matrices, of the topological theory. On the
other hand, the supersymmetric partition function on $\mathbb{B}$ with
topology ${\cal M}_{g,p}$ can be computed using localization technique as
briefly summarized in Appendix A.2.

$\quad\textrm{TFT}_{\pm}[{\cal T}_{\rm rank\;0}]$ | ${\cal T}_{\rm rank\;0}$
---|---
${\cal Z}^{{\cal M}_{g,p}}_{\rm TFT_{\pm}}(s)$ | BPS partition function ${\cal Z}^{\mathbb{B}}_{{\cal T}_{\rm rank\;0}}\big{|}_{\nu\rightarrow\pm 1,m=0}(s)$
| with (topology of $\mathbb{B}$) = ${\cal M}_{g,p}$
Spin or non-spin | Equation (14)
Rank $N$ | Witten index
| Bethe vacua $\\{\vec{z}_{\alpha}\\}_{\alpha=0}^{N-1}$
Simple objects | or
| BPS loop operators $\\{{\cal O}^{\pm}_{\alpha}(\vec{z})\\}_{\alpha=0}^{N-1}$
$(S^{\pm}_{0\alpha})^{-2}$ | ${\cal H}_{\alpha}(m=0,\nu\rightarrow\pm 1;s=-1)$
$T^{\pm}_{\alpha\beta}$ (only for non-spin) | $\delta_{\alpha\beta}(\frac{{\cal F}_{\alpha}}{{\cal F}_{\alpha=0}})\big{|}_{\nu\rightarrow\pm 1,m=0}$
$(T^{2})^{\pm}_{\alpha\beta}$ | $\delta_{\alpha\beta}(\frac{{\cal F}_{\alpha}}{{\cal F}_{\alpha=0}})^{2}\big{|}_{\nu\rightarrow\pm 1,m=0,s=-1}$
$S^{\pm}_{00}$ | $\bigl{|}{\cal Z}^{S^{3}_{b}}_{{\cal T}_{\rm rank\;0}}(m=0,\nu\rightarrow\pm 1)\bigr{|}$
$W^{\pm}_{\beta}(\alpha)$ | ${\cal O}^{\pm}_{\alpha}(\vec{z}_{\beta})|_{\nu\rightarrow\pm 1,m=0}$
$\max_{\alpha}(-\log|S^{\pm}_{0\alpha}|)$ |  $F$ (three-sphere free energy)
Table 1: Basic dictionaries in (rank 0 SCFT)/(non-unitary TQFTs)
correspondence. $S^{\pm}_{\alpha\beta}$ and $T^{\pm}_{\alpha\beta}$ are
modular matrices of ${\rm TFT}_{\pm}[{\cal T}_{\rm rank\;0}]$. We define
$W_{\beta}(\alpha):=\frac{S_{\alpha\beta}}{S_{0\beta}}=\langle\beta|{\cal
O}^{A}_{\alpha}|\beta\rangle$, see (212), from which one can compute S-matrix
$S_{\alpha\beta}=W_{\beta}(\alpha)W_{0}(\beta)S_{00}$. ${\cal H}_{\alpha}$ and
${\cal F}_{\alpha}$ are handle gluing and fibering operator at the $\alpha$-th
Bethe-vacuum respectively, see (200) and (203). ${\cal O}_{\alpha=0}$
corresponds to the trivial loop and $W_{\beta}(0)=1$. The rank of a TFT (not
to be confused with the rank of its associated 3D ${\cal N}=4$ SCFT) is
defined as the dimension of ${\cal H}(\mathbb{T}^{2})$ for non-spin case while
is defined as the dimension of ${\cal H}_{--}(\mathbb{T}^{2})$, i.e. Hilbert-
space in NS-NS sector, for spin case. Similarly, the $S$ and $T$ matrices
(resp. $S$ and $T^{2}$) for non-spin TQFT case (resp. spin TQFT case) are
modular matrices acting on ${\cal H}(\mathbb{T}^{2})$ (resp. ${\cal
H}_{--}(\mathbb{T}^{2})$). Simple objects are in one-to-one with a basis of
${\cal H}(\mathbb{T}^{2})$ (resp. ${\cal H}_{--}(\mathbb{T}^{2})$) for non-
spin TQFT case (resp. spin TQFT case). We refer readers to Appendix A.2 for a
general review on the modular data of non-spin and spin TQFTs.

For a ${\rm TQFT}$ its rank, i.e. the size of modular matrices, is equal to
the ground state degeneracy on the two torus. For a supersymmetric field
theory, the degeneracy is equal to the Witten index. The dictionaries for
$S_{0\alpha}^{2},T_{\alpha\beta}$ and $W_{\beta}(\alpha)$ simply follow from
comparison between (217) and (204). The Bethe-vacuum corresponding to the
trivial simple object, $\alpha=0$, can be determined by requiring that

$\displaystyle\textrm{Trivial object $\alpha=0$ : }\frac{1}{\sqrt{|{\cal
H}_{\alpha=0}(m=0,\nu\rightarrow\pm 1)|}}=S^{\pm}_{00}=|{\cal
Z}^{S^{3}_{b}}(m=0,\nu\rightarrow\pm 1)|\;.$ (17)

In topological field theories simple objects (labeled by $\alpha$) are loop
operators, while in supersymmetric field theories $\alpha$ labels the types
(gauge and flavor charge) of loop operators. Therefore, we expect that for
each Bethe-vacuum $\alpha$ there is a corresponding supersymmetric loop
operator ${\cal O}_{\alpha}(\vec{z})$, see around (204):

$\displaystyle\textrm{(Bethe vacua)-to-(loop operators) map :
}\vec{z}_{\alpha}\rightarrow{\cal O}^{\pm}_{\alpha}\;.$ (18)

The trivial object $\alpha=0$ corresponds to the identity operator. The map
will be determined by requiring the following consistency conditions

$\displaystyle\begin{split}&W^{\pm}_{0}(\alpha)=\frac{S^{\pm}_{\alpha
0}}{S^{\pm}_{00}}=\sqrt{\frac{{\cal H}_{\alpha=0}(m=0,\nu\rightarrow\pm
1)}{{\cal H}_{\alpha}(m=0,\nu\rightarrow\pm 1)}}\;\textrm{ and
}W^{\pm}_{0}(\alpha)={\cal
O}^{\pm}_{\alpha}(\vec{z}_{\alpha=0})\big{|}_{\nu\rightarrow\pm 1,m=0}\;,\\\
&\Rightarrow{\cal O}^{\pm}_{\alpha}(\vec{z}_{\alpha=0})=\sqrt{\frac{{\cal
H}_{\alpha=0}(m=0,\nu\rightarrow\pm 1)}{{\cal
H}_{\alpha}(m=0,\nu\rightarrow\pm 1)}}\;.\end{split}$ (19)

The dictionary for $F$ in the last line is one of the most non-trivial and
interesting statements in this paper. It says that

$\displaystyle\begin{split}&F[{\cal T}_{\rm rank\;0}]\\\
&=-\log|S_{0\alpha_{*}}|{\rm\;of\;TFT}_{\pm}[{\cal T}_{\rm
rank\;0}]\quad\textrm{($\alpha_{*}$ is chosen such that
$|S_{0\alpha_{*}}|\leq|S_{0\alpha}|$ for other $\alpha$)}\;.\end{split}$ (20)

$F$ is the free energy on round three-sphere, which is the quantity appearing
in the F-theorem and is a proper measure of degree of freedom. The relation is
surprising since it relates the quantity ($F$) at the superconformal point,
$\nu=0$, to the quantity ($S_{0\alpha_{*}}$) in the degenerate limits $\nu=\pm
1$.

One possible explanation for the dictionary above is as follows. In general,
$S_{0\alpha_{*}}$ in a topological field theory computes the three-sphere
partition function with an insertion of loop operator ${\cal
O}_{\alpha_{*}}^{\Gamma=\textrm{(unknot)}}$ of type $\alpha_{*}$ along the
unknot in $S^{3}$. In the rank 0 theory ${\cal T}_{\rm rank\;0}$, on the other
hand, there is a flavor vortex loop operator associated to the $U(1)$ axial
flavor symmetry. The loop operator is known to act on the three-sphere
partition as a difference operator shifting the parameter $\nu$ Dimofte:2011ju
; Kapustin:2012iw ; Drukker:2012sr

$\displaystyle\textrm{(flavor vortex loop)${}_{\pm}$ of charge $\pm 1$
}\;\longleftrightarrow\;\exp(\pm\partial_{\nu})\;.$ (21)

This means that the $S^{3}$ partition function at the conformal point can be
identified with the $S^{3}$ partition function with an insertion of the vortex
loop operators of charge $+1$($-1$) in TFT- (TFT+)222If we turn on the
squashing parameter $b$, the partition function at $\nu=-1$ with the flavor
vortex loop is given as $\mathcal{Z}^{S^{3}_{b}+{\cal O}_{\rm
flavor\;vortex}}(m=0,\nu=-1)=\mathcal{Z}^{S^{3}_{b}}\left(m=i\pi(b-\frac{1}{b}),\nu=0\right)$.
Interestingly, the partition function is actually independent $b$
Gaiotto:2019mmf ; Bullimore:2020jdq  and
$|\mathcal{Z}^{S^{3}_{b}}\left(m=i\pi(b-\frac{1}{b}),\nu=0\right)|=|\mathcal{Z}^{S^{3}_{b}}\left(m=i\pi(b-\frac{1}{b}),\nu=0\right)|_{b=1}=|\mathcal{Z}^{S^{3}_{b}}\left(m=0,\nu=0)\right|=e^{-F}$.

$\displaystyle\begin{split}&|{\cal
Z}^{S^{3}}(\nu=0)|=\exp(\partial_{\nu})\cdot|{\cal Z}^{S^{3}}(\nu=-1)|=|{\cal
Z}^{S^{3}+{\cal O}_{\rm flavor\;vortex}}(\nu=-1)|\\\ &\Rightarrow|{\cal
Z}^{S^{3}}(\nu=0)|=|S_{0{\alpha=\textrm{(flavor vortex)}_{+}}}\textrm{ of
TFT}_{-}|\;.\end{split}$ (22)

Hence if one identify $\alpha=(\textrm{flavor vortex})_{+}$
($\alpha=(\textrm{flavor vortex})_{-}$) of TFT- (TFT+) with
$\alpha=\alpha_{*}$ in (20), then the dictionary follows. Actually, according
to F-maximization (188), we expect that

$\displaystyle\begin{split}&|\mathcal{Z}^{S^{3}}(\nu=0)|=|S_{0\alpha=\textrm{(flavor
vortex)}_{+}}\textrm{ of TFT}_{-}|<|S_{00}\textrm{ of
TFT}_{-}|=|\mathcal{Z}^{S^{3}}(\nu=-1)|\;,\\\
&|\mathcal{Z}^{S^{3}}(\nu=0)|=|S_{0\alpha=\textrm{(flavor
vortex)}_{-}}\textrm{ of TFT}_{+}|<|S_{00}\textrm{ of
TFT}_{+}|=|\mathcal{Z}^{S^{3}}(\nu=1)|\;.\end{split}$ (23)

It is compatible with the desired identification, $(\textrm{flavor vortex
loop})=(\alpha_{*}\textrm{in }\eqref{S-alpha})$. The property above is also
compatible with the fact that TFT± are non-unitary since they violate the
unitarity condition (216).

The argument above gives circumstantial evidence but not a full proof for the
dictionary on $F$. The dictionary will be confirmed explicitly in section 3
with infinitely many examples. We leave general proof or disproof of the
dictionary for future work.

###  2.3 Application: lower bounds on $F$

Here we derive interesting lower bounds on $F$ for rank 0 SCFTs using the
correspondence introduced in the previous subsection.

One immediate and interesting consequence of the dictionaries in Table 1 is

$\displaystyle F>-\log\frac{1}{\sqrt{\textrm{Witten index}}}\;,\;\;\textrm{for
all interacting ${\cal N}\geq 4$ SCFT ${\cal T}_{\rm rank\;0}$ of rank
$0$}\;.$ (24)

This follows from the following fact

$\displaystyle\sum_{\alpha=0}^{N-1}(S_{0\alpha})^{2}=1\;\Rightarrow\;\textrm{min}_{\alpha}|S_{0\alpha}|=|S_{0\alpha_{*}}\textrm{
in \eqref{S-alpha}}|<\frac{1}{\sqrt{N}}\;,$ (25)

combined with the dictionaries for Witten index and $F$. More interestingly,
using the dictionary, one can prove following

$\displaystyle
F\geq-\log\sqrt{\frac{5-\sqrt{5}}{10}}=0.642965\;,\;\;\textrm{for any
interacting ${\cal N}\geq 4$ SCFT ${\cal T}_{\rm rank\;0}$ of rank $0$}\;.$
(26)

The bound is saturated by the minimal ${\cal T}_{\rm min}$ theory which will
be introduced in section 3. The inequality above follows from the following
fact combined with the dictionary for $F$

$\displaystyle|S_{0\alpha_{*}}\textrm{in
\eqref{S-alpha}}|\leq\sqrt{\frac{5-\sqrt{5}}{10}}=0.525731\quad\textrm{for all
non-unitary TFT}\;.$ (27)

Thanks to (25), we only need to check the inequality for non-unitary TQFTs up
to rank 3 since and $\frac{1}{\sqrt{4}}=0.5<\sqrt{\frac{5-\sqrt{5}}{10}}$. Let
us first consider rank 2 case. Let $x=S_{00}^{2}$ and $y=S_{01}^{2}$ where
$S_{\alpha\beta}$ is the S-matrix of a non-unitary TQFT. Then the two positive
real numbers $x$ and $y$ should satisfy followings

$\displaystyle\textrm{GSD}_{g=0}=x+y=1\;,\quad\textrm{GSD}_{g=2}=\frac{1}{x}+\frac{1}{y}=k\in\mathbb{Z}_{>0}\;.\
$ (28)

Here $\textrm{GSD}_{g}$ denotes the ground state degeneracy on genus $g$
Riemann surface, see (218). One can solve the equations and we have two
solutions

$\displaystyle x=\frac{1}{2}\pm\frac{1}{2}\sqrt{\frac{k-4}{k}}\;,\quad
y=\frac{1}{2}\mp\frac{1}{2}\sqrt{\frac{k-4}{k}}\;.$ (29)

Imposing the conditions, $x,y\in\mathbb{R}_{>0}$ and $x>y$ (non-unitarity
condition, see (216)), we have only one solution for each $k>4$. As the
natural number $k$ increases, $y^{1/2}=|S_{0\alpha_{*}}|$ in the solution
decreases. Thus, we have

$\displaystyle\begin{split}&|S_{0\alpha_{*}}|=y^{1/2}\leq(y^{1/2}\textrm{ at
}k=5)=\sqrt{\frac{5-\sqrt{5}}{10}}\;,\\\ &\textrm{ for any rank 2 non-unitary
TQFT}\;.\end{split}$ (30)

Now let us move on to the rank 3 case. Let $x=S_{00}^{2},y=S_{01}^{2}$ and
$z=S_{02}^{2}=S_{0\alpha_{*}}^{2}$. Then, the three positive real numbers
should satisfy

$\displaystyle\begin{split}&\textrm{GSD}_{g=0}=x+y+z=1\;,\quad\textrm{GSD}_{g=2}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=k_{1}\in\mathbb{Z}_{>0}\;,\\\
&\textrm{GSD}_{g=3}=\frac{1}{x^{2}}+\frac{1}{y^{2}}+\frac{1}{z^{2}}=k_{2}\in\mathbb{Z}_{>0}\;.\end{split}$
(31)

One can confirm that any solution to equations above satisfying the non-
unitarity conditions, $\textrm{min}(x,y)\geq z$ and $\textrm{max}(x,y)>z$,
have following property333Since
$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\leq\frac{3}{z}$,
$\frac{1}{x^{2}}+\frac{1}{y^{2}}+\frac{1}{z^{2}}\leq\frac{3}{z^{2}}$ and thus
$z\leq\textrm{min}(\frac{3}{k_{1}},\sqrt{\frac{3}{k_{2}}})$, it is enough to
check that $z^{1/2}\leq\frac{1}{2}$ for only finitely many cases $(0<k_{1}\leq
12$ and $0<k_{2}\leq 48)$.

$\displaystyle\begin{split}&|S_{0\alpha_{*}}|=z^{1/2}\leq(z^{1/2}\textrm{ at
}k_{1}=10\textrm{ and
}k_{2}=36)=\frac{1}{2}<\sqrt{\frac{5-\sqrt{5}}{10}}\;,\\\ &\textrm{ for any
rank 3 non-unitary TQFT}\;.\end{split}$ (32)

We are not certain if there is a rank 3 non-unitary TQFT saturating the bound
$|S_{0\alpha_{*}}|=\frac{1}{2}$. The results in (25), (30) and (32) imply the
bound in (27), from which the conclusion in (26) follows.

##  3 Examples

In this section, we introduce infinitely many examples of (2+1)D
$\mathcal{N}=4$ interacting rank 0 superconformal field theories ${\cal
T}_{\rm rank\;0}$. Using the dictionary in Table 1, we compute the set
$\\{|S^{\pm}_{0\alpha}|\\}$ for non-unitary TQFTs $\textrm{TFT}_{\pm}[{\cal
T}_{\rm rank\;0}]$. We also independently compute the three-sphere free energy
$F$ by performing the localization integral for ${\cal Z}^{S^{3}_{b}}$ at
$b=1,m=0,\nu=0$ and confirm the non-trivial dictionary for $F$. See Table 2
for the summary.

${\cal T}_{\rm rank\;0}$ | TFT${}_{\pm}[{\cal T}_{\rm rank\;0}]$ | Set of $\\{|S^{\pm}_{0\alpha}|\\}$ | $\exp(-F)$
---|---|---|---
${\cal T}_{\rm min}$ | (Lee-Yang) | {$\sqrt{\frac{5+\sqrt{5}}{10}}$ , $\sqrt{\frac{5-\sqrt{5}}{10}}$ } | $\sqrt{\frac{5-\sqrt{5}}{10}}$
$(U(1)_{1}+H)$ | $\textrm{Gal}_{d}(SU(2)_{6})/\mathbb{Z}_{2}^{f}$ | $\\{2\zeta^{1}_{6},2\zeta^{3}_{6}\\}$ | $2\zeta^{1}_{6}$
| $(\textrm{with }d=\zeta_{6}^{3})$ |  |
$SU(2)_{k}^{\frac{1}{2}\oplus\frac{1}{2}}$ | $\textrm{Gal}_{d}(SU(2)_{4|k|-2})/\mathbb{Z}_{2}^{f}$ | $\big{\\{}2\zeta^{2n-1}_{4|k|-2}\big{\\}}_{n=1}^{|k|}$ | $2\zeta_{4|k|-2}^{1}$
($|k|>1$) | $(\textrm{with }d=\zeta_{4|k|-2}^{2|k|-1})$ |  |
$T[SU(2)]_{k_{1},k_{2}}$ | See the caption | $\big{\\{}(\frac{1}{\sqrt{2}}\zeta_{|k_{1}k_{2}-1|-2}^{n})^{\otimes 2}\big{\\}}_{n=1}^{|k_{1}k_{2}-1|-1}$ | $\frac{1}{\sqrt{2}}\zeta_{|k_{1}k_{2}-1|-2}^{1}$
$\frac{T[SU(2)]}{SU(2)^{\rm diag}_{|k|=3}}$ | (Lee-Yang)${}^{\otimes 2}\otimes U(1)_{2}$ | $\big{\\{}\frac{1}{\sqrt{10}}^{\otimes 4},\frac{5+\sqrt{5}}{10\sqrt{2}}^{\otimes 2},\frac{5-\sqrt{5}}{10\sqrt{2}}^{\otimes 2}\big{\\}}$ | $\frac{5-\sqrt{5}}{10\sqrt{2}}$
$\frac{T[SU(2)]}{SU(2)^{\rm diag}_{|k|=4}}$ | $\frac{\textrm{Gal}_{\zeta_{10}^{7}}(SU(2)_{10})\times SU(2)_{2}}{\mathbb{Z}_{2}^{\rm diag}}$ | $\big{\\{}\frac{1}{2},\frac{1}{2\sqrt{3}}^{\otimes 5},\frac{3+\sqrt{3}}{12}^{\otimes 2},\frac{3-\sqrt{3}}{12}^{\otimes 2}\big{\\}}$ | $\frac{3-\sqrt{3}}{12}$
$\frac{T[SU(2)]}{SU(2)^{\rm diag}_{|k|=5}}$ | Gal${}_{d}\big{(}(G_{2})_{3}\big{)}\otimes U(1)_{-2}$ | $\big{\\{}\frac{1}{\sqrt{6}}^{\otimes 2},\frac{1}{\sqrt{14}}^{\otimes 6},$ | $\sqrt{\frac{5}{84}-\frac{1}{4\sqrt{21}}}$
| $\left(d=\sqrt{\frac{5}{84}+\frac{1}{4\sqrt{21}}}\,\right)$ | $\sqrt{\frac{5}{84}\pm\frac{1}{4\sqrt{21}}}^{\,\otimes 2}\big{\\}}$ |
|  | $\big{\\{}\frac{1}{\sqrt{2|k|-4}}^{\otimes(|k|-3)},\frac{1}{\sqrt{2|k|+4}}^{\otimes(|k|+1)}$ | $\frac{1}{\sqrt{8|k|-16}}$
$\frac{T[SU(2)]}{SU(2)^{\rm diag}_{|k|\geq 6}}$ | ? |  $(\frac{1}{\sqrt{8|k|-16}}+\frac{1}{\sqrt{8|k|+16}})^{\otimes 2}$ , |  $-\frac{1}{\sqrt{8|k|+16}}$
|  |  $(\frac{1}{\sqrt{8|k|-16}}-\frac{1}{\sqrt{8|k|+16}})^{\otimes 2}$} |
Table 2: Non-unitary TQFTs from rank 0 ${\cal N}=4$ SCFTs.
$\textrm{Gal}_{d}(\textrm{TFT})$ denotes a Galois conjugate of an unitary
topological field theory TFT with $S_{00}(\textrm{Gal}_{d}[\textrm{TFT}])=d$.
We define $\zeta_{k}^{n}:=\sqrt{\frac{2}{k+2}}\sin\frac{n\pi}{k+2}$. For the
rank 0 SCFT ${\cal T}_{\rm
rank\;0}=T[SU(2)]_{k_{1},k_{2}}:=\frac{T[SU(2)]}{SU(2)^{C}_{k_{1}}\times
SU(2)^{H}_{k_{2}}}$ with $|k_{1}k_{2}-1|>3$ and
$\textrm{min}(|k_{1}|,|k_{2}|)>2$, the corresponding non-unitary TQFTs are
$\textrm{TFT}_{+}=\big{[}\textrm{Gal}_{\zeta_{|k_{1}k_{2}-1|-2}^{|k_{2}|}}(SU(2)_{|k_{1}k_{2}-1|-2})\big{]}\otimes
U(1)_{2}$ and
$\textrm{TFT}_{-}=\big{[}\textrm{Gal}_{\zeta_{|k_{1}k_{2}-1|-2}^{|k_{1}|}}(SU(2)_{|k_{1}k_{2}-1|-2})\big{]}\otimes
U(1)_{2}$. For ${\cal T}_{\rm rank\;0}=\frac{T[SU(2)]}{SU(2)^{\rm
diag}_{|k|\geq 6}}$, we could not identify their associatec TFTs with
previously known non-unitary TQFTs.

###  3.1 The minimal $\mathcal{N}=4$ SCFT ${\cal T}_{\rm min}$

####  3.1.1 SUSY enhancement

In Gang:2018huc , it was claimed that

$\displaystyle\begin{split}&\textrm{(3D $\mathcal{N}=2$ gauge theory,
$U(1)_{k=-3/2}$ coupled to a chiral multiplet $\Phi$ of charge $+1$)}\\\
&\xrightarrow{\quad\textrm{at IR}\quad}(\textrm{3D $\mathcal{N}=4$
superconformal field theory $\mathcal{T}_{\rm min}$})\;.\end{split}$ (33)

As a quick evidence for the SUSY enhancement, there are following two gauge
invariant monopole operators in the theory

$\displaystyle\hat{\phi}\hat{\phi}|\mathfrak{m}=+1\rangle\;,\quad\hat{\psi}^{*}|\mathfrak{m}=-1\rangle\;,$
(34)

with following quantum numbers Borokhov:2002cg ; Benini:2011cma ; Gang:2018huc

$\displaystyle\begin{split}&A=-\mathfrak{m}=-1,\;R=\frac{1-R_{\Phi}}{2}|\mathfrak{m}|+2R_{\Phi}=1,\;j=1\;,\;\Delta=2\;,\\\
&A=-\mathfrak{m}=1,\;R=\frac{1-R_{\Phi}}{2}|\mathfrak{m}|+(1-R_{\Phi})=1,\;j=1\;,\;\Delta=2\;.\end{split}$
(35)

Here $A$ is the charge of $U(1)_{\rm top}$ and R is the charge of the
superconformal $U(1)$ R-symmetry; $j\in\frac{\mathbb{Z}}{2}$ and $\Delta$ are
the Lorentz spin and the conformal dimension respectively. We use the fact
that Gang:2018huc

$\displaystyle R_{\Phi}:=(\textrm{Superconformal $U(1)$ R-symmetry charge of
$\Phi$})=\frac{1}{3}\;,$ (36)

which can be determined by the F-maximization Jafferis:2010un . Here
$|\mathfrak{m}\rangle\in{\cal H}_{\rm rad}(S^{2})$ (the radially quantized
Hilbert-space on $S^{2}$) is a half BPS bare monopole operator. The bare
monopole operator has a $U(1)$ gauge charge
$-\frac{3}{2}\mathfrak{m}+\frac{1}{2}|\mathfrak{m}|$ and should be dressed by
excitations ($\hat{\phi},\hat{\psi}$ and their complex conjugations) of matter
fields to be gauge-invariant. The dressed monopole operators are $1/4$ BPS
local operators. According to the classification in Cordova:2016emh , the
monopole operators above, if they survive at the IR superconformal point, must
belong to an extra SUSY-current multiplet of the 3D ${\cal N}=2$
superconformal algebra. The multiplet consists of conformal primaries of the
following quantum numbers as well as their conformal descendants,

$\displaystyle[(R,j,\Delta)=(0,\frac{1}{2},\frac{3}{2})]\xrightarrow{\quad
Q,\bar{Q}\quad}[(R,j,\Delta)=(\pm 1,1,2)]\xrightarrow{\quad
Q,\bar{Q}\quad}[(R,j,\Delta)=(0,\frac{3}{2},\frac{5}{2})]\;.$ (37)

Here $Q:=Q_{1}+iQ_{2}$ and $\bar{Q}:=Q_{1}-iQ_{2}$ are the ${\cal N}=2$
supercharges. The local operators in the top component with
$(R,j,\Delta)=(0,\frac{3}{2},\frac{5}{2})$ correspond to the conserved current
for the extra supersymmetry, whose existence guarantees the SUSY enhancement.

Further, it was claimed in Gang:2018huc  that the infra-red (IR)
superconformal field theory (SCFT) $\mathcal{T}_{\rm min}$ is the minimal 3D
$\mathcal{N}=4$ SCFT having the smallest three-sphere free-energy $F$ and the
smallest non-zero stress-energy tensor central charge $C_{T}$ whose exact
values are Gang:2019jut ; Gang:2018huc

$\displaystyle\begin{split}F({\cal T}_{\rm
min})&=-\log\sqrt{\frac{5-\sqrt{5}}{10}}\\\ &\simeq 0.642965\;,\\\
\frac{C_{T}(\mathcal{T}_{\rm min})}{C_{T}(\textrm{free theory with single
$\Phi$})}&=\frac{8}{26}\left(8-\frac{5\sqrt{5+2\sqrt{5}}}{\pi}\right)\\\
&\simeq 0.992549\;.\end{split}$ (38)

There is no vacuum moduli space in the minimal theory and thus the minimal
SCFT is of rank 0.

##### Superconformal index

Applying the localization results in Kim:2009wb ; Imamura:2011su  (see also
Appendix A of Aharony:2013dha ), the superconformal index for ${\cal T}_{\rm
min}$ can be written as

$\displaystyle\begin{split}&{\cal I}_{{\cal T}_{\rm min}}^{\rm
sci}\big{(}q,\eta,\nu;s=1\big{)}=\sum_{\mathfrak{m}\in\mathbb{Z}}\oint_{|a|=1}\\!\frac{da}{2\pi
ia}\,q^{\frac{|\mathfrak{m}|}{6}}(a(-1)^{\mathfrak{m}})^{-\frac{3\mathfrak{m}}{2}-\frac{|\mathfrak{m}|}{2}}(\eta
q^{\frac{\nu}{2}})^{-\mathfrak{m}}\textrm{P.E.}[f_{\rm
single}(q,a;\mathfrak{m})]\\\ &\textrm{with }f_{\rm
single}(q,a;\mathfrak{m}):=\frac{q^{\frac{1}{6}+\frac{|\mathfrak{m}|}{2}}a}{1-q}-\frac{q^{\frac{5}{6}+\frac{|\mathfrak{m}|}{2}}a^{-1}}{1-q}\;.\end{split}$
(39)

In the above, we use the superconformal $R$ charge of $\Phi$ given in (36). At
the conformal point, $\nu=0$, the index becomes

$\displaystyle{\cal I}_{{\cal T}_{\rm min}}^{\rm
sci}\big{(}q,u,\nu=0;s=1\big{)}=1-q+\left(\eta+\frac{1}{\eta}\right)q^{3/2}-2q^{2}+\left(\eta+\frac{1}{\eta}\right)q^{5/2}-2q^{3}+\ldots\;.$
(40)

The terms in $q^{3/2}$ come from the monopole operators in (35) and the index
is compatible with the claimed SUSY enhancement Evtikhiev:2017heo . On the
other hand, the index at the non-conformal point $\nu=\pm 1$ is

$\displaystyle\begin{split}&{\cal I}_{{\cal T}_{\rm min}}^{\rm
sci}\big{(}q,\eta,\nu,s=1\big{)}\big{|}_{\nu\rightarrow\pm 1}\\\
&=1+\left(-1+\eta^{\mp
1}\right)q+\left(-2+\eta+\frac{1}{\eta}\right)q^{2}+\left(-2+\eta+\frac{1}{\eta}\right)q^{3}+\ldots\;.\end{split}$
(41)

As anticipated from the superconformal multiplet analysis in (16), the index
becomes ($q$-independent) 1 in the degenerate limits, $\nu\rightarrow\pm 1$
and $\eta\rightarrow 1$. This reconfirm that the ${\cal N}=4$ theory is of
rank 0 and gives a strong signal that a topological field theory emerges in
the limits.

####  3.1.2 Lee-Yang TQFT in degenerate limits

Here we claim that

$\displaystyle{\rm TFT}_{\pm}[{\cal T}_{\rm min}]=(\textrm{Lee-Yang TQFT})$
(42)

with explicit checks of the dictionaries in Table 1. The non-unitary TQFT has
following modular data

$\displaystyle
S=\begin{pmatrix}\sqrt{\frac{1}{10}\left(\sqrt{5}+5\right)}&-\sqrt{\frac{1}{10}\left(5-\sqrt{5}\right)}\\\
-\sqrt{\frac{1}{10}\left(5-\sqrt{5}\right)}&-\sqrt{\frac{1}{10}\left(\sqrt{5}+5\right)}\end{pmatrix}\;,\quad
T=\begin{pmatrix}1&0\\\ 0&\exp(-\frac{2\pi i}{5})\end{pmatrix}\;.$ (43)

##### Squashed three-sphere partition function

The squashed three-sphere partition function for the minimal theory is
($\hbar=2\pi ib^{2},b\in\mathbb{R}$)

$\displaystyle\begin{split}&\mathcal{Z}^{S^{3}_{b}}_{{\cal T}_{\rm
min}}(b,m,\nu)=\int\\!\frac{dZ}{\sqrt{2\pi\hbar}}\,\,e^{-\frac{Z^{2}+2Z\left(m+(i\pi+\frac{\hbar}{2})\nu\right)}{2\hbar}}\psi_{\hbar}(Z)\;.\end{split}$
(44)

Here $\psi_{\hbar}$ is a special function called the quantum dilogarithm, for
which readers are referred to Appendix D. The partition function in the limit
$b\rightarrow 1$ was studied in Gang:2019jut  and one can check that

$\displaystyle\begin{split}&\Big{|}\mathcal{Z}^{S^{3}_{b}}_{{\cal T}_{\rm
min}}(b,m=0,\nu\rightarrow\pm 1)\Big{|}\xrightarrow{\quad b\rightarrow
1\quad}\sqrt{\frac{1}{10}\left(\sqrt{5}+5\right)}+\sum_{n=2}^{\infty}s_{n}(1-b)^{n}\;,\\\
&\textrm{with }s_{n}=0\textrm{ up to arbitrary higher order
$n$}\;.\end{split}$ (45)

That the partition function becomes independent on the squashing parameter $b$
in the degenerate limit is a strong signal that the theory becomes
topological. Furthermore, the partition function is identical to the $S^{3}$
partition function ($S_{00}$, see (43)) of the Lee-Yang TQFT,

$\displaystyle\Big{|}\mathcal{Z}^{S^{3}_{b}}_{{\cal T}_{\rm
min}}(b,m=0,\nu=\pm 1)\Big{|}=\Big{|}\mathcal{Z}^{S^{3}}_{\rm Lee-
Yang}\Big{|}\;.$ (46)

The free energy $F=-\log\big{|}\mathcal{Z}^{S^{3}_{b}}_{{\cal T}_{\rm
min}}(b=1,m=0,\nu=0)\big{|}$ of the minimal theory, given in (38), nicely
matches with $\max_{\alpha}(-\log|S_{0\alpha}|)$, see (43) for the S-matrix,

$\displaystyle\left(F\textrm{ of }{\cal T}_{\rm
min}\right)=\left(\max_{\alpha}(-\log|S_{0\alpha}|)\textrm{ of Lee-Yang
TQFT}\right)\;,$ (47)

which confirms the dictionary for $F$ in Table 1.

##### Perturbative invariants ${\cal S}_{n}^{\alpha}$

The integrand in (44) can be expanded as

$\displaystyle\begin{split}&\log{\cal
I}_{\hbar}(Z,m,\nu)=\log\left(e^{-\frac{Z^{2}+2Z\left(m+(i\pi+\frac{\hbar}{2})\nu\right)}{2\hbar}}\psi_{\hbar}(Z)\right)\\\
&\xrightarrow{\quad\hbar\rightarrow 0\quad}\frac{1}{\hbar}{\cal
W}_{0}(Z,m,\nu)+{\cal W}_{1}(Z,m,\nu)+\ldots\quad\textrm{with }\\\ &{\cal
W}_{0}=\textrm{Li}_{2}(e^{-Z})-\frac{Z^{2}}{2}-Z(m+i\pi\nu)\;,\quad{\cal
W}_{1}=-\frac{1}{2}\log(1-e^{-Z})-\frac{Z\nu}{2}\;.\end{split}$ (48)

There are two Bethe-vacua (191) determined by the following algebraic equation

$\displaystyle\textrm{Bethe-vacua of ${\cal T}_{\rm
min}$}\;:\;\Big{\\{}z\;:\;\frac{(z-1)e^{-m-i\pi\nu}}{z^{2}}=1\Big{\\}}\;.$
(49)

In the degenerate limits, $m=0$ and $\nu\rightarrow\pm 1$, the two Bethe-vacua
approach following values

$\displaystyle
z_{\alpha=0}\rightarrow\frac{1}{2}\left(\sqrt{5}-1\right)\;,\quad
z_{\alpha=1}\rightarrow\frac{1}{2}\left(-\sqrt{5}-1\right)\;.$ (50)

Perturbative invariants (193) of two saddle points associated to the two
Bethe-vacua in the degenerate limits are

$\displaystyle\begin{split}&{\cal
S}^{\alpha=0}_{0}\rightarrow\frac{7\pi^{2}}{30}\;,\quad{\cal
S}_{1}^{\alpha=0}\rightarrow-\frac{1}{2}\log\left(\frac{5-\sqrt{5}}{2}\right)\;,\quad{\cal
S}_{2}^{\alpha=0}\rightarrow-\frac{7}{120}\;,\\\ &{\cal
S}^{\alpha=1}_{0}\rightarrow-\frac{17\pi^{2}}{30}\;,\quad{\cal
S}_{1}^{\alpha=1}\rightarrow-\frac{1}{2}\log\left(\frac{5+\sqrt{5}}{2}\right)\;,\quad{\cal
S}_{2}^{\alpha=1}\rightarrow-\frac{7}{120}\;,\\\ &{\cal S}^{\alpha}_{n\geq
3}\rightarrow 0\;.\end{split}$ (51)

That this is compatible with the expected properties in (195) is a highly non-
trivial evidence for emergence of topological theory in the degenerate limits.

##### Fibering and Handle gluing

Using the formulae in (203) and (200) combined with the above computation of
${\cal S}^{\alpha}_{n}$, we have

$\displaystyle\begin{split}&\bigg{\\{}{\cal F}_{\alpha}(m=0,\nu\rightarrow\pm
1,s=-1)\bigg{\\}}_{\alpha=0,1}\longrightarrow\;\left\\{\exp\left(-\frac{7i\pi}{60}\right),\exp\left(\frac{17i\pi}{60}\right)\right\\}\;,\\\
&\bigg{\\{}{\cal H}_{\alpha}(m=0,\nu\rightarrow\pm
1,s=-1)\bigg{\\}}_{\alpha=0,1}\longrightarrow\;\bigg{\\{}\frac{5-\sqrt{5}}{2},\frac{5+\sqrt{5}}{2}\bigg{\\}}\;.\end{split}$
(52)

Since the $1/\sqrt{{\cal H}_{\alpha=0}}$ is equal to $|{\cal Z}^{S^{3}_{b}}|$
at $\nu=\pm 1$ in (45), the $z_{\alpha=0}$ is indeed the Bethe-vacuum
corresponding to the trivial object according to the criterion in (17). The
computations above also confirm the dictionary for $S_{0\alpha}^{-2}$ and
$T_{\alpha\beta}$ in Table 1, see (43) for modular matrices of Lee-Yang TQFT.

##### Supersymmetric loop operator

For a $U(1)$ gauge theory, the supersymmetric dyonic loop operator ${\cal
O}_{(p,q)}$ of (electric charge, magnetic charge)=$(p,q)$ is

$\displaystyle{\cal O}_{(p,q)}=z^{p}(1-z^{-1})^{q}\;.$ (53)

The consistency condition in (19) is met when we choose the (Bethe
vacua)-to-(loop operators) map as follow

$\displaystyle{\cal O}_{\alpha=0}=(\textrm{identity operator})\;,\quad{\cal
O}_{\alpha=1}={\cal O}_{(p,q)=(1,0)}\;.$ (54)

Then, using the dictionary in Table 1

$\displaystyle\begin{split}&W_{\beta=0,1}(0)=1\;,\quad
W_{\beta=0}(1)=z_{0}=\frac{1}{2}(\sqrt{5}-1)\;,\quad
W_{\beta=1}(1)=z_{1}=\frac{1}{2}(-\sqrt{5}-1)\;.\end{split}$ (55)

From ${\cal W}_{\beta}(\alpha)$, one can compute the S-matrix using the
formula $S_{\alpha\beta}=S_{00}W_{\beta}(\alpha)W_{0}(\beta)$, and confirm
that it is identical to the S-matrix of the Lee-Yang TQFT given in (43).

###  3.2 $U(1)_{1}+H$ : SUSY enhancement ${\cal N}=3\rightarrow{\cal N}=5$

####  3.2.1 SUSY enhancement

We define

$\displaystyle\begin{split}(U(1)_{k}+H)&:=(\textrm{3D ${\cal N}=4$ $U(1)$
gauge theory with CS level $k$}\\\ &\qquad\;\textrm{ coupled to a
hypermultiplet of charge $+1$})\;.\end{split}$ (56)

For non-zero $k$, the theory has ${\cal N}=3$ supersymmetry instead of ${\cal
N}=4$ since the CS term breaks some of the ${\cal N}=4$ supersymmetry. The
${\cal N}=3$ theory has the $U(1)_{\rm top}$ flavor symmetry associated to the
dynamical $U(1)$ gauge theory. As pointed out in Garozzo:2019ejm ;
Beratto:2020qyk

$\displaystyle\begin{split}\textrm{For $k=1$},\;&\textrm{the $(U(1)_{k}+H)$
has enhanced ${\cal N}=5$ supersymmetry at IR}\\\ &\textrm{and the resulting
IR SCFT is of rank 0 }.\end{split}$ (57)

The $U(1)_{\rm top}=SO(2)_{\rm top}$ symmetry becomes the $U(1)$ axial
symmetry, which is a subgroup of $SO(4)\subset SO(5)$ R-symmetry, in the
supersymmetry enhancement.

$\displaystyle SO(3)_{R}\times SO(2)_{\rm
top}\xrightarrow{\quad\textrm{RG}\quad}SO(5)_{R}\;.$ (58)

For $|k|=1$, there are two BPS monopole operators whose quantum numbers are

$\displaystyle\begin{split}&A=+1,\;j=\frac{1}{2},\;R=1,\;\Delta=\frac{3}{2}\;\textrm{
and}\\\ &A=-1,\;j=\frac{1}{2},\;R=1,\;\Delta=\frac{3}{2}\;.\end{split}$ (59)

Here $A$ is the charge of $U(1)_{\rm top}$ and $R\in\frac{\mathbb{Z}}{2}$ is
the spin of $SO(3)$ R-symmetry. $j$ and $\Delta$ are the Lorentz spin and the
conformal dimension respectively. The BPS operators belong to extra SUSY-
current multiplet Cordova:2016emh  of ${\cal N}=3$ superconformal algebra,
which consists of the following conformal primaries and their descendants

$\displaystyle\begin{split}&[(R,j,\Delta)=(0,0,1)]\xrightarrow{\quad\mathbf{Q}\quad}[(R,j,\Delta)=(1,\frac{1}{2},\frac{3}{2})]\xrightarrow{\quad\mathbf{Q}\quad}\\\
&[(R,j,\Delta)=(1,1,2)]\oplus[(R,j,\Delta)=(0,0,2)]\xrightarrow{\quad\mathbf{Q}\quad}[(R,j,\Delta)=(0,\frac{3}{2},\frac{5}{2})]\;.\end{split}$
(60)

Here $\mathbf{Q}=(Q_{1},Q_{2},Q_{3})$ are the ${\cal N}=3$ supercharges. The
local operators in the top component of the supermultiplet with
$(R,j,\Delta)=(0,\frac{3}{2},\frac{5}{2})$ correspond to the conserved current
for extra supersymmetry, and thus the existence of the multiplet implies the
supersymmetry enhancement Garozzo:2019ejm ; Beratto:2020qyk .

##### Superconformal index

Using the localization summarized in A.1, the index is given as

$\displaystyle{\cal I}^{\rm sci}_{U(1)_{k}+H}(q,\eta,\nu;s=1)$
$\displaystyle=$ $\displaystyle\sum_{\mathfrak{m}}\oint_{|a|=1}\frac{da}{2\pi
ia}q^{\frac{|\mathfrak{m}|}{4}}((-1)^{\mathfrak{m}}a)^{k\mathfrak{m}}(\eta
q^{\frac{1}{2}\nu})^{-\mathfrak{m}}\textrm{P.E.}[f_{\rm
single}(q,\mathbf{a},\eta;\mathfrak{m})]\;,$ (61)

where $\eta$ is a fugacity of $U(1)_{\rm top}$ and

$\displaystyle f_{\rm
single}(q,\mathbf{a},\eta;\mathfrak{m}):=\frac{q^{\frac{1}{4}+\frac{1}{2}|\mathfrak{m}|}(a+\frac{1}{a})}{1-q}-\frac{q^{\frac{3}{4}+\frac{1}{2}|\mathfrak{m}|}(a+\frac{1}{a})}{1-q}\;.$
(62)

Using the above expression, the index can be evaluated and we find444The same
computation was done in Beratto:2020qyk ; Garozzo:2019ejm .

$\displaystyle{\cal I}^{\rm sci}_{U(1)_{k}+H}(q,\eta,\nu;s=1)$
$\displaystyle=\begin{cases}1+q^{1/2}+\left(-\eta-\frac{1}{\eta}-1\right)q+\left(\eta+\frac{1}{\eta}+2\right)q^{3/2}+\ldots,\quad|k|=1\\\
1+q^{1/2}-q+\ldots\;,\quad|k|>1\end{cases}$ (63)

The term $(-\eta-\frac{1}{\eta})q$ comes from the monopole operators (59) in
extra SUSY-current multiplet and implies the SUSY enhancement. Note that the
SUSY enhancement occurs only at $|k|=1$.

In the degenerate limit $\nu=\pm 1$ and $\eta=1$, the index becomes

$\displaystyle{\cal I}^{\rm sci}_{U(1)_{k}+H}(q,\eta=1,\nu=\pm 1;s=1)$
$\displaystyle=\left\\{\begin{array}[]{lr}1+\left(1-\eta^{\mp
1}\right)q^{1/2}+\left(\eta^{\mp 1}-1\right)q+\ldots\big{|}_{\eta=1}=1\;,&\
\quad(|k|=1)\\\ \textrm{Non-trivial power series in
}q^{1/2}\;,&\quad(|k|>1)\\\ \end{array}\right..$ (66)

It is compatible the expectation that the theory is a ${\cal N}=4$ (actually
${\cal N}=5$) SCFT of rank 0 when $|k|=1$. It also implies that there emerge
non-unitary TQFTs in the degenerate limits only when $|k|=1$. The non-unitary
TQFTs are expected to be fermionic according to (14).

####  3.2.2 Non-unitary TQFTs in degenerate limits

Here we extract modular data of $\textrm{TFT}_{\pm}[U(1)_{k=1}+H]$ by
computing various supersymmetric partition functions.

##### Squashed three-sphere partition function

It can be written as (see Appendix A.1)

$\displaystyle\begin{split}&{\cal
Z}^{S^{3}_{b}}_{U(1)_{k}+H}(b,m,\nu)=\int\\!\frac{\text{d}Z}{\sqrt{2\pi\hbar}}\,\,{\cal
I}_{\hbar}(Z,m,\nu)\;\textrm{ with }\\\ &{\cal
I}_{\hbar}(Z,m,\nu)=\exp\left(\frac{kZ^{2}}{2\hbar}\right)\exp\left(-\frac{ZW}{\hbar}\right)\bigg{|}_{W=m+(\pi
i+\frac{\hbar}{2})\nu}\\!\\!\prod_{\epsilon_{1}\in\\{\pm
1\\}}\Psi_{\hbar}\Big{[}\epsilon_{1}Z+\frac{\pi
i}{2}+\frac{\hbar}{4}\Big{]}\;.\end{split}$ (67)

Here $\Psi_{\hbar}(X):=\psi_{\hbar}(X)\exp(\frac{X^{2}}{4\hbar})$ as defined
in (185). In a round sphere limit ($b=1$) with $k=1$ and $m=0$, the integral
reduces to (using Appendix D)

$\displaystyle\begin{split}{\cal
Z}^{S^{3}}_{U(1)_{1}+H}(b=1,m=0,\nu)=\frac{e^{i\delta}}{4\pi}\int\text{d}Z\frac{e^{\frac{kZ^{2}}{4\pi
i}}e^{-Z\nu}}{\cosh(Z/2)}\;.\end{split}$ (68)

Here $e^{i\delta}$ is an unimportant phase factor sensitive to local
counterterms. For $\nu=0$, this integration can be exactly evaluated by
applying the residue theorem to the integral along the contour depicted in Fig
1:

$\infty$$-\infty$$\times$Re$Z$Im$Z$$\pi i$$C_{1}$$C_{2}$$A_{1}$$C_{3}$ Figure
1: A contour for the evaluation of (68). There is a simple pole at $Z=\pi i$
inside the contour.

$\displaystyle\int_{-\infty}^{\infty}\text{d}Z\frac{e^{\frac{Z^{2}}{4\pi
i}}}{\cosh(Z/2)}$
$\displaystyle+\int_{\infty}^{-\infty}\text{d}Z\frac{e^{\frac{Z^{2}}{4\pi
i}}e^{\frac{Z}{2}e^{\frac{\pi i}{4}}}}{i\sinh(Z/2)}+\pi i(-2ie^{\frac{\pi
i}{4}})=2\pi i(-2ie^{\frac{\pi i}{4}})\;,$
$\displaystyle\Rightarrow\int_{-\infty}^{\infty}\text{d}Z\frac{e^{\frac{Z^{2}}{4\pi
i}}}{\cosh(Z/2)}$ $\displaystyle=2\pi e^{\frac{\pi i}{4}}+e^{-\frac{\pi
i}{4}}\int_{-\infty}^{\infty}\text{d}Z\frac{e^{\frac{Z^{2}}{4\pi
i}}e^{\frac{Z}{2}}}{\sinh(Z/2)}$ $\displaystyle=2\pi e^{\frac{\pi
i}{4}}+\frac{e^{-\frac{\pi
i}{4}}}{2}\Bigg{(}\int_{-\infty}^{\infty}\text{d}Z\frac{e^{\frac{Z^{2}}{4\pi
i}}e^{\frac{Z}{2}}}{\sinh(Z/2)}-\int_{-\infty}^{\infty}\text{d}Z\frac{e^{\frac{Z^{2}}{4\pi
i}}e^{-\frac{Z}{2}}}{\sinh(Z/2)}\Bigg{)}$ $\displaystyle=2\pi e^{\frac{\pi
i}{4}}+e^{-\frac{\pi
i}{4}}\int_{-\infty}^{\infty}\text{d}Ze^{\frac{Z^{2}}{4\pi i}}$
$\displaystyle=2\pi(e^{\frac{\pi i}{4}}+e^{-\frac{\pi i}{2}})=4\pi
e^{-\frac{\pi i}{8}}\sin\Big{(}\frac{\pi}{8}\Big{)}\;.$ (69)

Here, the residue at the simple pole $Z=\pi i$ is $-2ie^{\frac{\pi i}{4}}$.
The first, second, and third term in the first line comes from the path
$C_{1}$, $C_{2}+C_{3}$, and an arc $A_{1}$ respectively. At the third
equality, we have used the changing variable as $Z\rightarrow-Z$ from the last
term in the second line.

Likewise, the integration for $\nu=\pm 1$ can be computed exactly in a similar
way by using the same contour and we found

$\displaystyle\int_{-\infty}^{\infty}\text{d}Z\frac{e^{\frac{Z^{2}}{4\pi
i}}\,e^{-Z}}{\cosh(Z/2)}=4\pi
e^{-\frac{5\pi}{8}}\sin\Big{(}\frac{3\pi}{8}\Big{)}\;.$ (70)

Restoring the overall factor $1/(4\pi)$ in (68), we finally have

$\displaystyle\begin{split}&\exp(-F)=\Big{|}{\cal
Z}^{S^{3}_{b}}_{U(1)_{1}+H}(b=1,m=0,\nu=0)\Big{|}=\sin\left(\frac{\pi}{8}\right)=(4+2\sqrt{2})^{-1/2}\;,\\\
&(S_{00}\textrm{ of TFT}_{\pm})=\Big{|}{\cal
Z}^{S^{3}_{b}}_{U(1)_{1}+H}(b=1,m=0,\nu\rightarrow\pm
1)\Big{|}=\sin\left(\frac{3\pi}{8}\right)=(4-2\sqrt{2})^{-1/2}\;.\end{split}$
(71)

The partition functions in the degenerate limits $\nu\rightarrow\pm 1$ are
actually independent on the squashing parameter $b$, as we will check it
perturbatively in (75), and equality in the 2nd line holds for arbitrary
$b\in\mathbb{R}$.

##### Perturbative invariants ${\cal S}_{n}^{\alpha}$

The integrand in (67) at $k=1$, after a shift $Z\to Z+\frac{1}{2}(\pi
i+\frac{\hbar}{2})$ of dummy integral variable, can be expanded as

$\displaystyle\begin{split}&\log{\cal I}_{\hbar}(Z,m,\nu)|_{Z\rightarrow
Z+\frac{1}{2}(\pi i+\frac{\hbar}{2})}\xrightarrow{\quad\hbar\rightarrow
0\quad}\frac{1}{\hbar}{\cal W}_{0}(Z,m,\nu)+{\cal
W}_{1}(Z,m,\nu)+\ldots\quad\textrm{with }\\\ &{\cal
W}_{0}(m=0)=\textrm{Li}_{2}(e^{-Z+i\pi})+\textrm{Li}_{2}(e^{Z})+Z^{2}+\pi
i(1-\nu)Z+c_{0}\pi^{2}\;,\\\ &{\cal W}_{1}(m=0)=\frac{1}{2}\left(-i\pi\nu-\nu
Z+Z-\log\left(1-e^{Z}\right)\right)+ic_{1}\pi\;,\end{split}$ (72)

where $c_{0}$ and $c_{1}$ are $Z$-independent rational numbers. There are two
Bethe-vacua (191) determined by a following algebraic equation

$\displaystyle\textrm{Bethe-vacua of ($U(1)_{1}+H$) at
$m=0$}\;:\;\Big{\\{}z\;:\;\frac{(-1)^{\nu}z(z+1)}{z-1}=1\Big{\\}}\;.$ (73)

In the degenerate limits, $m=0$ and $\nu\rightarrow\pm 1$, the two Bethe-vacua
approach following values

$\displaystyle z_{\alpha=0}\rightarrow\left(\sqrt{2}-1\right),\quad
z_{\alpha=1}\rightarrow\left(-\sqrt{2}-1\right)\;.$ (74)

Perturbative invariants (193) of two saddle points associated to the two
Bethe-vacua in the degenerate limits are

$\displaystyle\begin{split}&{\cal
S}^{\alpha=0}_{0}\rightarrow-\frac{11}{12}\pi^{2}\;,\quad{\cal
S}_{1}^{\alpha=0}\rightarrow-\frac{\pi
i}{8}-\frac{1}{2}\log\left(4-2\sqrt{2}\right)\;,\\\ &{\cal
S}^{\alpha=1}_{0}\rightarrow\frac{1}{12}\pi^{2}\;,\quad{\cal
S}_{1}^{\alpha=1}\rightarrow-\frac{\pi
i}{8}-\frac{1}{2}\log\left(4+2\sqrt{2}\right)\;,\\\ &\textrm{Im}[{\cal
S}^{\alpha}_{n=2}],\;{\cal S}^{\alpha}_{n\geq 3}\rightarrow 0\;.\end{split}$
(75)

That this is compatible with the expected properties in (195) is a highly non-
trivial evidence for emergence of topological theory in the degenerate limits.

##### Fibering and Handle gluing

Using the formulae in (203) and (200) combined with the above computation of
${\cal S}^{\alpha}_{n}$, we have

$\displaystyle\bigg{\\{}{\cal F}_{\alpha}(m=0,\nu\rightarrow\pm
1,s=-1)\bigg{\\}}_{\alpha=0,1}\longrightarrow\;\left\\{\exp\left(\frac{11i\pi}{24}\right),\exp\left(-\frac{i\pi}{24}\right)\right\\}\;,$
$\displaystyle\bigg{\\{}{\cal H}_{\alpha}(m=0,\nu\rightarrow\pm
1,s=-1)\bigg{\\}}_{\alpha=0,1}\longrightarrow\left\\{\left(4-2\sqrt{2}\right),\left(4+2\sqrt{2}\right)\right\\}.$
(76)

Since the $1/\sqrt{{\cal H}_{\alpha=0}}$ is equal to $|{\cal Z}^{S^{3}_{b}}|$
at $\nu=\pm 1$ in (71), the $z_{\alpha=0}$ is indeed the Bethe-vacuum
corresponding to the trivial object according to the criterion in (17).

##### Supersymmetric loop operators

The consistency condition in (19) is met when we choose the (Bethe
vacua)-to-(loop operators) map as follows:

$\displaystyle{\cal O}_{\alpha=0}=(\textrm{identity operator})\;,\quad{\cal
O}_{\alpha=1}={\cal O}_{(p,q)=(1,0)}\;.$ (77)

Then, using the dictionary in Table 1,

$\displaystyle\begin{split}&W_{\beta=0,1}(0)=1\;,\quad
W_{\beta=0}(1)=z_{0}=\sqrt{2}-1\;,\quad
W_{\beta=1}(1)=z_{1}=-\sqrt{2}-1\;.\end{split}$ (78)

From ${\cal W}_{\beta}(\alpha)$, one can compute the S-matrix using the
formula $S_{\alpha\beta}=S_{00}W_{\beta}(\alpha)W_{0}(\beta)$ and the result
is

$\displaystyle S=\begin{pmatrix}\sin\frac{3\pi}{8}&\sin\frac{\pi}{8}\\\
\sin\frac{\pi}{8}&-\sin\frac{3\pi}{8}\\\ \end{pmatrix}\;.$ (79)

Since the TQFTs, TFT${}_{\pm}[U(1)_{1}+H]$, are spin TQFTs, only the modular
$T^{2}$ matrix is well-defined and according to the dictionary in Table 1

$\displaystyle T^{2}=\begin{pmatrix}1&0\\\ 0&-1\\\ \end{pmatrix}\;.$ (80)

The modular data ($S$ and $T^{2}$) of the spin non-unitary TQFT,
TFT${}_{\pm}[U(1)_{1}+H]$, is identical to that of
$\textrm{Gal}_{d}(SU(2)_{6})/\mathbb{Z}_{2}^{f}$ with
$d=\frac{1}{2}\sin\left(\frac{3\pi}{8}\right)$.

###  3.3 $SU(2)_{k}^{\frac{1}{2}\oplus\frac{1}{2}}$ : ${\cal N}=5$ theory

The theory is defined as

$\displaystyle\begin{split}&\left(SU(2)_{k}^{\frac{1}{2}\oplus\frac{1}{2}}\right)\;\;\\\
&:=\textrm{$SU(2)$ gauge theory coupled to a half hypermultiplet and a half
twisted-hypermultiplet }\\\ &\textrm{ \quad\; in fundamental representations
with Chern-Simons level $k$}\;.\end{split}$ (81)

The theory is can be regarded as a special case of $O(M)\times Sp(2N)$ type
quiver theories (with $M=N=1$) which have ${\cal N}=5$ supersymmetry
Hosomichi:2008jb .

####  3.3.1 IR phases

##### Superconformal index

The superconformal index (163) of the ${\cal N}=5$ theory is

$\displaystyle\begin{split}&{\cal I}^{\rm
sci}_{SU(2)_{k}^{\frac{1}{2}\oplus\frac{1}{2}}}(q,\eta,\nu;s=1)\\\
&=\sum_{\mathfrak{m}=0}^{\infty}\oint_{|a|=1}\frac{da}{2\pi
ia}\Delta(\mathfrak{m},a)a^{2k\mathfrak{m}}q^{\frac{|\mathfrak{m}|}{2}}\textrm{P.E.}\big{[}f_{\rm
single}(q,\eta,a;\nu,\mathfrak{m})\big{]}\;.\end{split}$ (82)

Here we define

$\displaystyle\begin{split}&f_{\rm
single}(q,\eta,a;\nu,\mathfrak{m}):=\frac{q^{\frac{1}{4}+\frac{|\mathfrak{m}|}{2}}}{1+q^{\frac{1}{2}}}(a+a^{-1})\left((\eta
q^{1/2\nu})^{\frac{1}{2}}+(\eta q^{1/2\nu})^{-\frac{1}{2}}\right)\;,\\\
&\Delta(\mathfrak{m},a):=\frac{1}{\textrm{Sym}(\mathfrak{m})}q^{-|\mathfrak{m}|}(1-a^{2}q^{|\mathfrak{m}|})(1-a^{-2}q^{|\mathfrak{m}|})\;,\\\
&\textrm{with }\textrm{Sym}(\mathfrak{m}):=\begin{cases}2\;\quad\textrm{if
$\mathfrak{m}=0$}\;,\\\ 1\;\quad\textrm{if
$\mathfrak{m}>0$}\;.\end{cases}\end{split}$ (83)

Using the formula, one can compute the superconformal index and check that

$\displaystyle\begin{split}&{\cal I}^{\rm
sci}_{SU(2)_{k}^{\frac{1}{2}\oplus\frac{1}{2}}}(q,\eta,\nu=0;s=1)\\\
&=\begin{cases}\infty\;\quad\textrm{if $k=0$}\;\\\ 1\;\quad\textrm{if
$|k|=1$}\;\\\
1+q^{\frac{1}{2}}+\left(-1-\frac{1}{\eta}-\eta\right)q+\left(2+\eta+\frac{1}{\eta}\right)q^{\frac{3}{2}}+\ldots\;\;\;\textrm{if
$|k|\geq 2$}\;.\end{cases}\end{split}$ (84)

The higher order terms (represented by $\ldots$) depend on $k$ for $|k|\geq
2$. From the computation, one can determine the basic property of theory
appearing at the IR. The triviality of the index for $|k|=1$ implies that the
theory has a mass gap and the IR physics is described by an unitary TQFT. The
divergence of the index is a signal of emergence of a free chiral theory that
decouples with the other part of the theory Yaakov:2013fza . The non-
triviality of the index implies that the theory flows to a superconformal
field theory. In summary, from the index computation we conclude that

$\displaystyle\begin{split}&SU(2)_{k}^{\frac{1}{2}\oplus\frac{1}{2}}\xrightarrow{\rm\quad
at\;IR\quad}\begin{cases}\textrm{Contains decoupled free
chirals}\;\quad\textrm{if $|k|=0$}\;,\\\ \textrm{Unitary
TQFT}\;\quad\textrm{if $|k|=1$}\;,\\\ \textrm{3D $\mathcal{N}=5$
SCFT}\;\;\;\textrm{if $|k|\geq 2$}\;.\end{cases}\end{split}$ (85)

In the degenerate limits, $\nu\rightarrow\pm 1$ and $\eta\rightarrow 1$, the
index becomes (for $|k|\geq 2$)

$\displaystyle\begin{split}&{\cal I}^{\rm
sci}_{SU(2)_{k}^{\frac{1}{2}\oplus\frac{1}{2}}}(q,\eta,\nu=\pm
1;s=1)=1+(1-\eta^{\mp 1})q^{1/2}-(\eta^{\mp 1}-1)q+(2-\eta^{\pm 1}-\eta^{\mp
2})q^{3/2}+\ldots\;,\\\ &{\cal I}^{\rm
sci}_{SU(2)_{k}^{\frac{1}{2}\oplus\frac{1}{2}}}(q,\eta=1,\nu=\pm
1;s=1)=1\;.\end{split}$ (86)

It implies that the theory is of rank 0 and non-unitary TQFTs,
$\textrm{TFT}_{\pm}[SU(2)_{k}^{\frac{1}{2}\oplus\frac{1}{2}}]$, emerges when
$|k|\geq 2$ in the limits. In addition, we expect that they are spin TQFTs
according to (14). The
$\textrm{TFT}_{\pm}[SU(2)_{k}^{\frac{1}{2}\oplus\frac{1}{2}}]$ in the Table 2
is indeed a spin TQFT since it is given by a $\mathbb{Z}_{2}^{f}$ quotient
(fermionic anyon condensation) of a bosonic TQFT.

####  3.3.2 Non-unitary TQFTs in degenerate limits

##### Squashed three-sphere partition function

The partition function of the ${\cal N}=5$ theory is

$\displaystyle\begin{split}&{\cal
Z}^{S^{3}_{b}}_{SU(2)_{k}^{\frac{1}{2}\oplus\frac{1}{2}}}(b,m,\nu)=\int\frac{\text{d}Z}{\sqrt{2\pi\hbar}}{\cal
I}_{\hbar}(Z,m,\nu)\;\textrm{ with }\\\ &{\cal
I}_{\hbar}(Z,m,\nu)=\frac{1}{2}\big{(}2\sinh(Z)\big{)}\bigg{(}2\sinh\Big{(}\frac{2\pi
iZ}{\hbar}\Big{)}\bigg{)}\exp\left(\frac{kZ^{2}}{\hbar}\right)\\\
&\qquad\qquad\qquad\times\prod_{\epsilon_{1},\epsilon_{2}\in\\{\pm
1\\}}\Psi_{\hbar}\Big{[}\epsilon_{1}Z+\epsilon_{2}\frac{m+\nu(i\pi+\frac{\hbar}{2})}{2}+\frac{\pi
i}{2}+\frac{\hbar}{4}\Big{]}\;.\end{split}$ (87)

At $b=1$ and $m=0$, the localization integral is simplified as

$\displaystyle{\cal
Z}^{S^{3}_{b}}_{SU(2)_{k}^{\frac{1}{2}\oplus\frac{1}{2}}}(b=1,m=0,\nu)=\frac{e^{\frac{\pi
i}{12}}}{2\pi}\int\text{d}Z\frac{\sinh^{2}(Z)}{\cosh(Z)+\cosh(i\pi\nu)}\exp\left(\frac{kZ^{2}}{2\pi
i}\right).$ (88)

At $\nu=0,\pm 1$, the partition function is exactly computable and we obtain

$\displaystyle\begin{split}&\exp(-F)=\Big{|}{\cal
Z}^{S^{3}_{b}}_{SU(2)_{k}^{\frac{1}{2}\oplus\frac{1}{2}}}(b=1,m=0,\nu=0)\Big{|}=\sqrt{\frac{2}{|k|}}\sin\left(\frac{\pi}{4|k|}\right)\;,\\\
&(S_{00}\textrm{ of TFT}_{\pm})=\Big{|}{\cal
Z}^{S^{3}_{b}}_{SU(2)_{k}^{\frac{1}{2}\oplus\frac{1}{2}}}(b=1,m=0,\nu\rightarrow\pm
1)\Big{|}=\sqrt{\frac{2}{|k|}}\sin\left(\frac{\pi(2|k|-1)}{4|k|}\right)\;.\end{split}$
(89)

##### Bethe-vacua and Handle gluing operators in the degenerate limits

Since there is a $\mathbb{Z}_{2}$ symmetry $(m,\nu)\leftrightarrow(-m,-\nu)$,
we only consider the limit $\nu\rightarrow 1$. In the limit $\nu\rightarrow
1$, the asymptotic expansion coefficients ${\cal W}_{0}$ and ${\cal W}_{1}$ of
the integrand are

$\displaystyle\begin{split}&\log{\cal
I}_{\hbar}(Z,m,\nu)\xrightarrow{\quad\hbar\rightarrow
0\quad}\frac{1}{\hbar}{\cal W}_{0}(Z,m,\nu)+{\cal
W}_{1}(Z,m,\nu)+O(\hbar)\;\textrm{ with }\\\
&\mathcal{W}_{0}(Z,m,\nu=1)=-\frac{\pi^{2}}{2}+\frac{\pi
im}{2}+\frac{m^{2}}{4}\pm 2\pi
iZ+(k+1)Z^{2}+\sum_{\epsilon_{1},\epsilon_{2}\in\\{\pm
1\\}}\text{Li}_{2}(e^{-\epsilon_{1}Z-\epsilon_{2}\frac{m}{2}})\;,\\\
&\mathcal{W}_{1}(Z,m,\nu=1)=\frac{1}{2}\Big{(}\pi
i+\frac{m}{2}-\log(1-e^{\frac{m}{2}-Z})-\log(1-e^{\frac{m}{2}+Z})\Big{)}+\log(\sinh(Z))\;.\end{split}$
(90)

The Bethe vacua equation at $\nu=1$ is

$\displaystyle\exp(\partial_{Z}\mathcal{W}_{0}(Z,m,\nu=1))\bigg{|}_{Z\rightarrow\log(z),m\rightarrow\log(\eta)}=\frac{(\sqrt{\eta}-z)(\sqrt{\eta}z+1)}{(\sqrt{\eta}+z)(\sqrt{\eta}z-1)}z^{2k}=1\;.$
(91)

In the degenerate limit $\eta\rightarrow 1$, the equation simplifies as
$z^{2k}=-1$ and there are $|k|$ Bethe-vacua after taking into account of the
Weyl $\mathbb{Z}_{2}$ quotient, $z\leftrightarrow 1/z$,

$\displaystyle\textrm{Bethe-vacua :
}z_{\alpha}=(-1)^{\frac{2k-2\alpha+1}{2|k|}}\;,\quad\alpha=1,\cdots,|k|\;.$
(92)

Now, the handle gluing in the degenerate limit, $m=0,\nu=1$, is given by

$\displaystyle\mathcal{H}(z)$ $\displaystyle=\exp\left(-2{\cal
S}_{1}^{\alpha}\right)|_{Z\rightarrow\log(z),\nu\rightarrow 1,m\rightarrow
0}=\frac{1}{4}e^{-2\mathcal{W}_{1}}\partial_{Z}\partial_{Z}\mathcal{W}_{0}\Big{|}_{Z\rightarrow\log(z),\nu\rightarrow
1,m\rightarrow 0}$ $\displaystyle=\frac{2kz}{(z+1)^{2}}\;.$ (93)

The factor $1/4$ comes from $1/|\textrm{Weyl}(SU(2))|^{2}$, see (194). By
plugging the eq.(92) in the eq.(93), we have

$\displaystyle\left(\mathcal{H}_{\alpha}\textrm{ of
}{SU(2)_{k}^{\frac{1}{2}\oplus\frac{1}{2}}}\right)$ $\displaystyle=$
$\displaystyle\Bigg{(}\sqrt{\frac{2}{|k|}}\sin\left(\frac{\pi(2\alpha-1)}{4|k|}\right)\Bigg{)}^{-2},\quad\alpha=1,\ldots,|k|\;.$
(94)

The set of $\\{|S_{0\alpha}|=(\mathcal{H}_{\alpha})^{-1/2}\\}$ is identical to
the set $\\{|S_{0\alpha}|\\}$ of the $SU(2)_{4k-2}/\mathbb{Z}_{2}^{f}$ theory.
It implies that the non-unitary TQFT
$\textrm{TFT}_{\pm}[SU(2)^{\frac{1}{2}\oplus\frac{1}{2}}_{k}]$ is a Galois
conjugate of $SU(2)_{4|k|-2}/\mathbb{Z}_{2}^{f}$ with $S_{00}$ in (89) .

###  3.4 $T[SU(2)]_{k_{1},k_{2}}$ and $T[SU(2)]_{k_{1},k_{2}}/\mathbb{Z}_{2}$

$T[SU(2)]$ is the 3D theory living on the S duality domain wall in 4D
$\mathcal{N}=4$ SYM Gaiotto:2008ak . The theory is the 3d $\mathcal{N}=4$ SQED
with two fundamental hyper-multiplets. Let the four $\mathcal{N}=2$ chiral
fields in the two $\mathcal{N}=4$ hyper-multiplets be
$q_{1},q_{2},q_{3},q_{4}$ and the adjoint $\mathcal{N}=2$ chiral field in the
$\mathcal{N}=4$ vector multiplet be $\phi_{0}$. The theory has
$SU(2)^{H}\times SU(2)^{C}$ flavor symmetry at the IR as well as the $SO(4)$
R-symmetry. The charge assignments for chiral fields under the Cartan
subalgebra of the gauge and global symmetries are:

| $q_{1}$ | $q_{2}$ | $q_{3}$ | $q_{4}$ | $\phi_{0}$
---|---|---|---|---|---
$U(1)_{\text{gauge}}$ | $1$ | $1$ | $-1$ | $-1$ | $0$
$U(1)^{H}$ | $1$ | $-1$ | $1$ | $-1$ | $0$
$U(1)_{\text{axial}}$ | $\frac{1}{2}$ | $\frac{1}{2}$ | $\frac{1}{2}$ | $\frac{1}{2}$ | $-1$
$U(1)^{C}$ | $0$ | $0$ | $0$ | $0$ | $0$
Table 3: Charge assignment in $T[SU(2)]$ theory. $U(1)^{H},U(1)^{C}$ and
$U(1)_{\rm axial}$ denote the Cartans of $SU(2)^{H},SU(2)^{C}$ and axial
$U(1)\subset SO(4)_{R}$ symmetry respectively.

By gauging the two $SU(2)$s with non-zero Chern-Simons level $k_{1}$ and
$k_{2}$, we obtain infinitely many rank 0 3D $\mathcal{N}=4$ SCFTs which will
be denoted as

$\displaystyle\begin{split}&T[SU(2)]_{k_{1},k_{2}}:=\frac{T[SU(2)]}{SU(2)^{H}_{k_{1}}\times
SU(2)^{C}_{k_{2}}}\\\ &:=\left(\textrm{Gauging $SU(2)^{H}\times SU(2)^{C}$ of
$T[SU(2)]$ with Chern-Simons levels $k_{1}$ and $k_{2}$}\right)\;.\end{split}$
(95)

As argued in Gang:2018huc , the gauging does not break the supersymmetry down
to ${\cal N}=3$ thanks to the nilpotent property of the moment map operators,
$\vec{\mu}^{H}$ and $\vec{\mu}^{C}$, of the two $SU(2)$s.

The theory has $\mathbb{Z}^{H}_{2}\times\mathbb{Z}^{C}_{2}$ one-form symmetry
originating from the center symmetry $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ of
the $SU(2)^{H}\times SU(2)^{C}$ gauge group. The discrete one-form symmetry
has ’t Hooft anomaly characterized by the following bulk action555The anomaly
can be interpreted as a dependence of the partition function ${\cal Z}^{{\cal
M}_{3}=\partial X_{4}}$ on the choice of a 4-manifold $X_{4}$ having ${\cal
M}_{3}$ as a boundary. The difference between two partition functions, ${\cal
Z}^{{\cal M}_{3}=\partial X_{4}}$ and ${\cal Z}^{{\cal M}_{3}=\partial
Y_{4}}$, with two choices of 4-manifolds, $X_{4}$ and $Y_{4}$, is determined
by the bulk action $S_{\rm anom}$ as $\frac{{\cal Z}^{{\cal M}_{3}=\partial
X_{4}}}{{\cal Z}^{{\cal M}_{3}=\partial Y_{4}}}=e^{iS_{\rm anom}[{\cal
M}_{4}]}$. Here ${\cal M}_{4}=X_{4}\bigcup\overline{Y_{4}}$ is a closed
4-manifold obtained by gluing $X_{4}$ and orientation reversed $Y_{4}$ along
the common boundary ${\cal M}_{3}$.

$\displaystyle S_{\rm anom}=\pi\int_{{\cal M}_{4}}\left(k_{1}\frac{{\cal
P}(w_{2}^{H})}{2}+k_{2}\frac{{\cal P}(w_{2}^{H})}{2}+w_{2}^{H}\cup
w_{2}^{C}\right)\quad\textrm{(mod $2\pi$)}\;.$ (96)

Here $w_{2}^{H}$ ($w_{2}^{C}$) is the 2nd Stiefel-Whitney class, valued in
$H^{2}({\cal M}_{4},\mathbb{Z}_{2})$, of the
$SO(3)_{H}=(SU(2)_{H})/\mathbb{Z}_{2}$ and
$SO(3)_{C}=SU(2)_{C}/\mathbb{Z}_{2}$ bundle respectively. ${\cal P}$ is the
Pontryagin square operation,

$\displaystyle{\cal P}\;:\;\;H^{2}({\cal M}_{4},\mathbb{Z}_{2})\rightarrow
H^{4}({\cal M}_{4},\mathbb{Z}_{2})\;,$ (97)

which satisfies ${\cal P}(w_{2})=w_{2}^{2}\;(\textrm{mod }2)$. On spin
manifold ${\cal M}_{4}$, the $\frac{1}{2}{\cal P}(\omega)\in\mathbb{Z}$. The
first two terms in (96) come from the Chern-Simons action of two $SU(2)$ gauge
fields Benini:2017dus  while the last term is from the anomaly polynomial of
$T[SU(2)]$ theory Gang:2018wek . From the anomaly polynomial, one can confirm
that the following $\mathbb{Z}_{2}$ one-form symmetry is anomaly free

$\displaystyle\textrm{Anomaly free one-form $\mathbb{Z}_{2}$
symmetry}\;:\begin{cases}\mathbb{Z}_{2}^{H}\;,\quad k_{1}\in
2\mathbb{Z}\;\textrm{ and }\;k_{2}\in 2\mathbb{Z}+1\>,\\\
\mathbb{Z}_{2}^{C}\;,\quad k_{2}\in 2\mathbb{Z}\;\textrm{ and }\;k_{1}\in
2\mathbb{Z}+1\>,\\\ \mathbb{Z}_{2}^{\rm
diag}\subset\mathbb{Z}^{C}_{2}\times\mathbb{Z}^{H}_{2}\;,\quad\textrm{otherwise}\;.\end{cases}$
(98)

The $\mathbb{Z}_{2}$ one-form symmetry can be gauged and we define

$\displaystyle\begin{split}&T[SU(2)]_{k_{1},k_{2}}/\mathbb{Z}_{2}\\\
&:=\left(\textrm{Theory after gauging the anomaly free one-form
$\mathbb{Z}_{2}$ symmetry in $T[SU(2)]_{k_{1},k_{2}}$}\right)\;.\end{split}$
(99)

####  3.4.1 IR phases

##### Superconformal index

Index of the theory $T[SU(2)]_{k_{1},k_{2}}$ (or
$T[SU(2)]_{k_{1},k_{2}}/\mathbb{Z}_{2}$) is

$\displaystyle\begin{split}&{\cal I}^{\rm sci}(q,\eta,\nu;s=1)\\\
&=\sum_{\mathfrak{m}_{1},\mathfrak{m}_{2}}\oint_{|a_{1}|=1,|a_{2}|=1}\left(\prod_{i=1}^{2}\frac{\Delta(\mathfrak{m}_{i},a_{i})da_{i}}{2\pi
ia_{i}}(a_{i}(-1)^{\mathfrak{m}_{i}})^{2k_{i}\mathfrak{m}_{i}}\right){\cal
I}^{\rm
sci}_{T[SU(2)]}(a_{1},a_{2},\eta,\nu;\mathfrak{m}_{1},\mathfrak{m}_{2})\;.\end{split}$
(100)

Here the generalized superconformal index for $T[SU(2)]$ theory is

$\displaystyle\begin{split}&{\cal I}^{\rm
sci}_{T[SU(2)]}(a_{1},a_{2},\eta,\nu;\mathfrak{m}_{1},\mathfrak{m}_{2})={\cal
I}^{\rm
sci}_{T[SU(2)]}(a_{1},a_{2},\eta,\nu=0;\mathfrak{m}_{1},\mathfrak{m}_{2})\big{|}_{\eta\rightarrow\eta
q^{\frac{\nu}{2}}}\;\textrm{with}\\\ &{\cal I}^{\rm
sci}_{T[SU(2)]}(a_{1},a_{2},\eta,\nu=0;\mathfrak{m}_{1},\mathfrak{m}_{2})\\\
&=\sum_{\mathfrak{n}}\oint_{|u|=1}\frac{du}{2\pi
iu}((-1)^{\mathfrak{m}_{2}}a_{2})^{-2\mathfrak{n}}((-1)^{\mathfrak{n}}u)^{-2\mathfrak{m}_{2}}(q^{\frac{1}{2}}\eta^{-1})^{\frac{1}{2}(|\mathfrak{m}_{1}+\mathfrak{n}|+|\mathfrak{m}_{1}-\mathfrak{n}|)}\textrm{P.E.}[f_{\rm
single}]\;,\\\ &\textrm{where}\\\ &f_{\rm
single}(a_{1},a_{2},\eta,u;\mathfrak{m}_{1},\mathfrak{n})=\frac{q^{\frac{1}{4}}\sqrt{\eta}-q^{\frac{3}{4}}\sqrt{\eta^{-1}}}{1-q}q^{\frac{1}{2}|\mathfrak{m}_{1}+\mathfrak{n}|}\left(a_{1}u+\frac{1}{a_{1}u}\right)\\\
&\qquad\qquad\qquad\qquad\qquad\qquad+\frac{q^{\frac{1}{4}}\sqrt{\eta}-q^{\frac{3}{4}}\sqrt{\eta^{-1}}}{1-q}q^{\frac{1}{2}|\mathfrak{m}_{1}-\mathfrak{n}|}\left(\frac{a_{1}}{u}+\frac{u}{a_{1}}\right)+\frac{q^{\frac{1}{2}}}{1-q}\left(\frac{1}{\eta}-\eta\right)\;.\end{split}$
(101)

From Dirac quantization conditions, following monopole fluxes are allowed

$\displaystyle\mathfrak{n},\mathfrak{m}_{1},\mathfrak{m}_{2}\in\frac{1}{2}\mathbb{Z}\;\textrm{
with }\mathfrak{n}\pm\mathfrak{m}_{1}\in\mathbb{Z}\;.$ (102)

In the above formula, however, we are only summing over following monopole
fluxes

$\displaystyle\textrm{for $T[SU(2)]_{k_{1},k_{2}}$ theory :
}\mathfrak{n},\mathfrak{m}_{1},\mathfrak{m}_{2}\in\mathbb{Z}\;,$ (103)

since we are summing over $SU(2)$ bundles, i.e. $SO(3)$ bundles with trivial
$w_{2}$. The localization saddle point (168) with $SU(2)$ monopole flux
$\mathfrak{m}$ has non-trivial $w_{2}$ if and only if

$\displaystyle\mathfrak{m}\in\mathbb{Z}+\frac{1}{2}\;.$ (104)

For the theory (99) after gauging the $\mathbb{Z}_{2}$ one-form symmetry, we
also need to sum over gauge bundle with non-trivial $w_{2}^{\mathbb{Z}_{2}}$.
For the superconformal index for the $T[SU(2)]_{k_{1},k_{2}}/\mathbb{Z}_{2}$
theory (99), the summation range of monopole fluxes are

$\displaystyle\textrm{for $T[SU(2)]_{k_{1},k_{2}}/\mathbb{Z}_{2}$ theory :
}\begin{cases}\mathfrak{n}+\mathfrak{m}_{1},2\mathfrak{m}_{1},\mathfrak{m}_{2}\in\mathbb{Z}\;,\quad
k_{1}\in 2\mathbb{Z}\;\textrm{ and }\;k_{2}\in 2\mathbb{Z}+1\>,\\\
\mathfrak{n},\mathfrak{m}_{1},2\mathfrak{m}_{2}\in\mathbb{Z}\;,\quad k_{2}\in
2\mathbb{Z}\;\textrm{ and }\;k_{1}\in 2\mathbb{Z}+1\>,\\\
\mathfrak{n}+\mathfrak{m}_{1},2\mathfrak{m}_{1},\mathfrak{m}_{1}-\mathfrak{m}_{2}\in\mathbb{Z}\;,\quad\textrm{otherwise}\;.\end{cases}$
(105)

From the formulae in (100),(101),(103),(105), one can compute the
superconformal indices and check followings

$\displaystyle\begin{split}&{\cal I}^{\rm sci}(q,\eta,\nu=0;s=1)\textrm{ of
}T[SU(2)]_{k_{1},k_{2}}\;(\textrm{or
}T[SU(2)]_{k_{1},k_{2}}/\mathbb{Z}_{2})\\\ &=\begin{cases}\textrm{Non-trivial
power series in $q^{1/2}$}\;\;\;\textrm{if $|k_{1}k_{2}-1|>3$ and
min$(|k_{1}|,|k_{2}|)>1$}\>,\\\ 0\;\quad\textrm{if }|k_{1}k_{2}-1|=1\>,\\\
1\>\quad\textrm{if }|k_{1}k_{2}-1|=3\textrm{ or }(\textrm{$|k_{1}k_{2}-1|>3$
and min$(|k_{1}|,|k_{2}|)=1$})\>,\\\ 1\;(\textrm{or }2)\quad\textrm{if
}|k_{1}k_{2}-1|=2\>,\\\ \infty\>\quad\textrm{if
}|k_{1}k_{2}-1|=0\>.\end{cases}\end{split}$ (106)

For the case when $|k_{1}k_{2}-1|=1$, the index vanishes and it implies that
SUSY is spontaneously broken. For the case when $|k_{1}k_{2}-1|=2$, on the
other hand, the index for $T[SU(2)]_{k_{1}k_{2}}$ is just $1$ while the index
for $T[SU(2)]_{k_{1}k_{2}}/\mathbb{Z}_{2}$ is surprisingly $2$. It implies
that theory $T[SU(2)]_{k_{1}k_{2}}$ has a mass gap and flows to a topological
theory and the UV $\mathbb{Z}_{2}$ one-form symmetry decouples at IR, i.e. the
$\mathbb{Z}_{2}$ does not act faithfully on any IR observables. It means that
the IR TQFT actually does not have the $\mathbb{Z}_{2}$ symmetry. The index
for $T[SU(2)]_{k_{1}k_{2}}/\mathbb{Z}_{2}$ becomes $2$ just because we perform
the gauging of the decoupled (so absent) one-form symmetry by hand. At the
level of $S^{2}\times S^{1}$ partition function, the one-form gauging
procedure is

$\displaystyle{\cal Z}^{S^{2}\times
S^{1}}_{\textrm{TFT}/\mathbb{Z}_{2}}=\sum_{[\beta_{2}]\in H^{2}(S^{2}\times
S^{1},\mathbb{Z}_{2})}{\cal Z}^{S^{2}\times S^{1}}_{\rm TFT}([\beta_{2}])\;.$
(107)

Here the $[\beta_{2}]\in H^{2}(S^{2}\times
S^{1},\mathbb{Z}_{2})=\mathbb{Z}_{2}$ is the background 2-form
$\mathbb{Z}_{2}$ flat connections coupled to the one-form symmetry.
Alternatively, the RHS can be written as

$\displaystyle{\cal Z}^{S^{2}\times S^{1}}_{\rm TFT}+{\cal Z}^{S^{2}\times
S^{1}+{\cal O}^{[S^{1}]}_{\alpha_{\mathbb{Z}_{2}}}}_{\rm TFT}\;,$ (108)

where first term is the $S^{2}\times S^{1}$ partition function (with trivial
$[\beta_{2}]$) and the 2nd term is the partition function with insertion of
loop operator along the $[S^{1}]$, the generator of $H_{1}(S^{2}\times
S^{1},\mathbb{Z}_{2})=H^{2}(S^{2}\times S^{1},\mathbb{Z}_{2})$, with the
$\mathbb{Z}_{2}$ symmetry generating anyon $\alpha_{\mathbb{Z}_{2}}$. For
general TFT with a faithful $\mathbb{Z}_{2}$ one-form symmetry, the first term
is just $1$ while the 2nd term vanishes and the total partition function
${\cal Z}^{S^{2}\times S^{1}}_{\textrm{TFT}/\mathbb{Z}_{2}}$ becomes just 1 as
expected. When the $\mathbb{Z}_{2}$ one-form symmetry is decoupled at IR,
however, the 2nd term is also 1 since the anyon $\alpha_{\mathbb{Z}_{2}}$
actually becomes the trivial operator, ${\cal O}_{\alpha=0}$, at IR. So the
result, ${\cal Z}^{S^{2}\times S^{1}}_{\textrm{TFT}/\mathbb{Z}_{2}}=2$, is an
artifact due to the “gauging” of the “absent” $\mathbb{Z}_{2}$ symmetry. As we
will see below, one can actually confirm that the $\mathbb{Z}_{2}$ symmetry
act trivially on Bethe-vacua of $T[SU(2)]_{k_{1},k_{2}}$ theory for the case
when $|k_{1}k_{2}-1|=2$. In summary, from the index computation, we can
conclude that

$\displaystyle\begin{split}&T[SU(2)]_{k_{1},k_{2}}\\\
&\xrightarrow{\rm\;at\;IR\;}\begin{cases}\textrm{Non-trivial ${\cal N}=4$ SCFT
}\>\;\;\;\textrm{if $|k_{1}k_{2}-1|>3$ and min$(|k_{1}|,|k_{2}|)>1$}\>,\\\
\textrm{SUSY broken}\>\quad\textrm{if }|k_{1}k_{2}-1|=1\>,\\\ \textrm{Unitary
TQFT}\>\quad\textrm{if }|k_{1}k_{2}-1|=3\textrm{ or
}(\textrm{$|k_{1}k_{2}-1|>3$ and min$(|k_{1}|,|k_{2}|)=1$}\>,\\\
\textrm{Unitary TQFT with decoupled $\mathbb{Z}_{2}$ }\;\quad\textrm{if
}|k_{1}k_{2}-1|=2\>,\\\ \textrm{Decoupled free chirals}\;\quad\textrm{if
}|k_{1}k_{2}-1|=0\>.\end{cases}\end{split}$ (109)

In the degenerate limits ($\nu\rightarrow\pm 1$ and $\eta\rightarrow 1$), on
the other hand, the indices are (when $|k_{1}k_{2}-1|>3$ and
min$(|k_{1}|,|k_{2}|)>1$)

$\displaystyle\begin{split}&\textrm{for $T[SU(2)]_{k_{1},k_{2}}$}\textrm{
theory },\\\ &{\cal I}^{\rm sci}\left(q,\eta,\nu=\pm
1;s=1\right)=(\textrm{non-trivial power series in $q$})\;\\\ &\textrm{ and
}{\cal I}^{\rm sci}\left(q,\eta=1,\nu=\pm 1;s=1\right)=1\;,\end{split}$ (110)

while

$\displaystyle\begin{split}&\textrm{for
}T[SU(2)]_{k_{1},k_{2}}/\mathbb{Z}_{2}\textrm{ theory },\\\ &{\cal I}^{\rm
sci}\left(q,\eta,\nu=\pm 1;s=1\right)=\begin{cases}\textrm{non-trivial power
series in $q^{1/2}$}\;\;\;\textrm{if $k_{1}k_{2}\in 4\mathbb{Z}+1$}\;.\\\
\textrm{non-trivial power series in
$q$}\;\;\;\textrm{otherwise}\;.\end{cases}\\\ &\textrm{ and }{\cal I}^{\rm
sci}\left(q,\eta=1,\nu=\pm 1;s=1\right)=1\;.\end{split}$ (111)

The computation implies that the IR theories are ${\cal N}=4$ SCFTs of rank 0
and non-unitary TQFTs

$\displaystyle\textrm{TFT}_{\pm}\big{[}T[SU(2)_{k_{1},k_{2}}]\big{]}\textrm{
or }\textrm{TFT}_{\pm}\big{[}T[SU(2)_{k_{1},k_{2}}]/\mathbb{Z}_{2}\big{]}$
(112)

emerge in the degenerate limits, $\eta\rightarrow 1$ and $\nu\rightarrow\pm
1$. According to the criterion in (14), we further expect that
$\textrm{TFT}_{\pm}\big{[}T[SU(2)]_{k_{1},k_{2}}/\mathbb{Z}_{2}\big{]}$ is a
spin TQFT when $k_{1}k_{2}\in 4\mathbb{Z}+1$. The non-unitary TQFTs associated
to the ${\cal N}=4$ SCFT before the $\mathbb{Z}_{2}$ one-form symmetry gauging
are given in Table 2 and the TQFTs after the gauging are

$\displaystyle\begin{split}&\textrm{TFT}_{\pm}\big{[}T[SU(2)]_{k_{1},k_{2}}/\mathbb{Z}_{2}\big{]}=\frac{\textrm{TFT}_{\pm}\big{[}T[SU(2)]_{k_{1},k_{2}}\big{]}}{\mathbb{Z}_{2}}\\\
&=\frac{\textrm{Gal}_{d_{\pm}}(SU(2)_{|k_{1}k_{2}-1|-2})\otimes
U(1)_{2}}{\mathbb{Z}_{2}}\;,\quad\textrm{with
}d_{+}=\zeta_{|k_{1}k_{2}-1|-2}^{|k_{2}|},\;d_{-}=\zeta_{|k_{1}k_{2}-1|-2}^{|k_{1}|}\;.\end{split}$
(113)

Here $\mathbb{Z}_{2}$ is the anomaly free $\mathbb{Z}_{2}$ one-form symmetry
in $\textrm{Gal}_{d_{\pm}}(SU(2)_{|k_{1}k_{2}-1|-2})\otimes U(1)_{2}$. The
topological theory has $\mathbb{Z}^{SU(2)}_{2}\times\mathbb{Z}^{U(1)}_{2}$
one-form symmetry and the following $\mathbb{Z}_{2}$ one-form symmetry is non-
anomalous

$\displaystyle\textrm{Anomaly free one-form $\mathbb{Z}_{2}$
symmetry}\;:\begin{cases}\mathbb{Z}_{2}^{SU(2)}\;\quad\textrm{if
}k_{1}k_{2}\in 2\mathbb{Z}+1\;,\\\ \mathbb{Z}_{2}^{\rm
diag}\;\quad\textrm{otherwise}\>.\end{cases}$ (114)

When $k_{1}k_{2}\in 4\mathbb{Z}_{2}+1$, the one-form $\mathbb{Z}_{2}$ symmetry
is fermionic and the theory after the $\mathbb{Z}_{2}$ quotient becomes a spin
TQFT,

$\displaystyle\frac{\textrm{Gal}_{d_{\pm}}(SU(2)_{|k_{1}k_{2}-1|-2})\otimes
U(1)_{2}}{\mathbb{Z}_{2}}\;\textrm{ is a spin TQFT when }k_{1}k_{2}\in
4\mathbb{Z}+1\;.$ (115)

It confirms the criterion (14) combined with the superconformal index
computation (111).

####  3.4.2 Non-unitary TQFTs in degenerate limits

##### Squashed three-sphere partition function

The partition function $\mathcal{Z}_{(k_{1},k_{2})}^{S_{b}^{3}}(b,m,\nu)$ of
the $T[SU(2)]_{k_{1},k_{2}}$ theory is

$\displaystyle\begin{split}&\mathcal{Z}_{(k_{1},k_{2})}^{S_{b}^{3}}(b,m,\nu)=\int\frac{dX_{1}dX_{2}dZ}{(2\pi\hbar)^{3/2}}{\cal
I}_{\hbar}(X_{1},X_{2},Z,m,\nu)\;,\\\ &\textrm{ with }{\cal
I}_{\hbar}(X_{1},X_{2},Z,m,\nu)={\cal I}_{\hbar}^{\rm
vec}(X_{1},X_{2})\times{\cal
I}_{\hbar}^{T[SU(2)]}(X_{1},X_{2},Z,m,\nu)\;.\end{split}$ (116)

Here ${\cal I}_{\hbar}^{\rm vec}$ is the contribution from the vector
multiplet for the $SU(2)^{H}_{k_{1}}\times SU(2)^{C}_{k_{2}}$ gauging:

$\displaystyle{\cal I}_{\hbar}^{\rm
vec}(Z_{1},Z_{2})=\exp\left(\frac{k_{1}X_{1}^{2}+k_{2}X_{2}^{2}}{\hbar}\right)\prod_{i=1}^{2}\frac{1}{2}\bigg{(}2\sinh
X_{i}\bigg{)}\left(2\sinh\frac{2\pi iX_{i}}{\hbar}\right)\;.$ (117)

${\cal I}^{T[SU(2)]}_{\hbar}$ is the contribution from the $T[SU(2)]$ theory
whose squashed three-sphere partition function is

$\displaystyle\begin{split}&\mathcal{Z}_{T[SU(2)]}^{S_{b}^{3}}(b,X_{1},X_{2},m,\nu)=\int\\!\frac{dZ}{\sqrt{2\pi\hbar}}\,\,{\cal
I}_{\hbar}^{T[SU(2)]}(X_{1},X_{2},Z,m,\nu),\textrm{ where}\\\ &{\cal
I}_{\hbar}^{T[SU(2)]}(X_{1},X_{2},Z,m,\nu)=\prod_{\epsilon_{1,2}=\pm
1}\Psi_{\hbar}\left(\epsilon_{1}Z+\epsilon_{2}X_{1}+\frac{m+\nu(i\pi+\frac{\hbar}{2})}{2}+\frac{\pi
i}{2}+\frac{\hbar}{4}\right)\\\
&\qquad\quad\times\Psi_{\hbar}\left(-2\frac{m+\nu(i\pi\\!\\!+\\!\\!\frac{\hbar}{2})}{2}+\pi
i+\frac{\hbar}{2}\right)\exp\Bigl{(}-\frac{2ZX_{2}}{\hbar}-\frac{\left(\pi
i+\frac{\hbar}{2}\right)}{\hbar}\frac{m+\nu(i\pi\\!\\!+\\!\\!\frac{\hbar}{2})}{2}\Bigr{)}\;.\end{split}$
(118)

One of the non-trivial consistency checks is the mirror property

$\displaystyle\mathcal{Z}_{T[SU(2)]}^{S_{b}^{3}}(b,X_{1},X_{2},m=0,\nu)=\mathcal{Z}_{T[SU(2)]}^{S_{b}^{3}}(b,X_{2},X_{1},m=0,-\nu)\;,$
(119)

which we have checked numerically for various values of $X_{1}$, $X_{2}$, and
$\nu$. Particularly for $b=1,m=0,\nu=0$, we have Gulotta:2011si ;
Benvenuti:2011ga ; Nishioka:2011dq  (see also Appendix C.1)

$\displaystyle\begin{split}\mathcal{Z}^{S_{b}^{3}}_{T[SU(2)]}(b=1,X_{1},X_{2},m=0,\nu=0)=\frac{e^{\frac{2\pi
i}{3}}}{2}\frac{\sin(\frac{X_{1}X_{2}}{\pi})}{\sinh(X_{1})\sinh(X_{2})}\;.\end{split}$
(120)

As in (88), this localization (116) also simplified at $b=1$, $m=0$, and is
exactly computable at $\nu=0,\pm 1$ (see Appendix C.2).

$\displaystyle\begin{split}\exp(-F)&=\Big{|}{\cal
Z}^{S^{3}_{b}}_{(k_{1},k_{2})}(b=1,m=0,\nu=0)\Big{|}=\sqrt{\frac{1}{|k_{1}k_{2}-1|}}\sin\left(\frac{\pi}{|k_{1}k_{2}-1|}\right)\;,\\\
(S_{00}\textrm{ of TFT}_{\pm})&=\Big{|}{\cal
Z}^{S^{3}_{b}}_{(k_{1},k_{2})}(b=1,m=0,\nu\rightarrow\pm 1)\Big{|}\\\
&=\bigg{\\{}\begin{array}[]{ll}\sqrt{\frac{1}{|k_{1}k_{2}-1|}}\sin\left(\frac{\pi|k_{2}|}{|k_{1}k_{2}-1|}\right),\quad\nu\rightarrow+1\;,\\\
\sqrt{\frac{1}{|k_{1}k_{2}-1|}}\sin\left(\frac{\pi|k_{1}|}{|k_{1}k_{2}-1|}\right),\quad\nu\rightarrow-1\;.\end{array}\end{split}$
(121)

##### Bethe-vacua and Handle gluing operators in the degenerate limits

The asymptotic expansions, ${\cal W}_{0}$ and ${\cal W}_{1}$, of the
localization integral are

$\displaystyle\begin{split}&\log{\cal
I}_{\hbar}(X_{1},X_{2},Z,m,\nu)\xrightarrow{\quad\hbar\rightarrow
0\quad}\frac{1}{\hbar}{\cal W}_{0}(X_{1},X_{2},Z,m,\nu)+{\cal
W}_{1}(X_{1},X_{2},Z,m,\nu)+O(\hbar)\;,\\\ &\textrm{where}\\\ &{\cal
W}_{0}=(k_{1}+1)X_{1}^{2}+k_{2}X_{2}^{2}\pm 2\pi iX_{1}\pm 2\pi
iX_{2}-2X_{2}Z+Z^{2}+\sum_{\epsilon_{1},\epsilon_{2}=\pm
1}\textrm{Li}_{2}(e^{\epsilon_{1}X_{1}+\epsilon_{2}Z-\frac{m+i\pi\nu}{2}-\frac{\pi
i}{2}})\;,\\\ &{\cal
W}_{1}=\frac{i\pi(\nu^{2}-\nu+1)}{2}+\frac{2m\nu-m}{4}+\frac{1}{4}\sum_{\epsilon_{1},\epsilon_{2}=\pm
1}(\nu-1)\log\left(1+e^{\epsilon_{1}X_{1}+\epsilon_{2}Z-\frac{m+i\pi\nu}{2}-\frac{\pi
i}{2}}\right)\\\
&\quad\quad\;-\frac{\nu}{2}\log\left(1+e^{m+i\pi\nu}\right)+\log(\sinh
X_{1})+\log(\sinh X_{2})\;.\end{split}$ (122)

In the ${\cal W}_{0}$ above, we have ignored terms which are independent on
$X_{1},X_{2}$ and $Z$. By extremizing the twisted superpotential

$\displaystyle\exp(\partial_{X_{1}}{\cal W}_{0})=\exp(\partial_{X_{2}}{\cal
W}_{0})=\exp(\partial_{Z}{\cal W}_{0})=1\;,$ (123)

we have following Bethe-vacua equations

$\displaystyle\begin{split}&\frac{x_{1}^{2k_{1}}(wx_{1}+iz)(wx_{1}z+i)}{(wz+ix_{1})(w+ix_{1}z)}=\frac{x_{2}^{2k_{2}}}{z^{2}}=\frac{(wz+ix_{1})(wx_{1}z+i)}{x_{2}^{2}(wx_{1}+iz)(w+ix_{1}z)}=1\;,\\\
&\textrm{where
}x_{1}=e^{X_{1}},\;x_{2}=e^{X_{2}},\;z=e^{Z},\;w=e^{\frac{m+i\pi\nu}{2}}\;.\end{split}$
(124)

At generic choice of $w$, there are $2\times\big{|}(|k_{1}k_{2}-1|-1)\big{|}$
Bethe-vacua,

$\displaystyle\\{(x_{1},x_{2},z)=((x^{*}_{1})_{n,\pm},(x^{*}_{2})_{n,\pm},z^{*}_{n,\pm})\\}_{n=1}^{\big{|}|k_{1}k_{2}-1|-1\big{|}}\;,$
(125)

after removing the unphysical solutions, which are invariant under a non-
trivial subgroup of the Weyl $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ acting as
$x_{i}\rightarrow 1/x_{i}$, and quotienting by the Weyl group. The one-form
symmetry $\mathbb{Z}^{H}_{2}\times\mathbb{Z}^{C}_{2}$, on the other hand, acts
on the Bethe-vacua in the following way

$\displaystyle\mathbb{Z}_{2}^{H}\;:\;x_{1}\rightarrow\pm
x_{1}\;,\quad\mathbb{Z}_{2}^{C}\;:\;x_{2}\rightarrow\pm x_{2}\;.$ (126)

When $|k_{1}k_{2}-1|=2$, the anomaly free $\mathbb{Z}_{2}=\mathbb{Z}_{2}^{\rm
diag}\subset\mathbb{Z}_{2}^{H}\times\mathbb{Z}_{2}^{C}$ act trivially (modulo
Weyl $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$) on the Bethe-vacua, i.e.

$\displaystyle x_{1}^{*},x_{2}^{*}\in\\{i,-i\\}\;.$ (127)

This explains why we obtain $2$ (instead of $1$) in the index computation of
$T[SU(2)]_{k_{1},k_{2}}/\mathbb{Z}_{2}$ theory when $|k_{1}k_{2}-1|=2$, see
the paragraph below (106).

The handle gluing operator is

$\displaystyle\mathcal{H}(x_{1},x_{2},z;m,\nu)=\frac{1}{16}\exp(-2{\cal
W}_{1})\det_{i,j}\left(\partial_{i}\partial_{j}{\cal W}_{0}\right)\;.$ (128)

The set of the handle gluing operators evaluated at the Bethe-vacua in the
degenerate limits are

$\displaystyle\bigg{\\{}{\cal
H}\big{(}(x^{*}_{1})_{n,\pm},(x^{*}_{2})_{n,\pm},(z^{*})_{n,\pm}\big{)}\bigg{\\}}_{n=1}^{\big{|}|k_{1}k_{2}-1|-1\big{|}}\xrightarrow{\quad(m,\nu)\rightarrow(0,\pm
1)\quad}\bigg{\\{}\frac{|k_{1}k_{2}-1|}{\sin^{2}\left(\frac{n\pi}{|k_{1}k_{2}-1|}\right)}^{\otimes
2}\bigg{\\}}_{n=1}^{\big{|}|k_{1}k_{2}-1|-1\big{|}}\;.$ (129)

###  3.5 ${T[SU(2)]}/{SU(2)^{\rm diag}_{k}}$ and ${T[SU(2)]}/{``PSU(2)^{\rm
diag}_{k}"}$

Let us consider

$\displaystyle\frac{T[SU(2)]}{SU(2)^{\rm diag}_{k}}=\left(\textrm{Gauging
diagonal $SU(2)^{\rm diag}$ of $T[SU(2)]$ with Chern-Simons level
$k$}\right)\;.$ (130)

Thanks to the nilpotency of the moment map $\vec{\mu}^{\rm diag}$ for the
$SU(2)^{\rm diag}$ flavor symmetry, the theory remains ${\cal N}=4$ theory
even after the gauging with non-zero $k$ Gang:2018huc ; Garozzo:2019ejm .

The theory has a $\mathbb{Z}_{2}$ one-form symmetry which corresponds to the
center group of the gauged $SU(2)^{\rm diag}$ symmetry. The ’t Hooft anomaly
polynomial for the one-form symmetry is

$\displaystyle S_{\rm anom}=\pi\int_{{\cal M}_{4}}\left(k\frac{{\cal
P}(w_{2})}{2}\right)\quad\textrm{(mod $2\pi$)}\;.$ (131)

Thus, the one-form symmetry is anomalous for $k=({\rm odd})$ while non-
anomalous for $k=({\rm even})$. For odd $k$, the theory can be tensored with a
topological theory $U(1)_{2}=SU(2)_{1}$, which also has anomalous
$\mathbb{Z}_{2}$ one-form symmetry, and the diagonal $\mathbb{Z}_{2}$ one-form
symmetry becomes non-anomalous. We define

$\displaystyle\begin{split}&\frac{T[SU(2)]}{``PSU(2)^{\rm diag}_{k}"}\\\
&:=\begin{cases}\textrm{Gauging the $\mathbb{Z}_{2}$ one-form symmetry of
$\frac{T[SU(2)]}{SU(2)^{\rm diag}_{k}}$ },\quad{\rm even\;}k\;,\\\
\textrm{Gauging the (diagonal) $\mathbb{Z}_{2}$ one-form symmetry of
$\left(\frac{T[SU(2)]}{SU(2)^{\rm diag}_{k}}\otimes U(1)_{2}\right)$
},\quad{\rm odd\;}k\;.\\\ \end{cases}\end{split}$ (132)

####  3.5.1 IR phases

##### Superconformal index

The superconformal index of the $\frac{T[SU(2)]}{SU(2)^{\rm diag}_{k}}$ (or
$\frac{T[SU(2)]}{``PSU(2)^{\rm diag}_{k}"}$) theory is

$\displaystyle\begin{split}&{\cal I}^{\rm sci}(q,\eta,\nu;s=1)\\\
&=\sum_{\mathfrak{m}}\oint_{|a|=1}\frac{\Delta(\mathfrak{m},a)da}{2\pi
ia}\,(a(-1)^{\mathfrak{m}})^{2k\mathfrak{m}}\,{\cal I}^{\rm
sci}_{T[SU(2)]}(a,a,\eta,\nu;\mathfrak{m},\mathfrak{m})\;,\end{split}$ (133)

where ${\cal I}^{\rm sci}_{T[SU(2)]}$ is the index of $T[SU(2)]$ theory given
in (101). From the Dirac quantization conditions, the following monopole
fluxes are allowed

$\displaystyle\mathfrak{n},\mathfrak{m}\in\frac{\mathbb{Z}}{2}\;\textrm{ with
}\mathfrak{n}\pm\mathfrak{m}\in\mathbb{Z}\;.$ (134)

The summation range of monopole fluxes is

$\displaystyle\begin{split}&\textrm{for $\frac{T[SU(2)]}{SU(2)^{\rm
diag}_{k}}$ \;\; : \;\; }\mathfrak{n},\mathfrak{m}\in\mathbb{Z}\;,\\\
&\textrm{for $\frac{T[SU(2)]}{``PSU(2)^{\rm diag}_{k}"}$ \;\; : \;\;
}\mathfrak{n},\mathfrak{m}\in\frac{\mathbb{Z}}{2}\;\textrm{ with
}\mathfrak{n}-\mathbf{m}\in\mathbb{Z}\;.\end{split}$ (135)

From superconformal index computation Garozzo:2019ejm ; Beratto:2020qyk ,

$\displaystyle\begin{split}&{\cal I}^{\rm sci}(q,\eta,\nu=0;s=1)\textrm{ for
$\frac{T[SU(2)]}{SU(2)^{\rm diag}_{k}}$ or $\frac{T[SU(2)]}{``PSU(2)^{\rm
diag}_{k}"}$ }\\\ &=\begin{cases}\textrm{1}\;,\quad|k|<2\;,\\\
\infty\;,\quad|k|=2\;,\\\ \textrm{non-trivial power series in
$q^{1/2}$}\;,\quad|k|>2\;.\ \\\ \end{cases}\end{split}$ (136)

we expect that

$\displaystyle\begin{split}&\textrm{$\frac{T[SU(2)]}{SU(2)^{\rm diag}_{k}}$ or
$\frac{T[SU(2)]}{``PSU(2)^{\rm diag}_{k}"}$ }\\\ &\xrightarrow{\quad\textrm{at
IR}\quad}\begin{cases}\textrm{Unitary topological field
theory}\;\quad\textrm{if $|k|<2$}\;,\\\ \textrm{Decoupled free
chirals}\;\quad\textrm{if $|k|=2$}\;,\ \\\ \textrm{3D $\mathcal{N}=4$
SCFT}\;\quad\textrm{if $|k|>2$}\;.\ \\\ \end{cases}\end{split}$ (137)

##### Superconformal indices in degenerate limits

In the degenerate limits ($\nu\rightarrow\pm 1$ and $\eta\rightarrow 1$), on
the other hand, the indices are (when $|k|>2$)

$\displaystyle\begin{split}&\textrm{for $\frac{T[SU(2)]}{SU(2)_{k}^{\rm
diag}}$}\textrm{ theory },\\\ &{\cal I}^{\rm sci}\left(q,\eta,\nu=\pm
1;s=1\right)=(\textrm{non-trivial power series in $q$})\;\\\ &\textrm{ and
}{\cal I}^{\rm sci}\left(q,\eta=1,\nu=\pm 1;s=1\right)=1\;,\end{split}$ (138)

while

$\displaystyle\begin{split}&\textrm{for }\frac{T[SU(2)]}{``PSU(2)^{\rm
diag}_{k}"}\textrm{ theory },\\\ &{\cal I}^{\rm sci}\left(q,\eta,\nu=\pm
1;s=1\right)=\begin{cases}\textrm{non-trivial power series in
$q^{1/2}$}\;\;\;\textrm{if $k\in 4\mathbb{Z}$}\;,\\\ \textrm{non-trivial power
series in $q$}\;\;\;\textrm{otherwise}\;,\end{cases}\\\ &\textrm{ and }{\cal
I}^{\rm sci}\left(q,\eta=1,\nu=\pm 1;s=1\right)=1\;.\end{split}$ (139)

The computation implies that non-unitary TQFTs,
$\textrm{TFT}_{\pm}[\frac{T[SU(2)]}{SU(2)_{k}^{\rm diag}}]$ and
$\textrm{TFT}_{\pm}[\frac{T[SU(2)]}{``PSU(2)^{\rm diag}_{k}"}]$, emerge in the
degenerate limits and that $\textrm{TFT}_{\pm}[\frac{T[SU(2)]}{``PSU(2)^{\rm
diag}_{k}"}]$ are spin TQFTs when $k\in 4\mathbb{Z}$. From Table 2, one can
see that $\textrm{TFT}_{\pm}[\frac{T[SU(2)]}{``PSU(2)^{\rm diag}_{k}"}]$ at
$|k|=4$ are indeed spin TQFTs.

####  3.5.2 Non-unitary TQFTs in degenerate limits

##### Squashed three-sphere partition function

The squashed three-sphere partition function of the ${T[SU(2)]}/{SU(2)^{\rm
diag}_{k}}$ theory is realized as

$\displaystyle\mathcal{Z}_{\text{diag}_{k}}^{S_{b}^{3}}(b,m,\nu)\equiv\frac{1}{2}\int\frac{\text{d}X}{\sqrt{2\pi\hbar}}\big{(}2\sinh(X)\big{)}\bigg{(}2\sinh\Big{(}\frac{2\pi
iX}{\hbar}\Big{)}\bigg{)}e^{\frac{kX^{2}}{\hbar}}\mathcal{Z}^{S_{b}^{3}}_{T[SU(2)]}(b,X,X,m,\nu)\;,$

where $\mathcal{Z}^{S_{b}^{3}}_{T[SU(2)]}$ is given in (118). As in (88), this
localization (LABEL:Diagona_gauging_partition_function) also simplified at
$b=1$, $m=0$, and is exactly computable at $\nu=0,\pm 1$ (see Appendix C.3).

$\displaystyle\begin{split}&\exp(-F)=\Big{|}{\cal
Z}^{S^{3}_{b}}_{\text{diag}_{k}}(b=1,m=0,\nu=0)\Big{|}=\frac{1}{\sqrt{8(|k|-2)}}-\frac{1}{\sqrt{8(|k|+2)}}\;,\\\
&(S_{00}\textrm{ of TFT}_{\pm})=\Big{|}{\cal
Z}^{S^{3}_{b}}_{\text{diag}_{k}}(b=1,m=0,\nu\rightarrow\pm
1)\Big{|}=\frac{1}{\sqrt{8(|k|-2)}}+\frac{1}{\sqrt{8(|k|+2)}}\;.\end{split}$
(141)

##### Bethe-vacua and Handle gluing operators in the degenerate limits

Similar to (122), the asymptotic expansions $\mathcal{W}_{0}$ and
$\mathcal{W}_{1}$ of the localization integral are

$\displaystyle\begin{split}&\mathcal{W}_{0}=(k+1)X^{2}\pm 2\pi
iX-2XZ+Z^{2}+\sum_{\epsilon_{1},\epsilon_{2}=\pm
1}\text{Li}_{2}(e^{\epsilon_{1}X+\epsilon_{2}Z-\frac{m+i\pi\nu}{2}-\frac{\pi
i}{2}})\;,\\\ &\mathcal{W}_{1}=\frac{\pi
i(\nu^{2}-\nu+1)}{2}+\frac{2m\nu-m}{4}+\frac{1}{4}\sum_{\epsilon_{1},\epsilon_{2}=\pm
1}(\nu-1)\log\big{(}e^{\epsilon_{1}X+\epsilon_{2}Z-\frac{m+i\pi\nu}{2}-\frac{\pi
i}{2}}\big{)}\\\
&\qquad-\frac{\nu}{2}\log(1+e^{m+i\pi\nu})+\log(\sinh(X))\;.\end{split}$ (142)

We have ignored terms which are independent on $X$ and $Z$ in the expression
for $\mathcal{W}_{0}$ given above. By extremizing the twisted superpotential

$\displaystyle\exp(\partial_{X}\mathcal{W}_{0})=\exp(\partial_{Z}\mathcal{W}_{0})=1\;,$
(143)

we have the Bethe-vacua equations

$\displaystyle\begin{split}&\frac{x^{2k}(wx+iz)(i+wxz)}{z^{2}(x-iwz)(iw-
xz)}=\frac{(x-iwz)(i+wxz)}{x^{2}(z-iwx)(w+ixz)}=1\;,\\\ &\text{where}\quad
x=e^{X},\,z=e^{Z}\textrm{ and }w=e^{\frac{m+\pi i\nu}{2}}\;.\end{split}$ (144)

At generic choice of $w$, there are $2\times(2|k|+2)$ Bethe-vacua,

$\displaystyle{(x,z)=(x^{*}_{n,\pm},z^{*}_{n,\pm})}_{n=1}^{2|k|+2}\;,$ (145)

after removing the unphysical solutions, which are invariant under
$x_{i}\rightarrow 1/x_{i}$, and quotienting by the Weyl group. The handle
gluing operator is

$\displaystyle\mathcal{H}(x,z;m,\nu)=\frac{1}{4}\exp(-2\mathcal{W}_{1})\det_{i,j}(\partial_{i}\partial_{j}\mathcal{W}_{0})\;.$
(146)

The set of the handle gluing operators evaluated at the Bethe-vacua in the
degenerate limits are

$\displaystyle\begin{split}&\bigg{\\{}{\cal
H}\big{(}x^{*}_{n,\pm},z^{*}_{n,\pm}\big{)}\bigg{\\}}_{n=1}^{2|k|+2}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\xrightarrow{\quad(m,\nu)\rightarrow(0,\pm
1)\quad}\\\
&\bigg{\\{}\frac{1}{\sqrt{2A_{k}}}^{\\!\\!\\!\\!\\!\\!\otimes(|k|-3)}\\!\\!\\!\\!\\!,\\!\frac{1}{\sqrt{2B_{k}}}^{\\!\\!\\!\\!\\!\\!\otimes(|k|+1)}\\!\\!\\!\\!,\Big{(}\frac{1}{\sqrt{8A_{k}}}\\!\\!+\\!\\!\frac{1}{\sqrt{8B_{k}}}\Big{)}^{\otimes
2}\\!\\!,\Big{(}\frac{1}{\sqrt{8A_{k}}}\\!\\!-\\!\\!\frac{1}{\sqrt{8B_{k}}}\Big{)}^{\otimes
2}\bigg{\\}}\;,\end{split}$ (147)

where we define $A_{k}:=|k|-2$, $B_{k}:=|k|+2$.

For $|k|>2$, as concluded in (137), the theory $T[SU(2)]/SU(2)^{\rm diag}_{k}$
lands on 3D ${\cal N}=4$ SCFT of rank 0 at the end of RG. According to the
dictionary in Table 1, the above set should be equal to the set of
$\\{S_{0\alpha}^{2}\\}$ for a non-unitary TQFT for the case when $|k|>2$.

#####  $|k|=3$ case

The set of handle gluing (and $S_{00}$ in (141)) is identical to the set of
$\\{|S_{0\alpha}^{-2}|\\}$ (and $S_{00}$) for the (Lee-Yang)$\otimes$(Lee-
Yang)$\otimes U(1)_{2}$. From the computation, we arrive the conclusion in
Table 2.

#####  $|k|=4$ case

The set is identical to the set of $\\{|S_{0\alpha}^{-2}|\\}$ for
$\left(SU(2)_{10}\times SU(2)_{2}\right)/{\mathbb{Z}_{2}^{\rm diag}}$, see
(259). Combined with the computation of $S_{00}$ in (141), we arrive the
conclusion in Table 2.

#####  $|k|=5$ case

The set of handle gluing is identical to the set of $\\{|S_{0\alpha}^{-2}|\\}$
for $(G_{2})_{3}\times U(1)_{-2}$, where $G_{2}$ is a exceptional group with
dimension 14. Combined with the computation of $S_{00}$ in (141), we arrive
the conclusion in Table 2.

#####  $|k|>6$ case

For the cases, we could not identify
$\textrm{TFT}_{\pm}\left[\frac{T[SU(2)]}{SU(2)^{\rm diag}_{k}}\right]$ with
previously known non-unitary TQFTs in the literature. It would be an
interesting future work to better understand this novel series of non-unitary
TQFTs.

###  3.6 Dualities among rank 0 theories

In this paper, we introduce a pair of non-unitary TQFTs, TFT${}_{\pm}[{\cal
T}_{\rm rank0}]$, as an invariant of 3D rank 0 ${\cal N}=4$ SCFT ${\cal
T}_{\rm rank0}$. One natural question is

$\displaystyle\begin{split}&\textrm{Q : How powerful are TFT${}_{\pm}[{\cal
T}_{\rm rank0}]$s in distinguishing ${\cal T}_{\rm rank0}$? }\\\
&\quad\quad\textrm{i.e.\ are TFT${}_{\pm}[{\cal T}_{\rm rank0}]$s different
for different ${\cal T}_{\rm rank0}$s?}\end{split}$ (148)

In this subsection, we provide evidences that the answer is affirmative. This
is by examination of the duality between two ${\cal T}_{\rm rank\;0}$s whose
associated non-unitary TQFTs are identical. Having common non-unitary TQFTs in
the degenerate limits, $m=0$ and $\nu=\pm 1$, automatically implies that
equivalence of all supersymmetric partition function ${\cal Z}^{\mathbb{B}}$
in the limits. Here we will confirm that the equivalence still holds at
general values of $m$ and $\nu$.

##### Duality among $(U(1)_{|k|=1}+H)$,
$SU(2)_{|k|=2}^{\frac{1}{2}\oplus\frac{1}{2}}$ and
$\frac{T[SU(2)]_{k_{1}=3,k_{2}=3}}{\mathbb{Z}_{2}}$

According to Table 2, the non-unitary TQFTs associated to the 3 theories are
all identical to the following theory

$\displaystyle\textrm{TFT}_{\pm}=\textrm{Gal}_{d=\sin\left(\frac{3\pi}{8}\right)}(SU(2)_{6})/\mathbb{Z}_{2}^{f}\;.$
(149)

Using the explicit formulas in (61),(82),(83),(100),(101) and (105), one can
confirm that the superconformal indices for the 3 theories are all equal to

$\displaystyle\begin{split}&{\cal I}^{\rm sci}(q,\eta,\nu=1;s=1)\\\
&=1+q^{\frac{1}{2}}-\left(1+\eta+\frac{1}{\eta}\right)q+\left(2+\eta+\frac{1}{\eta}\right)q^{\frac{3}{2}}-\left(2+\eta+\frac{1}{\eta}\right)q^{2}+\ldots\;.\end{split}$
(150)

One can check that computations of various other supersymmetric partition
functions also support the duality.

##### Duality between $({\cal T}_{\rm min})^{\otimes 2}\otimes U(1)_{2}$ and
$\frac{T[SU(2)]}{SU(2)^{\rm diag}_{|k|=3}}$

One can check that Gang:2018huc ; Garozzo:2019ejm

$\displaystyle\begin{split}&\left({\cal I}^{\rm sci}(q,\eta,\nu=0;s=1)\textrm{
of }\frac{T[SU(2)]}{SU(2)_{|k|=3}^{\rm diag}}\textrm{ given in \eqref{index
for T[SU(2)]k1k2-2},\eqref{index for T[SU(2)]k},\eqref{index for
T[SU(2)]k-2}}\right)\\\ &=\left({\cal I}^{\rm sci}(q,\eta,\nu=0;s=1)\textrm{
of }\frac{T[SU(2)]}{``PSU(2)^{\rm diag}_{|k|=3}"}\textrm{ given in
\eqref{index for T[SU(2)]k1k2-2},\eqref{index for T[SU(2)]k},\eqref{index for
T[SU(2)]k-2}}\right)\\\ &=\bigg{(}{\cal I}^{\rm sci}(q,\eta,\nu=0;s=1)\textrm{
of }{\cal T}_{\rm min}\textrm{ given in \eqref{sci for Tmin}}\bigg{)}^{2}\;\\\
&=1-2q+2\left(\eta+\frac{1}{\eta}\right)q^{3/2}-3q^{2}+\left(2+\eta^{2}+\frac{1}{\eta^{2}}\right)q^{3}-4\left(\eta+\frac{1}{\eta}\right)q^{7/2}+\ldots\;.\\\
\end{split}$ (151)

From the index computation, it is tempting to identify both
$\frac{T[SU(2)]}{SU(2)_{|k|=3}^{\rm diag}}$ and $\frac{T[SU(2)]}{``PSU(2)^{\rm
diag}_{|k|=3}"}$ with $({\cal T}_{\rm min})^{\otimes 2}$. But this cannot be
true since $\frac{T[SU(2)]}{SU(2)_{|k|=3}^{\rm diag}}$ has the one-form
$\mathbb{Z}_{2}$ symmetry while the other two theories do not. Further, the
Witten indices of the three theories do not match:

$\displaystyle\begin{split}&\left(\textrm{Witten index of $({\cal T}_{\rm
min})^{\otimes 2}$}\right)=(\textrm{Witten index of $({\cal T}_{\rm
min})$})^{2}=4\;,\\\ &\left(\textrm{Witten index of
$\frac{T[SU(2)]}{``PSU(2)^{\rm diag}_{|k|=3}"}$}\right)=4\;,\;\textrm{but
}\left(\textrm{Witten index of $\frac{T[SU(2)]}{SU(2)^{\rm
diag}_{|k|=3}}$}\right)=8\;.\end{split}$

From the computation of superconformal indices, Witten indices, and one-form
$\mathbb{Z}_{2}$ symmetry matching, we propose that

$\displaystyle\frac{T[SU(2)]}{SU(2)^{\rm diag}_{|k|=3}}=({\cal T}_{\rm
min})^{\otimes 2}\otimes U(1)_{2}\;\;\;\textrm{at IR}.$ (152)

The additional $U(1)_{2}$ theory does not contribute to the superconformal
index since there is no non-trivial local operator in the topological sector.
The additional topological sector, however, doubles the Witten index since it
has two ground states on two-torus and provide the one-form $\mathbb{Z}_{2}$
symmetry. The proposal is also compatible with following fact

$\displaystyle\textrm{TFT}_{\pm}\bigg{[}\frac{T[SU(2)]}{SU(2)^{\rm
diag}_{|k|=3}}\bigg{]}=\left(\textrm{TFT}_{\pm}[{\cal T}_{\rm
min}]\right)^{\otimes 2}\otimes U(1)_{2}\;,$ (153)

which is obvious from Table 2. Using the duality, we can also confirm that

$\displaystyle\begin{split}&\frac{T[SU(2)]}{``PSU(2)^{\rm
diag}_{|k|=3}"}:=\left(\frac{T[SU(2)]}{SU(2)^{\rm diag}_{|k|=3}}\otimes
U(1)_{2}\right)/\mathbb{Z}^{\rm diag}_{2}=\left(({\cal T}_{\rm min})^{\otimes
2}\otimes U(1)_{2}\otimes U(1)_{2}\right)/\mathbb{Z}^{\rm diag}_{2}\\\
&=({\cal T}_{\rm min})^{\otimes 2}\otimes\frac{U(1)_{2}\otimes
U(1)_{2}}{\mathbb{Z}^{\mathop{\rm diag}\nolimits}_{2}}\simeq({\cal T}_{\rm
min})^{\otimes 2}\;.\end{split}$ (154)

The theory $(U(1)_{2}\otimes U(1)_{2})/\mathbb{Z}_{2}^{\rm diag}$ is an
almost-trivial theory, whose partition function on any closed 3-manifold is a
pure phase factor. Throughout this paper, we have ignored the overall phase
factor of the partition function and thus will also ignore such a decoupled
almost trivial theory. The duality above has natural interpretation in terms
of the 3D-3D correspondence for once-punctured torus bundles Terashima:2011qi
; Gang:2013sqa , for which readers are referred to Appendix B for details.

##### Duality between $SU(2)_{|k|=3}^{\frac{1}{2}\oplus\frac{1}{2}}$ and
$\frac{T[SU(2)]}{"PSU(2)^{\rm diag}_{|k|=4}"}$

The superconformal index is

$\displaystyle\begin{split}&\left({\cal I}^{\rm sci}(q,\eta,\nu=0;s=1)\textrm{
of }\frac{T[SU(2)]}{SU(2)_{|k|=4}^{\rm diag}}\right)\\\
&=1-q+\left(\eta+\frac{1}{\eta}\right)q^{3/2}-q^{2}-2\left(\eta+\frac{1}{\eta}\right)q^{5/2}+2\left(3+\eta^{2}+\frac{1}{\eta^{2}}\right)q^{3}+\ldots\;,\\\
&\left({\cal I}^{\rm sci}(q,\eta,\nu=0;s=1)\textrm{ of
}\frac{T[SU(2)]}{``PSU(2)^{\rm diag}_{|k|=4}"}\right)\\\
&=1+q^{1/2}-\left(1+\eta+\frac{1}{\eta}\right)q+\left(2+\eta+\frac{1}{\eta}\right)q^{3/2}-q^{2}-\left(\eta+\frac{1}{\eta}\right)\left(2+\eta+\frac{1}{\eta}\right)q^{5/2}+\ldots\;.\end{split}$
(155)

Two indices are different unlike in the $|k|=3$ case. Surprisingly, the index
computation shows that the two theories actually have different amount of
supersymmetries Evtikhiev:2017heo :

$\displaystyle\begin{split}&\frac{T[SU(2)]}{SU(2)_{|k|=4}^{\rm diag}}\textrm{
has ${\cal N}=4$ SUSY}\;,\\\ &\frac{T[SU(2)]}{``PSU(2)^{\rm
diag}_{|k|=4}"}\textrm{ has ${\cal N}=5$ SUSY}\;.\end{split}$ (156)

We can thus conclude that supersymmetry is enhanced under the $\mathbb{Z}_{2}$
one-form symmetry gauging in (132). We further claim that the ${\cal N}=5$
theory is actually dual to the following theory with manifest ${\cal N}=5$
supersymmetry:

$\displaystyle\left(\frac{T[SU(2)]}{``PSU(2)^{\rm
diag}_{|k|=4}"}\right)=\left(SU(2)_{|k|=3}^{\frac{1}{2}\oplus\frac{1}{2}}\textrm{
in \eqref{SU(2)k1/2+1/2}}\right)\;\textrm{ at IR}.$ (157)

The proposal can be checked using the superconformal index and various other
supersymmetric partition functions. Neither theories in the duality has any
one-form $\mathbb{Z}_{2}$ symmetry. The duality is also compatible with Table
2 since

$\displaystyle\begin{split}&\textrm{TFT}_{\pm}\bigg{[}\frac{T[SU(2)]}{``PSU(2)^{\rm
diag}_{|k|=4}"}\bigg{]}=\textrm{TFT}_{\pm}\bigg{[}\frac{T[SU(2)]}{SU(2)^{\rm
diag}_{|k|=4}}\bigg{]}/\mathbb{Z}_{2}\\\
&=\left(\frac{\textrm{Gal}_{\zeta_{10}^{7}}(SU(2)_{10})\otimes
SU(2)_{2}}{\mathbb{Z}_{2}^{\rm
diag}}\right)/\mathbb{Z}_{2}=\left(\frac{\textrm{Gal}_{\zeta_{10}^{7}}(SU(2)_{10})}{\mathbb{Z}_{2}}\right)\otimes\left(\frac{SU(2)_{2}}{\mathbb{Z}_{2}}\right)\\\
&\simeq\left(\frac{\textrm{Gal}_{\zeta_{10}^{7}}(SU(2)_{10})}{\mathbb{Z}_{2}}\right)=\textrm{TFT}_{\pm}\bigg{[}SU(2)_{|k|=3}^{\frac{1}{2}\oplus\frac{1}{2}}\bigg{]}\;.\end{split}$
(158)

In the last line, we again ignore the almost trivial spin TQFT
$\frac{SU(2)_{2}}{\mathbb{Z}_{2}}$. In the second line, we use the following
fact

$\displaystyle\left(\frac{\textrm{TFT}_{1}\otimes\textrm{TFT}_{2}}{\mathbb{Z}_{2}^{\rm
diag}}\right)/\mathbb{Z}_{2}=\left(\frac{\textrm{TFT}_{1}}{\mathbb{Z}_{2}}\right)\otimes\left(\frac{\textrm{TFT}_{2}}{\mathbb{Z}_{2}}\right)\;,$
(159)

which holds for any two TQFTs, TFT1 and TFT2, which have non-anomalous
$\mathbb{Z}_{2}$ one-form symmetries. The theory ${T[SU(2)]}/{``PSU(2)^{\rm
diag}_{|k|=4}"}$ has yet another dual description with only manifest ${\cal
N}=2$ supersymmetry which is expected from the geometrical aspects of the
3D-3D correspondence for a once-punctured torus bundle Terashima:2011qi ;
Gang:2013sqa . The ${\cal N}=2$ dual is presented in Appendix B.

##  4 Discussion

There are several interesting questions we want to address in future.

##### Relation with Rozansky-Witten theory

One well-known method of constructing a topological theory from a 3D ${\cal
N}=4$ SCFT is using topological twisting as studied by Rozansky and Witten in
Rozansky:1996bq . An ${\cal N}=4$ SCFT ${\cal T}$ has $SU(2)_{L}\times
SU(2)_{R}$ R-symmetry and we can consider a pair of topological twisted
theories, $\textrm{RW}_{+}[{\cal T}]$ and $\textrm{RW}_{-}[{\cal T}]$, using
the $SU(2)_{L}$ or $SU(2)_{R}$ in the twisting respectively. It would be
interesting to clarify the exact relation between the pair of topological
twisted theories, $\textrm{RW}_{\pm}[{\cal T}_{\rm rank\;0}]$, and our
$\textrm{TFT}_{\pm}[\mathcal{T}_{\rm rank0}]$ for rank 0 SCFT ${\cal T}_{\rm
rank\;0}$.666For a 3D ${\cal N}=4$ SCFT ${\cal T}$ of non-zero rank, the
topological twisted theories $\textrm{RW}_{\pm}[{\cal T}]$ are not genuine
TQFTs since they do not obey the standard axioms of TQFT. Nevertheless,
interesting 3-manifold/knot invariants can be studied using the
$\textrm{RW}_{\pm}[{\cal T}]$ Gukov:2020lqm .

Two theories, $\textrm{RW}[{\cal T}_{\rm rank\;0}]$ and $\textrm{TFT}[{\cal
T}_{\rm rank\;0}]$, have the same ground state degeneracy $\textrm{GSD}_{g}$
for all $g\geq 0$ since the partitial topological twisting on
$\Sigma_{g}\times S^{1}$ using the $U(1)\subset SU(2)_{R}$ (or $U(1)\subset
SU(2)_{L}$ ) symmetry is actually equivalent to the full topological twisting
on the 3-manifold using the $SU(2)_{R}$ ($SU(2)_{L}$) symmetry Gukov:2016gkn ;
Bullimore:2018jlp . But two theories cannot be the same since one (RW) is
unitary while the other (TFT) is non-unitary. From the comparison, we
naturally conjecture that

$\displaystyle\textrm{TFT}_{\pm}[{\cal T}_{\rm rank\;0}]\textrm{ is a Galois
conjugate of }\textrm{RW}_{\pm}[{\cal T}_{\rm rank\;0}]\;.$ (160)

##### 3D non-unitary TQFTs from 4D ${\cal N}=2$ SCFTs

In Dedushenko:2018bpp , the authors constructed 3D non-unitary TQFTs from some
4D ${\cal N}=2$ Argyres-Douglas theories. The construction is somewhat similar
to our construction of TFT, but the precise relation is not clear. In our
construction, semi-simple non-unitary TQFTs appear in 3D SCFTs of rank 0,
while their examples after dimension reduction to 3D are not of rank 0. It
would be interesting to clarify for which classes of 4D ${\cal N}=2$ SCFTs
their construction works, and to see if we can apply their construction in
classification of 4D ${\cal N}=2$ SCFTs.

##### Are all non-unitary TQFTs correspond to rank 0 ${\cal N}=4$ SCFTs?

In our paper, we assign a pair of non-unitary TQFTs to 3D rank 0 ${\cal N}=4$
SCFTs. But it is not clear if all non-unitary TQFTs can be constructed in this
way. There are exotic non-unitary TQFTs which cannot be related to any unitary
TQFT via Galois conjugation, as found at rank 6 in rowell2006quantum . If the
proposed relation in (160) is true, such exotic non-unitary TQFTs cannot be
realized from rank 0 ${\cal N}=4$ SCFTs.

###### Acknowledgements.

We thank Hee-Cheol Kim for helpful discussions and comments at various stages
of the project. The research of DG is supported by the NRF Grant
2019R1A2C2004880 and by the New Faculty Startup Fund from Seoul National
University. DG also acknowledges support by the appointment to the JRG program
at the APCTP through the Science and Technology Promotion Fund and Lottery
Fund of the Korean Government, as well as support by the Korean Local
Governments, Gyeongsangbuk-do Province, and Pohang City. KL is supported in
part by KIAS Individual Grant PG006904 and by the National Research Foundation
of Korea Grant NRF-2017R1D1A1B06034369. MS is supported in part by National
Research Foundation of Korea grant NRF-2019R1A2C2004880 and in part by
scholarship from Hyundai Motor Chung Mong-Koo Foundation. MS also thanks for
hospitality of APCTP during the visits. MY is supported in part by JSPS
KAKENHI Grant-in-Aid for Scientific Research (Nos. 19H00689, 19K03820,
20H05850, 20H05860).

##  Appendix A Some reviews

In order to help better understand the correspondence in (13), we briefly
review basic relevant aspects of (2+1)D non-unitary topological field theory
and supersymmetric partition functions of 3D superconformal field theories.

###  A.1 Localization on 3D ${\cal N}\geq 3$ gauge theories

In this paper, we consider following 4 types of supersymmetric partition
functions of 3D ${\cal N}=4$ SCFT ${\cal T}$ of rank 0 theory

$\displaystyle\begin{split}&{\cal I}^{\rm sci}_{{\cal
T}}(q,\eta,\nu;s)\;:\;\textrm{Superconformal index
\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Kim:2009wb,Imamura:2011su}{\@@citephrase{(}}{\@@citephrase{)}}}}\;,\\\
&{\cal Z}^{S^{3}_{b}}_{{\cal T}}(b,m,\nu)\;:\;\textrm{Squashed three-sphere
partition function \cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Jafferis:2010un,Hama:2011ea}{\@@citephrase{(}}{\@@citephrase{)}}}}\;,\\\
&{\cal I}^{\Sigma_{g}}_{{\cal T}}(\eta,\nu;s)\;:\;\textrm{(Topologically)
twisted indices on $\Sigma_{g}$ \cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Gukov:2015sna,Benini:2015noa,Benini:2016hjo,Closset:2016arn}{\@@citephrase{(}}{\@@citephrase{)}}}
}\;,\\\ &{\cal Z}^{{\cal M}_{g,p}}_{{\cal T}}(m,\nu;s)\;:\;\textrm{Twisted
partition function on ${\cal M}_{g,p}$ \cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Gang:2019juz,Alday:2012au,Imamura:2012rq,Closset:2017zgf,Closset:2018ghr}{\@@citephrase{(}}{\@@citephrase{)}}}}\;.\end{split}$
(161)

These 4 types of partition functions are not totally exclusive.

$\displaystyle\begin{split}&{\cal Z}^{S^{3}_{b}}(b=1,m,\nu)={\cal Z}^{{\cal
M}_{g=0,p=1}}(m,\nu;s=1)\;,\\\ &{\cal
I}^{\Sigma_{g}}(\eta,\nu;s)\big{|}_{\eta=e^{m}}={\cal Z}^{{\cal
M}_{g,p=0}}(m,\nu;s)\;.\end{split}$ (162)

In the partition function, $m$ (resp. $\eta$) is the real mass parameter
(resp. fugacity) associated to the axial $U(1)$ symmetry while $\nu$ is the
R-symmetry mixing parameter (5). Rank 0 ${\cal N}=4$ SCFT cannot have any
flavor symmetry commuting with ${\cal N}=4$ supersymmetries and thus the BPS
partition functions cannot be further refined.

3D ${\cal N}=4$ SCFTs can appear as IR fixed points of 3D quantum field
theories. Since there could be supersymmetry enhancement along the RG flow, we
do not need to start from a UV theory with manifest ${\cal N}=4$ symmetry. For
the exact computation of supersymmetric partition using localization, however,
the UV theory should have at least ${\cal N}=2$ supersymmetry. In this paper,
we study several examples of ${\cal N}=4$ rank 0 SCFTs which appears as IR
fixed points of ${\cal N}\geq 3$ supersymmetric theories. In the below, we
summarize localization formulae for the BPS partition functions introduced
above for ${\cal N}\geq 3$ gauge theories. In localization computations,
${\cal N}\geq 3$ gauge theories have several advantages over ${\cal N}=2$
gauge theories. The local Lagrangian density of an ${\cal N}=3$ gauge theory
is uniquely determined by the choice of gauge group $G$, its Chern-Simons
levels $\vec{k}$ and matter contents (hypermultiplets and twisted
hypermultilplets in unitary representations of $G$). When the CS levels are
all zero, i.e. $\vec{k}=\vec{0}$, the theory has ${\cal N}=4$ supersymmetry.
For ${\cal N}\geq 3$ gauge theories, the R-symmetry is non-abelian ($SO(3)$),
and thus the IR R-symmetry is uniquely fixed (i.e. is not mixed with other
abelian flavor symmetries) and we do not need to perform the F-maximization
Jafferis:2010un . The localization for general ${\cal N}\geq 2$ theories can
be done in a similar way but with some more complications.

##### Superconformal index

The superconformal index for a 3D $\mathcal{N}=4$ SCFT is defined as

$\displaystyle\mathcal{I}^{\rm
sci}(q,\eta,\nu;s):=\begin{cases}\textrm{Tr}_{\mathcal{H}_{\rm
rad}(S^{2})}(-1)^{2j_{3}}q^{\frac{R_{\nu}}{2}+j_{3}}\eta^{A}\;\;,\quad
s=1\;,\\\ \textrm{Tr}_{\mathcal{H}_{\rm
rad}(S^{2})}(-1)^{R_{\nu}}q^{\frac{R_{\nu}}{2}+j_{3}}\eta^{A}\;\;,\quad
s=-1\;.\end{cases}$ (163)

Here the trace is taken over the radially quantized Hilbert-space
$\mathcal{H}_{\rm rad}(S^{2})$ on $S^{2}$ whose elements are local operators.
$j_{3}\in\frac{\mathbb{Z}}{2}$ is the Lorentz spin, the Cartan of $SO(3)$
isometry on the $S^{2}$. The parameter $q$ plays role as an
$\Omega$-deformation parameter. Only BPS operators satisfying following
relation contribute to the index

$\displaystyle\Delta=R+R^{\prime}+j_{3}\;,$ (164)

where $\Delta$ is the conformal dimension. The index alternatively can be
regarded as a partition function on $(S^{2}\times S^{1})={\cal M}_{g=0,p=0}$
with a fixed metric, background electric fields coupled to $U(1)_{R_{\nu}}$
and axial $U(1)$ symmetry and spin-structure along the $S^{1}$. The indices at
different $\nu$ are simply related as follows

$\displaystyle\begin{split}&\mathcal{I}^{\rm
sci}(q,\eta,\nu;s=1)=\mathcal{I}^{\rm
sci}(q,\eta,\nu=0;s=1)\big{|}_{\eta\rightarrow\eta q^{\frac{\nu}{2}}}\;,\\\
&\mathcal{I}^{\rm sci}(q,\eta,\nu;s=-1)=\mathcal{I}^{\rm
sci}(q,\eta,\nu=0;s=-1)\big{|}_{\eta\rightarrow\eta(-q^{\frac{1}{2}})^{\nu}}\;.\end{split}$
(165)

The two indices with different choices of the spin structure are related to
each other in the following way

$\displaystyle\mathcal{I}^{\rm sci}\left(q,\eta,\nu=\pm
1;s=1\right)=\mathcal{I}^{\rm sci}\left(q,\eta,\nu=\pm
1;s=-1\right)\big{|}_{q^{\frac{1}{2}}\rightarrow-q^{\frac{1}{2}}}\;.$ (166)

Using localization, the superconformal index at $\nu=0$ is given as

$\displaystyle\begin{split}&{\cal I}^{\rm sci}(q,\eta,\nu=0;s=1)\\\
&=\sum_{\mathbf{m}}\oint_{|a_{i}|=1}\left(\prod_{i=1}^{\textrm{rank}G}\frac{da_{i}}{2\pi
ia_{i}}\right)\Delta_{G}(\mathbf{m},\mathbf{a};q)q^{\epsilon_{0}(\mathfrak{n})}{\cal
I}_{0}^{cs}(\mathbf{m},\mathbf{a})\textrm{P.E.}[f_{\rm
single}(q,\mathbf{a},\eta;\mathbf{m})]\;.\end{split}$ (167)

In the localization, the saddle points are parametrized by
$\\{\mathfrak{m}_{i},a_{i}\\}_{i=1}^{\textrm{rank}(G)}$,

$\displaystyle\begin{split}&\frac{1}{2\pi}\int_{S^{2}}F=\mathbf{m},\;\exp\left(i\int_{S^{1}}A\right)=\mathbf{a}\;\;\textrm{and
}\sigma=\frac{\mathbf{m}}{2}\\\ &\textrm{ with
}\mathbf{m}:=\sum_{i}\mathfrak{m}_{i}\mathbf{h}^{i}\textrm{ and
}\mathbf{a}:=\exp(\mathbf{A}):=\exp\left(\sum_{i}(\log
a_{i})\mathbf{h}^{i}\right)\;.\end{split}$ (168)

Here $\sigma$ is the adjoint real scalar in the ${\cal N}=2$ vector multiplet
and $\\{\mathbf{h}^{i}\\}$ is a normalized basis of Cartan subalgebra of $G$.
For $G=U(N)$ or $SU(2)$ case, the basis is chosen as

$\displaystyle\begin{split}&G=U(N),\quad\mathbf{h}^{i}:=\textrm{diag}\\{0,\ldots,\stackrel{{\scriptstyle
i-\text{th}}}{{1}},\ldots,0\\}\;,\\\
&G=SU(2),\quad\mathbf{h}=\mathbf{h}^{i=1}=\textrm{diag}\\{1,-1\\}\;.\end{split}$
(169)

$\Delta_{G}$ is the contribution from ${\cal N}=2$ vector multiplet

$\displaystyle\begin{split}&\Delta_{G}(\mathbf{m},\mathbf{a};q):=\frac{1}{\textrm{Sym}(\mathbf{m})}\prod_{\lambda\in\Lambda^{+}_{\rm
adj}}q^{-\frac{|\lambda(\mathbf{m})|}{2}}\left(1-q^{\frac{1}{2}|\lambda(\mathbf{m})|}e^{\lambda(\mathbf{A})}\right)\left(1-q^{\frac{1}{2}|\lambda(\mathbf{m})|}e^{-\lambda(\mathbf{A})}\right)\;.\end{split}$
(170)

The monopole flux $\mathbf{m}$ in (168) breaks the gauge group $G$ to its
subgroup $H(\mathbf{m})$

$\displaystyle H(\mathbf{m}):=\\{h\in G\;:\;[h,\mathbf{m}]=0\\}\;,$ (171)

and $\textrm{Sym}(\mathbf{m})$ is the order of the Weyl group of the subgroup,

$\displaystyle\textrm{Sym}(\mathbf{m}):=|\textrm{Weyl}(H(\mathbf{m}))|\;.$
(172)

$\Lambda_{\rm adj}^{+}$ is the set of positive roots of $G$.

$\displaystyle\begin{split}&G=U(N),\quad\\{\lambda(\mathbf{m})\;:\;\lambda\in\Lambda^{+}_{\rm
adj}\\}=\\{\mathfrak{m}_{i}-\mathfrak{m}_{j}\;:\;0<i<j\leq N\\}\;,\\\
&G=SU(2),\quad\\{\lambda(\mathbf{m})\;:\;\lambda\in\Lambda^{+}_{\rm
adj}\\}=\\{\mathfrak{m}_{1}-\mathfrak{m}_{2}\\}\;.\end{split}$ (173)

The single particle index is

$\displaystyle f_{\rm
single}(q,\mathbf{a},\eta;\mathbf{m})=\sum_{\Phi}\sum_{\beta\in\rho_{R_{\Phi}}}\left(\frac{q^{\frac{1}{2}\Delta_{\Phi}+\frac{1}{2}|\beta(\mathbf{m})|}e^{\beta(\mathbf{A})}\eta^{q_{A}(\Phi)}}{1-q}-\frac{q^{\frac{1}{2}(2-\Delta_{\Phi})+\frac{1}{2}|\beta(\mathbf{m})|}e^{-\beta(\mathbf{A})}\eta^{-q_{A}(\Phi)}}{1-q}\right)\;.$
(174)

Here the the summation is over ${\cal N}=2$ chiral multiplets $\Phi$ in the
representation of $R_{\Phi}$ under the gauge group $G$. $\rho_{R}$ is the set
of weights of the representation $R$. $q_{A}(\Phi)$ and $\Delta_{\Phi}$ are
the axial $U(1)$ symmetry and the conformal dimension of the chiral field
$\Phi$ respectively. An ${\cal N}=4$ hypermultiplet consists of two chiral
multiplets with gauge charges $R$ and $\overline{R}$, $q_{A}=\frac{1}{2}$ and
$\Delta=\frac{1}{2}$. The adjoint chiral multiplet in ${\cal N}=4$ vector
multiplet has $q_{A}=-1$ and $\Delta=1$. If one wants to introduce a Chern-
Simons interaction in an ${\cal N}=4$ gauge theory, it will break the ${\cal
N}=4$ supersymmetry down to ${\cal N}=3$ symmetry. In the case, the R-symmetry
is broken to $SO(3)$ and thus we cannot introduce the fugacity $\eta$ for the
$U(1)$ axial symmetry. The Casimir energy $\epsilon_{0}$ is

$\displaystyle\epsilon_{0}=\frac{1}{2}\big{(}\partial_{q}f_{\rm
single}\big{)}\big{|}_{q,\eta\rightarrow
1}=\sum_{\Phi}\sum_{\beta\in\rho_{R_{\Phi}}}\frac{(1-\Delta_{\Phi})|\beta(\mathbf{m})|}{4}\;.$
(175)

${\cal I}_{0}^{cs}(\mathbf{m},\mathbf{a})$ is the contribution from the
classical CS term

$\displaystyle\begin{split}&U(N)_{k}\;:\;\prod_{i=1}^{N}(u_{i}(-1)^{\mathfrak{m}_{i}})^{k\mathfrak{m}_{i}}\;,\\\
&SU(2)_{k}\;:\;(u(-1)^{\mathfrak{m}})^{2k\mathfrak{m}}\;.\end{split}$ (176)

For a $U(N)$ dynamical gauge group, there is a $U(1)$ topological symmetry
whose Noether current is

$\displaystyle j_{\rm
top}^{\mu}=-\epsilon^{\mu\nu\rho}\textrm{Tr}(F_{\nu\rho})\;.$ (177)

The fugacity $a$ and its background monpole flux $\mathfrak{m}_{a}$ for the
topological symmetry can be introduced by including the following term to
${\cal I}_{0}^{cs}$

$\displaystyle\left(a(-1)^{\mathfrak{m}_{a}}\right)^{-\sum_{i=1}^{N}\mathfrak{m}_{i}}\left(\prod_{i=1}^{N}u_{i}(-1)^{\mathfrak{m}_{i}}\right)^{-\mathfrak{m}_{a}}\;.$
(178)

The monopole flux $\mathbf{m}$ should satisfy the following Dirac quantization
conditions

$\displaystyle\begin{split}&\lambda(\mathbf{m})\in\mathbb{Z},\;\;\forall\lambda\in\Lambda^{+}_{\rm
adj}\textrm{ and}\\\ &\beta(\mathbf{m})\in\mathbb{Z},\;\;\forall\beta\in
R_{\Phi}\;.\end{split}$ (179)

There could be additional constraints on the monopole fluxes depending on the
global structure of the ${\cal N}\geq 3$ gauge theories as we have seen in
(103) and (105). In the localization summation, we need to sum over monopole
flux $\mathbf{m}$ modulo the redundant Weyl symmetry of $G$.

##### Squashed three-sphere partition function ${\cal Z}^{S^{3}_{b}}(b,m,\nu)$

This is a partition function on $S^{3}={\cal M}_{g=0,p=1}$ with the following
metric

$\displaystyle
ds^{2}(S^{3}_{b})=|dz|^{2}+|dw|^{2}\;,\quad(z,w)\in\mathbb{C}^{2}\textrm{ are
subject to }b^{-2}|z|^{2}+b^{2}|w|^{2}=1\;.$ (180)

To preserve some supercharges, a background field coupled to the
$U(1)_{R_{\nu}}$ symmetry is properly turned on. Using localization, the
partition function can be given in the following integral form

$\displaystyle{\cal
Z}^{S^{3}_{b}}(b,m,\nu)=\int\left(\prod_{i=1}^{\textrm{rank}(G)}\frac{dZ_{i}}{\sqrt{2\pi\hbar}}\right)\Delta_{G}(\mathbf{Z};\hbar){\cal
I}_{\hbar}(\mathbf{Z},m,\nu)\;,\quad\hbar:=2\pi ib^{2}\;.$ (181)

Here $\\{Z_{i}\\}_{i=1}^{\textrm{rank}(G)}$ parametrizes the Cartan subalgebra
of $G$.

$\displaystyle\mathbf{Z}=\sum_{i=1}^{\textrm{rank}(G)}Z_{i}\mathbf{h}^{i}\in\textrm{(Cartan
subalgebra of $G$)}\;,$ (182)

and $\Delta_{G}(\mathbf{Z})$ is the contribution from the ${\cal N}=2$ vector
multiplet associated to the gauge group $G$

$\displaystyle\Delta_{G}(\mathbf{Z};\hbar):=\frac{1}{|\textrm{Weyl}(G)|}\prod_{\lambda\in\Lambda^{+}_{\rm
adj}}\bigg{[}4\sinh\left(\frac{1}{2}\lambda\cdot\mathbf{Z}\right)\sinh\left(\frac{\pi
i}{\hbar}\lambda\cdot\mathbf{Z}\right)\bigg{]}\;.$ (183)

$|\textrm{Weyl}(G)|$ is the order of the Weyl group of $G$.

The integrand ${\cal I}_{\hbar}$ is determined by gauge group, matter contents
and Chern-Simons levels of the ${\cal N}=3$ gauge theory as follows:
$\bullet$ An ${\cal N}=2$ chiral multiplet in a representation $R$ under $G$
with $U(1)$ axial charge $q_{A}$ and conformal dimension $\Delta$ contributes

$\displaystyle\begin{split}&\prod_{\Phi}\prod_{\beta\in\rho_{R_{\Phi}}}\Psi_{\hbar}\left(\beta\cdot\mathbf{Z}+q_{A}\big{(}m+(i\pi+\frac{\hbar}{2})\nu\big{)}+(i\pi+\frac{\hbar}{2})\Delta\right)\;.\end{split}$
(184)

We define $\Psi_{\hbar}$ as

$\displaystyle\Psi_{\hbar}(X)$
$\displaystyle:=\psi_{\hbar}(X)\exp{\left(\frac{X^{2}}{4\hbar}\right)}\;,$
(185)

with $\psi_{\hbar}(x)$ being the non-compact quantum dilogarithm function. (We
refer to D for details of the definition and basic properties of the
function.) An ${\cal N}=4$ hypermultiplet consists of two ${\cal N}=2$ chiral
multiplets with gauge charge $R$ and $\overline{R}$, $q_{A}=\frac{1}{2}$ and
$\Delta=\frac{1}{2}$. The adjoint ${\cal N}=2$ chiral multiplet in a ${\cal
N}=4$ vector multiplet has $q_{A}=-1$ and $\Delta=1$.
$\bullet$ Chern-Simons term of gauge $G$ of level $k$ contributes the
following term to the integrand

$\displaystyle\exp\left(\frac{k}{2\hbar}\textrm{Tr}(\mathbf{Z}^{2})\right)\;.$
(186)

The real mass $m$ (FI parameter) and the R-symmetry mixing parameter $\nu$ of
the $U(1)$ topological symmetry for $G=U(N)$ are introduced by adding the
following term to the integrand

$\displaystyle\exp\left(-\frac{W\textrm{Tr}(\mathbf{Z})}{\hbar}\right)\bigg{|}_{W=m+(i\pi+\frac{\hbar}{2})\nu}\;.$
(187)

The partition function at $b=1$, which corresponds to round three-sphere,
enjoys interesting properties. Firstly, its free-energy is maximized at the
superconformal R-charge choice, i.e.

$\displaystyle F_{\nu=0}>F_{\nu\neq 0}\;,\quad\textrm{where
}F_{\nu}:=-\log|{\cal Z}^{S^{3}_{b}}(b=1,m=0,\nu)|\;.$ (188)

Secondly, the round sphere free-energy $F$ at conformal point

$\displaystyle F=-\log|{\cal Z}^{S^{3}_{b}}(b=1,m=0,\nu=0)|\;,$ (189)

always monotonically decreases under the RG flow. So the quantity $F$ can be
regarded as a proper measure of degrees of freedom.

##### Perturbative expansion of squashed three-sphere partition function
integral

One can consider formal perturbative expansion of the localization integral in
an asymptotic limit $\hbar\rightarrow 0$, to obtain infinitely many 3D SCFT
invariants. In the limit, the integrand ${\cal I}_{\hbar}$ can be
perturbatively expanded in the following form

$\displaystyle\log{\cal
I}_{\hbar}(\vec{Z},m,\nu)\xrightarrow{\quad\hbar\rightarrow
0\quad}\sum_{n=0}^{\infty}\hbar^{n-1}{\cal W}_{n}(\vec{Z},n,\nu)\;.$ (190)

The leading part ${\cal W}_{0}$ corresponds to the twisted superpotential. By
extremizing the twisted superpotential, we obtain Bethe-vacua

$\displaystyle\textrm{Bethe-vacua :
}\frac{\bigg{\\{}\vec{z}\;:\;\left(\exp(\partial_{Z_{i}}{\cal
W}_{0})\big{|}_{\vec{Z}\rightarrow\log\vec{z}}\right)=1\;,\;w\cdot\vec{z}\neq\vec{z}\;\;\;\forall\;\textrm{non-
trivial
}w\in\textrm{Weyl}(G)\bigg{\\}}_{i=1}^{\textrm{rank}(G)}}{\textrm{Weyl}(G)}\;.$
(191)

Here $\textrm{Weyl}(G)$ is the Weyl group of gauge group $G$. A Bethe-vacuum
$\vec{z}_{\alpha}$ can be promoted to a saddle point
$\vec{Z}_{\alpha}=\log\vec{z}_{\alpha}$ of the localization integral by
properly shifting ${\cal W}_{0}$ as follows,

$\displaystyle{\cal W}_{0}^{\vec{n}_{\alpha}}={\cal W}_{0}+2\pi
i\sum_{i}n_{\alpha}^{i}Z_{i},\;\;n_{\alpha}^{i}\in\mathbb{Z}\;\textrm{is
chosen such that }\partial_{Z_{i}}{\cal
W}^{\vec{n}_{\alpha}}_{0}|_{\vec{Z}\rightarrow\log\vec{z}_{\alpha}}=0\;.$
(192)

Then we can consider formal perturbative expansion of the localization
integral around the saddle point

$\displaystyle\begin{split}&|\textrm{Weyl}(G)|\times\int\prod_{i=1}^{\textrm{rank}(G)}\frac{d(\delta
Z_{i})}{\sqrt{2\pi\hbar}}\exp\left(\frac{1}{\hbar}{\cal
W}_{0}^{\vec{n}_{\alpha}}(\vec{Z}^{\alpha}+\delta\vec{Z},m,\nu)+\sum_{n=1}^{\infty}\hbar^{n-1}{\cal
W}_{n}(\vec{Z}+\delta\vec{Z},,m,\nu)\right)\\\
&\xrightarrow{\quad\hbar\rightarrow
0\quad}\exp\left(\sum_{n=0}^{\infty}\hbar^{n-1}{\cal
S}^{\alpha}_{n}(m,\nu)\right)\;.\end{split}$ (193)

The factor $|\textrm{Weyl}(G)|$ is multiplied since that many saddle points,
which all give the same perturbative expansion, collapse into a single Bethe-
vacuum after the Weyl quotient. The perturbative expansion can be formally
computed by performing Gaussian integrals Dimofte:2012qj ; Gang:2019jut . For
example,

$\displaystyle{\cal S}^{\alpha}_{0}={\cal
W}_{0}^{\vec{n}_{\alpha}}(\vec{Z}_{\alpha})\;,\quad{\cal
S}_{1}^{\alpha}=-\frac{1}{2}\log\left(\det_{i,j}\frac{\partial^{2}{\cal
W}_{0}}{\partial Z_{i}\partial
Z_{j}}\right)\bigg{|}_{\vec{Z}=\vec{Z}_{\alpha}}+{\cal
W}_{1}(\vec{Z}_{\alpha})+\log|\textrm{Weyl}(G)|\;.$ (194)