aainstitutetext: School of Physics, Korea Institute for Advanced Study,
85 Hoegiro, Dongdaemun-gu, Seoul 02455, Republic of Korea.bbinstitutetext:
Department of Physics and Astronomy $\&$ Center for Theoretical Physics,
Seoul National University, 1 Gwanak-ro, Seoul 08826, Republic of
Korea.ccinstitutetext: Asia Pacific Center for Theoretical Physics (APCTP),
Pohang 37673, Republic of Korea.ddinstitutetext: Department of Physics, Pohang
University of Science and Technology (POSTECH),
Pohang 37673, Republic of Korea.

# Infrared Phases of 3D Class R Theories

Sunjin Choi b,c Dongmin Gang c,d Hee-Cheol Kim<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study the IR phases of 3D class R theories associated with closed non-
hyperbolic 3-manifolds. Non-hyperbolic 3-manifolds can be obtained by
performing Dehn fillings on 1-cusped hyperbolic 3-manifolds along exceptional
slopes. In 3D-3D correspondence, the ‘exceptional’ Dehn filling corresponds to
the gauging of an $SU(2)$ flavor symmetry in a superconformal field theory
associated with a 1-cusped 3-manifold with ‘small’ Chern-Simons levels. With
several explicit examples, we analyze various interesting non-perturbative IR
phenomena (such as spontaneous SUSY breaking, generation of mass gap and
supersymmetry enhancement) from the ‘exceptional’ gaugings. Interestingly,
distinguished features of the IR phases can be captured by simple topological
properties of non-hyperbolic 3-manifolds. We also find that 3D class R
theories associated with certain classes of atoroidal non-hyperbolic
3-manifolds always exhibit supersymmetry enhancement at low energy and
actually flow to 3D rank-0 $\mathcal{N}=4$ SCFTs with trivial vacuum moduli
space.

††preprint: KIAS-P22046

##  1 Introduction and Summary

Recently, considerable progress has been made in understanding lower
dimensional physics from compactifications of higher dimensional
superconformal field theories (SCFTs). In particular, plenty of 4D
$\mathcal{N}=2$ SCFTs, called class S theories, are engineered by
compactifications of 6D $\mathcal{N}=(2,0)$ SCFTs on punctured Riemann
surfaces, and non-perturbative physics of the 4D theories such as dualities
and enhanced symmetries at infrared have been studied in great detail through
the compactifications Gaiotto:2009we ; Gaiotto:2009hg . Similarly, the
compactification of the 6D SCFTs living on M5-branes wrapped around hyperbolic
three-manifolds $M$ leads to a family of 3D $\mathcal{N}=2$ theories called 3D
class R theories Dimofte:2011ju ; Dimofte:2011py . This construction allows us
to understand intricate networks of 3D dualities and also to predict new
dualities based on geometric operations on the internal 3-manifolds.

More recently, we have revisited the class R construction of 3D theories and
extended it to incorporate non-hyperbolic 3-manifolds into the construction
Cho:2020ljj . Unlike the compactification on hyperbolic 3-manifolds, which
engineers gapless phases at low energy, the 3D field theories constructed from
non-hyperbolic manifolds can give rise to a wide variety of strongly coupled
IR phases including gapped phases, SUSY-broken phases as well as conformal
phases. For example, a certain family of non-hyperbolic 3-manifolds, which we
can use in the class R construction to engineer 3D topological field theories
(TFTs), was proposed in Cho:2020ljj . The proposal was verified by explicitly
constructing modular S- and T-matrices for the TFTs in terms of topological
invariants of the non-hyperbolic 3-manifolds, which can be computed using
various partition functions of complex $SL(2,\mathbb{C})$ Chern-Simons
theories on the 3-manifolds. This establishes a new correspondence between 3D
TFTs and non-hyperbolic 3-manifolds.

Especially, when the modular data from the internal 3-manifold is non-unitary,
the infrared phase of the corresponding 3D theory is conjectured to become an
SCFT embedding a non-unitary subsector described by that modular data
Cho:2020ljj . Additionally, this class of IR SCFTs are believed to enjoy SUSY
enhancement from $\mathcal{N}=2$ to $\mathcal{N}=4$, and they have neither
Coulomb nor Higgs branch of vacua, i.e. they are of rank-0 $\mathcal{N}=4$
SCFTs. The subsector equipped with the non-unitary modular structure can be
isolated in a degenerate limit where supersymmetric partition functions
compute contributions only from the Coulomb or Higgs branch chiral ring
operators and their descendants, which are empty in this case since the SCFT
is of rank-0. Therefore, in the degenerate limit, the relevant physical
observables are loop operators and their correlation functions, which together
define the non-unitary modular tensor category (MTC) associated with the non-
unitary sub-sector. This novel relation between 3D $\mathcal{N}=4$ rank-0
SCFTs and non-unitary TFTs was extensively studied in Gang:2021hrd .

In this paper, we study the relation between topological invariants of closed
non-hyperbolic 3-manifolds and infrared phases of 3D class R theories. We
focus our attention on the non-hyperbolic 3-manifolds constructed by so-called
exceptional Dehn fillings from 1-cusped hyperbolic 3-manifolds. The
3-manifolds of this class have been classified in mathematical literature (see
for instance 2018arXiv181211940D ) based on the topological properties of
2-cycles embedded in the 3-manifolds, which is summarized in Fig. 1.
Surprisingly, we find that the mathematical classification can be repackaged
in physics as a systematic characterization of low-energy phases of the class
R theories.

Our main aim in this paper is to develop a geometric scheme to identify IR
phases of 3D class R theories in terms of topological properties of the
associated non-hyperbolic 3-manifolds. As we will see, this task will also
formulate a concrete dictionary between topological invariants of the internal
non-hyperbolic 3-manifolds and physical observables in the 3D class R
theories. The dictionary provides a number of criteria to characterize IR
phases of the 3D theories without detailed analysis of strongly coupled low-
energy dynamics which in general is notoriously difficult.

The main tools we will use are the 3D index and the irreducible
$SL(2,\mathbb{C})$ flat connections of the non-hyperbolic 3-manifolds. These
are defined as the partition function of a complex $SL(2,\mathbb{C})$ Chern-
Simons (CS) theory on the non-hyperbolic 3-manifold at Chern-Simons level
$k=0$ and irreducible flat connections in the complex CS theory respectively.
The convergence of the 3D index and the reality of the irreducible flat
connections turn out to capture the existence of certain topological 2-cycles
(more precisely, separating essential 2-spheres or tori), which plays a
central role in the classification of 3-manifolds in mathematics. Field
theoretically, the 3D index corresponds to a supersymmetric index counting the
number of local BPS operators, and the irreducible flat connections are
related to discrete vacua (or Bethe vacua) of the 3D field theory with
supersymmetric massive deformations. These are protected under renormalization
group (RG) flows by supersymmetry, so we can use them for a systematic
exploration of rich structure of low-energy phases. We first discuss what are
the possible IR phases from non-hyperbolic closed 3-manifolds via the class R
construction, and then propose geometric criteria on how to catalog the IR
phases in terms of the properties of the 3D index and irreducible flat
connections of the 3-manifolds. This will give a novel field theoretic
interpretation of the mathematical classification of closed 3-manifolds.
Utilizing the geometric criteria, we will identify all possible IR phases of
3D class R theories constructed from non-hyperbolic 3-manifolds obtained by
exceptional Dehn fillings on
$S^{3}\backslash\mathbf{4}_{1},S^{3}\backslash\mathbf{5}_{2},m003,m006,m007$
and $m009$. This provides a concrete confirmation for the geometric criteria
we propose.

The organization for the rest of this paper is as follows: In Section 2, we
review how to obtain closed non-hyperbolic 3-manifolds by exceptional Dehn
fillings. We also introduce a mathematical classification of closed non-
hyperbolic 3-manifolds, and explain how to understand it in terms of
properties of 3D indices and irreducible flat connections of the 3-manifolds.
In Section 3, we discuss IR phases of 3D class R theories constructed from
closed non-hyperbolic 3-manifolds. We explain how to identify precise IR
phases using supersymmetric partition functions, which can be computed from
topological data of the associated 3-manifolds. Section 4 presents various
examples of 3D class R theories and explicit analysis of their infrared
phases.

##  2 Space of non-hyperbolic 3-manifolds

This section provides a brief introduction to closed non-hyperbolic
3-manifolds with a focus on the construction using exceptional Dehn fillings
from 1-cusped hyperbolic 3-manifolds, and their topological features in terms
of basic properties of the 3D index and $SL(2,\mathbb{C})$ flat connections.
We also give some background mathematical materials relevant to the
discussions in this paper.

###  2.1 Exceptional Dehn fillings on 1-cusped hyperbolic 3-manifolds

Let us first explain the notations and a method for constructing closed non-
hyperbolic 3-manifolds. We start with the hyperbolic 3-manifolds with a single
torus boundary, which are often called 1-cusped hyperbolic 3-manifolds. We
shall use the following notation:

$\displaystyle\begin{split}&N\;:\;\textrm{1-cusped hyperbolic
3-manifold}\;,\\\ &pA+qB\in H_{1}(\partial
N,\mathbb{Z})=\mathbb{Z}\oplus\mathbb{Z}\;:\;\textrm{primitive boundary
1-cycle}\;.\end{split}$ (1)

In this paper, the 1-cusped hyperbolic 3-manifold $N$’s are labeled by the
nomenclature used in SnapPy SnapPy  as

$\displaystyle N=m003,\,m004,\,m006,\ldots,$ (2)

There is no canonical basis choice for boundary 1-cycles $A$ and $B$ in
$H_{1}(\partial N,\mathbb{Z})$ of a generic 3-manifold $N$. We find it
convenient to choose the particular basis used in SnapPy which we denote by

$\displaystyle A_{\rm SP},B_{\rm SP}\;:\;\textrm{a basis of $H_{1}(\partial
N,\mathbb{Z})$ chosen by SnapPy}\;.$ (3)

There is however a choice of canonical basis for $H_{1}(\partial
N,\mathbb{Z})$, which are called meridian $\mu$ and longitude $\lambda$, for a
special type of 3-manifold $N=S^{3}/\mathcal{K}$ engineered by a complement of
a knot $\mathcal{K}$ in $S^{3}$.

$\displaystyle\mu,\;\lambda\;:\;\textrm{a canonical basis of $H_{1}(\partial
N,\mathbb{Z})$ when $N=S^{3}\backslash{\cal K}$}\;.$ (4)

The closed 3-manifolds we are interested in can be constructed from the
1-cusped 3-manifolds using so-called Dehn fillings. Dehn filling is a natural
topological operation on a 1-cusped 3-manifold that produces a new 3-manifold
without boundary. It is a simple operation that glues a solid torus to the
boundary torus in a cusped 3-manifold along a primitive cycle $pA+qB$. Here,
the gluing can be performed by identifying the primitive cycle $pA+qB$ with
the shrinkable boundary $S^{1}$ of the solid torus. So the boundary of the
1-cusped 3-manifold is closed after the operation. We denote the closed
3-manifold resulting from the Dehn filling as

$\displaystyle\begin{split}N_{pA+qB}&:=(\textrm{3-manifold obtained by Dehn
filling on $N$ along $pA+qB$})\;,\\\ &:=(N\cup(\textrm{solid-
torus}))/\sim\;,\\\ &\quad\textrm{where }(pA+qB)\sim(\textrm{shrinkable
boundary $S^{1}$ of solid-torus})\;.\end{split}$ (5)

The primitive cycle $pA+qB$ (up to an overall sign) is called a slope. We
remark that according to the well-known Thurston’s hyperbolic Dehn surgery
theorem, $N_{pA+qB}$ is always hyperbolic except only finitely many slopes
called exceptional slopes.

$\displaystyle\textrm{A slope $pA+qB\in H_{1}(\partial N,\mathbb{Z})$ is
called exceptional if $N_{pA+qB}$ is non-hyperbolic}.$ (6)

The exceptional Dehn fillings drastically alter geometrical structure of the
3-manifolds and can produce a large class of closed 3-manifolds with rich
geometrical/topological structures. Actually 7 geometries among Thurston’s 8
geometries can be realized on non-hyperbolic 3-manifolds thurston2014three .

###  2.2 Topological types of non-hyperbolic manifolds

Figure 1: Two classifications of closed non-hyperbolic 3-manifolds: (Left)
according to the existence of essential 2-sphere or 2-torus and (Right) based
on the properties of 3D index and $SL(2,\mathbb{C})$ irreducible flat
connections on the 3-manifolds. The relation between two classifications are
illustrated in the two diagrams. The 3-manifolds of finite/complex types are
always hyperbolic and thus do not appear here. Among prime/torodial
3-manifolds, it is of finite type if all essential tori are non-separating and
of infinite type otherwise. The 3-manifolds of infinite type can be further
divided into two categories depending on whether it is real or complex but the
refined division is not drawn in the diagram. Among toroidal SFS, toroidal
3-fibered Seifert-spaces, i.e.
$S^{2}((p_{1},q_{1})(p_{2},q_{2})(p_{3},q_{3}))$ with
$\sum_{i}\frac{q_{i}}{p_{i}}=0$, are of finite and real while
$\Sigma_{g>1}\times S^{1}$ are of infinite type. We could not fully identify
the type of other toroidal SFS and leave them in a black color. SOL manifolds
are always finite/real/unitary while
$S^{2}((p_{1},q_{1})(p_{2},q_{2})(p_{3},q_{3}))$ could be either
finite/real/unitary or finite/real/non-unitary as in (17).

####  2.2.1 Conventional mathematical classification

The left diagram of Fig. 1 summarizes a conventional classification of closed
non-hyperbolic 3-manifolds according to the existence of embedded essential
2-sphere or embedded essential torus. An embedded 2-sphere is essential if it
does not bound a 3-ball. See 2018arXiv181211940D  for details on the
classification and related mathematical materials.

Depending on the existence of embedded essential 2-spheres, closed 3-manifolds
can be divided into two classes: reducible (if exists) or irreducible. When
the essential sphere is separating, the 3-manifold can be given as a connected
sum of the two separated manifolds. So the “connected sum” implies
“reducible”. The only exceptional 3-manifold, which is reducible but not
connected sum, is $S^{2}\times S^{1}$, which is regarded as a lens space
$L(0,1)$ in this paper. The prime 3-manifold means the 3-manifold which can
not be given by a connected sum of two manifolds neither of which is a
3-sphere $S^{3}$.1113-sphere can be regarded as an identity under the connect
sum operation $\sharp$, i.e. $M\sharp S^{3}=M$ for all $M$.  Depending on the
existence of the essential torus, 3-manifolds can further be divided into two
classes: toroidal (if exists) or atoroidal.

All hyperbolic 3-manifolds are prime and atoroidal. Among non-hyperbolic
3-manifolds, the only prime/atoroidal manifolds are small Seifert fibered
spaces (small SFS). There are two kinds of small Seifert fibered manifolds,
one is lens space and the other is 3-fibered Seifert spaces
$S^{2}\left((p_{1},q_{1})(p_{2},q_{2}),(p_{3},q_{3})\right)$ over $S^{2}$
which is not toroidal, i.e. $\sum_{i}\frac{q_{i}}{p_{i}}\neq 0$.

$\displaystyle\textrm{Prime/atoroidal }\begin{cases}\textrm{Hyperbolic }\\\
\textrm{Small Seifert fibered }\begin{cases}\textrm{Lens space $L(p,q)$ }\\\
S^{2}\left((p_{1},q_{1})(p_{2},q_{2}),(p_{3},q_{3})\right)\big{|}_{\sum_{i}\frac{q_{i}}{p_{i}}\neq
0}\end{cases}\end{cases}$ (7)

When $\sum_{i}\frac{q_{i}}{p_{i}}=0$, the manifold
$S^{2}\left((p_{1},q_{1})(p_{2},q_{2}),(p_{3},q_{3})\right)$ is toroidal and
corresponds to a Seifert-toroidal manifold. The manifold
$S^{2}\left((p_{1},q_{1}),(p_{2},q_{2}),(p_{3},q_{3})\right)$ can be described
by a Dehn surgery along a link with 4 components depicted as in Fig. 2. Here,
$(p_{i},q_{i})$ are coprime with $p_{i}>0$.

Figure 2: Dehn surgery description of 3-fibered Seifert manifold
$S^{2}\left((p_{1},q_{1}),(p_{2},q_{2}),(p_{3},q_{3})\right)$.

SOL manifolds refer to torus bundles $\mathbb{T}^{2}\times_{\varphi}S^{1}$
with Anosov monodromy $\varphi$. The torus bundles are defined with a
monodromy matrix $\varphi\in SL(2,\mathbb{Z})$ and an equivalence relation
given by

$\displaystyle\mathbb{T}^{2}\times_{\varphi}S^{1}=\big{(}\mathbb{T}^{2}\times[0,1]\big{)}/\big{(}(x,0)\sim(\varphi(x),1)\big{)}\;,$
(8)

where $x$ is a point in the two-torus $\mathbb{T}^{2}$ and $\varphi(x)$ is the
image of $x$ under the mapping class group element $\varphi$. $\varphi$ is
called a Anosov monodromy if $|\textrm{Tr}(\varphi)|>2$.

Graph manifolds generally refer to prime non-hyperbolic 3-manifolds that do
not include hyperbolic pieces in JSJ decomposition jaco1979seifert ;
johannson2006homotopy . In the usual definition, the space of graph manifolds
includes all prime Seifert-fibered spaces and SOL manifolds. However, graph
manifolds in this paper refer only to graph manifolds which are neither
Seifert-fibered spaces nor SOL manifolds. Similarly, the space of SFS’s
generally includes lens spaces, but our terminology for Seifert-fibered spaces
in this paper refers only to Seifert-fibered spaces which are not lens spaces.

####  2.2.2 Classification using 3D index and $SL(2,\mathbb{C})$ flat
connections

We shall propose a new classification scheme for closed 3-manifolds based on
the basic properties of 3D index Dimofte:2011py ; Gang:2018wek  and
irreducible $SL(2,\mathbb{C})$ flat connections as follows

  * •

Def : Infinite/Finite/Empty
i. A closed 3-manifold $M$ is of empty type if the 3D index vanishes, i.e.
${\cal I}_{M}(q)=0$.
ii. A non-empty closed 3-manifold $M$ is of finite (resp. infinite) type if
${\cal I}_{M}(q)$ converges (diverges) as a formal Laurent series.

  * •

Def : Complex/Real
i. A $SL(2,\mathbb{C})$ flat connection $\rho$ is
$\begin{cases}\textrm{real\,, \ \ if \ Tr}(\rho(a))\in\mathbb{R}\ \textrm{for
all}\ a\in\pi_{1}M.\\\ \textrm{complex\,, \ \ otherwise.}\end{cases}$
ii. A closed 3-manifold $M$ is $\begin{cases}\textrm{real\,, \ \ if all
irreducible flat}\ SL(2,\mathbb{C})\ \textrm{connections are real.}\\\
\textrm{complex\,, \ \ otherwise.}\end{cases}$

The 3D index is a 3-manifold invariant introduced as an $SL(2,\mathbb{C})$
Chern-Simons partition function with a quantized level $k=0$. The invariant
for 3-manifolds $N$ with torus boundaries was firstly studied in
Dimofte:2011py  and then generalized to cover closed 3-manifold $M$ in
Gang:2018wek . The index, ${\cal I}_{N}(\vec{m},\vec{e};q)$ or ${\cal
I}_{M}(q)$, gives a formal Laurent polynomial in $q^{1/2}$ for each
$(\vec{m},\vec{e})\in\mathbb{Z}^{2n}$ where $n$ is the number of boundary
tori. Using a state-integral model developed in Dimofte:2011py ; Gang:2018wek
; Gang:2018gyt , the index is schematically given in the following form

$\displaystyle\mathcal{I}(q)=\sum_{\vec{s}\in\mathbb{Z}^{T}}{\cal
I}(\vec{s};q)\;.$ (9)

Here $T$ is a positive integer and we suppress the dependence on
$(\vec{m},\vec{e})$ for simplicity. The ${\cal I}(\vec{s};q)$ for each
$\vec{s}\in\mathbb{Z}^{T}$ is $q^{\epsilon(\vec{s})}\times(\textrm{power
series in $q^{1/2}$ starting with $I_{0}(\vec{s})+O(q^{1/2})$})$ with
$\epsilon(\vec{s}),I_{0}(\vec{s})\in\mathbb{Z}/2$. The precise definition of
the 3D index and some examples will be presented in Section 2.3. We say the 3D
index converges (as a formal series in $q^{1/2}$) when one needs to sum over
only finitely many $\vec{s}\in\mathbb{Z}^{T}$ to obtain the invariant up to
${\cal O}(q^{\frac{\rm max}{2}})$ for arbitrary
$\textrm{max}\in\mathbb{Z}_{+}$.

We define the (adjoint) irreducible flat connections as follows:222We remark
that our definition of the irreducibility of the $SL(2,\mathbb{C})$ flat
connection is somewhat different from the definition used in many other math
literature. In the math literature, irreducible means (fundamental)
irreducible and our irreducible flat connections are called non-Abelian flat
connection. Our notion of irreducibility is relevant in the study of
$SL(2,\mathbb{C})$ Chern-Simons theory and class R theories.

  * •

Def : Irreducibility of flat connection
A $SL(2,\mathbb{C})$ flat connection $\rho\in\textrm{Hom}[\pi_{1}M\rightarrow
SL(2,\mathbb{C})]/\sim$ is called irreducible if
$\textrm{Stab}(\rho):=\\{g\in SL(2,\mathbb{C})\;:\;[g,\rho(a)]=0\quad\forall
a\in\pi_{1}M\\}$ is a finite subgroup of $SL(2,\mathbb{C})$.

It turns out that the only possible finite subgroup here is the
$\mathbb{Z}_{2}$ center symmetry. We also regard two homomorphisms
$\rho_{1},\rho_{2}\in\textrm{Hom}[\pi_{1}M\rightarrow SL(2,\mathbb{C})]$ as
the same $SL(2,\mathbb{C})$ flat connection if their characters are all
identical. Namely, we have the following equivalence relation:

$\displaystyle\rho_{1}\sim\rho_{2}\quad\textrm{if
$\textrm{Tr}(\rho_{1}(a))=\textrm{Tr}(\rho_{2}(a))$ for all
$a\in\pi_{1}M$}\;.$ (10)

We define a set of irreducible flat connections as

$\displaystyle\chi^{SL(2,\mathbb{C})}_{\rm irred}(M):=\\{\textrm{a set of
inequivalent irreducible $SL(2,\mathbb{C})$ flat connections on $M$}\\}\;.$
(11)

As we will see in the next section, this set of irreducible flat connections
will play a crucial role in the study of 3D class R theories.

We are particularly interested in the closed 3-manifolds of finite and real
type since they enjoy the correspondence with topological field theories
$\textrm{TFT}[M]$ established in Cho:2020ljj  (see also Cui:2021lyi  and
Cui:2021yes ). A closed 3-manifold $M$ of finite and real type is non-
hyperbolic, and the class R construction for two M5-branes wrapped around such
3-manifold $M$ provides a 3D field theory, which flows at low energy to either
a unitary topological field theory (TQFT) or a (unitary) superconformal field
theory (SCFT) containing a sub-sector realizing a non-unitary TQFT structure.
The details will be discussed in the next section. We can distinguish the non-
hyperbolic 3-manifolds in terms of the TQFT structures of the associated 3D
class R theories. We define

  * •

Def : Unitary/Non-unitary
$M$ of finite/real type is of $\begin{cases}\text{unitary type, \ if TFT$[M]$
is unitary.}\\\ \text{non-unitary type, \ if TFT$[M]$ is non-
unitary.}\end{cases}$

In this definition, the TFT$[M]$ denotes the IR topological theory arising
from the class R theory when the 3-manifold $M$ is of unitary type, while it
means the sub-sector of the class R theory admitting a non-unitary TQFT
structure at low energy when $M$ is of non-unitary type. We note that unlike
other properties above, the unitarity/non-unitarity is defined only for the
3-manifolds of finite and real type.

####  2.2.3 Relation between two classifications

Now let us relate our new classification scheme to the conventional
classification of closed 3-manifolds. The relation is summarized in Fig. 1.

##### Lens space $\Rightarrow$ Empty type

As we will see in Section 2.3, the empty type 3-manifolds can alternatively be
characterized by the emptiness of irreducible $SL(2,\mathbb{C})$ flat
connections:

$\displaystyle\begin{split}&\textrm{$M$ is of empty type if there are no
irreducible $SL(2,\mathbb{C})$ flat connections on $M$}\\\
&\qquad\qquad\qquad\qquad\qquad i.e.\;\;\chi_{\rm
irred}^{SL(2,\mathbb{C})}(M)=\emptyset\;.\end{split}$ (12)

Lens spaces $L(p,q)$ are obviously of empty type since its fundamental group
is Abelian, $\pi_{1}(L(p\neq 0,q))=\mathbb{Z}_{p}$ and
$\pi_{1}(L(0,1))=\mathbb{Z}$, and thus all $SL(2,\mathbb{C})$ flat connections
are reducible.

##### No separating essential sphere and torus $\Rightarrow$ 3D index
converges (i.e. empty or finite)

The convergence of the 3D index has been extensively studied in
Garoufalidis:2013axa ; Garoufalidis:2016ckn  using an interesting relationship
between the 3D index and the normal surface counting. The main result of these
studies is that the 3D indices for cusped 3-manifolds only converge when the
3-manifolds are irreducible and atoroidal. These mathematical studies have
focused on 3-manifolds with torus boundaries and hence the conclusion is not
directly applicable to 3D indices for closed 3-manifolds. But as we will see
in Section 2.3, their basic logic can be applicable even to closed 3-manifolds
via the 3D-3D correspondence. Motivated by these circumstantial evidences and
confirmation with explicit examples in Appendix A, we claim that

$\displaystyle\begin{split}&\textrm{If a closed 3-manifold $M$ is
irreducible/atoroidal}\;\\\ &\Rightarrow\textrm{3D index ${\cal I}_{M}(q)$
converges as a formal series in $q^{1/2}$ }\ .\end{split}$ (13)

In addition to these irreducible/atoroidal 3-manifolds, we found that
following reducible or toroidal 3-manifolds have convergent 3D indices.

$\displaystyle\begin{split}&\textrm{$S^{2}\times S^{1}$, Torus bundle
$\mathbb{T}^{2}\times_{\varphi}S^{1}$ and
$S^{2}((p_{1},q_{1}),(p_{2},q_{2}),(p_{3},q_{3}))|_{\sum_{i=1}^{3}q_{i}/p_{i}=0}$}\\\
&:\textrm{Reducible or toroidal closed 3-manifolds having convergent 3D
indices}\;.\end{split}$ (14)

One characteristic property of these manifolds is that the essential spheres
and tori in these manifolds are non-separating. So, we naturally speculate
that closed 3-manifolds with no separating essential sphere and torus have
convergent 3D indices. For $S^{2}\times S^{1}$ (a lens space), as we saw
above, the index vanishes, which is trivially compatible with the above
statement.

#####  $\exists$ Separating essential sphere or torus $\Rightarrow$ Infinity

In the case, we expect that the 3D index diverges. We will give a physical
reasoning for this in Section 2.3 and confirm it with many explicit examples
in Appendix A.

#####  $\mathbb{T}^{2}\times_{\varphi}S^{1}$ or
$S^{2}\left((p_{1},q_{1})(p_{2},q_{2}),(p_{3},q_{3})\right)$ $\Rightarrow$
Finite and Real

Two basic examples of finite/real 3-manifolds are
$S^{2}\left((p_{1},q_{1}),(p_{2},q_{2}),(p_{3},q_{3})\right)$ and
$\mathbb{T}^{2}\times_{\varphi}S^{1}$. The fundamental group of the 3-fibered
Seifert 3-manifold is

$\displaystyle\begin{split}&\pi_{1}\left(S^{2}\left((p_{1},q_{1}),(p_{2},q_{2}),(p_{3},q_{3})\right)\right)\\\
&=\langle x_{1},x_{2},x_{3},h\;:\;x_{i}^{p_{i}}\cdot
h^{q_{i}}=1\;(i=1,2,3),\;x_{1}x_{2}x_{3}=1,\;\textrm{$h$ is
central}\rangle\;,\end{split}$ (15)

and that of the torus bundle is

$\displaystyle\pi_{1}(\mathbb{T}^{2}\times_{\varphi}S^{1})=\langle
x,y,h\;:\;h^{-1}xh=x^{a}y^{c},\;h^{-1}yh=x^{b}y^{d}\rangle,\;\textrm{when
}\varphi=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\;.$ (16)

Here $(x,y)$ are the two generators of the fundamental group of the fiber
$\mathbb{T}^{2}$ while $h$ represents the loop along the base $S^{1}$. The
relations in the fundamental group represent how the two generators transform
under the mapping class group element $\varphi$. For these 3-manifolds of
finite/real, it turns out to be to-apper

$\displaystyle\begin{split}&\textrm{TFT}[S^{2}\left((p_{1},q_{1}),(p_{2},q_{2}),(p_{3},q_{3})\right)]\textrm{
is }\begin{cases}\textrm{unitary},\quad q_{i}\in\pm 1(\textrm{mod\;}p_{i})\\\
\textrm{non-unitary},\quad\textrm{otherwise}\end{cases}\\\ &\textrm{and}\\\
&\textrm{TFT}[\mathbb{T}^{2}\times_{\varphi}S^{1}]\textrm{ is always
unitary}\;.\end{split}$ (17)

###  2.3 3D index

We now review the 3D index ${\cal I}_{M}(q)$ in details. We introduce two
seemingly unrelated interpretations of the invariant: one is the partition
function of the complex $SL(2,\mathbb{C})$ Chern-Simons theory Dimofte:2012qj
; Gang:2017cwq  and the other is weighted sum over the normal surfaces
Garoufalidis:2013axa ; Garoufalidis:2016ckn . As we will see, the 3D-3D
correspondence provides a unified picture for these two interpretations.

##### 3D index as a weighted sum over surfaces

In a series of mathematical papers Garoufalidis:2013axa ; Garoufalidis:2016ckn
, they found intricate relation between the 3D index for cusped 3-manifolds
and the normal surfaces. In the 3D-3D correspondence, the 3D index counts BPS
local operators of the 3D class R theories. In the M-theoretic set-up of the
3D-3D correspondence, the BPS local operators originate from BPS M2-branes
wrapped on surfaces of the internal 3-manifold Gadde:2013wq , and it explains
the mathematician’s observations. In the M-theoretical picture, essential
surfaces play special roles. M2-branes can wrap an essential surface $\Sigma$
arbitrarily many times. They correspond to the local operators ${\cal
O}_{\Sigma}$, ${\cal O}_{\Sigma}^{2},{\cal O}_{\Sigma}^{3},\cdots$ and their
contribution to the index is of the form $\sum_{n}(\pm q^{k_{\Sigma}})^{n}$
where $k_{\Sigma}$ is a half-integer determined by the topology of $\Sigma$.
It means that ${\cal O}_{\Sigma}$ corresponds to $1/2$ BPS chiral primary
operators. From the analysis in Garoufalidis:2013axa ; Garoufalidis:2016ckn ,
it turns out that essential surface of Euler characteristic $\chi$ contributes
following terms to the index

$\displaystyle\sum_{n}(-q^{1/2})^{-\chi
n}\left(1+O(q^{1/2})\right)\;:\;\textrm{contributions from essential surface
to 3D index}\ .$ (18)

These terms make the index diverges when the essential surface is 2-sphere
($\chi=2$) or 2-torus ($\chi=0$). Although there is no similar mathematical
analysis for the 3D index of closed 3-manifolds, the M-theoretic picture is
still valid and one would expect the essential sphere and torus in a closed
3-manifold make its index diverge. From direct computations, we check that the
index indeed diverges for all reducible or toroidal closed 3-manifolds except
for the cases when the essential spheres and tori are non-separating.

Such exceptional cases include $S^{2}\times S^{1}$,
$\mathbb{T}^{2}\times_{\varphi}S^{1}$ and toroidal
$S^{2}((p_{1},q_{1}),(p_{2},q_{2}),(p_{3},q_{3}))$ whose 3D indices are simply
$0,1$ and $1\,(\textrm{or }2)$ respectively. Note that the 3D indices for
these exceptions are $q$-independent. We do not have a clear understanding,
but the observation suggests that the separating essential (instead of merely
essential) spheres or tori make the index divergent for closed 3-manifold
cases (unlike cupsed 3-manifold cases). We will come back to this issue below
using another physical interpretation of the 3D index.

##### 3D index as partition function of $SL(2,\mathbb{C})$ CS theory

The 3D index for a 3-manifold $M$ computes the partition function of an
$SL(2,\mathbb{C})$ Chern-Simons theory on $M$ at Chern-Simons level $k=0$
which takes a path-integral expression

$\displaystyle\begin{split}&{\cal
I}_{M}(q)=\int\frac{[D\mathcal{A}D\overline{{\cal
A}}]}{(\textrm{gauge})}\exp\bigg{(}\pi i(k+is)CS[\mathcal{A};M]+\pi
i(k-is)CS[\overline{\mathcal{A}};M]\bigg{)}\bigg{|}_{k=0,s=\frac{4\pi}{\log
q}}\;,\\\ &\textrm{where
}CS[\mathcal{A};M]=\frac{1}{8\pi^{2}}\int_{M}\textrm{Tr}\left(\mathcal{A}d\mathcal{A}+\frac{2}{3}\mathcal{A}^{3}\right)\;,\end{split}$
(19)

where $\mathcal{A}$ is the $SL(2,\mathbb{C})$ gauge field and $s$ is the
imaginary part of the Chern-Simons level, which exists for a complex gauge
group. The path-integral can be evaluated exactly using so-called state-
integral model based on a Dehn surgery description $M=N_{pA+qB}$ and an ideal
triangulation of $N$ Dimofte:2012qj ; Gang:2017cwq .

The index can be factorized into holomorphic/anti-holomorphic blocks in the
following way Beem:2012mb :

$\displaystyle{\cal
I}_{M}(q)=\frac{1}{2}\sum_{[\rho]\in\chi^{SL(2,\mathbb{C})}_{\rm
irred}(M)}B^{\rho}(q^{\frac{1}{2}})B^{\bar{\rho}}(q^{-\frac{1}{2}})\;.$ (20)

Here $\chi^{SL(2,\mathbb{C})}_{\rm irred}(M)$ is the set of irreducible
$SL(2,\mathbb{C})$ flat connections on $M$ as defined in (11). $B^{\rho}$ is
the holomorphic block of the complex CS theory defined as

$\displaystyle B^{\rho}(q):=\int_{\Gamma^{\rho}}\frac{[D{\cal A}]}{(\rm
gauge)}e^{-\frac{4\pi^{2}}{\hbar}CS[{\cal A},M]}\;\;\textrm{with
$q=e^{\hbar}$}\;,$ (21)

where $\Gamma^{\rho}$ is the Lefschetz thimble in the functional space of
${\cal A}$ associated to the flat connection $\rho$. $B^{\bar{\rho}}$ is the
holomorphic block for the complex conjugate flat connection $\bar{\rho}$.

The factorization formula makes sense only when the set
$\chi^{SL(2,\mathbb{C})}_{\rm irred}$ is finite, i.e. when it contains a
finite number of elements, and otherwise we expect the index to diverge. For
example, a 3-manifold given by a connected sum, which means $M=M_{1}\sharp
M_{2}$ and none of the $M_{i}$’s is a 3-sphere, always has an infinite number
of inequivalent irreducible flat connections.333It simply follows from the
fact that $\pi_{1}(M_{1}\sharp M_{2})=\pi_{1}(M_{1})*\pi_{1}(M_{2})$ where $*$
is the free product of groups. Let $\rho_{1}$ and $\rho_{2}$ be (not
necessarily irreducible) homomorphisms of $\pi_{1}(M_{i})\rightarrow
SL(2,\mathbb{C})$ with $\textrm{Im}[\rho_{i}]\neq\\{\pm 1\\}=(\textrm{Center
subgroup $\mathbb{Z}_{2}$})$. We define
$\rho_{g}\in\textrm{Hom}(\pi_{1}M\rightarrow SL(2,\mathbb{C}))$ by
$\rho_{g}(a)=\rho_{1}(a)$ for $a\in\pi_{1}(M_{1})\subset\pi_{1}M$ and
$\rho_{g}(b)=g\rho_{2}(b)g^{-1}$ for $b\in\pi_{1}(M_{2})\subset\pi_{1}M$ with
$g\in SL(2,\mathbb{C})$. Then the set $\\{\rho_{g}\\}_{g\in SL(2,\mathbb{C})}$
contains infinitely many inequivalent irreducible flat $SL(2,\mathbb{C})$
connections on $M$. This is compatible with the fact that 3-manifolds given by
a connected sum are always of infinite type.

Now let us try to understand the behavior of the 3D index using the partition
function representation. First, the factorization formula of the 3D index
implies (12). For example, the 3-manifold $S^{2}\times S^{1}$ does not admit
any irreducible $SL(2,\mathbb{C})$ flat connections since its fundamental
group is $\mathbb{Z}$ and is Abelian. This implies that $\chi_{\rm
irred}^{SL(2,\mathbb{C})}=\emptyset$ and thus the 3D index vanishes.

In the Cardy limit $q\rightarrow 1$ (or $\hbar\rightarrow 0$), the holomorphic
blocks are expanded as

$\displaystyle\begin{split}&B^{\rho}(q^{\frac{1}{2}})\xrightarrow{\quad\hbar=\log
q\rightarrow
0\quad}\exp\left(-\frac{4\pi^{2}}{\hbar}CS[\rho]+\ldots\right)\;,\\\
&B^{\bar{\rho}}(q^{-\frac{1}{2}})\xrightarrow{\quad\hbar=\log q\rightarrow
0\quad}\exp\left(\frac{4\pi^{2}}{\hbar}\overline{CS[\rho]}+\ldots\right)\;,\\\
&\Rightarrow
B^{\rho}(q^{\frac{1}{2}})B^{\bar{\rho}}(q^{-\frac{1}{2}})\xrightarrow{\quad\hbar=\log
q\rightarrow
0\quad}\exp\left(-\frac{8\pi^{2}i}{\hbar}\textrm{Im}[CS[\rho]]+\ldots\right)\;.\end{split}$
(22)

This shows that the asymptotic behavior of the 3D index essentially depends on
whether the 3-manifold is of complex or real type. A 3-manifold of complex
type has a complex flat connection $\rho$ with non-zero
$\textrm{Im}[CS[\rho]]$ and this gives an exponentially growing contribution
to the index in the Cardy limit.444In 3-manifolds, there is always a pair of
complex flat connections, $\rho$ and $\bar{\rho}$. One of them gives
exponentially growing contribution to the index in the $\hbar\in
e^{i\delta}\mathbb{R}\rightarrow 0$ limit unless $\delta=0$ or $\pi$. On the
other hand, for 3-manifolds of finite/real type, the 3D index becomes finite
and $\hbar$-independent in the asymptotic limit $\hbar\rightarrow 0$ (or
equivalently $q\rightarrow 1$) since $\textrm{Im}[CS[\rho]]=0$. This partly
explains why the 3D indices for $\mathbb{T}^{2}\times_{\varphi}S^{1}$ and
$S^{2}((p_{i},q_{i}))|_{i=1}^{3}$ are finite and $q$-independent. Indeed these
3-manifolds have only a finite number of irreducible $SL(2,\mathbb{C})$ flat
connections which are all real.

Next, the partition function for a 3-manifold $\Sigma_{g}\times S^{1}$ is
expected to compute the ($q$-independent) dimension of Hilbert space of the
$SL(2,\mathbb{C})$ Chern-Simons theory on the genus $g$ Riemann surface
$\Sigma_{g}$. The Hilbert space can be obtained by quantizing the phase space
$P(\Sigma_{g})$ defined as

$\displaystyle P(\Sigma_{g})=\chi^{SL(2,\mathbb{C})}_{\rm
irred}(\Sigma_{g}):=(\textrm{set of inequivalent irreducible
$SL(2,\mathbb{C})$ flat connections on $\Sigma_{g}$})\;.$ (23)

The phase space is empty for $g=0$ since $\pi_{1}(\Sigma_{g=0,1})$ is Abelian,
whereas for $g>1$, it is a non-compact Kähler manifold of complex dimension
$(g-1)$. We thus assert that

$\displaystyle{\cal I}_{\Sigma_{g}\times
S^{1}}(q)=\begin{cases}0\;,\quad\text{for }g=0\;,\\\ \textrm{$q$-independent
infinity}\;,\quad\text{for }g>1\;.\end{cases}$ (24)

This behavior is expected since the 3-manifolds $\Sigma_{g=0}\times S^{1}$ do
not contain separating essential 2-sphere and torus, which we will call ${\rm
SES}_{g=0}$ and ${\rm SES}_{g=1}$ respectively, and they have no irreducible
$SL(2,\mathbb{C})$ flat connections, while there are separating essential
2-tori and infinitely many irreducible flat connections in $\Sigma_{g>1}\times
S^{1}$.555The separating essential 2-torus is given by a direct product of the
$S^{1}$ in $S^{1}\times\Sigma_{g}$ and a separating non-contractible $S^{1}$
in $\Sigma_{g}$. The separating non-contractible 1-cycle exists only when
$g>1$.

Combining the above discussions with experimental evidences, we propose that
the 3D index of a 3-manifold $M$

$\displaystyle{\cal I}_{M}(q)=\begin{cases}\textrm{$1+O(q^{1/2})$ in $q$
expansion}&\ \Longleftrightarrow\ \textrm{No ${\rm SES}_{g=0,g=1}$ and
Complex}\\\ 0&\ \Longleftrightarrow\chi^{SL(2,\mathbb{C})}_{\rm
irred}=\emptyset\\\ |\cap_{\rho}\textrm{Inv}(\rho)|\in\\{1,2\\}&\
\Longleftrightarrow\ \textrm{No ${\rm SES}_{g=0,g=1}$ and Real}\\\
\textrm{Divergent}&\ \Longleftrightarrow\exists\;{\rm SES}_{g=0}\textrm{ or
}{\rm SES}_{g=1}\end{cases}$ (25)

We have tested this proposal for a large number of 3-manifolds by explicit
calculations. Some concrete examples will be presented below and we also refer
to Appendix A for more detailed computations.

For hyperbolic 3-manifolds, we expect the 3D index to converge as a
$q^{1/2}$-series since hyperbolic 3-manifolds are irreducible and atoroidal.
Also, the 3D index of a hyperbolic 3-manifold should show exponentially
growing behavior in the Cardy limit since it always contains at least one
complex flat connection, say $\rho_{\rm hyp}$, associated with the hyperbolic
structure. For lens spaces, as explained above, we expect
$\mathcal{I}_{M}(q)=0$ since they have no irreducible flat connection. This is
not manifest at all in the actual computations of the 3D indices using the
state-integral model, but after delicate cancellations, one can see that the
3D indices vanish for these cases. For reducible or toroidal cases, the 3D
index diverges if some of essential spheres or tori are separating.

The 3-manifolds in the 3rd line of the above proposal are of finite and real
type. For these 3-manifolds, it is argued in Cho:2020ljj  that there is an
associated topological field theory ${\rm TFT}[M]$ and the 3D index computes
the partition function of ${\rm TFT}[M]$ on $S^{2}\times S^{1}$. Note that for
topological theories, the $S^{2}\times S^{1}$ partition function is always $1$
since there are no local excitations contributing to the partition function.
However, we observe that the index sometimes becomes $2$ instead. This is
puzzling, but we can understand this as follows. Let us first introduce some
mathematical definitions. The cohomology group
$H^{1}(M,\mathbb{Z}_{2})=\textrm{Hom}(\pi_{1}M\rightarrow\mathbb{Z}_{2})$
naturally acts on the elements in $\chi^{SL(2,\mathbb{C})}_{\rm irred}$ as
follows

$\displaystyle\begin{split}&\eta\in
H^{1}(M,\mathbb{Z}_{2})\;:\;\rho\in\chi^{SL(2,\mathbb{C})}_{\rm
irred}\quad\longrightarrow\quad\rho\otimes\eta\in\chi^{SL(2,\mathbb{C})}_{\rm
irred}\\\ &\textrm{where}\;(\rho\otimes\eta)(a):=\rho(a)\eta(a)\;.\end{split}$
(26)

The invariant subgroup $\textrm{Inv}(\rho)\subset H^{1}(M,\mathbb{Z}_{2})$ is
then defined as

$\displaystyle\textrm{Inv}(\rho):=\\{\eta\in
H^{1}(M,\mathbb{Z}_{2})\;:\;\rho\otimes\eta\sim\rho\\}\;.$ (27)

Recall from equivalence (10) that two homomorphisms $\rho_{1}$ and $\rho_{2}$
are considered identical if their characters are the same. Among finite/real
3-manifolds, we find that the following 3-fibered Seifert spaces have non-
trivial $\cap_{\rho}\textrm{Inv}(\rho)$ as

$\displaystyle
S^{2}((2,1),(2,1),(p,q))\;:\;\cap_{\rho}\textrm{Inv}(\rho)=\mathbb{Z}_{2}\;.$
(28)

In Cho:2020ljj , it is claimed that the ${\rm TFT}[M]$ can be obtained from a
progenitor TQFT, denoted as $\widetilde{{\rm TFT}}[M]$, by gauging the 1-form
symmetry $H^{1}(M,\mathbb{Z}_{2})$. But sometimes a subgroup of the 1-form
symmetry acts trivially on all observables and thus decouples in IR. This
decoupling happens when the subgroup of $H^{1}(M,\mathbb{Z}_{2})$ trivially
acts on $\chi^{SL(2,\mathbb{C})}_{\rm irred}$. We note that
$\cap_{\rho}\textrm{Inv}(\rho)$ is a subgroup of $H^{1}(M,\mathbb{Z}_{2})$
trivially acting on the $\chi^{SL(2,\mathbb{C})}_{\rm irred}$. In the case
when $\cap_{\rho}\textrm{Inv}(\rho)=\mathbb{Z}_{2}$, the 3D index counts $2$
instead of $1$ since the ground state of ${\rm TFT}[M]$ on $S^{2}$ comes with
multiplicity $2$. An example of this type of decoupled 1-form symmetry will be
presented in Section 4.4.

##### Example : $M=(S^{3}\backslash\mathbf{4}_{1})_{P\mu+Q\lambda}$.

$(S^{3}\backslash\mathbf{4}_{1})_{P\mu+Q\lambda}$ denotes a closed 3-manifold
obtained by a Dehn surgery along the figure-eight knot ($\mathbf{4}_{1}$) with
slope $P/Q$. The 3D index can be written as

$\displaystyle\mathcal{I}_{(S^{3}\backslash\mathbf{4}_{1})_{P\mu+Q\lambda}}\big{(}q\big{)}:=\\!\sum_{(m,e)\in\mathbb{Z}^{2}}\\!\\!\\!\\!\\!K(m,e;P,Q;q)\text{
}\mathcal{I}_{S^{3}\backslash\mathbf{4_{1}}}\big{(}m,e;q\big{)}\ ,$ (29)

where $K(m,e;P,Q;q)$ is a kernel of $SO(3)$ type Gang:2018wek  given by

$\displaystyle K$
$\displaystyle:=\frac{1}{2}(-1)^{Rm+2Se}\Big{(}\delta_{Pm+2Qe,0}(x^{\frac{Rm+2Se}{2}}+x^{-\frac{Rm+2Se}{2}})-\delta_{Pm+2Qe,-2}-\delta_{Pm+2Qe,2}\Big{)}\
.$

Here the integers $R,S\in\mathbb{Z}$ are determined by a condition $QR-
PS=1$.666This condition cannot fix $(R,S)$ uniquely. However, the index will
not depend on this freedom. The 3D index for the figure-eight knot complement
is

$\displaystyle{\cal
I}_{S^{3}\backslash\mathbf{4}_{1}}(m,e;q)=\sum_{e_{2}\in\mathbb{Z}}{\cal
I}^{c}_{\Delta}(m-e_{2},m+e-e_{2}){\cal I}^{c}_{\Delta}(e-e_{2},-e_{2})\;.$
(31)

Here ${\cal I}^{c}_{\Delta}(m,e)$ is the tetrahedron index in charge basis
defined as follows.

$\displaystyle\mathcal{I}_{\Delta}(m,u):=\prod_{r=0}^{\infty}\frac{1-q^{r-\frac{m}{2}+1}u^{-1}}{1-q^{r-\frac{m}{2}}u}=\sum_{e\in\mathbb{Z}}{\cal
I}^{c}_{\Delta}(m,e)u^{e}\;.$ (32)

The index ${\cal I}_{S^{3}\backslash\mathbf{4}_{1}}(m,e)$ gets contributions
from the surface in the knot complement along the 1-cycle $2e\mu+m\lambda$ at
the boundary Garoufalidis:2016ckn . In the knot complement, there are embedded
essential once-punctured Klein bottles ending on the 1-cycle $\pm
4\mu\pm\lambda$ at the boundary. These essential surfaces can be detected by
looking at the leading term of the 3D indices in the $q$ expansion as follows.

$\displaystyle\begin{split}&{\cal I}_{S^{3}\backslash\mathbf{4}_{1}}(\pm 1,\pm
2;q)=-q^{1/2}+q^{5/2}+4q^{7/2}+\cdots\;,\\\ &{\cal
I}_{S^{3}\backslash\mathbf{4}_{1}}(\pm 2,\pm 4;q)=q-q^{4}-2q^{5}+\cdots\;,\\\
&{\cal I}_{S^{3}\backslash\mathbf{4}_{1}}(\pm 3,\pm
6;q)=-q^{3/2}+q^{11/2}+2q^{13/2}+\cdots\;.\end{split}$ (33)

From these results, one can confirm that ${\cal
I}_{S^{3}\backslash\mathbf{4}_{1}}(m=\pm k,e=\pm 2k;q)$ starts with
$(-q^{1/2})^{k}$, which can be identified with the contribution from a BPS
M2-brane winding around the essential surface $k$-times. Note that the once-
punctured Klein bottle has Euler characteristic $-1$ which is compatible with
(18). For comparison, one can compute the index with $(m,e)=k(\pm 1,\pm 1)$ as

$\displaystyle\begin{split}&{\cal I}_{S^{3}\backslash\mathbf{4}_{1}}(\pm 1,\pm
1;q)=-q-q^{2}+2q^{3}+\cdots\;,\\\ &{\cal
I}_{S^{3}\backslash\mathbf{4}_{1}}(\pm 2,\pm
2;q)=2q^{3}+2q^{4}+2q^{5}+\cdots\;,\\\ &{\cal
I}_{S^{3}\backslash\mathbf{4}_{1}}(\pm 3,\pm
3;q)=-q^{5}-3q^{6}-4q^{7}+\cdots\;.\end{split}$ (34)

One may notice that unlike the above cases, the lowest power in $q^{1/2}$ is
not linearly increasing. This implies the absence of embedded essential
surfaces ending on boundary 1-cycle $\pm 2\mu\pm\lambda$.

After the Dehn filling, the 3D index becomes

$\displaystyle{\cal
I}_{(S^{3}\backslash\mathbf{4}_{1})_{P\mu+Q\lambda}}(q)=\begin{cases}0\;,\quad(P,Q)=(\pm
1,0)\\\ 1\;,\quad P/Q=0,\pm 1,\pm 2\textrm{ and }\pm 3\\\
(\sum_{k=0}^{\infty}1)-2q^{2}-2q^{3}-4q^{4}-4q^{5}-6q^{6}+\cdots\;,\quad
P/Q=\pm 4\\\ \textrm{Power series in $q^{1/2}$ starting with }1+{\cal
O}(q^{1/2})\;,\quad\textrm{otherwise}\\\ \end{cases}$ (35)

This is indeed consistent with our proposal in (25) for the 3D index and with
the following topological facts 2018arXiv181211940D .

$\displaystyle(S^{3}\backslash\mathbf{4}_{1})_{P\mu+Q\lambda}=\begin{cases}S^{3}\;,\quad(P,Q)=(\pm
1,0)\\\ \mathbb{T}^{2}\times_{\varphi}S^{1}\textrm{ with
}\varphi=\begin{pmatrix}2&1\\\ 1&1\end{pmatrix}\;,\quad P/Q=0\\\
\textrm{Atoroidal SFS $S^{2}((2,1),(3,1),(7,-6))$}\;,\quad P/Q=\pm 1\\\
\textrm{Atoroidal SFS $S^{2}((2,1),(4,1),(5,-4))$}\;,\quad P/Q=\pm 2\\\
\textrm{Atoroidal SFS $S^{2}((3,1),(3,1),(4,-3))$}\;,\quad P/Q=\pm 3\\\
\textrm{Graph}\;,\quad P/Q=\pm 4\\\
\textrm{Hyperbolic}\;,\quad\textrm{otherwise}\end{cases}$ (36)

The divergent term $(\sum_{k=0}^{\infty}1)$ for $P/Q=\pm 4$ comes from
$(-q^{1/2})^{k}$ in ${\cal I}_{S^{3}\backslash\mathbf{4}_{1}}(\pm k,\pm 2k;q)$
after the Dehn filling in (29). We note that this term can be regarded as
contributions from BPS M2-branes winding around the essential torus in
$(S^{3}\backslash\mathbf{4}_{1})_{\pm 4\mu+\lambda}$. Here, the essential
torus comes from the once-punctured Klein bottle in the knot complement. The
puncture is closed after the Dehn filling. Then the puncture-closed essential
surface has Euler characteristic $\chi=0$ and it can therefore give the
divergent contribution in the 3D index as noticed in (18).

##  3 IR phases of class R theory

In this section, we discuss low-energy phases of 3D class R theories arising
from compactifications of the 6d $\mathcal{N}=(2,0)$ theory on M5-branes
wrapped around closed 3-manifolds. We particularly focus on the 3D theories
from the $A_{1}$-type 6D (2,0) theory. When the 3-manifold $M$ is hyperbolic,
the low-energy theories become 3D $\mathcal{N}=2$ superconformal field
theories, which have been extensively studied in Dimofte:2011ju ;
Dimofte:2011py . See also Gang:2018wek  for more discussions. On the other
hand, the low-energy phases for the 3D theories engineered by non-hyperbolic
3-manifolds have much richer possibilities. For example, as discussed in
Cho:2020ljj , compactifications on non-hyperbolic 3-manifolds can lead to 3D
theories in gapped phases described by TQFTs and also to SCFTs embedding a
sub-sector related to a non-unitary TQFT structure.

More specifically, we study how the IR phases of class R theories change under
the exceptional Dehn fillings, and what kinds of IR phases can appear from the
class R theories engineered by closed non-hyperbolic 3-manifolds.
Interestingly, the exceptional Dehn filling corresponds to gauging of $SU(2)$
symmetry with a small Chern-Simons (CS) level. The small CS level means that
the theory becomes strongly coupled and thus the IR phase can be drastically
changed after the gauging due to wild non-perturbative quantum effects. This
is in a nice parallelism with the 3-manifold side story, drastic change of
geometrical structures under the exceptional Dehn filling. We will make the
parallelism more precise. We will use basic features of 3-manifolds explained
in the previous section and determine the IR phases of class R theories. The
main result is summarized in (49). Using the result, one can easily determine
the IR phases of class R theories without analyzing the non-perturbative
quantum effects. The emergent IR phases after the exceptional gaugings turn
out to be very rich. One can find SUSY breaking phases, gapped topological
phases, gapless phases, IR phases with SUSY enhancement and etc, which is also
compatible with the mathematical fact that non-hyperbolic manifolds covers 7
geometries out of Thurston’s 8 geometries. We will provide concrete examples
for these phases.

###  3.1 3D class R theory from 6D (2,0) theory

Let $M$ be a closed (i.e. without boundary) 3-manifold. Then the 3D class R
theory ${\cal T}[M]$ associated to the $M$ is defined as follows:

$\displaystyle(\textrm{6D $A_{1}$ (2,0) theory on $\mathbb{R}^{1,2}\times
M$})\xrightarrow{\qquad\textrm{size}(M)\rightarrow 0\qquad}(\textrm{3D ${\cal
T}[M]$ theory on $\mathbb{R}^{1,2}$})\;.$ (37)

In the compactification, we perform a topological twisting along the internal
3-manifold using a $SO(3)$ subgroup of $SO(5)$ R-symmetry of the 6D theory.
The topological twisting preserves 4 supercharges and thus the low energy
theory under the compactification becomes a 3D ${\cal N}=2$ supersymmetric
theory. The 3D theory has $SO(2)=U(1)$ R-symmetry originated from the
$SO(2)\subset SO(3)\times SO(2)\subset SO(5)$ subgroup of the 6D R-symmetry
preserved in the topological twisting. For some cases, the ${\cal T}[M]$ can
have accidental Abelian flavor symmetries and the $U(1)$ R-symmetry can be
mixed with them in IR. To distinguish it, we denote the R-symmetry originated
from the 6D theory by $U(1)_{R_{\rm geo}}$ and its charge by $R_{\rm geo}$:

$\displaystyle U(1)_{R_{\rm geo}}\subset SO(2)_{R_{\rm geo}}\times
SO(3)\subset(\textrm{$SO(5)$ R-symmetry of the 6D theory})\;.$ (38)

In this paper, we consider the 3D class R theory $\mathcal{T}[M]$ proposed in
Dimofte:2011ju ; Gang:2018wek , which will be briefly reviewed in the next
subsection. The theory is sometimes denoted as ${\cal T}_{\rm irred}[M]$ since
it only sees an irreducible sub-sector of the 6D compactification
Chung:2014qpa . So our $\mathcal{T}[M]$ is different from $\mathcal{T}_{\rm
full}[M]$ (which is expected to see the full sector) studied in Pei:2015jsa ;
Gukov:2017kmk ; Eckhard:2019jgg . Unlike $\mathcal{T}_{\rm irred}[M]$, we
still do not have 3D field theoretic descriptions for general
$\mathcal{T}_{\rm full}[M]$ theories. It is even not sure whether
$\mathcal{T}_{\rm full}[M]$ for a general $M$ (especially for a hyperbolic
$M$) can be described by a genuine 3D theory instead of the 6D theory
compactified on $M$ Gang:2018wek . As we will see below, $\mathcal{T}_{\rm
irred}[M]$ for non-hyperbolic 3-manifolds have richer IR phases than
$\mathcal{T}_{\rm full}[M]$. For example, the spontaneously SUSY breaking
phase can not be realized in $\mathcal{T}_{\rm full}[M]$ since there always
exists a Bethe-vacuum, i.e. SUSY vacuum on $\mathbb{R}^{2}\times S^{1}$, in
the theory, which corresponds to the trivial $SL(2,\mathbb{C})$ flat
connection on $M$ Dimofte:2010tz .

###  3.2 Field theoretic construction

We start with a brief review of the field theoretic construction of the ${\cal
T}[M]$ theory proposed in Dimofte:2011ju ; Gang:2018wek . For simplicity, we
focus on the 3-manifolds $M$ given by

$\displaystyle M=N_{pA+B}\;,$ (39)

where $N$ is a 1-cusped hyperbolic 3-manifold. Then the corresponding 3D
theory ${\cal T}[M]$ can be obtained by gauging certain global symmetries in
the 3D class R theory ${\cal T}[N,A;B]$ associated to $N$

$\displaystyle{\cal T}[M=N_{pA+B}]=\begin{cases}\frac{{\cal
T}[N,A;B]}{``SO(3)"_{p}},\quad{\rm odd\;}A\\\ \frac{{\cal
T}[N,A;B]}{SU(2)_{p}},\quad{\rm even\;}A\ .\end{cases}$ (40)

Here, ${\cal T}[N,A;B]$ is the 3D theory constructed by Dimofte-Gaiotto-Gukov
Dimofte:2011ju  based on an ideal triangulation of $N$ (see section 2.2 of
Gang:2018wek  for a review with some improvements). $``SO(3)"_{p}$ and
$SU(2)_{p}$ in the denominators mean particular gauging procedures which we
will now explain.

First, the 3D theory ${\cal T}[N,A;B]$ depends on the choice of the boundary
1-cycle $A$ and always has $U(1)$ flavor symmetry, say $U(1)_{A}$, associated
to the boundary 1-cycle. The theory can be schematically described as follows:

$\displaystyle\begin{split}{\cal T}[N,A;B]&=\left(U(1)^{r}_{K}\;\textrm{gauge
theory coupled to $S$ chiral multiplets of charge $Q$}\right)\\\
&\quad+(\textrm{deformed by superpotential }{\cal
W}=\sum_{I=1}^{S-\tilde{F}-1}{\cal O}_{I})\ ,\end{split}$ (41)

where $K$ is a symmetric $(r\times r)$ matrix representing the mixed Chern-
Simons levels and $Q$ is an $(S\times r)$ matrix representing the gauge
charges of the chirial multiplets. The number of chrial multiplets denoted by
$S$ is equal to the number of tetrahedra in the ideal triangulation. The $r$,
$K$, $Q$ and $\\{{\cal O}\\}_{I=1}^{S-\tilde{F}-1}$ are determined by the
ideal triangulation of $N$ and choice of the boundary 1-cycle $A$. Before the
superpotential deformation, the Abelian gauge theory has $U(1)^{S}$ flavor
symmetry, which is broken to $U(1)^{\tilde{F}+1}$ after the deformation. The
$U(1)^{\tilde{F}+1}$ always contains the $U(1)_{A}$, i.e. $U(1)_{A}\subset
U(1)^{\tilde{F}+1}$. The $U(1)_{A}$ symmetry is expected to exist from the 6D
construction of ${\cal T}[N,A;B]$ theory777For a 1-cusped hyperbolic
3-manifold $N$, the 6D set-up includes a codimension-two defect which gives an
$SU(2)_{A}$ symmetry. The $SU(2)_{A}$ symmetry is broken to the $U(1)_{A}$
symmetry for a generic choice of $A$ due to a symmetry breaking
superpotential. The superpotential is forbidden when the cycle $A$ is non-
closable. We shall refer to Gang:2018wek  for details. , while the other
symmetries are accidental symmetry, which are not manifest in the UV 6D set-
up. The existence of the additional flavor symmetry is closely related to the
existence of so-called hard internal edges Dimofte:2011ju  in the ideal
triangulation of $N$ as pointed out in Gang:2018wek .

In this paper, we always choose the 1-cycle $A$ to be a non-closable cycle
defined as

$\displaystyle\textrm{$A\in H_{1}(\partial N,\mathbb{Z})$ is called {\it non-
closable} if $N_{A}$ is of empty type}\;.$ (42)

It then follows that the $U(1)_{A}$ symmetry of ${\cal T}[N,A;B]$ is enhanced
to $SU(2)$ flavor symmetry which we denote by $SU(2)_{A}$ Gang:2018wek :

$\displaystyle\textrm{$A$ is non-closable }\Rightarrow\textrm{${\cal
T}[N,A;B]$ has a $SU(2)_{A}\supset U(1)_{A}$ flavor symmetry}\;.$ (43)

Also, we define an oddness/evenness of the 1-cycle $A\in H_{1}(\partial
N,\mathbb{Z})$ as follows

$\displaystyle\begin{split}A\in H_{1}(\partial N,\mathbb{Z})\textrm{ is called
}\begin{cases}\textrm{even},\quad A\in\textrm{Ker}[i_{*}:H_{1}(\partial
N,\mathbb{Z})\rightarrow H_{1}(N,\mathbb{Z}_{2})]\\\
\textrm{odd},\quad\textrm{otherwise}\ .\end{cases}\end{split}$ (44)

If the cycle $A$ is odd, the global structure of the flavor symmetry is
$SO(3)$ (i.e. all local operators have integer spin under the $SU(2)_{A}$).

The choice of the other 1-cycle $B$ then affects the background CS level for
the $SU(2)$ flavor symmetry. In order for the CS level to be properly
quantized, we need to choose two 1-cycles ($A$ and $B$) to have different
oddness/evenness. For example, when $N=S^{3}\backslash{\cal K}$ is a knot
complement in $S^{3}$ we can choose

$\displaystyle A=\mu,\quad B=\lambda\;,\quad\textrm{for
$N=S^{3}\backslash{\cal K}$}\;.$ (45)

In this case, the meridian $\mu$ is always non-closable and odd while
longitude $\lambda$ is always even.

In order to obtain the $\mathcal{T}[M]$ theory, we gauge the $SU(2)_{A}$
flavor symmetry of the ${\cal T}[N,A;B]$ theory. The gauging should be
performed with a Chern-Simons coupling at level $p$ for the $SU(2)_{A}$ flavor
symmetry. For even $A$ cases, the gauging is straightforward and one obtains
the $T[M]$ theory as

$\displaystyle\begin{split}&{\cal T}[N_{pA+B}]=\frac{{\cal
T}[N,A;B]}{SU(2)_{p}}=(\textrm{Gauging $SU(2)_{A}$ of ${\cal T}[N,A;B]$ with
additional CS level $p$})\;.\end{split}$ (46)

For odd $A$ cases, on the other hand, the gauging of the $SU(2)_{A}$ flavor
symmetry is more involved. As explained above, when $A$ is odd, the gauge
group is $SO(3)$ rather than $SU(2)$. This means that the $\mathbb{Z}_{2}$
1-form symmetry coming from the $\mathbb{Z}_{2}$ center symmetry of the
$SU(2)$ symmetry must also be gauged. However, the $\mathbb{Z}_{2}$ 1-form
symmetry has non-trivial ‘t Hooft anomaly for odd $p$. In this case, we first
need to tensor the ${\cal T}[N,A;B]$ theory with a topological theory to
cancel the $\mathbb{Z}_{2}$ anomaly, and then we gauge the $\mathbb{Z}_{2}$
1-form symmetry. We thus propose for an odd $A$ that the $\mathcal{T}[M]$
theory is constructed as

$\displaystyle\begin{split}&\mathcal{T}[N_{pA+B}]=\frac{{\cal
T}[N,A;B]}{``SO(3)"_{p}}\\\
&\qquad\qquad\;:=\begin{cases}\frac{(\textrm{Gauging $SU(2)_{A}$ of ${\cal
T}[N,A;B]$ with additional CS level $p$})}{\mathbb{Z}_{2}},\quad\textrm{even
$p$}\\\ \frac{(\textrm{Gauging $SU(2)_{A}$ of ${\cal T}[N,A;B]$ with
additional CS level $p$})\otimes U(1)_{-2}}{\mathbb{Z}^{\rm
diag}_{2}},\quad\textrm{odd $p$}\ .\end{cases}\end{split}$ (47)

Here $/\mathbb{Z}_{2}$ denotes the gauging of the $\mathbb{Z}_{2}$ 1-form
symmetry. Note that when $p$ is odd, we tensor the theory with the $U(1)_{-2}$
theory, which is a topological theory with an anomalous $\mathbb{Z}_{2}$
1-form symmetry, such that the tensored theory has an anomaly free 1-form
symmetry $\mathbb{Z}_{2}^{\rm diag}$, and this symmetry is finally gauged. For
odd $p$, the above relation implies that

$\displaystyle\frac{{\cal T}[N,A;B]}{SU(2)_{p}}={\cal T}[N_{pA+B}]\otimes
U(1)_{2}\;.$ (48)

After the $SU(2)$ or $``SO(3)"$ gauging, the final ${\cal T}[M]$ has
$U(1)^{F}$ flavor symmetry, which is a subgroup of $U(1)^{\tilde{F}}$ symmetry
commuting with the gauged $SU(2)$ symmetry.

###  3.3 IR phases (Main Proposal)

We now discuss low energy physics of the 3D field theory $\mathcal{T}[M]$ for
a closed 3-manifold $M$. The IR phase of this theory is entirely determined by
geometric data of the 3-manifold $M$. We thus approach the problem of the
classification of the IR phases of the class R theories by examining the
geometric properties of the corresponding 3-manifolds.

For example, the IR phases associated with 3-manifolds of type $M=N_{pA+B}$ at
large $p$ can be rather easily determined. As explained in the previous
subsection, the corresponding UV field theory can be constructed by gauging
the $SU(2)$ flavor symmetry of the SCFT $\mathcal{T}[N,A;B]$ with a Chern-
Simons coupling at level $p$. When the level $p$ is large, the quantum effect
of the gauging is suppressed at $1/|p|\ll 1$, and this implies that the theory
${\cal T}[M]$ after the gauging is expected to flow to a SCFT at low energy.
Indeed, the stress-energy central charge of the ${\cal T}[M]$ theory
approaches to that of the original theory $\mathcal{T}[N,A;B]$ as $|p|$
increases. This is compatible with the Thurston’s hyperbolic Dehn surgery
theorem saying that $M=N_{pA+B}$ is hyperbolic for a large enough $|p|$. Note
that ${\cal T}[M]$ for a hyperbolic 3-manifold $M$ always flows to a non-
trivial SCFT.

However, on the other hand when $|p|$ is small, the $SU(2)$ gauging leads to
strongly interacting theories showing various interesting non-perturbative
phenomena at low energy. Thus, the analysis of the IR phases becomes much more
complicated for these cases. It turns out that ${\cal T}[M]$ theories for
generic closed 3-manifolds, which involve such strong interactions, can flow
to non-trivial IR phases other than SCFT phases.

Based on the characteristics of BPS partition functions related to a closed
3-manifold $M$, which we will explain in details in the next subsection, we
propose that

Main proposal

$\displaystyle{\cal T}[M]=\begin{cases}\textrm{$\mathcal{N}=2$
SCFT},&\textrm{for a finite/complex type $M$}\\\ \textrm{Spontaneous SUSY
breaking phase},&\textrm{for an empty type $M$}\\\ \textrm{Unitary
TQFT},&\textrm{for a finite/real/unitary type $M$}\\\ \textrm{$\mathcal{N}=4$
(rank-0) SCFT},&\textrm{for a finite/real/non-unitary type $M$}\\\
\textrm{${\cal N}=2$ SCFT $+$ accidental sym/CPO},&\textrm{for an infinite
type $M$}\end{cases}$ (49)

More precise statements in this proposal are as follows:

  * •

When the closed 3-manifold $M$ is of finite and complex type, $M$ is
hyperbolic and the $\mathcal{T}[M]$ theory flows to a $\mathcal{N}=2$
superconformal field theory.

  * •

When $M$ is of empty type, supersymmetry (SUSY) of the $\mathcal{T}[M]$ theory
is spontaneously broken at low energy.

  * •

When $M$ is of finite, real, and unitary type, the low energy theory of
$\mathcal{T}[M]$ is in a gapped phase described by a (unitary) topological
field theory.

  * •

When $M$ is of finite, real, and non-unitary type, the $\mathcal{T}[M]$ theory
flows to a superconformal field theory that includes sub-sectors realizing a
non-unitary TQFT structure. The IR theory also enjoys an $\mathcal{N}=4$ SUSY
enhancement, but has no Coulomb and Higgs branches, which means that it is a
rank-0 theory.

  * •

When $M$ is of infinite type, the low energy phase of $\mathcal{T}[M]$ is
rather subtle. The 3D index in this case diverges, which is an unexpected
behavior for a consistent IR phase. A natural mechanism to remedy this is the
emergence of an accidental flavor symmetry at low energy. R-symmetry can mix
with this accidental flavor symmetry, and the 3D superconformal index takes an
appropriate index expression, capturing a finite degeneracy of BPS states for
a given charge, only if it is computed with the correct IR R-symmetry. The
divergence of the 3D index is due to a wrong choice of the IR superconformal
R-symmetry. Geometrically, this divergence comes from the contributions of
separating essential surface $\Sigma$ with a genus $g\leq 1$. The essential
surface corresponds to a $1/2$ BPS chiral primary operator (CPO) in
$\mathcal{T}[M]$ theory with $R_{\rm geo}=(g-1)$. To satisfy the unitarity
bound $R_{\rm IR}\,(\textrm{the correct IR R-charge})\geq\frac{1}{2}$, the CPO
should be charged under the accidental flavor symmetry. Since there is no such
a CPO in $\mathcal{T}[N,A;B]$888Otherwise, the 3D index for a 1-cupsed
hyperbolic 3-manifold $N$ should diverge. But we know that the 3D index always
converges Garoufalidis:2013axa ., the CPO should correspond to a gauge-
invariant monopole operator of the $SU(2)$ gauge symmetry of $\mathcal{T}[M]$
in (40). This operator is generically $1/4$ BPS operator but we expect that it
becomes $1/2$ BPS operator when $M$ is of infinite type. In summary, we
propose IR phase of this type is a $\mathcal{N}=2$ SCFT with accidental flavor
symmetry, and the $SU(2)$ monopole operators become 1/2 BPS CPOs, which are
charged under the accidental symmetry.

We will now explain how to read off the IR phases by inspecting BPS partition
functions, and then utilize it to concrete examples. This will illustrate the
physics behind our proposal and also provide supporting evidences for it.

###  3.4 Strategy for reading off IR phases

To verify our proposal (49) from the UV description of ${\cal T}[M]$, reviewed
in Section 3.2, we provide a general method to determine the IR phases using
the properties of various supersymmetric partition functions.

##### BPS partition functions

We consider the following three types of BPS partition functions

$\displaystyle\begin{split}&{\cal
I}_{q}(u_{i},\nu_{i})\;:\;\textrm{Superconformal index (or 3D index)}\;,\\\
&{\cal Z}^{(s=\pm 1)}_{{\cal M}_{g,p}}(m_{i},\nu_{i})\;:\;\textrm{Twisted
partition function on ${\cal M}_{g,p}$}\;,\\\ &{\cal
Z}_{b}(m_{i},\nu_{i})\;:\;\textrm{Squashed 3-sphere partition
function}\;.\end{split}$ (50)

Here, $m_{i}$ and $u_{i}$ with the index $i=1,\cdots,F$ are respectively real
masses and fugacities for the flavor symmetry $U(1)^{F}\subset G_{F}$ of
${\cal T}[M]$. The UV $U(1)_{R}$ R-symmetry can mix, at low energy, with the
Abelian components of the flavor symmetry, and this mixing is captured by the
mixing parameters $\nu_{i}$ as

$\displaystyle R_{\vec{\nu}}=R_{\nu_{i}=0}+\sum_{i=1}^{F}J_{i}\nu_{i}\;,$ (51)

where $R_{\nu_{i}=0}$ stands for a reference R-charge and $J_{i}$’s denote the
Abelian flavor charges. One linear combination, $\vec{\nu}_{\rm geo}$, gives
the $U(1)_{R_{\rm geo}}$ symmetry in (38), i.e. $R_{\vec{\nu}_{\rm
geo}}=R_{\rm geo}$.

The superconformal index is a Witten index counting supersymmetric (or BPS)
local operators in a 3D SCFT that can be defined as a trace over the Hilbert
space in radial quantization

${\cal I}_{q}(u_{i},\nu_{i})=\textrm{Tr}_{\mathcal{H}_{\rm
rad}(S^{2})}\bigg{[}(-1)^{R_{\vec{\nu}}}q^{\frac{R_{\vec{\nu}}}{2}+j_{3}}\prod_{i}u_{i}^{J_{i}}\bigg{]}\
.$ (52)

Here, $q$ is the fugacity for a combination of R-charge $R_{\vec{\nu}}$ and
the angular momentum $j_{3}$ commuting with a pair of supercharges preserved
by the local operators. Via SUSY localization, the index can be computed and
the result can be expressed in the following form Kim:2009wb ; Imamura:2011su

$\displaystyle\begin{split}{\cal
I}_{q}^{N_{p}}(u_{i},\nu_{i})&=\sum_{m_{x}}\oint_{|u_{x}|=1}\frac{du_{x}}{2\pi
iu_{x}}\Delta(m_{x},u_{x}){\cal
I}^{N}_{q}(m_{x},u_{x},u_{i},\nu_{i})u_{x}^{2pm_{x}}\ .\end{split}$ (53)

Here ${\cal I}^{N}_{q}(m_{x},u_{x},u_{i},\nu_{i})$ is the superconformal index
for ${\cal T}[N,A;B]$ theory. $(m_{x},u_{x})$ are (monopole flux, fugacity)
coupled to the $SU(2)$ gauge symmetry, and $(u_{i},\nu_{i})|_{i=1}^{F}$ are
the (fugacities, R-symmetry mixing parameters) associated with $U(1)^{F}$
flavor symmetry. $\Delta(m_{x},u_{x})$ is the measure factor coming from the
$SU(2)$ vector multiplet given by

$\displaystyle\Delta(m_{x},u_{x}):=(-1)^{2m_{x}}(q^{m_{x}/2}u_{x}-q^{-m_{x}/2}u_{x}^{-1})(q^{m_{x}/2}u^{-1}_{x}-q^{-m_{x}/2}u_{x})\times\begin{cases}\frac{1}{2},\;m_{x}=0\\\
1,\;m_{x}\neq 0\end{cases}\ .$ (54)

The summation range of the $m_{x}$ will be discussed below.

One notices that the index at general $\nu_{i}$ can be obtained by shifting
$u_{i}\to(-q^{1/2})^{\nu_{i}}u_{i}$ in the index at $\vec{\nu}=0$ as

$\displaystyle{\cal I}_{q}(u_{i},\nu_{i})={\cal
I}_{q}(u_{i},\nu_{i}=0)|_{u_{i}\rightarrow(-q^{1/2})^{\nu_{i}}u_{i}}\;.$ (55)

The BPS operators of the IR SCFT can be counted correctly only if we choose
the right mixing parameters $\nu_{i}$, which can be determined by the
F-extremization in Jafferis:2010un .

We can also tune $\nu_{i}$’s to compute the spectrum of a sub-sector in which
we are particularly interested. For example, certain IR phases exhibit
$\mathcal{N}=4$ SUSY enhancement and contain interesting sub-sectors
consisting of holomorphic functions on the Coulomb or Higgs branch. In this
case, the parameter $\nu$ for the axial R-charge $J_{\rm Axial}$ of the
$SU(2)_{L}\times SU(2)_{R}$ R-symmetry, which is a flavor charge from the
point of view of the $\mathcal{N}=2$ superconformal algebra, can be tuned to
compute the Hilbert series of the Coulomb or Higgs branch operators as
discussed in Razamat:2014pta . We will see some examples below.

Next, ${\cal Z}^{(s=\pm 1)}_{{\cal M}_{g,p}}$ is the twisted partition
function on ${\cal M}_{g,p}$, the $S^{1}$ bundle of degree $p$ over genus $g$
Riemann surface $\Sigma_{g}$ Closset:2017zgf ; Closset:2018ghr . The $s$ in
the superscript represents the choice of spin-structure along the $S^{1}$. We
choose $s=+1$ ($s=-1$) as periodic (anti-periodic) boundary condition for
fermionic fields. For odd $p$, only $s=1$ choice is allowed while both choices
are allowed for even $p$. Especially for $p=0$, the partition function
corresponds to the twisted index $I^{(s)}_{g}(u_{i}=e^{m_{i}},\nu_{i}):={\cal
Z}^{(s)}_{{\cal M}_{g,p=0}}(m_{i},u_{i})$ on a genus-$g$ Riemann surface
$\Sigma_{g}$ Gukov:2015sna ; Benini:2015noa ; Benini:2016hjo ; Closset:2016arn
.

The twisted index counts supersymmetric ground states with signs on a
topologically twisted surface $\Sigma_{g}$, i.e.

$\displaystyle\begin{split}&I^{(s=1)}_{g}(u_{i},\nu_{i})=\textrm{Tr}_{\mathcal{H}(\Sigma_{g};R_{\vec{\nu}})}\bigg{[}(-1)^{2j_{3}}\prod_{i}u_{i}^{J_{i}}\bigg{]}\;,\\\
&I^{(s=-1)}_{g}(u_{i},\nu_{i})=\textrm{Tr}_{\mathcal{H}(\Sigma_{g};R_{\vec{\nu}})}\bigg{[}(-1)^{R_{\vec{\nu}}}\prod_{i}u_{i}^{J_{i}}\bigg{]}\;.\end{split}$
(56)

Here ${\cal H}(\Sigma_{g};R_{\vec{\nu}})$ is Hilbert space on $\Sigma_{g}$
with a topological twisting using the $U(1)_{R_{\vec{\nu}}}$ symmetry. Unlike
other BPS partition functions, the mixing parameters $\nu_{i}$ in the twisted
indices (with $g\neq 1$) can take only discrete values satisfying the
following Dirac quantization condition:

$\displaystyle R_{\vec{\nu}}(g-1)\in\mathbb{Z}\;.$ (57)

The condition is always satisfied at $\vec{\nu}=\vec{\nu}_{\rm geo}$ since the
$U(1)_{R_{\rm geo}}$ is a compact subgroup of a $SO(5)$ symmetry as in (38).
At $g=1$, the twisted index is a $(u_{i},\nu_{i})$-independent number, which
is equal to the Witten index on 3-torus Kim:2010mr ; Intriligator:2013lca .
When $g=0$ and $p=1$, the twisted partition function is identical to the round
3-sphere partition function,

$\displaystyle{\cal Z}^{(s=1)}_{{\cal M}_{g=0,p=1}}={\cal Z}_{b=1}\;.$ (58)

The twisted partition function can also be written as follows Closset:2017zgf

$\displaystyle{\cal Z}^{(s)}_{{\cal M}_{g,p}}=\sum_{a\in{\cal S}_{\rm
BE}}({\cal H}^{(s)}_{\alpha}(m_{i},\nu_{i}))^{g-1}({\cal
F}^{(s)}_{\alpha}(m_{i},\nu_{i}))^{p}\;.$ (59)

Here $\alpha$ labels so-called Bethe-vacua of the theory and ${\cal
H}^{(s)}_{\alpha}$, ${\cal F}^{(s)}_{\alpha}$ are handle-gluing/fibering
operators respectively. Note that the ${\cal F}^{(s)}$ is only well-defined up
to an overall sign factor, i.e. only $({\cal F}^{(s)})^{2}$ is well-defined,
for $s=-1$ since the twisted partition function is defined only for $p\in
2\mathbb{Z}$. Especially for $p=0$, the twisted indices are given as

$\displaystyle I_{g}^{(s=\pm
1)}(u_{i},\nu_{i})\big{|}_{u_{i}=e^{m_{i}}}=\sum_{a\in{\cal S}_{\rm BE}}({\cal
H}^{(s)}_{\alpha}(m_{i},\nu_{i}))^{g-1}\;.$ (60)

Lastly, $\mathcal{Z}_{b}$ is the partition function of a 3D $\mathcal{N}=2$
gauge theory on a squashed three-sphere $S^{3}_{b}$ with a squashing parameter
$b$. Using the Coulomb branch localization in Hama:2011ea , the squashed
$S^{3}_{b}$ partition function for the ${\cal T}[N,A;B]/SU(2)_{p}$ can be
written as

$\displaystyle\mathcal{Z}^{N_{p}}_{b}(m_{i},\nu_{i})$
$\displaystyle:=(\textrm{Squashed 3-sphere partition function of ${\cal
T}[N,A;B]/SU(2)_{p}$})$ (61)
$\displaystyle=\frac{1}{2}\int_{\mathbb{R}^{r+1}}\frac{dXd^{r}Z}{(2\pi\hbar)^{\frac{r+1}{2}}}{\cal
I}^{N_{p}}_{\hbar}(X,\vec{Z};W_{i})|_{W_{i}=m_{i}+(i\pi+\frac{\hbar}{2})\nu_{i},\;\hbar=2\pi
ib^{2}}\;,$

where $X$ and $\vec{Z}$ parametrize the Coulomb branches of the $SU(2)$ and
$U(1)^{r}$ vector multiplets, see (41), in the theory respectively. The
integrand $\mathcal{I}_{\hbar}$ is the collection of 1-loop contributions from
the vector and the chiral multiplets on the Coulomb branch. The contribution
from a chiral multiplet is given by a special function called quantum
dilogarithm faddeev1994quantum . The function will be denoted by
$\psi_{\hbar}(X)$. We follow the convention used in Gang:2019jut  and refer to
the paper for the definition and some basic properties for the function.

$\displaystyle\psi_{\hbar}(Z)\;:\;\textrm{quantum dilogarithm function}.$ (62)

Interestingly, the $S^{3}_{b}$ partition function depends only on a
holomorphic combination, $\vec{m}+(i\pi+\frac{\hbar}{2})\vec{\nu}$, of
$\vec{m}$ and $\vec{\nu}$. As a consequence of this holomorphic property, one
finds

$\displaystyle{\cal Z}_{b}(\vec{m},\vec{\nu})={\cal
Z}_{b}\left(\vec{m}+(i\pi+\frac{\hbar}{2})(\vec{\nu}-\vec{\nu}_{0}),\vec{\nu}_{0}\right)\;.$
(63)

This partition function has overall factor ambiguity of the following form due
to the local counter-terms,

$\displaystyle\exp\left(\frac{k^{ij}_{FF}W_{i}W_{j}+2k^{i}_{FR}W_{i}(i\pi+\frac{\hbar}{2})+k_{RR}(i\pi+\frac{\hbar}{2})^{2}}{2\hbar}\right)\bigg{|}_{W_{i}=m_{i}+(i\pi+\frac{\hbar}{2})\nu_{i}}\;,$
(64)

with properly normalized mixed CS levels $k_{FF},k_{FR},k_{RR}$ between
$U(1)^{F}$ flavor symmetry and $U(1)_{R}$ symmetry.

The above BPS partition functions are not all independent but related to each
other in a sophisticated way. For later use, we will review the relationship
between the squashed 3-sphere partition function and the twisted indices
discussed in Gang:2019jut . To relate the two partition functions, we consider
the following asymptotic expansion of the integrand ${\cal I}_{\hbar}$ in the
limit $\hbar\rightarrow 0$

$\displaystyle\log{\cal I}^{N_{p}}_{\hbar}\xrightarrow{\quad\hbar\rightarrow
0\quad}\frac{1}{\hbar}{\cal W}_{0}(X,\vec{Z};m_{i},\nu_{i})+{\cal
W}_{1}(X,\vec{Z};m_{i},\nu_{i})+\cdots\;.$ (65)

The leading part ${\cal W}_{0}$ corresponds to the twisted superpotential. In
order to compute the perturbative series ${\cal W}_{n}$, one needs to use
following asymptotic expansion of the quantum dilogarithm,

$\displaystyle\log\psi_{\hbar}(X)\xrightarrow{\quad\hbar\rightarrow
0\quad}\sum_{n=0}^{\infty}\frac{B_{n}}{n!}\textrm{Li}_{2-n}(e^{-Z})\hbar^{n-1}\;.$
(66)

Here $B_{n}$ are Bernoulli numbers with $B_{1}=+\frac{1}{2}$.

The Bethe vacua are the solutions to the Bethe equations that extremize the
twisted superpotential as

$\mathcal{S}_{\textrm{BE}}=\left\\{(x,\vec{z})_{\alpha}:\exp\left.\left(\partial_{\vec{Y}}\mathcal{W}_{0}(\vec{Y};m_{i},\nu_{i})\right)\right|_{\vec{Y}=\log\vec{y}_{\alpha}}=\vec{1},\;\textrm{triv.
isotr.}\right\\}/\mathbb{Z}_{2}^{\rm Weyl}\ ,$ (67)

where $\vec{Y}:=(X,\vec{Z})$ and $\vec{y}:=(x,\vec{z})$ and $\alpha$ labels a
Bethe vacuum. ‘triv. isotr.’ means that the Bethe solutions invariant under
the Weyl group $\mathbb{Z}_{2}$ should be discarded. The Weyl symmetry acts on
$(x,\vec{z})$ as

$\displaystyle\mathbb{Z}_{2}^{\rm
Weyl}\;:\;(x,\vec{z})\leftrightarrow(1/x,\vec{\tilde{z}})\;.$ (68)

In general, $\vec{\tilde{z}}$ is a non-trivial function of $\vec{z}$. However,
when the $SU(2)_{A}$ symmetry is manifest in the field theory description of
${\cal T}[N,A;B]$, $\vec{\tilde{z}}$ is just $\vec{z}$. In any case, we should
always discard Bethe-vacua with $x=1/x$, i.e. $x=\pm 1$. The handle-
gluing/fibering operators $\mathcal{H}^{(s)}_{\alpha}$/${\cal
F}^{(s)}_{\alpha}$ in (59) with $s=-1$ for each Bethe vacuum
$\vec{z}_{\alpha}$ can be computed as

$\displaystyle{\cal
F}_{\alpha}(m_{i},\nu_{i}):=\mathcal{F}^{(s=-1)}_{\alpha}(m_{i},\nu_{i})=\exp\left(i\frac{S_{0}^{\alpha}(m_{i},\nu_{i})-m_{i}\partial_{m_{i}}S_{0}^{\alpha}(m_{i},\nu_{i})}{2\pi}\right)\
,$ (69) $\displaystyle{\cal
H}_{\alpha}(m_{i},\nu_{i}):=\mathcal{H}^{(s=-1)}_{\alpha}(m_{i},\nu_{i})=\exp\left(-2S_{1}^{\alpha}(m_{i},\nu_{i})\right)\;.$

To avoid clutter, we omit the superscript $(s)$ in ${\cal H}_{\alpha}$ and
${\cal F}_{\alpha}$ when $s=-1$. We define

$\displaystyle
S_{0}^{\alpha}(m_{i},\nu_{i})=(\mathcal{W}_{0}(\vec{Y};m_{i},\nu_{i})-2\pi
i\vec{n}_{\alpha}\cdot\vec{Y})|_{\vec{Y}=\log\vec{y}_{\alpha}}\ ,$ (70)
$\displaystyle
S_{1}^{\alpha}(m_{i},\nu_{i})=\left.\left(\mathcal{W}_{1}(\vec{Y};m_{i},\nu_{i})-\frac{1}{2}\log\det\left(\frac{\partial^{2}\mathcal{W}_{0}}{\partial\vec{Y}\partial\vec{Y}}\right)\right)\right|_{\vec{Y}=\log\vec{y}_{\alpha}}\
,$

with

$\partial_{\vec{Y}}\mathcal{W}_{0}(\vec{Y};m_{i},\nu_{i})|_{\vec{Z}=\log\vec{y}_{\alpha}}=2\pi
i\vec{n}_{\alpha}\ .$ (71)

Due to the overall factor ambiguity of $\mathcal{Z}_{b}$ in (64), the
fibering/handle-gluing operators have following overall factor ambiguities

$\displaystyle\begin{split}&\exp\left(-\frac{i}{4\pi}(k_{FF}^{ij}m_{i}m_{j}-\nu_{i}\nu_{j})-\frac{i\pi}{4}(2k_{FR}^{i}\nu_{i}+k_{RR})\right)\textrm{
for }\mathcal{F}_{\alpha}\;,\\\
&\exp\left(-k_{FF}^{ij}\nu_{i}(m_{j}+i\pi\nu_{j})-k^{i}_{FR}(m_{i}+2i\pi\nu_{i})-i\pi
k_{RR}\right)\textrm{ for }\mathcal{H}_{\alpha}\;.\end{split}$ (72)

As commented around (59), only $({\cal F}_{\alpha}^{(s=-1)})^{2}$ is well-
defined and the above relation for ${\cal F}_{\alpha}$ should be understood as
$({\cal
F}_{\alpha}^{(s=-1)})^{2}=\exp\left(i\frac{S_{0}-m\partial_{m}S_{0}}{\pi}\right)$
for general cases. But for some cases, the twisted partition function
$\mathcal{Z}^{(s)}_{{\cal M}_{g,p\in 2\mathbb{Z}}}$ does not depend on the
spin-structure choices. In the case, the above relation should be understood
as ${\cal F}^{(s=+1)}={\cal
F}^{(s=-1)}=\exp\left(i\frac{S_{0}-m\partial_{m}S_{0}}{2\pi}\right)$.

The partition function $\mathcal{Z}_{b}$ at $b=1$, which corresponds to a
round 3-sphere, i.e. ${\cal Z}^{(s=1)}_{{\cal M}_{(g,p)=(1,1)}}$, can also be
written as a sum over the Bethe-vacua,

$\mathcal{Z}_{b=1}(\vec{m}=0,\vec{\nu})=\mathcal{Z}_{b=1}\left(\vec{m}=2\pi
i(\vec{\nu}-\vec{\nu}_{0}),\vec{\nu}_{0}\right)=\sum_{\alpha}\frac{\mathcal{F}_{\alpha}(\vec{m}=2\pi
i(\vec{\nu}-\vec{\nu}_{0}),\vec{\nu}_{0})}{\mathcal{H}_{\alpha}(\vec{m}=2\pi
i(\vec{\nu}-\vec{\nu}_{0}),\vec{\nu}_{0})}\;.$ (73)

The first equality follows from the relation in (63) with $\hbar=2\pi i$.
Here, $\vec{\nu}_{0}$ denotes a special choice of $\nu$ satisfying the
condition

$\displaystyle R_{\vec{\nu}_{0}}+2j_{3}\in 2\mathbb{Z}\;,$ (74)

for all local BPS operators Gang:2019jut . Namely, the superconformal index at
$\vec{\nu}=\vec{\nu}_{0}$ will only contain the terms with integer power of
$q$, which is the fugacity conjugate to $\frac{R_{\vec{\nu}}}{2}+j_{3}$. It is
argued in Gang:2019jut  that $({\cal H}^{(s=1)},{\cal F}^{(s=1)})=({\cal
H}^{(s=-1)},{\cal F}^{(s=-1)})$ at $\vec{\nu}=\vec{\nu}_{0}$ for non-trivial
SCFTs, and we thus have the formula given in (73) in this case. The Bethe sum
formula makes it much easier to compute the round 3-sphere partition function
${\cal Z}_{b=1}$ than the Coulomb branch integral formula.

The $S^{3}$ partition function $\mathcal{Z}_{b=1}$ can be used to obtain the
exact superconformal R-symmetry $R_{\rm IR}=R_{\vec{\nu}_{\rm IR}}$ of IR CFT.
This is the F-extremization principle in Jafferis:2010un  which can be
summarized as

$\displaystyle\textrm{F-extremization : }F(\vec{\nu}):=-\log|{\cal
Z}_{b=1}(m_{i}=0,\vec{\nu})|\textrm{ is extremized at
$\vec{\nu}=\vec{\nu}_{\textrm{IR}}$.}$ (75)

##### Effect of $\mathbb{Z}_{2}$ 1-form symmetry gauging

So far we have studied the BPS partition functions for ${\cal
T}[N,A;B]/SU(2)_{p}$ theory. The theory is identical to ${\cal T}[N_{pA+B}]$
for even $A$, but the two theories are different for odd $A$. To study the BPS
partition functions of ${\cal T}[N_{pA+B}]$ theory for odd $A$, one needs to
understand the effect of gauging 1-form $\mathbb{Z}_{2}$ symmetry on the BPS
partition functions. For this, we shall use the general formula given by

$\displaystyle\mathcal{Z}[{\cal T}/{\mathbb{Z}_{2}}\textrm{ on
}\mathcal{B}]=\frac{|H^{0}(\mathcal{B},\mathbb{Z}_{2})|}{|H^{1}(\mathcal{B},\mathbb{Z}_{2})|}\sum_{[\beta_{2}]\in
H^{2}(\mathcal{B},\mathbb{Z}_{2})}{\cal Z}[\mathcal{T}\textrm{ on }{\cal
B};[\beta_{2}]]\;.$ (76)

Here ${\cal T}$ is a 3D field theory with non-anomalous $\mathbb{Z}_{2}$ one-
form symmetry and ${\cal T}/\mathbb{Z}_{2}$ is the theory after gauging the
symmetry. ${\cal B}$ is a general curved background and ${\cal
Z}[\mathcal{T}\textrm{ on }{\cal B};[\beta_{2}]]$ is the partition function of
${\cal T}$ on ${\cal B}$ with background flat $\mathbb{Z}_{2}$ 2-form
$[\beta_{2}]$ coupled to the 1-form symmetry.

By applying the formula to ${\cal T}=\frac{{\cal T}[N,A;B]}{SU(2)_{p}}$ and
${\cal B}=S^{3}_{b}$ (squashed 3-sphere), one has

$\displaystyle(\mathcal{Z}_{b}(m_{i},\nu_{i})\textrm{ for }{\cal
T}[N_{pA+B}]\textrm{ with odd $A$})=\begin{cases}2{\cal
Z}_{b}^{N_{p}}(m_{i},\nu_{i})\;,\quad p\in 2\mathbb{Z}\\\ \sqrt{2}{\cal
Z}_{b}^{N_{p}}(m_{i},\nu_{i})\;,\quad p\in 2\mathbb{Z}+1\end{cases}$ (77)

Here we use the fact that
$|H^{1}(S^{3},\mathbb{Z}_{2})|=|H^{2}(S^{3},\mathbb{Z}_{2})|=1$,
$|H^{0}(S^{3},\mathbb{Z}_{2})|=2$ and
$\mathcal{Z}[U(1)_{-2}\textrm{ on }S^{3}_{b}]=\frac{1}{\sqrt{2}}$. Since the
two partition functions are related to each other by an overall numerical
factor, the gauging does not affect the F-extremization.

Gauging the 1-form $\mathbb{Z}_{2}$ symmetry affects the summation range of
$m_{x}$ (when $A$ is odd) in the superconformal index in (53) as

$\displaystyle\begin{cases}m_{x}\in\mathbb{Z}_{\geq 0}\;,\quad\textrm{for
}\frac{{\cal T}[N,A;B]}{SU(2)_{p}}\\\ m_{x}\in\frac{1}{2}\mathbb{Z}_{\geq
0}\;,\quad\textrm{for }{\cal T}[N_{pA+B}]=\frac{{\cal
T}[N,A;B]}{``SO(3)"_{p}}\end{cases}$ (78)

For odd $A$, all local operators have integer spins under the $SU(2)$ gauge
symmetry and the Dirac quantization requires that $m_{x}\in\mathbb{Z}/2$
instead of $m_{x}\in\mathbb{Z}$. The above summation range reflects the fact
that $SO(3)$ principle bundle on $S^{2}$ admits half-integer monopole fluxes,
$m_{x}\in\mathbb{Z}/2$, while $SU(2)$ bundle only admits integer monopole
fluxes, $m_{x}\in\mathbb{Z}$.

The superconformal index is a BPS partition function on ${\cal B}=S^{2}\times
S^{1}$ and $H^{2}(\mathcal{B},\mathbb{Z}_{2})=\mathbb{Z}_{2}=\\{1,-1\\}$.
Therefore, unlike in higher dimensions, 3D superconformal index is sensitive
to the gauging of 1-form symmetry. The index obtained by summing over all
$m_{x}\in\mathbb{Z}_{\geq 0}$ can be identified with ${\cal Z}[{\cal
T}\textrm{ on }S^{2}\times S^{1};[\beta_{2}]=1]$ in (76) while the index from
all $m_{x}\in\mathbb{Z}_{\geq 0}+\frac{1}{2}$ is ${\cal Z}[{\cal T}\textrm{ on
}S^{2}\times S^{1};[\beta_{2}]=-1]$. For odd $p$, the $\mathbb{Z}_{2}$ gauging
does not affect the superconformal index since there is no contribution from
$m_{x}\in\mathbb{Z}+\frac{1}{2}$. This is indeed expect from the relation in
(48). For some cases, the 1-form $\mathbb{Z}_{2}$ symmetry decouples at IR. In
that case, the theory ${\cal T}[N,A;B]/SU(2)_{p}$ is the same as ${\cal
T}[N,A;B]/``SO(3)"_{p}$, and the index gets the same contributions from
$m_{x}\in\mathbb{Z}$ and $m_{x}\in\mathbb{Z}+\frac{1}{2}$. Then, we need to
sum over only $m_{x}\in\mathbb{Z}$ for ${\cal T}[N_{pA+B}]$ theory, otherwise
the index will start with $2+O(q^{1/2})$.

The twisted partition function is also affected after gauging the
$\mathbb{Z}_{2}$ 1-form symmetry. For odd $p$, using the relation in (48), one
has

$\displaystyle\begin{split}&\left(\mathcal{S}_{\rm BE}\textrm{ of }\frac{{\cal
T}[N,A;B]}{SU(2)_{p}}\right)=\left(\mathcal{S}_{\rm BE}\textrm{ of
}\frac{{\cal T}[N,A;B]}{``SO(3)_{p}"}\right)^{\otimes 2}\;,\\\
&\bigg{\\{}{\cal H}_{\alpha}\textrm{ of }\frac{{\cal T}[N,A;B]}{SU(2)_{p\in
2\mathbb{Z}+1}}\bigg{\\}}=\bigg{\\{}2{\cal H}_{\alpha}\textrm{ of }\frac{{\cal
T}[N,A;B]}{``SO(3)"_{p}}\bigg{\\}}\times\bigg{\\{}1,1\bigg{\\}}\;,\\\
&\bigg{\\{}{\cal F}_{\alpha}\textrm{ of }\frac{{\cal T}[N,A;B]}{SU(2)_{p\in
2\mathbb{Z}+1}}\bigg{\\}}=\bigg{\\{}{\cal F}_{\alpha}\textrm{ of }{\cal
T}[(N)_{pA+B}]\bigg{\\}}\times\bigg{\\{}1,e^{\frac{\pi
i}{2}}\bigg{\\}}\;.\end{split}$ (79)

This follows from that the topological theory $U(1)_{2}$ theory has two simple
objects (or two Bethe-vacua) $\alpha=0$ and $\alpha=1$ with
$S_{00}=S_{01}=1/\sqrt{2}$ (i.e. ${\cal H}_{\alpha=0,1}=2$) and topological
spins $h_{\alpha=0}=0$ and $h_{\alpha=1}=\frac{1}{4}$ (i.e ${\cal
F}_{\alpha=1}/{\cal F}_{\alpha=0}=\exp(\frac{1}{4}\times 2\pi i)$).

For even $p$, the effect of the gauging is more subtle. The Bethe-vacua in
(67) can be divided into two classes, ${\cal S}_{\rm BE}=S^{(+1)}_{\rm
BE}\bigsqcup S^{(-1)}_{\rm BE}$,

$\displaystyle\left(\mathcal{S}^{Q=\pm 1}_{\rm BE}\textrm{ of }\frac{{\cal
T}[N,A;B]}{SU(2)_{p}}\right)=\left\\{(x,z)\in{\cal S}_{\rm
BE}\;:\;\exp\left(\frac{1}{2}\partial_{X}{\cal W}_{0}\right)=Q\right\\}\;.$
(80)

Then, the Bethe-vacua for the gauged theory are (when $p\in 2\mathbb{Z}$)

$\displaystyle\begin{split}&\left(\mathcal{S}_{\rm BE}\textrm{ of }{\cal
T}[N_{pA+B}]=\big{(}\frac{{\cal
T}[N,A;B]}{SU(2)_{p}}\big{)}/\mathbb{Z}_{2}\right)={\cal S}^{\rm
untwisted}_{\rm BE}\bigsqcup{\cal S}^{\rm twisted}_{\rm
BE}\quad\textrm{where}\\\ &{\cal S}^{\rm untwisted}_{\rm
BE}=\bigg{\\{}(x,\vec{z})\in{\cal S}_{\rm
BE}^{Q=1}\;:\;x+\frac{1}{x}\neq-(x+\frac{1}{x})\bigg{\\}}/\mathbb{Z}_{2}^{\rm
1-form}\textrm{ and }\\\ &{\cal S}^{\rm twisted}_{\rm
BE}=\bigg{\\{}(x,\vec{z})\in{\cal S}_{\rm
BE}^{Q=1}\;:\;x+\frac{1}{x}=-(x+\frac{1}{x})\bigg{\\}}^{\otimes
2}\;.\end{split}$ (81)

The $\mathbb{Z}_{2}$ 1-form symmetry acts as999The Bethe-vacua represents the
supersymmetric vacua on two torus $\mathbb{T}^{2}$. Since the
$H_{1}(\mathbb{T}^{2},\mathbb{Z})=\mathbb{Z}\oplus\mathbb{Z}$, one can
consider two $\mathbb{Z}_{2}$ 1-form symmetry charge operators , $Q_{(1,0)}$
and $Q_{(0,1)}$, supported on the two generators of the
$H_{1}(\mathbb{T}^{2},\mathbb{Z})$. The set ${\cal S}_{\rm BE}^{Q=1}$ selects
the Bethe-vacua with $Q_{(0,1)}=1$ while the $Q_{(1,0)}$ transforms a Bethe-
vacuum $(x,\vec{z})$ to another Bethe-vacuum $(-x,\vec{\tilde{z}})$.

$\displaystyle\mathbb{Z}_{2}^{\rm
1-form}\;:\;(x,\vec{z})\leftrightarrow(-x,\vec{\tilde{z}})\;.$ (82)

Like the $\mathbb{Z}^{\rm Weyl}_{2}$ case, the $\vec{\tilde{z}}$ is in general
a nontrivial function of $\vec{z}$. For the Bethe-vacua invariant under the
$\mathbb{Z}_{2}^{\rm 1-form}$ modulo the $\mathbb{Z}_{2}^{\rm Weyl}$, i.e
$x=\pm i$, we count them with multiplicity 2. For some cases, all the Bethe-
vacua in ${\cal S}_{\rm BE}$ are invariant under the $\mathbb{Z}_{2}^{\rm
1-form}$. This implies that 1-form symmetry decouples at IR. In this case, the
two theories ${\cal T}[N_{pA+B}]$ and ${\cal T}[\frac{{\cal
T}[N,A;B]}{SU(2)_{p}}]$ are identical and thus they have the same Bethe vacua.
The handle-gluing and fibering operators of the Bethe-vacua are

$\displaystyle\begin{split}&\left({\cal H}_{\alpha}\textrm{ of }{\cal
T}[N_{pA+B}]_{p\in
2\mathbb{Z}}\right)=\begin{cases}\frac{1}{4}\times\left({\cal
H}_{\alpha}\textrm{ of }\frac{{\cal
T}[N,A;B]}{SU(2)_{p}}\right),\;\;\alpha\in{\cal S}^{\rm untwisted}_{\rm BE}\\\
\left({\cal H}_{\alpha}\textrm{ of }\frac{{\cal
T}[N,A;B]}{SU(2)_{p}}\right),\;\;\alpha\in{\cal S}^{\rm twisted}_{\rm
BE}\end{cases}\\\ &\left({\cal F}_{\alpha}\textrm{ of }{\cal
T}[N_{pA+B}]_{p\in 2\mathbb{Z}}\right)=\left({\cal F}_{\alpha}\textrm{ of
}\frac{{\cal T}[N,A;B]}{SU(2)_{p}}\right)\;.\end{split}$ (83)

For some cases, only $({\cal F}_{\alpha})^{2}$ (instead of ${\cal
F}_{\alpha}$) is well-defined in the ${\cal T}[N_{pA+B}]$ theory since the
fibering operator for a Bethe-vacuum with $Q=+1$ can change its sign under the
action of $\mathbb{Z}^{\rm 1-form}_{2}$. It implies that ${\cal
F}_{\alpha}^{(s=-1)}\neq{\cal F}_{\alpha}^{(s=1)}$ and the twisted partition
functions $\mathcal{Z}_{{\cal M}_{g,p\in 2\mathbb{Z}}}^{(s)}$ depends on the
spin-structure choice.

##### 3D-3D relations for 3D index and ${\cal S}_{\rm BE}$

The 3D-3D relation tells us that the supersymmetric partition functions of the
${\cal T}[M]$ theory are related to the topological invariants of the
$SL(2,\mathbb{C})$ Chern-Simons theory on the associated 3-manifold $M$. For
example,

$\displaystyle\begin{split}&\big{(}{\cal
I}_{q}(u_{i}=1,\vec{\nu}=\vec{\nu}_{\rm geo})\textrm{ of ${\cal T}[M]$
theory}\big{)}=\begin{cases}{\cal
I}_{M}(q)\;,\quad\cap_{\rho}\textrm{Inv}(\rho)=1\\\ \frac{1}{2}{\cal
I}_{M}(q)\;,\quad\cap_{\rho}\textrm{Inv}(\rho)=\mathbb{Z}_{2}\end{cases}\\\
&\big{(}{\cal S}_{\rm BE}\textrm{ of ${\cal T}[M]$ theory}\big{)}\\\
&=\begin{cases}\big{\\{}[\rho]\in\chi^{SL(2,\mathbb{C})}_{\rm
irred}\;\textrm{counted with multiplicity
}|\textrm{Inv}(\rho)|\big{\\}}\;,\quad\cap_{\rho}\textrm{Inv}(\rho)=1\\\
\chi^{SL(2,\mathbb{C})}_{\rm
irred}\;,\quad\cap_{\rho}\textrm{Inv}(\rho)=\mathbb{Z}_{2}\;.\end{cases}\end{split}$
(84)

For the case when $\cap_{\rho}\textrm{Inv}(\rho)=\mathbb{Z}_{2}$, the
corresponding ${\cal T}[N,A;B]/SU(2)_{p}$ theory has a decoupled
$\mathbb{Z}_{2}^{\rm 1-form}$ symmetry and the 3D-3D relation should be
treated separately as discussed above.

Since the BPS partition functions are protected along RG-flows, they can be
used to scrutinize the physics of the low energy phases. We now explain a
number of possible scenarios for the IR phases of $\mathcal{T}[M]$ captured by
the BPS partition functions.

##### Spontaneous SUSY Breaking

We start with the IR phases with spontaneously broken supersymmetries. As
explained above, the superconformal index counts local BPS operators in radial
quantization. In particular, the identity operator corresponding to a unique
supersymmetric ground state in the radially quantized theory is the leading
contribution to the index, which cannot be cancelled by other operator
contributions due to the unitarity bounds. The absence of this SUSY ground
state and its index contribution therefore implies spontaneous SUSY breaking.
As a result, all the BPS states in this case are paired up to form long
multiplets and their contributions to the index are all cancelled. This means
the superconformal index becomes trivial, i.e. $\mathcal{I}_{q}=0$. Twisted
indices $I_{g}$ also vanish for all $g$ since there are fermionic zero modes
associated to the broken SUSYs, which implies that ${\cal S}_{\rm
BE}=\emptyset$ from (60). Therefore, the supersymmetry of $\mathcal{T}[M]$ is
spontaneously broken at low energy if and only if

$\displaystyle(\textrm{Spontaneous SUSY
breaking})\Leftrightarrow(\mathcal{I}_{q}=0)\Leftrightarrow(\mathcal{S}_{\rm
BE}=\emptyset)\;.$ (85)

The 3D-3D relation for the superconformal index in (84) implies that the
geometric invariant $\mathcal{I}_{M}(q)$ (or 3D index) becomes trivial in this
case. This means that the 3-manifold is of empty type. On the other hand, the
relation for ${\cal S}_{\rm BE}$ implies the 3-manifold does not have any
irreducible $SL(2,\mathbb{C})$ flat connection, which is another implication
that the 3-manifold is of empty type.

##### Gapped Phase

The gapped phase is another interesting IR phase of $\mathcal{T}[M]$. One
distinguished property of the gapped phase is that it has a unique ground
state on $S^{2}$ and all other local states are gapped out at low energy. From
this, we assert that a 3D ${\cal N}=2$ gauge theory has a mass gap only if

$\displaystyle{\cal I}_{q}(u_{i},\nu_{i})=1\;.$ (86)

The IR phase in this case is described by a unitary topological field theory
(TFT). This condition can be satisfied only when $M$ is of finite/real type.
Otherwise, the index at $u_{i}=1$ and $\vec{\nu}=\vec{\nu}_{\rm geo}$ is
divergent or it shows exponentially growing behavior in the Cardy limit
$q\rightarrow 1$ according to the 3D-3D relation (84) and the analysis below
(22). For a 3-manifold $M$ of finite and real type, the superconformal index
in the specialization always becomes just $1$. But it does not guarantee that
the superconformal index becomes 1 at general value of $(u_{i},\nu_{i})$. For
these finite/real 3-manifolds, one can assign a topological field theory ${\rm
TFT}[M]$ to the 3-manifold as studied in Cho:2020ljj . Furthermore, it was
claim that the ${\cal T}[M]$ has a mass gap only when the topological theory
is unitary. In that case, the superconformal index is
$(m_{i},\nu_{i})$-independent $1$, and the IR physics is described by the
unitary topological field theory ${\rm TFT}[M]$. For the case when the ${\rm
TFT}[M]$ is non-unitary, the corresponding IR phase will be discussed below.

The physics of a bosonic unitary TQFT can be described by a mathematical
framework called modular tensor category (MTC). Let us provide a simple
dictionary between MTC data and BPS partition functions that we can use to
identify the topological theory appearing at IR. Firstly, simple objects
(anyons) in TQFT are in one-to-one with Bethe-vacua.

$\displaystyle\alpha\;(\textrm{simple objects in
MTC})\quad\leftrightarrow\quad(x_{\alpha},\vec{z}_{\alpha})\in{\cal S}_{\rm
BE}\;.$ (87)

The modular S- and T-matrices in MTC can be computed using the handle-
gluing/fibering operators of $\mathcal{T}[M]$. We compute them as

$\displaystyle S_{0\alpha}^{2}=(\mathcal{H}_{\alpha}(\vec{\nu}_{\rm
geo}))^{-1}\;,\quad
T_{\alpha\alpha}/T_{00}=\mathcal{F}_{\alpha}(\vec{\nu}_{\rm
geo})/F_{\alpha=0}(\vec{\nu}_{\rm geo})\;.$ (88)

Here $\alpha=0$ corresponds to the trivial simple object in TQFT, and the
corresponding Bethe-vacuum $(x_{\alpha},\vec{z}_{\alpha})_{\alpha=0}$ is
chosen to satisfy

$\displaystyle|{\cal Z}_{b}|=S_{00}=\frac{1}{\sqrt{{\cal
H}_{\alpha=0}(\vec{\nu}_{\rm geo})}}\;.$ (89)

For a gapped phase, the squashed 3-sphere partition function is
$b$-independent, modulo the local counter-terms in (64), and its absolute
value should equal to the $S_{00}$ witten1989quantum . The above relations
make sense since the handle-gluing and fibering operators for a gapped phase
do not depend on the continuous real mass parameters $m_{i}$, and they satisfy

$\displaystyle|{\cal F}_{\alpha}|=1\textrm{ and }{\cal
H}_{\alpha}\in\mathbb{R}_{+}\textrm{ for all Bethe-vacua $\alpha$}\;$ (90)

in a proper choice of the local counter-terms
$(k_{FF}^{ij},K_{FR}^{i},K_{RR})$ in (72). The topological spin
$h_{\alpha}\in\mathbb{R}/\mathbb{Z}$ of an anyon can be read off from the
T-matrix as

$\displaystyle e^{2\pi ih_{\alpha}}=T_{\alpha\alpha}/T_{00}\;.$ (91)

For some cases, as seen below (83), only $({\cal F}_{\alpha})^{2}$ (instead of
${\cal F}_{\alpha}$) is well-defined. In that case, the partition functions on
${\cal M}_{g,p\in 2\mathbb{Z}}$ depend on the spin-structure choices, and thus
the topological theory is actually spin (fermionic) TQFT. For spin TQFTs, the
topological spin $h_{\alpha}$ is well defined only up to mod $1/2$ (instead of
$1$).

We can extract some constraints on the handle-gluing operators for a unitary
TQFTs from the BPS partition functions. Since there is a unique ground state
in a TQFT on $S^{2}$, the twisted index at $g=0$ and the handle-gluing
operators must satisfy

$\displaystyle I_{g=0}=\sum_{\alpha}({\cal H}_{\alpha})^{-1}=1\;.$ (92)

In addition, the modular S-matrix for a unitary TQFT possess a unitarity
condition providing an inequality among the handle-gluing operators as

$\displaystyle(\textrm{Unitarity})\;:\;|S_{00}|\leq|S_{0\alpha}|\;\Rightarrow\;{\cal
H}_{\alpha=0}\geq{\cal H}_{\alpha}\;.$ (93)

##### Rank-0 ${\cal N}=4$ SCFT Phase

Interestingly, the ${\rm TFT}[M]$ associated with finite/real 3-manifold $M$
can be non-unitary as found in Cho:2020ljj . In that case, we call the
3-manifold is of finite/real/non-unitary type. For that case, it is claimed
that the ${\cal T}[M]$ does not have a mass gap. Instead, the theory flows to
an ${\cal N}=2$ SCFT with a $U(1)$ flavor symmetry, which contains the non-
unitary TQFT structure in a sub-sector. Modular S- and T-matrices for the TQFT
can be obtained using the expressions in (88) and (89). One crucial difference
with the gapped case is that i) the ${\cal H}$, ${\cal F}$ as well as ${\cal
Z}_{b}$ depend on $(m,\nu)$ associated with the $U(1)$ symmetry, and the
modular matrices appear only at $m=0$ and $\nu=\nu_{\rm geo}$, and ii) the
modular S-matrix violates the unitarity condition in (93). Recently, it is
found that non-unitary TQFTs appear in degenerate limits of 3D ${\cal N}=4$
rank-0 SCFTs Gang:2021hrd . The degenerate limit is the Coulomb/Higgs branch
limits studied in Razamat:2014pta  to compute the Hilbert-series.

Motivated by these curious observations, we claim that the 3D $\mathcal{T}[M]$
theory associated with the finite/real/non-unitary 3-manifolds $M$ flows at
low energy to a rank-0 SCFT with an enhanced $\mathcal{N}=4$ superconformal
algebra.

In this case, as we will see with explicit examples below, the UV field theory
constructed by the method in Section 3.2 always has a $U(1)$ flavor symmetry,
which is an accidental flavor symmetry from the point of view of the class R
theory construction. Hence, we can turn on the real mass/fugacity $m/u$ and
the mixing parameter $\nu$ for the flavor symmetry in the BPS partition
function computations using the UV field theory description. For convenience,
we shall set the mixing parameter $\nu=1$ as the geometrical $R$-charge
choice, i.e. $\nu_{\rm geo}=1$. As we will explicitly check from the
F-extremizations on various examples, it turns out that the superconformal
R-charge $R_{\nu_{\rm IR}}$ in the low energy theory is always at $\nu_{\rm
IR}=1\pm 1$. Then, by a proper redefinition of the mixing parameter as
$\nu\leftarrow 2-\nu$ or $\nu\leftarrow\nu$, one can always set

$\displaystyle\nu_{\rm IR}=0\;,\quad\nu_{\rm geo}=1\;.$ (94)

Our claim is that the accidental $U(1)$ flavor symmetry becomes the $U(1)_{\rm
Axial}$ subgroup of the $SO(4)\simeq SU(2)_{L}\times SU(2)_{R}$ R-symmetry of
the enhanced $\mathcal{N}=4$ superconformal algebra in the low energy theory.
The charge $J_{\rm Axial}$ for this accidental symmetry is identified with the
axial combination of the enhanced R-charges as

$\displaystyle J_{\rm Axial}=(J^{3}_{L}-J^{3}_{R})\in\mathbb{Z}\;,$ (95)

where $J^{3}_{L/R}$ are the Cartan charges of $SU(2)_{L/R}$ normalized as
$J^{3}_{L/R}\in\frac{\mathbb{Z}}{2}$. The mixing parameter $\nu$ is then

$\displaystyle R_{\nu}=(J^{3}_{L}+J^{3}_{R})+\nu(J_{L}^{3}-J_{R}^{3})\;.$ (96)

To confirm the SUSY enhancement in the low energy theory, we first compute the
superconformal index using the R-charge of the IR $\mathcal{N}=2$
superconformal algebra. The additional SUSY current multiplets for the
enhanced ${\cal N}=4$ superconformal algebra contains a $1/4$ BPS operator
with $j_{3}=R=1$ and $\Delta=2$ (see, for example, Cordova:2016emh ), and this
operator contributes to the superconformal index at order $q^{3/2}$.
Therefore, the existence of the enhanced SUSY ensures the following pattern of
the index:

$\displaystyle{\cal I}_{q}(u,\nu=0)\ni-\left(u+\frac{1}{u}\right)q^{3/2}\;.$
(97)

The appearance of the above term is a strong evidence101010But it can not be a
proof since the term $-q^{3/2}$ could also come from a chiral primary operator
with $R=\Delta=3$. that the the theory has a SUSY enhancement to ${\cal N}=4$
and the accidental $U(1)$ symmetry becomes the axial $U(1)_{\rm Axial}$
subgroup of the enhanced R-symmetry.

We further conjecture that the resulting IR ${\cal N}=4$ SCFT is a rank-0
theory having no Higgs/Coulomb branches. The 3D ${\cal N}=4$ rank-0 SCFTs are
extensively studied in Gang:2021hrd  and they propose several non-trivial
relations among SUSY partition functions of the rank-0 SCFTs. We list some of
the relations which we will use in the next section,

$\displaystyle\bullet{\cal I}_{q}(u=1,\nu=\pm 1)=1\;,$ (98)
$\displaystyle\bullet|\mathcal{F}_{\alpha}(m,\nu=\pm
1)|=1,\;\mathcal{H}_{\alpha}(m,\nu=\pm 1)\in\mathbb{R}_{+}\textrm{ and
}\;\sum_{\alpha}\big{(}\mathcal{H}_{\alpha}(m=0,\nu=\pm 1)\big{)}^{-1}=1\;,$
(99) $\displaystyle\bullet\exists\textrm{ a Bethe-vacuum }\alpha=0\textrm{
such that }|\mathcal{Z}_{b=1}(m=0,\nu=\pm
1)|=\mathcal{H}^{-1/2}_{\alpha=0}(m=0,\nu=\pm 1)\;,$ (100)
$\displaystyle\bullet|\mathcal{Z}_{b=1}(m=0,\nu=0)|=\textrm{min}_{\alpha}[\mathcal{H}_{\alpha}(m=0,\nu=\pm
1)^{-1/2}]\;.$ (101)

The 1st equation implies that the theory has no Coulomb/Higgs branches since
the index in the degenerate limit, $u=1$ and $\nu\rightarrow\pm 1$, counts
Coulomb/Higgs branch operators and their descendants Razamat:2014pta . The 2nd
equation implies the emergence of a pair of TQFT structures in the degenerate
limits where $m=0$ and $\nu\rightarrow\pm 1$. This can be compared with the
relations (90) and (92) for unitary TQFTs. The 3rd one implies that the
emergent TQFT structure in the degenerate limit is of non-unitary type unlike
the gapped cases in (89). The emergence of non-unitary TQFT structures in the
${\cal N}=4$ rank-0 SCFTs was already observed in Gang:2021hrd  and dubbed as
(rank-0 SCFT)/(non-unitary TQFT) correspondence. The last equation is one of
(and the most surprising) dictionary of the correspondence proposed in that
work. Thus, the confirmation of the above equations not only provides a strong
evidence for the emergence of 3D ${\cal N}=4$ rank-0 theory at IR but also
non-trivial test for the dictionary.

##  4 Examples

In this section, we check the our proposal in (49) using non-hyperbolic
3-manifolds $M$ obtained by exceptional Dehn fillings on five simplest
1-cusped hyperbolic manifolds
($m003,m004=S^{3}\backslash\mathbf{4}_{1},m006,m007,m009$) and
$m015=S^{3}\backslash\mathbf{5}_{2}$.111111Dehn fillings on $m005$ and $m008$
always yield non-hyperbolic manifolds.

$M$ | Topological type | IR phase | UV theory
---|---|---|---
$(S^{3}\backslash\mathbf{4}_{1})_{p\mu+\lambda:0\leq|p|\leq 3}$ | Finite/Real/Unitary | Gapped/TQFT | Sec. 4.1.1
$(S^{3}\backslash\mathbf{4}_{1})_{p\mu+\lambda:|p|=4}$ | Infinite |  ${\cal N}=2$ SCFT with CPOs |
$(S^{3}\backslash\mathbf{5}_{2})_{p\mu+\lambda:1\leq p\leq 3}$ | Finite/Real/Non-unitary |  ${\cal N}=4$ rank-0 SCFT | Sec. 4.2.1
$(S^{3}\backslash\mathbf{5}_{2})_{p\mu+\lambda:p=0,4}$ | Infinite |  ${\cal N}=2$ SCFT with CPOs |
$(m003)_{pA_{\rm sp}+B_{\rm sp}:p=1,-2}$ | Finite/Real/Unitary | Gapped/TQFT | Sec. 4.3.1
$(m003)_{pA_{\rm sp}+B_{\rm sp}:p=0,-1}$ | Empty | SUSY broken |
$(m006)_{pA_{\rm sp}+B_{\rm sp}:p=0}$ | Empty | SUSY broken | Sec. 4.4.1
$(m006)_{pA_{\rm sp}+B_{\rm sp}:p=\pm 1}$ | Finte/Real/Unitary | Gapped/TQFT |
$(m006)_{pA_{\rm sp}+B_{\rm sp}:p=-2}$ | Finte/Real/Non-unitary |  ${\cal N}=4$ rank-0 SCFT |
$(m006)_{pA_{\rm sp}+B_{\rm sp}:p=-3}$ | Infinite |  ${\cal N}=2$ SCFT with CPOs |
$(m007)_{pA_{\rm sp}+B_{\rm sp}:p=0,2}$ | Finte/Real/Unitary | Gapped/TQFT | Sec. 4.5.1
$(m007)_{pA_{\rm sp}+B_{\rm sp}:p=-2,-1,1}$ | Finte/Real/Non-unitary |  ${\cal N}=4$ rank-0 SCFT |
$(m007)_{pA_{\rm sp}+B_{\rm sp}:p=-3}$ | Infinite |  ${\cal N}=2$ SCFT with CPOs |
$(m009)_{pA_{\rm sp}+B_{\rm sp}:-2\leq p\leq 2}$ | Finte/Real/Unitary | Gapped/TQFT | Sec. 4.6.1
$(m009)_{pA_{\rm sp}+B_{\rm sp}:|p|=3}$ | Infinite |  ${\cal N}=2$ SCFT with CPOs |
Table 1: IR phases of class R theories associated to non-hyperbolic
3-manifolds obtained by exceptional Dehn fillings on
$S^{3}\backslash\mathbf{4}_{1},S^{3}\backslash\mathbf{5}_{2},m003,m006,m007$
and $m009$.

###  4.1 Example :
$M=(S^{3}\backslash\mathbf{4}_{1})_{p\mu+\lambda}=(m004)_{pA_{\rm SP}+B_{\rm
SP}}$

####  4.1.1 Field theory

${\cal K}=\mathbf{4}_{1}$ is the figure-eight knot, the twist knot with two
half-twists. We denote its complement in 3-sphere by
$S^{3}\backslash\mathbf{4}_{1}$, which is a hyperbolic cusped 3-manifold. The
corresponding 3D class R theory is Dimofte:2011ju

$\displaystyle\begin{split}&{\cal
T}[(S^{3}\backslash\mathbf{4}_{1})_{p\mu+\lambda}]=\frac{{\cal
T}[S^{3}\backslash\mathbf{4}_{1},\mu;\lambda]}{``SO(3)"_{p}}\;,\textrm{ with
}\\\ &{\cal T}[S^{3}\backslash\mathbf{4}_{1},\mu;\lambda]=(U(1)_{0}\;{\cal
N}=2\textrm{ gauge theory coupled to two chirals of charge +1})\;.\end{split}$
(102)

The ${\cal T}[S^{3}\backslash\mathbf{4}_{1},\mu;\lambda]$ has
$SU(2)_{\Phi}\times U(1)_{\rm top}$ which is enhanced to $SU(3)$ at IR
Benini:2018umh ; Gaiotto:2018yjh ; Gang:2018wek . The $SU(2)_{A}$ subgroup in
(43) of $SU(3)$ we are gauging is chosen such that

$\displaystyle(\mathbf{3}\textrm{ of }SU(3))=(\textrm{adjoint of
}SU(2)_{A})\;.$ (103)

After gauging the $``SO(3)"$ symmetry with CS level $p$, the resulting ${\cal
T}[(S^{3}\backslash\mathbf{4}_{1})_{p\mu+\lambda}]$ is expected not to have
any continuous flavor symmetry for generic $p$.

####  4.1.2 SUSY partition functions

The UV field theory ${\cal T}[(S^{3}\backslash\mathbf{4}_{1})_{p\mu+\lambda}]$
above does not have any flavor symmetry, i.e. $\textrm{rank}(G_{F})=0$, and
the supersymmetric partition functions do not depend on real masses/fugacities
or R-symmetry mixing parameters.

##### Squashed 3-sphere partition function

Using localization, the partition function for the theory $\frac{{\cal
T}[S^{3}\backslash\mathbf{4}_{1},\mu;\lambda]}{SU(2)_{p}}$ is given as

$\displaystyle\begin{split}\mathcal{Z}^{(4_{1})_{p}}_{b}=&\frac{1}{2}\int\frac{dXdZ}{(2\pi\hbar)}{\cal
I}^{(4_{1})_{p}}_{\hbar}(X,Z)\quad\textrm{with}\\\ {\cal
I}^{(4_{1})_{p}}_{\hbar}(X,Z)=&(4\sinh X\sinh\frac{2\pi
iX}{\hbar})\psi_{\hbar}(Z+2X)\psi_{\hbar}(Z)\\\
&\times\exp\left(\frac{(2p+8)X^{2}+Z^{2}+8XZ-(2\pi
i+\hbar)(2X+Z)}{2\hbar}\right)\;.\end{split}$ (104)

Here, $\psi_{\hbar}$ is the quantum dilogarithm function in (62).

##### Superconformal index

The index is

$\displaystyle\begin{split}{\cal
I}_{q}^{(4_{1})_{p}}&=\sum_{(m_{x},m_{z})}\oint_{|u_{x}|=1}\frac{du_{x}}{2\pi
iu_{x}}\oint_{|u_{z}|=1}\frac{du_{z}}{2\pi iu_{z}}\Delta(m_{x},u_{x}){\cal
I}_{\Delta}(m_{z}+2m_{x},u_{z}u^{2}_{x})\\\ &\quad\quad\times{\cal
I}_{\Delta}(m_{z},u_{z})u_{x}^{(2p+8)m_{x}}u_{z}^{m_{z}}u_{x}^{4m_{z}}u_{z}^{4m_{x}}(-q^{1/2})^{-2m_{x}-m_{z}}\;.\end{split}$
(105)

Here $(m_{z},u_{z})$ and $(m_{x},u_{x})$ are (monopole flux, fugacity) coupled
to the $U(1)$ and $SU(2)$ gauge symmetry respectively. The ${\cal
I}_{\Delta}(m,u)$ is the tetrahedron index in (32) and $\Delta(m_{x},u_{x})$
is the measure factor, given in (54), coming from the $SU(2)$ vector
multiplet. The summation range of the monopole fluxes is, see (78),

$\displaystyle\begin{cases}m_{x}\in\mathbb{Z}_{\geq
0},\;m_{z}\in\mathbb{Z}\;,\quad\textrm{for }\frac{{\cal
T}[S^{3}\backslash\mathbf{4}_{1},\mu\;\lambda]}{SU(2)_{p}}\\\
m_{x}\in\frac{1}{2}\mathbb{Z}_{\geq 0},\;m_{z}\in\mathbb{Z}\;,\quad\textrm{for
}{\cal T}[(S^{3}\backslash\mathbf{4}_{1})_{p\mu+\lambda}]=\frac{{\cal
T}[S^{3}\backslash\mathbf{4}_{1},\mu\;\lambda]}{``SO(3)"_{p}}\ .\end{cases}$
(106)

##### Bethe-vacua and handle-gluing/fibering operator

Expanding the integrand ${\cal I}^{(4_{1})_{p}}_{\hbar}(X,Z)$ in the limit
$\hbar\rightarrow 0$, see (65) and (66), we have

$\displaystyle\begin{split}&{\cal
W}^{(4_{1})_{p}}_{0}(X,Z)=\textrm{Li}_{2}(e^{-Z-2X})+\textrm{Li}_{2}(e^{-Z})+(p+4)X^{2}+\frac{1}{2}Z^{2}+4XZ-i\pi(2X+Z)\pm
2\pi iX\;,\\\ &{\cal
W}^{(4_{1})_{p}}_{1}(X,Z)=-\frac{1}{2}(2X+Z+\log(1-e^{-2X-Z})+\log(1-e^{-Z}))+\log\left(2\sinh
X\right)\;.\end{split}$ (107)

Here we used the following asymptotic expansion (assuming
$\textrm{Im}(\hbar)\neq 0$)

$\displaystyle\log(2\sinh(\frac{2\pi
iX}{\hbar}))\xrightarrow{\quad\hbar\rightarrow 0\quad}\pm\frac{2\pi
iX}{\hbar}+(\textrm{exponentially suppressed})\;.$ (108)

The overall sign depends on the sign of $\textrm{Im}(\hbar)$ but this choice
is not important for our purpose. Extremizing the twisted superpotential
${\cal W}^{(4_{1})_{p}}_{0}$, we have

$\displaystyle\begin{split}&\left(\mathcal{S}_{\rm BE}\textrm{ of }\frac{{\cal
T}[S^{3}\backslash\mathbf{4}_{1},\mu;\lambda]}{SU(2)_{p}}\right)\\\
&=\left\\{(x,z)\;:\;(x^{p+2}z(x^{2}z-1))^{2}=1,\;\frac{x^{2}(1-z)(x^{2}z-1)}{z}=1\;,x^{2}\neq
1\right\\}/\mathbb{Z}^{\rm Weyl}_{2}\ .\end{split}$ (109)

The Weyl $\mathbb{Z}_{2}$ acts as

$\displaystyle\mathbb{Z}^{\rm
Weyl}_{2}\;:\;(x,z)\;\leftrightarrow\;(1/x,\tilde{z})\;.$ (110)

The handle-gluing and fibering operators are

$\displaystyle\begin{split}&\left({\cal H}_{\alpha}\textrm{ of }\frac{{\cal
T}[S^{3}\backslash\mathbf{4}_{1},\mu;\lambda]}{SU(2)_{p}}\right)=\frac{2x^{2}\left((p-4)x^{2}z^{2}-p+2\left(4x^{2}+1\right)z-4\right)}{\left(x^{2}-1\right)^{2}z}\bigg{|}_{(x,z)=(x^{(\alpha)},z^{(\alpha)})\in\mathcal{S}_{\rm
BE}}\;,\\\ &\left({\cal F}_{\alpha}\textrm{ of }\frac{{\cal
T}[S^{3}\backslash\mathbf{4}_{1},\mu;\lambda]}{SU(2)_{p}}\right)=\exp\left(\frac{i({\cal
W}_{0}-2\pi in^{(\alpha)}_{x}X-2\pi
in^{(\alpha)}_{z}Z)}{2\pi}\right)\bigg{|}_{(X,Z)=(\log x^{(\alpha)},\log
z^{(\alpha)})}\;.\end{split}$ (111)

Two integers $(n_{x}^{(\alpha)},n_{z}^{(\alpha)})$ for each Bethe-vacuum
$\alpha$ are chosen such that, see (71),

$\displaystyle\begin{split}&\partial_{X}({\cal W}_{0}-2\pi
in^{(\alpha)}_{x}X-2\pi in^{(\alpha)}_{z}Z)\big{|}_{(X,Z)=(\log
x^{(\alpha)},\log z^{(\alpha)})}=0\;,\\\ &\partial_{Z}({\cal W}_{0}-2\pi
in^{(\alpha)}_{x}X-2\pi in^{(\alpha)}_{z}Z)\big{|}_{(X,Z)=(\log
x^{(\alpha)},\log z^{(\alpha)})}=0\;.\end{split}$ (112)

The theory $\frac{{\cal
T}[S^{3}\backslash\mathbf{4}_{1},\mu;\lambda]}{SU(2)_{p}}$ has 1-form
$\mathbb{Z}_{2}$ symmetry originated from the $\mathbb{Z}_{2}$ center symmetry
of the gauged $SU(2)$ symmetry.

####  4.1.3 IR phases

According to 2018arXiv181211940D , the Dehn filled manifolds are

$\displaystyle(S^{3}\backslash\mathbf{4}_{1})_{p\mu+\lambda}=\begin{cases}\textrm{Torus
bundle with }\varphi=\begin{pmatrix}2&1\\\ 1&1\end{pmatrix}\;,\quad p=0\\\
\textrm{Atoroidal SFS $S^{2}((2,1),(3,1),(7,-6))$}\;,\quad p=\pm 1\\\
\textrm{Atoroidal SFS $S^{2}((2,1),(4,1),(5,-4))$}\;,\quad p=\pm 2\\\
\textrm{Atoroidal SFS $S^{2}((3,1),(3,1),(4,-3))$}\;,\quad p=\pm 3\\\
\textrm{Graph}\;,\quad p=\pm 4\\\ \textrm{Hyperbolic}\;,\quad|p|>4\
.\end{cases}$ (113)

According to (17), the topological field theory $\textrm{TFT}[M]$ for the
above Seifert fibered manifolds are all unitary. Combined with the
mathematical facts, our proposal (49) predicts that

$\displaystyle{\cal
T}[(S^{3}\backslash\mathbf{4}_{1})_{p\mu+\lambda}]=\begin{cases}\textrm{Unitary
TQFT}\;,\quad p=0,\pm 1,\pm 2,\pm 3\\\ \textrm{${\cal N}=2$ SCFT with $U(1)$
flavor symmetry and CPOs}\;,\quad p=\pm 4\\\ \textrm{$\mathcal{N}=2$
SCFT}\;,\quad|p|>4\ .\end{cases}$ (114)

Now let us carefully check the expected IR phases using the general methods
outlined in Section 3.4.

Using the formulas in (105) and (106), one can compute the superconformal
index and check that

$\displaystyle\begin{split}&\left({\cal I}_{q}\textrm{ of }\frac{{\cal
T}[S^{3}\backslash\mathbf{4}_{1},\mu;\lambda]}{SU(2)_{p}}\right)=\left({\cal
I}_{q}\textrm{ of }\frac{{\cal
T}[S^{3}\backslash\mathbf{4}_{1},\mu;\lambda]}{``SO(3)"_{p}}\right)\\\
&=\begin{cases}1\;,\quad p=0,\pm 1,\pm 2,\pm 3\\\
\left(\sum_{n=0}^{\infty}1\right)\times
q^{0}-q^{2}-q^{3}-2q^{4}-2q^{5}-3q^{6}+O(q^{7})\;,\quad p=\pm 4\\\
\textrm{Non-trivial power series in $q^{1/2}$ starting with }1+{\cal
O}(q^{1/2})\;,\quad|p|>4\ .\end{cases}\\\ \end{split}$ (115)

For $p=\pm 4$, the $\mathbb{Z}_{2}$ 1-form symmetry decouples at IR and we
need to sum over $m_{x}\in\mathbb{Z}$ (not $m_{x}\in\mathbb{Z}/2$) even for
the $``SO(3)"$ gauged theory. See the discussion below Eq. (78). The result is
compatible with the expectation in (114). For the $p=\pm 4$ cases, the index
gets contribution of the form $\left(1-q^{2m_{x}+1}+(\textrm{higher
order})\right)$ from each $m_{x}\in\frac{1}{2}\mathbb{Z}_{\geq 0}$ and the
coefficient of $q^{0}$ goes to infinity as we sum over all values of $m_{x}$.
The contributions are from 1/2 BPS $SO(3)$ monopole operators with $R_{\rm
geo}=0$. It implies that there should be an emergent Abelian symmetry and the
correct IR superconformal R-charge $R_{\rm IR}$ is given as a linear
combination of $R_{\rm geo}$ and the charge of the accidental $U(1)$ and the
monopole operators satisfy the unitarity bound $R_{\rm IR}\geq\frac{1}{2}$.

##### Unitary topological field theory when $p=0,\pm 1,\pm 2,\pm 3$

To probe the topological field theories in $p=0,\pm 1,\pm 2$ and $\pm 3$
cases, we compute the handle-gluing/fibering operators given in (111)

$\displaystyle\begin{split}&\textrm{ For }\frac{{\cal
T}[S^{3}\backslash\mathbf{4}_{1},\mu;\lambda]}{SU(2)_{p=0}},\quad\\{{\cal
H}_{\alpha}^{-1}\\}=\\{20^{-1},20^{-1},5^{-1},5^{-1},4^{-1},4^{-1}\\}\textrm{
and }\\\ &\qquad\qquad\qquad\qquad\qquad\\{{\cal F}_{\alpha}\\}=e^{\frac{\pi
i}{6}}\\{1,1,e^{2\pi i/5},e^{-2\pi i/5},1,-1\\}\;,\\\ &\textrm{ For
}\frac{{\cal T}[S^{3}\backslash\mathbf{4}_{1},\mu;\lambda]}{SU(2)_{p=\pm
1}},\quad\\{{\cal
H}_{\alpha}^{-1}\\}=\\{(\zeta_{5}^{1})^{2},(\zeta_{5}^{2})^{2},(\zeta_{5}^{3})^{2}\\}\times\\{1,1\\}\textrm{
and }\\\ &\qquad\qquad\qquad\qquad\qquad\\{{\cal F}_{\alpha}\\}=e^{\frac{5\pi
i}{28}}\\{1,e^{2\pi i/7},e^{-4\pi i/7}\\}\times\\{1,e^{\frac{\pi
i}{2}}\\}\;,\\\ &\textrm{ For }\frac{{\cal
T}[S^{3}\backslash\mathbf{4}_{1},\mu;\lambda]}{SU(2)_{p=\pm 2}},\quad\\{{\cal
H}^{-1}_{\alpha}\\}=\\{\frac{(\zeta_{3}^{1})^{2}}{2},\frac{(\zeta_{3}^{1})^{2}}{2},\frac{(\zeta_{3}^{2})^{2}}{2},\frac{(\zeta_{3}^{2})^{2}}{2},(\zeta_{3}^{1})^{2},(\zeta_{3}^{2})^{2}\\}\textrm{
and }\\\ &\qquad\qquad\qquad\qquad\qquad\\{{\cal F}_{\alpha}\\}=e^{\frac{23\pi
i}{120}}\\{1,-1,e^{-4\pi i/5},e^{\pi i/5},e^{5\pi i/8},e^{-7\pi i/40}\\}\;,\\\
&\textrm{ For }\frac{{\cal
T}[S^{3}\backslash\mathbf{4}_{1},\mu;\lambda]}{SU(2)_{p=\pm 3}},\quad\\{{\cal
H}^{-1}_{\alpha}\\}=\\{8^{-1},8^{-1},4^{-1}\\}\times\\{1,1\\}\textrm{ and }\\\
&\qquad\qquad\qquad\qquad\qquad\\{{\cal F}_{\alpha}\\}=e^{\frac{17\pi
i}{24}}\\{1,e^{-3\pi i/2},e^{-7\pi i/8}\\}\times\\{1,e^{\frac{\pi
i}{2}}\\}\;.\end{split}$ (116)

Here we defined

$\displaystyle\zeta_{k}^{m}=\left(S_{0,\alpha=m-1}\textrm{ of
}SU(2)_{k}\right)=\sqrt{\frac{2}{k+2}}\sin\left(\frac{\pi m}{k+2}\right)\;.$
(117)

Notice that $|{\cal F}_{\alpha}|=1$ and ${\cal H}_{\alpha}\in\mathbb{R}_{+}$
for all $\alpha$, which are another evidences for emergence of TQFT at IR, see
(90). Furthermore, one can check that

$\displaystyle\sum_{\alpha}\frac{1}{{\cal H}_{\alpha}}=1\;,$ (118)

which is also expected from (92). Using the above computations combined with
the relation in (88), one can determine the $S^{2}_{0\alpha}$ and
$T_{\alpha\alpha}/T_{00}$ of the IR topological field theories.

Now let us compute the twisted partition functions for ${\cal
T}[(S^{3}\backslash\mathbf{4}_{1})_{p\mu+\lambda}]=\frac{{\cal
T}[S^{3}\backslash\mathbf{4}_{1},\mu;\lambda]}{``SO(3)"_{p}}$ using the
relation in (80), (81) and (83). For odd $p$, the above computation is
compatible with the general expectation in the Eqn.(79) and one can see that

$\displaystyle\begin{split}&\textrm{ For }\frac{{\cal
T}[S^{3}\backslash\mathbf{4}_{1},\mu;\lambda]}{``SO(3)"_{p=\pm 1}},\;\\{{\cal
H}_{\alpha}^{-1}\\}=\\{2(\zeta_{5}^{1})^{2},2(\zeta_{5}^{2})^{2},2(\zeta_{5}^{3})^{2}\\}\textrm{
and }\\{{\cal F}_{\alpha}\\}=e^{\frac{5\pi i}{28}}\\{1,e^{2\pi i/7},e^{-4\pi
i/7}\\}\;,\\\ &\textrm{ For }\frac{{\cal
T}[S^{3}\backslash\mathbf{4}_{1},\mu;\lambda]}{SU(2)_{p=\pm 3}},\;\\{{\cal
H}^{-1}_{\alpha}\\}=\\{4^{-1},4^{-1},2^{-1}\\}\textrm{ and }\\{{\cal
F}_{\alpha}\\}=e^{\frac{17\pi i}{24}}\\{1,e^{-3\pi i/2},e^{-7\pi
i/8}\\}\;.\end{split}$ (119)

For even $p$, on the other hand, the Bethe-vacua in (109) can be divided into
two classes, ${\cal S}_{\rm BE}=S^{(+1)}_{\rm BE}\bigsqcup S^{(-1)}_{\rm BE}$,

$\displaystyle\left(\mathcal{S}^{Q=\pm 1}_{\rm BE}\textrm{ of }\frac{{\cal
T}[S^{3}\backslash\mathbf{4}_{1},\mu;\lambda]}{SU(2)_{p}}\right)=\left\\{(x,z)\in{\cal
S}_{\rm BE}\;:\;x^{p+2}z(x^{2}z-1)=Q\right\\}\;.$ (120)

Then, using the Eqns. in (80), (81), (83), one has

$\displaystyle\begin{split}&\textrm{ For }\frac{{\cal
T}[S^{3}\backslash\mathbf{4}_{1},\mu;\lambda]}{``SO(3)"_{p=0}},\\\ &\\{{\cal
H}_{\alpha}^{-1}\\}=\\{5^{-1},5^{-1},5^{-1},5^{-1},5^{-1}\\}\textrm{ and
}\\{{\cal F}_{\alpha}\\}=e^{\frac{\pi i}{6}}\\{1,e^{2\pi i/5},e^{2\pi
i/5},e^{-2\pi i/5},e^{-2\pi i/5}\\}\;,\\\ &\textrm{ For }\frac{{\cal
T}[S^{3}\backslash\mathbf{4}_{1},\mu;\lambda]}{SU(2)_{p=\pm 2}},\\\ &\\{{\cal
H}^{-1}_{\alpha}\\}=\\{2(\zeta_{3}^{1})^{2},2(\zeta_{3}^{2})^{2}\\}\textrm{
and }\\{({\cal F}_{\alpha})^{2}\\}=e^{\frac{23\pi i}{60}}\\{1,e^{2\pi
i/5}\\}\;.\end{split}$ (121)

For $p=0$, the last 4 Beth-vacua correspond to the ${\cal S}^{\rm twisted}$.
For $p=2$, the ${\cal F}_{\alpha}$ for Bethe-vacua with $Q=1$ changes its sign
under the $\mathbb{Z}_{2}^{\rm 1-form}$ action, and only $({\cal
F}_{\alpha})^{2}$ is well-defined after gauging the 1-form symmetry. It
implies that the IR TQFT is a fermionic (spin) TQFT.

###  4.2 Example :
$M=(S^{3}\backslash\mathbf{5}_{2})_{p\mu+\lambda}=(m015)_{(p-2)A_{\rm
SP}+B_{\rm SP}}$

####  4.2.1 Field theory

${\cal K}=\mathbf{5}_{2}$ is the twist knot with three half-twists. The field
theory is Gang:2018wek

$\displaystyle\begin{split}&{\cal
T}[(S^{3}\backslash\mathbf{5}_{2})_{p\mu+\lambda}]=\frac{{\cal
T}[S^{3}\backslash\mathbf{5}_{2},\mu;\lambda]}{``SO(3)_{p}"}\\\
&=\begin{cases}\frac{(U(1)_{-\frac{1}{2}}\times SU(2)_{p-4}\;{\cal
N}=2\;\textrm{gauge theory coupled to a chiral in
}\textrm{(Adj)}_{1})}{\mathbb{Z}_{2}}\;,\quad\textrm{even $p$}\\\
\frac{(U(1)_{-\frac{1}{2}}\times SU(2)_{p-4}\;{\cal N}=2\;\textrm{gauge theory
coupled to a chiral in }\textrm{(Adj)}_{1})\otimes U(1)_{-2}}{\mathbb{Z}^{\rm
diag}_{2}},\quad\textrm{odd $p$}\ .\end{cases}\end{split}$ (122)

$\textrm{Adj}_{q}$ means that the matter field is in the adjoint
representation of $SU(2)$ and has charge $q$ under $U(1)$. The theory has a
$U(1)_{\rm top}$ symmetry (which is an unexpected accidental symmetry in 6D
set-up (37)).

####  4.2.2 SUSY partition functions

##### Squashed 3-sphere partition function

The $S^{3}_{b}$ partition function of the theory,
$\frac{{\cal
T}[S^{3}\backslash\mathbf{5}_{2},\mu;\lambda]}{SU(2)_{p}}=\big{(}U(1)_{-1/2}\times
SU(2)_{p-4}+\Phi$ (in Adj+1)$\big{)}$, is given by ($\hbar:=2\pi ib^{2}$)

$\displaystyle\begin{split}&\mathcal{Z}_{b}^{(5_{2})_{p}}\left(W=m+(i\pi+\frac{\hbar}{2})\nu\right)\\\
&=\frac{1}{2}\int\frac{dXdZ}{2\pi\hbar}\left(4\sinh(X)\sinh(\frac{2\pi
iX}{\hbar})\right)\psi_{\hbar}(Z+2X)\psi_{\hbar}(Z-2X)\psi_{\hbar}(Z)\\\
&\quad\quad\times\exp\left(\frac{Z^{2}}{2\hbar}+(p-2)\frac{X^{2}}{\hbar}+\frac{Z(W-2\pi
i-\hbar)}{\hbar}\right)\;.\end{split}$ (123)

Here $m$ is the (rescaled) real mass, $m=b\times(\textrm{real mass})$, for the
$U(1)_{\rm top}$ symmetry, topological symmetry for the $U(1)$ gauge symmetry.
The real mass parameter can be identified with the FI parameter. $\nu$
parametrizes the mixing between $U(1)_{R}$ and $U(1)_{\rm top}$ and the
R-symmetry at $\nu=1$ corresponds to the geometrical R-symmetry in (38), i.e.
$R_{\nu=1}=R_{\rm geo}$.121212One can check that there is a Bethe-vacuum,
$\alpha=(\textrm{hyp})$, satisfying that
$\textrm{Im}[S^{(\textrm{hyp})}_{0}(m=0,\nu=1)]$, defined in (70), of ${\cal
T}[\frac{{\cal T}[S^{3}\backslash\mathbf{5}_{2},\mu;\lambda]}{SU(2)_{p}}]$ is
equal to the hyperbolic volume of
$(S^{3}\backslash\mathbf{5}_{2})_{p\mu+\lambda}$ when the Dehn filled manifold
is hyperbolic.

##### Superconformal index

The index is

$\displaystyle\begin{split}{\cal
I}_{q}^{(5_{2})_{p}}(u,\nu)&=\sum_{(m_{x},m_{z})}\oint_{|u_{x}|=1}\frac{du_{x}}{2\pi
iu_{x}}\oint_{|u_{z}|=1}\frac{du_{z}}{2\pi iu_{z}}\Delta(m_{x},u_{x}){\cal
I}_{\Delta}(m_{z}+2m_{x},u_{z}u_{x}^{2})\\\ &\quad\quad\times{\cal
I}_{\Delta}(m_{z}-2m_{x},u_{z}/u_{x}^{2}){\cal
I}_{\Delta}(m_{z},u_{z})u_{z}^{m_{z}}u_{x}^{(2p-4)m_{x}}(u(-q^{1/2})^{\nu-2})^{m_{z}}\;.\end{split}$
(124)

Here $(m_{z},u_{z})$ and $(m_{x},u_{x})$ are (monopole flux, fugacity) coupled
to the $U(1)$ and $SU(2)$ gauge symmetry respectively. $u$ is the fugacity for
the $U(1)_{\rm top}$ symmetry. The summation range of the monopole fluxes is

$\displaystyle\begin{cases}m_{x}\in\mathbb{Z}_{\geq
0},\;m_{z}\in\mathbb{Z}\;,\quad\textrm{for }\frac{{\cal
T}[S^{3}\backslash\mathbf{5}_{2},\mu;\lambda]}{SU(2)_{p}}\\\
m_{x}\in\frac{1}{2}\mathbb{Z}_{\geq 0},\;m_{z}\in\mathbb{Z}\;,\quad\textrm{for
}{\cal T}[(S^{3}\backslash\mathbf{5}_{2})_{p\mu+\lambda}]=\frac{{\cal
T}[S^{3}\backslash\mathbf{5}_{2},\mu;\lambda]}{``SO(3)_{p}"}\ .\end{cases}$
(125)

##### Bethe-vacua and handle-gluing/fibering operators

Expanding the integrand of ${\cal Z}_{b}^{(5_{2})_{p}}$ in the limit
$\hbar\rightarrow 0$, see (65) and (66), we have

$\displaystyle\begin{split}{\cal
W}^{(5_{2})_{p}}_{0}(X,Z;m,\nu)&=\textrm{Li}_{2}(e^{-Z-2X})+\textrm{Li}_{2}(e^{-Z+2X})+\textrm{Li}_{2}(e^{-Z})+(p-2)X^{2}+\frac{1}{2}Z^{2}\\\
&\quad+Z(m+i\pi\nu)\pm 2\pi iX\;,\\\ {\cal
W}^{(5_{2})_{p}}_{1}(X,Z;m,\nu)&=-\frac{1}{2}\left((2-\nu)Z+\log(1-e^{-2X-Z})+\log(1-e^{2X-Z})+\log(1-e^{-Z})\right)\\\
&\quad+\log\left(2\sinh X\right)\;.\end{split}$

Extremizing the twisted superpotential ${\cal W}^{(5_{2})_{p}}_{0}$, we have

$\displaystyle\begin{split}&\left(\mathcal{S}_{\rm BE}\textrm{ of }\frac{{\cal
T}[S^{3}\backslash\mathbf{5}_{2},\mu;\lambda]}{SU(2)_{p}}\right)\\\
&=\left\\{(x,z)\;:\;\frac{x^{2p-8}(1-x^{2}z)^{2}}{(x^{2}-z)^{2}}=1,\;\frac{e^{m+i\pi\nu}(z-1)(1-x^{2}z)(1-x^{-2}z)}{z^{2}}=1\;,x^{2}\neq
1\right\\}/\mathbb{Z}^{\rm Weyl}_{2}\ .\end{split}$ (126)

The Weyl $\mathbb{Z}^{\rm Weyl}_{2}$ acts as

$\displaystyle\mathbb{Z}^{\rm
Weyl}_{2}\;:\;(x,z)\;\leftrightarrow\;(1/x,z)\;.$ (127)

The handle-gluing and fibering operator are

$\displaystyle\begin{split}&\left({\cal H}_{\alpha}\textrm{ of }\frac{{\cal
T}[S^{3}\backslash\mathbf{5}_{2},\mu;\lambda]}{SU(2)_{p}}\right)\\\
&=\frac{2z^{-\nu-1}\left(x^{4}z(-p+2z+2)+x^{2}\left((p-2)z^{3}-(p-6)z+2(p-6)\right)+z(-p+2z+2)\right)}{\left(x^{2}-1\right)^{2}}\bigg{|}_{(x,z)=(x^{(\alpha)},z^{(\alpha)})}\;,\\\
&\left({\cal F}_{\alpha}\textrm{ of }\frac{{\cal
T}[S^{3}\backslash\mathbf{5}_{2},\mu;\lambda]}{SU(2)_{p}}\right)=\exp\left(\frac{i({\cal
W}^{(5_{2})_{p}}_{0}-2\pi in^{(\alpha)}_{x}X-2\pi
in^{(\alpha)}_{z}Z-mZ)}{2\pi}\right)\bigg{|}_{(X,Z)=(\log x^{(\alpha)},\log
z^{(\alpha)})}\;.\end{split}$ (128)

Two integers $(n_{x}^{(\alpha)},n_{z}^{(\alpha)})$ for each Bethe-vacuum
$\alpha$ are chosen as in (71).

####  4.2.3 IR phases

According to 2018arXiv181211940D , the Dehn filled manifolds are

$\displaystyle(S^{3}\backslash\mathbf{5}_{2})_{p\mu+\lambda}=\begin{cases}\textrm{Atoroidal
SFS $S^{2}((2,1),(3,1),(11,-9))$}\;,\quad p=1\\\ \textrm{Atoroidal SFS
$S^{2}((2,1),(4,1),(7,-5))$}\;,\quad p=2\\\ \textrm{Atoroidal SFS
$S^{2}((3,1),(1,3),(5,-3))$}\;,\quad p=3\\\ \textrm{Graph}\;,\quad p=0,4\\\
\textrm{Hyperbolic}\;,\quad|p-2|>2\ .\end{cases}$ (129)

According to (17), the topological field theory $\textrm{TFT}[M]$ for the
above Seifert fibered manifolds are all non-unitary. Combined with the
mathematical facts, our proposal (49) predicts that

$\displaystyle\begin{split}&{\cal
T}[(S^{3}\backslash\mathbf{5}_{2})_{p\mu+\lambda}]\\\
&=\begin{cases}\textrm{$\mathcal{N}=4$ rank-0 SCFT}\;,\quad p=1,2,3\\\
\textrm{${\cal N}=2$ SCFT with $U(1)$ flavor symmetry and CPOs}\;,\quad
p=0,4\\\ \textrm{$\mathcal{N}=2$ SCFT}\;,\quad|p-2|>2\
.\end{cases}\end{split}$ (130)

Interestingly, the ${\cal N}=2$ gauge theories in (122) for $p=1,2,3$ are
expected to have SUSY enhancement to ${\cal N}=4$. Below, we will confirm the
expected IR phases using the general methods outlined in Section 3.4.

##### SUSY enhancement when $p=1,2,3$

The superconformal indices, computed using the formulae in (124) and (125),
for $p=1,2,3$ are

$\displaystyle\begin{split}&\left({\cal I}_{q}(u;\nu=0)\textrm{ of
}\frac{{\cal
T}[S^{3}\backslash\mathbf{5}_{2},\mu;\lambda]}{SU(2)_{p=1}}\right)=\left({\cal
I}_{q}(u;\nu=0)\textrm{ of }\frac{{\cal
T}[S^{3}\backslash\mathbf{5}_{2},\mu;\lambda]}{``SO(3)"_{p=1}}\right)\\\
&=1-q-\left(u+\frac{1}{u}\right)q^{3/2}-2q^{2}-uq^{5/2}+\left(-1+\frac{1}{u^{2}}\right)q^{3}+\left(\frac{1}{u}-u\right)q^{7/2}+O\left(q^{9/2}\right)\
,\\\ &\left({\cal I}_{q}(u;\nu=0)\textrm{ of }\frac{{\cal
T}[S^{3}\backslash\mathbf{5}_{2},\mu;\lambda]}{SU(2)_{p=2}}\right)=\left({\cal
I}_{q}(u;\nu=0)\textrm{ of }\frac{{\cal
T}[S^{3}\backslash\mathbf{5}_{2},\mu;\lambda]}{``SO(3)"_{p=2}}\right)\\\
&=1-q-\left(u+\frac{1}{u}\right)q^{3/2}-2q^{2}-uq^{5/2}+\left(-1+\frac{1}{u^{2}}\right)q^{3}+\left(\frac{1}{u}-u\right)q^{7/2}+\frac{q^{4}}{u^{2}}+O\left(q^{9/2}\right)\
,\\\ &\left({\cal I}_{q}(u;\nu)\textrm{ of }\frac{{\cal
T}[S^{3}\backslash\mathbf{5}_{2},\mu;\lambda]}{SU(2)_{p=3}}\right)=\left({\cal
I}_{q}(u;\nu)\textrm{ of }\frac{{\cal
T}[S^{3}\backslash\mathbf{5}_{2},\mu;\lambda]}{``SO(3)"_{p=3}}\right)\\\
&=1-q-\left(\frac{1}{u}+u\right)q^{3/2}-2q^{2}-\left(\frac{1}{u}+u\right)q^{5/2}-2q^{3}-\left(\frac{1}{u}+u\right)q^{7/2}-2q^{4}+O(q^{5})\
.\end{split}$ (131)

The index for general R-charge mixing $\nu$ can be obtained by the relation in
(55). To compute the correct IR superconformal index, one needs to determine
the IR superconformal R-symmetry, $\nu_{\rm IR}$, using the F-maximization in
(75). Using the Bethe-sum formula of round 3-sphere partition function in (73)
along with (126) and (128), one can confirm that (see Fig. 3)

$\displaystyle|\mathcal{Z}_{b=1}^{(5_{2})_{p}}(m=0,\nu=0)|<|\mathcal{Z}_{b=1}^{(5_{2})_{p}}(m=0,\nu\neq
0)|\;,\quad\textrm{for $p=1,2,3$}\ .$ (132)

Figure 3: $|\mathcal{Z}_{b=1}^{(5_{2})_{p}}(m=0,\nu)|$ as a function on
$-1.2<\nu<1.2$ for $p=1,2,3$. The minimums are located at $\nu=0$ and its
values are
$\sqrt{\frac{2}{11}}\sin(\frac{\pi}{11}),\sqrt{\frac{1}{7}}\sin(\frac{\pi}{7})$
and $\sqrt{\frac{1}{5}}\sin(\frac{\pi}{5})$ for $p=1,2,3$ respectively.

So the $\nu=0$ corresponds to the correct IR superconformal R-charge according
to the F-maximization, i.e. $\nu_{\rm IR}=0$. Applying the formula in (73),
one can choose arbitrary $\nu_{0}\in 2\mathbb{Z}+1$ at which the
superconformal index contains only $q^{\textrm{integer}}$ terms. Putting back
the IR superconformal R-charge choice to the superconformal index, one can
confirm that the index satisfies the necessary condition in (97) for the SUSY
enhancement. In the degenerate limits $u=1$ and $\nu\rightarrow\pm 1$, the
superconformal indices become trivial, i.e.

$\displaystyle\begin{split}&\textrm{For $p=1,2,3$},\\\ &\left({\cal
I}_{q}(u=1;\nu=\pm 1)\textrm{ of }\frac{{\cal
T}[S^{3}\backslash\mathbf{5}_{2},\mu;\lambda]}{SU(2)_{p}}\textrm{ and
}\frac{{\cal
T}[S^{3}\backslash\mathbf{5}_{2},\mu;\lambda]}{``SO(3)"_{p}}\right)=1\;.\end{split}$
(133)

It is a strong evidence for emergence of non-unitary TQFTs in the degenerate
limits. To probe the emergent non-unitary TQFTs, one can compute the handle-
gluing and fibering operators. For $p=1$ and $3$, we have

$\displaystyle\begin{split}&\left(\\{{\cal
H}_{\alpha}^{-1}(m=0,\nu=+1)\\}\textrm{ of }\frac{{\cal
T}[S^{3}\backslash\mathbf{5}_{2},\mu;\lambda]}{SU(2)_{p=1}}\right)=\bigg{\\{}(\zeta_{9}^{5})^{2},(\zeta_{9}^{1})^{2},(\zeta_{9}^{2})^{2},(\zeta_{9}^{3})^{2},(\zeta_{9}^{4})^{2}\bigg{\\}}\times\bigg{\\{}1,1\bigg{\\}}\;.\\\
&\left(\\{{\cal F}_{\alpha}(m=0,\nu=+1)\\}\textrm{ of }\frac{{\cal
T}[S^{3}\backslash\mathbf{5}_{2},\mu;\lambda]}{SU(2)_{p=1}}\right)=e^{\frac{19\pi
i}{132}}\bigg{\\{}1,e^{\frac{2\pi i}{11}},e^{\frac{10\pi
i}{11}},e^{\frac{16\pi i}{11}},e^{\frac{20\pi
i}{11}}\bigg{\\}}\times\bigg{\\{}1,e^{\frac{\pi i}{2}}\bigg{\\}}\;.\\\
&\left(\\{{\cal H}^{-1}_{\alpha}(m=0,\nu=-1)\\}\textrm{ of }\frac{{\cal
T}[S^{3}\backslash\mathbf{5}_{2},\mu;\lambda]}{SU(2)_{p=1}}\right)=\bigg{\\{}(\zeta_{9}^{2})^{2},(\zeta_{9}^{1})^{2},(\zeta_{9}^{3})^{2},(\zeta_{9}^{4})^{2},(\zeta_{9}^{5})^{2}\bigg{\\}}\times\bigg{\\{}1,1\bigg{\\}}\;.\\\
&\left(\\{{\cal F}_{\alpha}(m=0,\nu=-1)\\}\textrm{ of }\frac{{\cal
T}[S^{3}\backslash\mathbf{5}_{2},\mu;\lambda]}{SU(2)_{p=1}}\right)=e^{\frac{43\pi
i}{132}}\bigg{\\{}1,e^{\frac{20\pi i}{11}},e^{\frac{18\pi
i}{11}},e^{\frac{8\pi i}{11}},e^{\frac{14\pi
i}{11}},\bigg{\\}}\times\bigg{\\{}1,e^{\frac{\pi i}{2}}\bigg{\\}}\;.\\\
&\left(\\{{\cal H}^{-1}_{\alpha}(m=0,\nu=\pm 1)\\}\textrm{ of }\frac{{\cal
T}[S^{3}\backslash\mathbf{5}_{2},\mu;\lambda]}{SU(2)_{p=3}}\right)=\bigg{\\{}\frac{(\zeta_{3}^{2})^{2}}{2},\frac{(\zeta_{3}^{2})^{2}}{2},\frac{(\zeta_{3}^{1})^{2}}{2},\frac{(\zeta_{3}^{1})^{2}}{2}\bigg{\\}}\times\bigg{\\{}1,1\bigg{\\}}\;.\\\
&\left(\\{{\cal F}_{\alpha}(m=0,\nu=\pm 1)\\}\textrm{ of }\frac{{\cal
T}[S^{3}\backslash\mathbf{5}_{2},\mu;\lambda]}{SU(2)_{p=3}}\right)=e^{-\frac{3\pi
i}{10}}\bigg{\\{}1,e^{\frac{\pi i}{2}},e^{\frac{\pi i}{10}},e^{\frac{8\pi
i}{5}}\bigg{\\}}\times\bigg{\\{}1,e^{\frac{\pi i}{2}}\bigg{\\}}\;.\end{split}$
(134)

It is compatible with the general expectation in (79). For $p=2$ case,

$\displaystyle\begin{split}\\\ &\left(\\{{\cal
H}^{-1}_{\alpha}(m=0,\nu=+1)\\}\textrm{ of }\frac{{\cal
T}[S^{3}\backslash\mathbf{5}_{2},\mu;\lambda]}{SU(2)_{p=2}}\right)\\\
&=\bigg{\\{}\frac{(\zeta_{5}^{3})^{2}}{2},\frac{(\zeta_{5}^{3})^{2}}{2},\frac{(\zeta_{5}^{1})^{2}}{2},\frac{(\zeta_{5}^{1})^{2}}{2},\frac{(\zeta_{5}^{5})^{2}}{2},\frac{(\zeta_{5}^{5})^{2}}{2},(\zeta_{5}^{2})^{2},(\zeta_{5}^{4})^{2},(\zeta_{5}^{6})^{2}\bigg{\\}}\;.\\\
&\left(\\{{\cal F}_{\alpha}(m=0,\nu=+1)\\}\textrm{ of }\frac{{\cal
T}[S^{3}\backslash\mathbf{5}_{2},\mu;\lambda]}{SU(2)_{p=2}}\right)\\\
&=e^{\frac{197\pi i}{168}}\bigg{\\{}1,e^{\pi i},e^{\frac{\pi
i}{7}},e^{\frac{8\pi i}{7}},e^{\frac{5\pi i}{7}},e^{\frac{12\pi
i}{7}},e^{\frac{47\pi i}{56}},e^{\frac{9\pi i}{8}},e^{\frac{15\pi
i}{56}}\bigg{\\}}\;,\\\ &\left(\\{{\cal
H}^{-1}_{\alpha}(m=0,\nu=-1)\\}\textrm{ of }\frac{{\cal