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

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

Extended SUSY gauge theories have rich structures in their vacuum moduli space, and one natural invariant is the rank, i.e. the complex dimension of the Coulomb branch. There have been numerous efforts in classifying SCFTs with 8 Qs for a given low rank in various space-time dimensions [15][16][17][18][19][20][21][22][23][24][25]. For 3D N = 4 SCFTs, however, the rank is in general not a duality-invariant concept since the Coulomb and Higgs branches are exchanged under the 3D mirror symmetry [26]. For this reason, we hereby modify the definition of rank as the maximum of the dimension of Coulomb branch and that of the Higgs branch. Another peculiar fact about 3D N = 4 theories is that there exist non-trivial interacting SCFTs of rank 0, as studied by two of the authors of this present paper in [27]. This is in contrast with the case of D ≥ 4, where it is often implicitly assumed that there is no non-trivial interacting rank 0 SCFTs with 8 Qs, so that the classification program starts with rank 1. (Recently, 4D/5D rank 0 SCFTs were found through a geometrical engineering but it is yet unclear if they are interacting SCFTs [28].) Note that most of the classification schemes in previous studies do not work for rank 0 cases since the existence of Coulomb or Higgs branch operators is an crucial assumption in the analysis.
In this paper, we initiate the classification of rank 0 3D N = 4 SCFTs by establishing the following correspondence: 3D N = 4 superconformal field theories of rank 0 ←→ A pair of 3D non-unitary topological quantum field theories (TQFTs) . (1.1) The non-unitary TQFTs emerge at particular choices of non-superconformal R-symmetry, ν = ±1 in (2.3), of rank 0 N = 4 SCFTs. In the limits, due to huge Bose/Fermi cancellations the unrefined superconformal index gets contributions only from Coulomb-branch or Higgsbranch operators and their descendants. For rank 0 theories, the index becomes trivial (i.e. 1) since there are no non-trivial Coulomb/Higgs branch operators. Other partition functions on various rigid supersymmetric Euclidean backgrounds also drastically simplify in the degenerate limits for rank 0 theories. Our correspondence says that the simplified partition functions are actually equal to the partition functions of corresponding non-unitary TQFTs on the same 3D spacetime. (See (2.7) for a precise statement.) Concrete dictionaries of the correspondence are given in Table 1. In the degenerate limits, as seen in the superconformal index case, contributions from local operators become unimportant for rank 0 theories and only non-local loop operators become relevant physical observables. Similarly, loop operators are the only physical observables in general TQFTs. Using the correspondence, one can map the problem of classifying rank 0 SCFTs to the classification of non-unitary TQFTs, which is much easier to handle. Mathematically, TQFTs are described by modular tensor categories (MTCs) and classification of MTCs has been studied intensively in the literature [29][30][31][32][33][34]. The most basic quantity characterizing a 3D CFT is the round threesphere free energy, usually denoted as F . The F always monotonically decreases under the renormalization group (RG) flow and thus is regarded as a proper measure of the degrees of freedom of 3D CFT [35][36][37]. In one of the most interesting and surprising dictionaries of the correspondence, the F of a rank 0 CFT is related to the modular S-matrix of a non-unitary TQFT in a very simple way as given in Table 2. Combining the dictionary and universal algebraic properties of the S-matrix, we derive following lower bound on F which should hold for any rank 0 SCFTs. Interestingly, the lower bound is saturated by the minimal theory studied in [27]. The correspondence is similar in spirit with the (4D N = 2 SCFT)/(2D chiral algebra) correspondence mentioned above. In both correspondences, non-unitary algebraic structures, chiral algebras on the one hand and modular tensor category on the other, appear as protected subsectors of unitary superconformal field theories. But there are several crucial differences. First, two theories in our correspondence are defined on the same 3D space-time while the 2D chiral algebra lives in the 2D subspace of 4D space-time of the SCFT. Secondly, basic physical objects are BPS local operators in the (4D SCFTs)/(2D chiral algebra) story while BPS non-local loop operators are basic objects in our correspondence. That non-local loop operators play crucial roles can be a great advantage of our classification approach over the conventional conformal bootstrap approaches, since the latter are based on correlation functions of local operators. We note that the 3D non-unitary TQFTs are sensitive to the global structure of the 3D rank 0 SCFTs and two theories in the correspondence share the same one-form symmetry as well as their 't Hooft anomalies. In 3D, the quantity F (unlike the stress-energy tensor central charge C T ) is sensitive to the global structure of CFT and the conformal bootstrap approach never give a constraint on F but only on C T , which is not a proper measure of the degrees of freedom in a strict sense [38].
The remaining part of paper is organized as follows. In section 2, we present the precise statement of the correspondence with several concrete dictionaries. As an application of the correspondence, we derive interesting lower bounds on F for rank 0 SCFTs. In section 3, we explicitly work out the correspondence in detail with several classes of infinitely many rank 0 SCFTs. The results are summarized in Table 2. In Appendix A, we give brief reviews on supersymmetric partition functions of 3D N ≥ 3 gauge theories and modular data of 3D TQFTs which are basic ingredients of the dictionaries. In other Appendices, we collect technical details and supplementary materials.

(Rank 0 SCFT)/(Non-unitary TQFTs) correspondence
In this section, we establish a correspondence between T rank 0 : a 3D N = 4 interacting SCFT with empty Coulomb and Higgs branches −−−→ TFT ± [T rank 0 ] : a pair of 3D non-unitary TQFTs . (2.1) The basic dictionaries for the correspondence are summarized in Table 1. 3D N = 4 SCFTs have SO(4) SU (2) L × SU (2) R R-symmetry. The Coulomb (Higgs) branch is parametrized by chiral primary operators charged under the SU (2) R (SU (2) L ) symmetry while neutral under the SU (2) L (SU (2) R ) symmetry. In our definition, the rank 0 theory can have mixed branches parametrized by chiral primaries charged under both of SU (2) L and SU (2) R . Rank 0 SCFTs in Section 3 with N = 5 supersymmetry actually have the mixed branches. Rank 0 theories cannot have a continuous flavor symmetry commuting with the SO(4) R-symmetry, since a flavor current multiplet contains Higgs-or Coulombbranch operators. By the same reasoning, the rank 0 theories cannot have SUSY more than N = 5.
Axial U (1) symmetry and R-symmetry mixing Let R ν and A be the two Cartan generators of the SO(4) R-symmetry: Here R and R are the Cartans of SU (2) L and SU (2) R respectively. In our convention, they are normalized as R, R ∈ 1 2 Z. In terms of an N = 2 subaglebra, R ν is the R-charge while A is the charge of a U (1) flavor symmetry (commuting with the N = 2 supersymmetry) called the axial U (1) symmetry. The mixing between the U (1) R-symmetry and the axial flavor symmetry is parametrized by ν. The IR superconformal R-symmetry corresponds to ν = 0.  We consider supersymmetric backgrounds B whose topologies are given as M g,p , a degree p bundle over a genus g Riemann surface Σ g : We refer readers to Appendix A.1 for a brief review on localizations on supersymmetric backgrounds. We can turn on the Ω-deformation parameter (sometimes called squashing parameter) only for g = 0. For even p, one can consider two supersymmetric backgrounds depending on the choice of the spin structure along the fiber [S 1 ] ∈ H 1 (M, Z 2 ). We denote s = +1 (s = −1) for the periodic (anti-periodic) boundary condition for fermionic fields along the S 1 . For p = 0, the partition function can be regarded as a version of the BPS index I(s) and its spin structure dependence can be interpreted as Tr H BPS (−1) 2j 3 , for s = 1 . (2.6) As BPS indices, they can be defined without overall phase factor ambiguities. For p = 0, on the other hand, local counterterms affect the phase factor of the partition function [39] and it is non-trivial to keep track of the local counterterms. Throughout this paper, we are for the most part interested in the absolute value of partition functions. For g = 1 and p = 0, the supersymmetric partition function is independent of all the parameters and is simply an integer number called the Witten index.
Emergence of non-unitary TQFT in the limits ν → ±1 As main result of the paper, we propose that for any rank 0 N = 4 SCFT T rank 0 we can associate a pair of non-unitary TQFTs denoted by TFT ± [T rank 0 ], satisfying the following relation The partition function Z Mg,p TFT of the topological theory depends only on the topological structure M g,p of B and (possibly) a choice of a spin-structure s on it. 1 We claim that in the degenerate limits i) the rigid supersymmetry partition function becomes independent on the squashing parameter b 2 (or Ω-deformation parameter q) and ii) it becomes the partition function of a non-unitary topological quantum field theory.
We call a topological quantum field theory a non-spin (or bosonic) TQFT when its partition function is independent on the choice of the spin structure, and a spin (or fermionic) TQFT otherwise. We propose that The non-unitary TFT ± [T rank 0 ] is a spin (fermionic) TQFT , if R ν=±1 + 2j 2 ∈ 2Z + 1 for some BPS local operators . or equivalently, if I sci (q, η, ν = ±1; s) contains q 1 2 (odd integer) terms . (2.8) Here I sci is the superconformal index defined in (A.3) and the index at ν = ±1 is in general a power series in q 1/2 since R ν=±1 + 2j 3 ∈ Z. From (2.6), we expect that the supersymmetric partition on M g,p=0 depends on the spin-structure s if the above condition is met. This is because that the condition R ν=±1 + 2j 3 ∈ 2Z + 1 implies that (−1) R ν=±1 = (−1) 2j 3 for some BPS local operators which acts on the Hilbert-space H BPS non-trivially. The above condition gives sufficient but not necessary condition for TFT ± to be fermionic. To see this, consider a rank 0 SCFT T rank 0 not satisfying the above condition whose associated non-unitary topological field theories, TFT ± [T rank 0 ], are bosonic. Then, T rank 0 ⊗ T spin top with an unitary fermionic topological field theory T spin top is still a rank 0 SCFT not satisfying the above condition since the decoupled topological sector does not contribute to the superconformal index. But its associated non-unitary TQFTs TFT ± [T rank 0 ⊗ T spin top ] = TFT ± [T rank 0 ] ⊗ T spin top are fermionic due to the decoupled unitary spin TQFT T spin top .
The proposal in (2.7) can be easily proven for the case when Z B is the superconformal index (A.3). In the degenerate limit, ν → 1 and η = 1, the index becomes (2.9) All unitary multiplets of 3D N = 4 superconformal algebra are listed in [40]. From the classification, it is not difficult to check that the index above gets contributions only from operators in a short multiplet denoted by [40]. The bottom state inside the multiplet corresponds to a Coulomb branch operator parametrizing the Coulomb branch. From the superconformal index in the degenerate limit, one can actually compute the Hilbert-series of the Coulomb branch [41]. Since we are considering a rank 0 theory T rank 0 with empty Coulomb branch, the index gets contributions only from the identity operator and becomes simply (q-independent) 1. Similarly, one can also confirm that the index becomes 1 in the other degenerate limit, ν → −1 and η = 1, since there is no Higgs branch. In summary, from the superconformal multiplet analysis, we conclude that I sci T rank 0 (q, η, ν; s) This proves the proposal in (2.7) for the case when Z B = (superconformal index) since Z S 2 ×S 1 = 1 for all topological theories.
We currently do not know the full proof of the proposal (2.7) for other supersymmetric partition functions Z B . As noticed in [34], however, the triviality of superconformal index for a non-conformal choice of R-symmetry is a strong evidence for the appearance of nonunitary TQFT, while the triviality of the index at the superconformal R-symmetry implies an emergence of a unitary TQFT at the infra-red fixed point. We explicitly test the proposal with infinitely many rank 0 SCFTs and various supersymmetric partition functions in section 3.

Dictionaries
In Table 1, we summarize basic dictionaries of the correspondence. The dictionary in the first line is the most basic one and other dictionaries (except for the last one for F ) follow from it. On the one hand, partition functions (with insertion of loop operators along the fiber S 1 ) of a topological field theory on the geometries M g,p are determined by the modular data, i.e. S and T matrices, of the topological theory. On the other hand, the supersymmetric partition function on B with topology M g,p can be computed using localization technique as briefly summarized in Appendix A.2.
For a TQFT its rank, i.e. the size of modular matrices, is equal to the ground state degeneracy on the two torus. For a supersymmetric field theory, the degeneracy is equal to the Witten index. The dictionaries for S 2 0α , T αβ and W β (α) simply follow from comparison between (A.57) and (A.44). The Bethe-vacuum corresponding to the trivial simple object, α = 0, can be determined by requiring that In topological field theories simple objects (labeled by α) are loop operators, while in supersymmetric field theories α labels the types (gauge and flavor charge) of loop operators. Therefore, we expect that for each Bethe-vacuum α there is a corresponding supersymmetric loop operator O α ( z), see around (A.44): (Bethe vacua)-to-(loop operators) map : The trivial object α = 0 corresponds to the identity operator. The map will be determined by requiring the following consistency conditions (2.13)  52), from which one can compute S-matrix S αβ = W β (α)W 0 (β)S 00 . H α and F α are handle gluing and fibering operator at the α-th Bethe-vacuum respectively, see (A.40) and (A. 43). O α=0 corresponds to the trivial loop and W β (0) = 1. The rank of a TFT (not to be confused with the rank of its associated 3D N = 4 SCFT) is defined as the dimension of H(T 2 ) for non-spin case while is defined as the dimension of H −− (T 2 ), i.e. Hilbert-space in NS-NS sector, for spin case. Similarly, the S and T matrices (resp. S and T 2 ) for non-spin TQFT case (resp. spin TQFT case) are modular matrices acting on H(T 2 ) (resp. H −− (T 2 )). Simple objects are in one-to-one with a basis of H(T 2 ) (resp. H −− (T 2 )) for non-spin TQFT case (resp. spin TQFT case). We refer readers to Appendix A.2 for a general review on the modular data of non-spin and spin TQFTs.
The dictionary for F in the last line is one of the most non-trivial and interesting statements in this paper. It says that (2.14) F is the free energy on round three-sphere, which is the quantity appearing in the F-theorem and is a proper measure of degree of freedom. The relation is surprising since it relates the quantity (F ) at the superconformal point, ν = 0, to the quantity (S 0α * ) in the degenerate limits ν = ±1. One possible explanation for the dictionary above is as follows. In general, S 0α * in a topological field theory computes the three-sphere partition function with an insertion of loop operator O Γ=(unknot) α * of type α * along the unknot in S 3 . In the rank 0 theory T rank 0 , on the other hand, there is a flavor vortex loop operator associated to the U (1) axial flavor symmetry. The loop operator is known to act on the three-sphere partition as a difference operator shifting the parameter ν [42][43][44] (flavor vortex loop) ± of charge ±1 ←→ exp(±∂ ν ) .
(2. 15) This means that the S 3 partition function at the conformal point can be identified with the S 3 partition function with an insertion of the vortex loop operators of charge +1(−1) (2.16) Hence if one identify α = (flavor vortex) + (α = (flavor vortex) − ) of TFT − (TFT + ) with α = α * in (2.14), then the dictionary follows. Actually, according to F-maximization (A.28), we expect that It is compatible with the desired identification, (flavor vortex loop) = (α * in (2.14)). The property above is also compatible with the fact that TFT ± are non-unitary since they violate the unitarity condition (A.56). The argument above gives circumstantial evidence but not a full proof for the dictionary on F . The dictionary will be confirmed explicitly in section 3 with infinitely many examples. We leave general proof or disproof of the dictionary for future work.

Application: lower bounds on F
Here we derive interesting lower bounds on F for rank 0 SCFTs using the correspondence introduced in the previous subsection.
One immediate and interesting consequence of the dictionaries in Table 1 is This follows from the following fact combined with the dictionaries for Witten index and F . More interestingly, using the dictionary, one can prove following Interestingly, the partition function is actually independent b [45,46] and The bound is saturated by the minimal T min theory which will be introduced in section 3. The inequality above follows from the following fact combined with the dictionary for F |S 0α * in (2.14)| ≤ 5 − √ 5 10 = 0.525731 for all non-unitary TFT . where S αβ is the S-matrix of a non-unitary TQFT. Then the two positive real numbers x and y should satisfy followings Here GSD g denotes the ground state degeneracy on genus g Riemann surface, see (A.58). One can solve the equations and we have two solutions Imposing the conditions, x, y ∈ R >0 and x > y (non-unitarity condition, see (A.56)), we have only one solution for each k > 4. As the natural number k increases, y 1/2 = |S 0α * | in the solution decreases. Thus, we have for any rank 2 non-unitary TQFT .

Examples
In this section, we introduce infinitely many examples of (2+1)D N = 4 interacting rank 0 superconformal field theories T rank 0 . Using the dictionary in Table 1, we compute the set {|S ± 0α |} for non-unitary TQFTs TFT ± [T rank 0 ]. We also independently compute the threesphere free energy F by performing the localization integral for Z S 3 b at b = 1, m = 0, ν = 0 and confirm the non-trivial dictionary for F . See Table 2 for the summary.
(     As a quick evidence for the SUSY enhancement, there are following two gauge invariant monopole operators in the theorŷ with following quantum numbers [27,47,48] Here A is the charge of U (1) top and R is the charge of the superconformal U (1) R-symmetry; j ∈ Z 2 and ∆ are the Lorentz spin and the conformal dimension respectively. We use the fact that [27] which can be determined by the F-maximization [49]. Here |m ∈ H rad (S 2 ) (the radially quantized Hilbert-space on S 2 ) is a half BPS bare monopole operator. The bare monopole operator has a U (1) gauge charge − 3 2 m+ 1 2 |m| and should be dressed by excitations (φ,ψ and their complex conjugations) of matter fields to be gauge-invariant. The dressed monopole operators are 1/4 BPS local operators. According to the classification in [40], the monopole operators above, if they survive at the IR superconformal point, must belong to an extra SUSY-current multiplet of the 3D N = 2 superconformal algebra. The multiplet consists of conformal primaries of the following quantum numbers as well as their conformal descendants,  Here Q := Q 1 + iQ 2 andQ := Q 1 − iQ 2 are the N = 2 supercharges. The local operators in the top component with (R, j, ∆) = (0, 3 2 , 5 2 ) correspond to the conserved current for the extra supersymmetry, whose existence guarantees the SUSY enhancement.
Further, it was claimed in [27] that the infra-red (IR) superconformal field theory (SCFT) T min is the minimal 3D N = 4 SCFT having the smallest three-sphere free-energy F and the smallest non-zero stress-energy tensor central charge C T whose exact values are [27,50]   There is no vacuum moduli space in the minimal theory and thus the minimal SCFT is of rank 0.
Superconformal index Applying the localization results in [51,52] (see also Appendix A of [53]), the superconformal index for T min can be written as with f single (q, a; m) := q (3.7) In the above, we use the superconformal R charge of Φ given in (3.4). At the conformal point, ν = 0, the index becomes The terms in q 3/2 come from the monopole operators in (3.3) and the index is compatible with the claimed SUSY enhancement [54]. On the other hand, the index at the non- As anticipated from the superconformal multiplet analysis in (2.10), the index becomes (q-independent) 1 in the degenerate limits, ν → ±1 and η → 1. This reconfirm that the N = 4 theory is of rank 0 and gives a strong signal that a topological field theory emerges in the limits.

Lee-Yang TQFT in degenerate limits
Here we claim that with explicit checks of the dictionaries in Table 1. The non-unitary TQFT has following modular data .

(3.11)
Squashed three-sphere partition function The squashed three-sphere partition function for the minimal theory is ( Here ψ is a special function called the quantum dilogarithm, for which readers are referred to Appendix D. The partition function in the limit b → 1 was studied in [50] and one can check that with s n = 0 up to arbitrary higher order n . (3.13) That the partition function becomes independent on the squashing parameter b in the degenerate limit is a strong signal that the theory becomes topological. Furthermore, the partition function is identical to the S 3 partition function (S 00 , see (3.11)) of the Lee-Yang TQFT, (3.14) of the minimal theory, given in (3.6), nicely matches with max α (− log |S 0α |), see (3.11) for the S-matrix, which confirms the dictionary for F in Table 1.

Perturbative invariants S α n
The integrand in (3.12) can be expanded as There are two Bethe-vacua (A.31) determined by the following algebraic equation Bethe-vacua of T min : z : In the degenerate limits, m = 0 and ν → ±1, the two Bethe-vacua approach following values Perturbative invariants (A.33) of two saddle points associated to the two Bethe-vacua in the degenerate limits are That this is compatible with the expected properties in (A.35) is a highly non-trivial evidence for emergence of topological theory in the degenerate limits.
Fibering and Handle gluing Using the formulae in (A.43) and (A.40) combined with the above computation of S α n , we have (3.20) (3.13), the z α=0 is indeed the Bethevacuum corresponding to the trivial object according to the criterion in (2.11). The computations above also confirm the dictionary for S −2 0α and T αβ in Table 1, see (3.11) for modular matrices of Lee-Yang TQFT.
Supersymmetric loop operator For a U (1) gauge theory, the supersymmetric dyonic loop operator O (p,q) of (electric charge, magnetic charge)=(p, q) is The consistency condition in (2.13) is met when we choose the (Bethe vacua)-to-(loop operators) map as follow Then, using the dictionary in Table 1 W β=0, From W β (α), one can compute the S-matrix using the formula S αβ = S 00 W β (α)W 0 (β), and confirm that it is identical to the S-matrix of the Lee-Yang TQFT given in (3.11).

SUSY enhancement
We define coupled to a hypermultiplet of charge +1) .

(3.24)
For non-zero k, the theory has N = 3 supersymmetry instead of N = 4 since the CS term breaks some of the N = 4 supersymmetry. The N = 3 theory has the U (1) top flavor symmetry associated to the dynamical U (1) gauge theory. As pointed out in [55,56] For k = 1, the (U (1) k + H) has enhanced N = 5 supersymmetry at IR and the resulting IR SCFT is of rank 0 . The U (1) top = SO(2) top symmetry becomes the U (1) axial symmetry, which is a subgroup of SO(4) ⊂ SO(5) R-symmetry, in the supersymmetry enhancement.
For |k| = 1, there are two BPS monopole operators whose quantum numbers are Here A is the charge of U (1) top and R ∈ Z 2 is the spin of SO(3) R-symmetry. j and ∆ are the Lorentz spin and the conformal dimension respectively. The BPS operators belong to extra SUSY-current multiplet [40] of N = 3 superconformal algebra, which consists of the following conformal primaries and their descendants (3.28) The local operators in the top component of the supermultiplet with (R, j, ∆) = (0, 3 2 , 5 2 ) correspond to the conserved current for extra supersymmetry, and thus the existence of the multiplet implies the supersymmetry enhancement [55,56].
Superconformal index Using the localization summarized in A.1, the index is given as where η is a fugacity of U (1) top and Using the above expression, the index can be evaluated and we find 4 The term (−η − 1 η )q comes from the monopole operators (3.27) in extra SUSY-current multiplet and implies the SUSY enhancement. Note that the SUSY enhancement occurs only at |k| = 1.
In the degenerate limit ν = ±1 and η = 1, the index becomes It is compatible the expectation that the theory is a N = 4 (actually N = 5) SCFT of rank 0 when |k| = 1. It also implies that there emerge non-unitary TQFTs in the degenerate limits only when |k| = 1. The non-unitary TQFTs are expected to be fermionic according to (2.8).

Non-unitary TQFTs in degenerate limits
Here we extract modular data of TFT ± [U (1) k=1 +H] by computing various supersymmetric partition functions.
Squashed three-sphere partition function It can be written as (see Appendix A.1) Here Ψ (X) := ψ (X) exp( X 2 4 ) as defined in (A.25). In a round sphere limit (b = 1) with k = 1 and m = 0, the integral reduces to (using Appendix D) Here e iδ is an unimportant phase factor sensitive to local counterterms. For ν = 0, this integration can be exactly evaluated by applying the residue theorem to the integral along the contour depicted in Fig 1: Figure 1. A contour for the evaluation of (3.34). There is a simple pole at Z = πi inside the contour.
Here, the residue at the simple pole Z = πi is −2ie πi 4 . The first, second, and third term in the first line comes from the path C 1 , C 2 + C 3 , and an arc A 1 respectively. At the third equality, we have used the changing variable as Z → −Z from the last term in the second line.
Likewise, the integration for ν = ±1 can be computed exactly in a similar way by using the same contour and we found Restoring the overall factor 1/(4π) in (3.34), we finally have The partition functions in the degenerate limits ν → ±1 are actually independent on the squashing parameter b, as we will check it perturbatively in (3.41), and equality in the 2nd line holds for arbitrary b ∈ R.
Perturbative invariants S α n The integrand in (3.33) at k = 1, after a shift Z → Z + 1 2 (πi + 2 ) of dummy integral variable, can be expanded as where c 0 and c 1 are Z-independent rational numbers. There are two Bethe-vacua (A.31) determined by a following algebraic equation Bethe-vacua of (U (1) 1 + H) at m = 0 : z : In the degenerate limits, m = 0 and ν → ±1, the two Bethe-vacua approach following values Perturbative invariants (A.33) of two saddle points associated to the two Bethe-vacua in the degenerate limits are That this is compatible with the expected properties in (A.35) is a highly non-trivial evidence for emergence of topological theory in the degenerate limits.
Fibering and Handle gluing Using the formulae in (A.43) and (A.40) combined with the above computation of S α n , we have , the z α=0 is indeed the Bethevacuum corresponding to the trivial object according to the criterion in (2.11).

Supersymmetric loop operators
The consistency condition in (2.13) is met when we choose the (Bethe vacua)-to-(loop operators) map as follows: Then, using the dictionary in Table 1, From W β (α), one can compute the S-matrix using the formula S αβ = S 00 W β (α)W 0 (β) and the result is Since the TQFTs, TFT ± [U (1) 1 + H], are spin TQFTs, only the modular T 2 matrix is well-defined and according to the dictionary in Table 1 T The modular data (S and T 2 ) of the spin non-unitary TQFT, The theory is defined as Here we define (3.49) Using the formula, one can compute the superconformal index and check that (3.50) The higher order terms (represented by . . .) depend on k for |k| ≥ 2. From the computation, one can determine the basic property of theory appearing at the IR. The triviality of the index for |k| = 1 implies that the theory has a mass gap and the IR physics is described by an unitary TQFT. The divergence of the index is a signal of emergence of a free chiral theory that decouples with the other part of the theory [58]. The non-triviality of the index implies that the theory flows to a superconformal field theory. In summary, from the index computation we conclude that Contains decoupled free chirals if |k| = 0 , (3.51) In the degenerate limits, ν → ±1 and η → 1, the index becomes (for |k| ≥ 2) ], emerges when |k| ≥ 2 in the limits. In addition, we expect that they are spin TQFTs according to Table 2 is indeed a spin TQFT since it is given by a Z f 2 quotient (fermionic anyon condensation) of a bosonic TQFT.

Non-unitary TQFTs in degenerate limits
Squashed three-sphere partition function The partition function of the N = 5 theory is (3.53) At b = 1 and m = 0, the localization integral is simplified as At ν = 0, ±1, the partition function is exactly computable and we obtain , Bethe-vacua and Handle gluing operators in the degenerate limits Since there is a Z 2 symmetry (m, ν) ↔ (−m, −ν), we only consider the limit ν → 1. In the limit ν → 1, the asymptotic expansion coefficients W 0 and W 1 of the integrand are (3.56) The Bethe vacua equation at ν = 1 is In the degenerate limit η → 1, the equation simplifies as z 2k = −1 and there are |k| Bethe-vacua after taking into account of the Weyl Z 2 quotient, z ↔ 1/z, Now, the handle gluing in the degenerate limit, m = 0, ν = 1, is given by (3.59) The factor 1/4 comes from 1/|Weyl(SU (2))| 2 , see (A.34). By plugging the eq.(3.58) in the eq.(3.59), we have (2)] is the 3D theory living on the S duality domain wall in 4D N = 4 SYM [59]. The theory is the 3d N = 4 SQED with two fundamental hyper-multiplets. Let the four N = 2 chiral fields in the two N = 4 hyper-multiplets be q 1 , q 2 , q 3 , q 4 and the adjoint N = 2 chiral field in the N = 4 vector multiplet be φ 0 . The theory has SU (2) H × SU (2) C flavor symmetry at the IR as well as the SO(4) R-symmetry. The charge assignments for chiral fields under the Cartan subalgebra of the gauge and global symmetries are: By gauging the two SU (2)s with non-zero Chern-Simons level k 1 and k 2 , we obtain infinitely many rank 0 3D N = 4 SCFTs which will be denoted as (2)] with Chern-Simons levels k 1 and k 2 .
As argued in [27], the gauging does not break the supersymmetry down to N = 3 thanks to the nilpotent property of the moment map operators, µ H and µ C , of the two SU (2)s. The theory has Z H 2 × Z C 2 one-form symmetry originating from the center symmetry The discrete one-form symmetry has 't Hooft anomaly characterized by the following bulk action 5 Here w H 2 (w C 2 ) is the 2nd Stiefel-Whitney class, valued in H 2 (M 4 , Z 2 ), of the SO(3) H = (SU (2) H )/Z 2 and SO(3) C = SU (2) C /Z 2 bundle respectively. P is the Pontryagin square operation, which satisfies P(w 2 ) = w 2 2 (mod 2). On spin manifold M 4 , the 1 2 P(ω) ∈ Z. The first two terms in (3.62) come from the Chern-Simons action of two SU (2) gauge fields [60] while the last term is from the anomaly polynomial of T [SU (2)] theory [61]. From the anomaly polynomial, one can confirm that the following Z 2 one-form symmetry is anomaly free Anomaly free one-form Z 2 symmetry : (3.66) Here the generalized superconformal index for T [SU (2)] theory is where f single (a 1 , a 2 , η, u; m 1 , n) = q n + m 1 , 2m 1 , m 2 ∈ Z , k 1 ∈ 2Z and k 2 ∈ 2Z + 1 , n, m 1 , 2m 2 ∈ Z , k 2 ∈ 2Z and k 1 ∈ 2Z + 1 , For the case when |k 1 k 2 − 1| = 1, the index vanishes and it implies that SUSY is spontaneously broken. For the case when |k 1 k 2 − 1| = 2, on the other hand, the index for (2)] k 1 k 2 has a mass gap and flows to a topological theory and the UV Z 2 one-form symmetry decouples at IR, i.e. the Z 2 does not act faithfully on any IR observables. It means that the IR TQFT actually does not have the Z 2 symmetry. The index for T [SU (2)] k 1 k 2 /Z 2 becomes 2 just because we perform the gauging of the decoupled (so absent) one-form symmetry by hand. At the level of S 2 × S 1 partition function, the one-form gauging procedure is Here the [β 2 ] ∈ H 2 (S 2 × S 1 , Z 2 ) = Z 2 is the background 2-form Z 2 flat connections coupled to the one-form symmetry. Alternatively, the RHS can be written as where first term is the S 2 × S 1 partition function (with trivial [β 2 ]) and the 2nd term is the partition function with insertion of loop operator along the [S 1 ], the generator of , with the Z 2 symmetry generating anyon α Z 2 . For general TFT with a faithful Z 2 one-form symmetry, the first term is just 1 while the 2nd term vanishes and the total partition function Z S 2 ×S 1 TFT/Z 2 becomes just 1 as expected. When the Z 2 one-form symmetry is decoupled at IR, however, the 2nd term is also 1 since the anyon α Z 2 actually becomes the trivial operator, O α=0 , at IR. So the result, Z S 2 ×S 1 TFT/Z 2 = 2, is an artifact due to the "gauging" of the "absent" Z 2 symmetry. As we will see below, one can actually confirm that the Z 2 symmetry act trivially on Bethe-vacua of T [SU (2)] k 1 ,k 2 theory for the case when |k 1 k 2 − 1| = 2. In summary, from the index computation, we can conclude that Decoupled free chirals if |k 1 k 2 − 1| = 0 . and I sci (q, η = 1, ν = ±1; s = 1) = 1 , non-trivial power series in q otherwise .
The non-unitary TQFTs associated to the N = 4 SCFT before the Z 2 one-form symmetry gauging are given in Table 2 and the TQFTs after the gauging are (3.79) Here Z 2 is the anomaly free Z 2 one-form symmetry in Gal one-form symmetry and the following Z 2 one-form symmetry is non-anomalous Anomaly free one-form Z 2 symmetry : Z (3.80) When k 1 k 2 ∈ 4Z 2 + 1, the one-form Z 2 symmetry is fermionic and the theory after the Z 2 quotient becomes a spin TQFT, It confirms the criterion (2.8) combined with the superconformal index computation (3.77).

(3.82)
Here I vec is the contribution from the vector multiplet for the SU (2) H k 1 × SU (2) C k 2 gauging: (2)] is the contribution from the T [SU (2)] theory whose squashed three-sphere partition function is (3.84) One of the non-trivial consistency checks is the mirror property which we have checked numerically for various values of X 1 , X 2 , and ν. Particularly for b = 1, m = 0, ν = 0, we have [62][63][64] (see also Appendix C.1) .
As in (3.54), this localization (3.82) also simplified at b = 1, m = 0, and is exactly computable at ν = 0, ±1 (see Appendix C.2). (3.88) In the W 0 above, we have ignored terms which are independent on X 1 , X 2 and Z. By extremizing the twisted superpotential we have following Bethe-vacua equations (3.90) This explains why we obtain 2 (instead of 1) in the index computation of T [SU (2)] k 1 ,k 2 /Z 2 theory when |k 1 k 2 − 1| = 2, see the paragraph below (3.72). The handle gluing operator is The set of the handle gluing operators evaluated at the Bethe-vacua in the degenerate limits are .
(3.95) (2)  Thanks to the nilpotency of the moment map µ diag for the SU (2) diag flavor symmetry, the theory remains N = 4 theory even after the gauging with non-zero k [27,55].

T [SU
The theory has a Z 2 one-form symmetry which corresponds to the center group of the gauged SU (2) diag symmetry. The 't Hooft anomaly polynomial for the one-form symmetry is Thus, the one-form symmetry is anomalous for k = (odd) while non-anomalous for k = (even). For odd k, the theory can be tensored with a topological theory U (1) 2 = SU (2) 1 , which also has anomalous Z 2 one-form symmetry, and the diagonal Z 2 one-form symmetry becomes non-anomalous. We define   From superconformal index computation [55,56], non-trivial power series in q 1/2 , |k| > 2 .  and I sci (q, η = 1, ν = ±1; s = 1) = 1 ,  ] are spin TQFTs when k ∈ 4Z.

(3.108)
We have ignored terms which are independent on X and Z in the expression for W 0 given above. By extremizing the twisted superpotential we have the Bethe-vacua equations where x = e X , z = e Z and w = e The set of the handle gluing operators evaluated at the Bethe-vacua in the degenerate limits are where we define A k := |k| − 2, B k := |k| + 2.
For |k| > 2, as concluded in (3.103), the theory T [SU (2)]/SU (2) diag k lands on 3D N = 4 SCFT of rank 0 at the end of RG. According to the dictionary in Table 1, the above set should be equal to the set of {S 2 0α } for a non-unitary TQFT for the case when |k| > 2.
where G 2 is a exceptional group with dimension 14. Combined with the computation of S 00 in (3.107), we arrive the conclusion in Table 2.

Dualities among rank 0 theories
In this paper, we introduce a pair of non-unitary TQFTs, TFT ± [T rank0 ], as an invariant of 3D rank 0 N = 4 SCFT T rank0 . One natural question is Q : How powerful are TFT ± [T rank0 ]s in distinguishing T rank0 ?
i.e. are TFT ± [T rank0 ]s different for different T rank0 s? (3.114) In this subsection, we provide evidences that the answer is affirmative. This is by examination of the duality between two T rank 0 s whose associated non-unitary TQFTs are identical.
Having common non-unitary TQFTs in the degenerate limits, m = 0 and ν = ±1, automatically implies that equivalence of all supersymmetric partition function Z B in the limits.
Here we will confirm that the equivalence still holds at general values of m and ν. According to Table 2, the non-unitary TQFTs associated to the 3 theories are all identical to the following theory   From the computation of superconformal indices, Witten indices, and one-form Z 2 symmetry matching, we propose that The additional U (1) 2 theory does not contribute to the superconformal index since there is no non-trivial local operator in the topological sector. The additional topological sector, however, doubles the Witten index since it has two ground states on two-torus and provide the one-form Z 2 symmetry. The proposal is also compatible with following fact which is obvious from Table 2. Using the duality, we can also confirm that (3.120) is an almost-trivial theory, whose partition function on any closed 3-manifold is a pure phase factor. Throughout this paper, we have ignored the overall phase factor of the partition function and thus will also ignore such a decoupled almost trivial theory. The duality above has natural interpretation in terms of the 3D-3D correspondence for once-punctured torus bundles [65,66], for which readers are referred to Appendix B for details.

Duality between SU (2)
The superconformal index is Two indices are different unlike in the |k| = 3 case. Surprisingly, the index computation shows that the two theories actually have different amount of supersymmetries [54]: has N = 5 SUSY .

(3.122)
We can thus conclude that supersymmetry is enhanced under the Z 2 one-form symmetry gauging in (3.98). We further claim that the N = 5 theory is actually dual to the following theory with manifest N = 5 supersymmetry: The proposal can be checked using the superconformal index and various other supersymmetric partition functions. Neither theories in the duality has any one-form Z 2 symmetry. The duality is also compatible with Table 2 since (3.124) In the last line, we again ignore the almost trivial spin TQFT SU (2) 2 Z 2 . In the second line, we use the following fact which holds for any two TQFTs, TFT 1 and TFT 2 , which have non-anomalous Z 2 one-form symmetries. The theory T [SU (2)]/"P SU (2) diag |k|=4 " has yet another dual description with only manifest N = 2 supersymmetry which is expected from the geometrical aspects of the 3D-3D correspondence for a once-punctured torus bundle [65,66]. The N = 2 dual is presented in Appendix B.

Discussion
There are several interesting questions we want to address in future.
Relation with Rozansky-Witten theory One well-known method of constructing a topological theory from a 3D N = 4 SCFT is using topological twisting as studied by Rozansky and Witten in [67]. An N = 4 SCFT T has SU (2) L × SU (2) R R-symmetry and we can consider a pair of topological twisted theories, RW + [T ] and RW − [T ], using the SU (2) L or SU (2) R in the twisting respectively. It would be interesting to clarify the exact relation between the pair of topological twisted theories, RW ± [T rank 0 ], and our TFT ± [T rank0 ] for rank 0 SCFT T rank 0 . 6 Two theories, RW[T rank 0 ] and TFT[T rank 0 ], have the same ground state degeneracy GSD g for all g ≥ 0 since the partitial topological twisting on Σ g × S 1 using the U (1) ⊂ SU (2) R (or U (1) ⊂ SU (2) L ) symmetry is actually equivalent to the full topological twisting on the 3-manifold using the SU (2) R (SU (2) L ) symmetry [69,70]. From the comparison, we naturally conjecture that If the conjecture is true, we need to explain how the BPS partition functions of rank-0 theories in the degenerate limits compute the partition functions of topologically twisted theories.

A Some reviews
In order to help better understand the correspondence in (2.7), we briefly review basic relevant aspects of (2+1)D non-unitary topological field theory and supersymmetric partition functions of 3D superconformal field theories.

A.1 Localization on 3D N ≥ 3 gauge theories
In this paper, we consider following 4 types of supersymmetric partition functions of 3D N = 4 SCFT T of rank 0 theory In the partition function, m (resp. η) is the real mass parameter (resp. fugacity) associated to the axial U (1) symmetry while ν is the R-symmetry mixing parameter (2.3). Rank 0 N = 4 SCFT cannot have any flavor symmetry commuting with N = 4 supersymmetries and thus the BPS partition functions cannot be further refined. 3D N = 4 SCFTs can appear as IR fixed points of 3D quantum field theories. Since there could be supersymmetry enhancement along the RG flow, we do not need to start from a UV theory with manifest N = 4 symmetry. For the exact computation of supersymmetric partition using localization, however, the UV theory should have at least N = 2 supersymmetry. In this paper, we study several examples of N = 4 rank 0 SCFTs which appears as IR fixed points of N ≥ 3 supersymmetric theories. In the below, we summarize localization formulae for the BPS partition functions introduced above for N ≥ 3 gauge theories. In localization computations, N ≥ 3 gauge theories have several advantages over N = 2 gauge theories. The local Lagrangian density of an N = 3 gauge theory is uniquely determined by the choice of gauge group G, its Chern-Simons levels k and matter contents (hypermultiplets and twisted hypermultilplets in unitary representations of G). When the CS levels are all zero, i.e. k = 0, the theory has N = 4 supersymmetry. For N ≥ 3 gauge theories, the R-symmetry is non-abelian (SO(3)), and thus the IR R-symmetry is uniquely fixed (i.e. is not mixed with other abelian flavor symmetries) and we do not need to perform the F-maximization [49]. The localization for general N ≥ 2 theories can be done in a similar way but with some more complications.

Superconformal index
The superconformal index for a 3D N = 4 SCFT is defined as Here the trace is taken over the radially quantized Hilbert-space H rad (S 2 ) on S 2 whose elements are local operators. j 3 ∈ Z 2 is the Lorentz spin, the Cartan of SO(3) isometry on the S 2 . The parameter q plays role as an Ω-deformation parameter. Only BPS operators satisfying following relation contribute to the index where ∆ is the conformal dimension. The index alternatively can be regarded as a partition function on (S 2 × S 1 ) = M g=0,p=0 with a fixed metric, background electric fields coupled to U (1) Rν and axial U (1) symmetry and spin-structure along the S 1 . The indices at different ν are simply related as follows I sci (q, η, ν; s = 1) = I sci (q, η, ν = 0; s = 1) η→ηq ν 2 , I sci (q, η, ν; s = −1) = I sci (q, η, ν = 0; s = −1) The two indices with different choices of the spin structure are related to each other in the following way Using localization, the superconformal index at ν = 0 is given as  Λ + adj is the set of positive roots of G. (A.13) The single particle index is (A.14) Here the the summation is over N = 2 chiral multiplets Φ in the representation of R Φ under the gauge group G. ρ R is the set of weights of the representation R. q A (Φ) and ∆ Φ are the axial U (1) symmetry and the conformal dimension of the chiral field Φ respectively. An N = 4 hypermultiplet consists of two chiral multiplets with gauge charges R and R, q A = 1 2 and ∆ = 1 2 . The adjoint chiral multiplet in N = 4 vector multiplet has q A = −1 and ∆ = 1. If one wants to introduce a Chern-Simons interaction in an N = 4 gauge theory, it will break the N = 4 supersymmetry down to N = 3 symmetry. In the case, the R-symmetry is broken to SO(3) and thus we cannot introduce the fugacity η for the U (1) axial symmetry. The Casimir energy 0 is , a) is the contribution from the classical CS term For a U (N ) dynamical gauge group, there is a U (1) topological symmetry whose Noether current is The fugacity a and its background monpole flux m a for the topological symmetry can be introduced by including the following term to I cs The monopole flux m should satisfy the following Dirac quantization conditions λ(m) ∈ Z, ∀λ ∈ Λ + adj and β(m) ∈ Z, ∀β ∈ R Φ .

(A.19)
There could be additional constraints on the monopole fluxes depending on the global structure of the N ≥ 3 gauge theories as we have seen in (3.69) and (3.71). In the localization summation, we need to sum over monopole flux m modulo the redundant Weyl symmetry of G.
Squashed three-sphere partition function Z S 3 b (b, m, ν) This is a partition function on S 3 = M g=0,p=1 with the following metric To preserve some supercharges, a background field coupled to the U (1) Rν symmetry is properly turned on. Using localization, the partition function can be given in the following integral form Here parametrizes the Cartan subalgebra of G.
and ∆ G (Z) is the contribution from the N = 2 vector multiplet associated to the gauge group G |Weyl(G)| is the order of the Weyl group of G. The integrand I is determined by gauge group, matter contents and Chern-Simons levels of the N = 3 gauge theory as follows: We define Ψ as Ψ (X) := ψ (X) exp The real mass m (FI parameter) and the R-symmetry mixing parameter ν of the U (1) topological symmetry for G = U (N ) are introduced by adding the following term to the integrand The partition function at b = 1, which corresponds to round three-sphere, enjoys interesting properties. Firstly, its free-energy is maximized at the superconformal R-charge choice, i.e.
Secondly, the round sphere free-energy F at conformal point always monotonically decreases under the RG flow. So the quantity F can be regarded as a proper measure of degrees of freedom.
Perturbative expansion of squashed three-sphere partition function integral One can consider formal perturbative expansion of the localization integral in an asymptotic limit → 0, to obtain infinitely many 3D SCFT invariants. In the limit, the integrand I can be perturbatively expanded in the following form The leading part W 0 corresponds to the twisted superpotential. By extremizing the twisted superpotential, we obtain Bethe-vacua Bethe-vacua : Weyl(G) . (A.31) Here Weyl(G) is the Weyl group of gauge group G. A Bethe-vacuum z α can be promoted to a saddle point Z α = log z α of the localization integral by properly shifting W 0 as follows, Then we can consider formal perturbative expansion of the localization integral around the saddle point n−1 S α n (m, ν) . (A.33) The factor |Weyl(G)| is multiplied since that many saddle points, which all give the same perturbative expansion, collapse into a single Bethe-vacuum after the Weyl quotient. The perturbative expansion can be formally computed by performing Gaussian integrals [50,82]. For example, The proposal in (2.7) implies the following highly non-trivial constraints on the perturbative invariants for rank 0 SCFTs, This follows from the fact that the squashed three-sphere partition function becomes bindependent in the degenerate limits, m = 0 and ν → ±1, modulo local counter terms which affect an overall factor of the following form (A.36)

Twisted indices and twisted partition functions
The twisted index is defined as Tr H(Σg;ν) (−1) Rν η A , s = −1 . (A.37) Here H (Σ g ; ν) is the Hilbert-space on Σ g with topological twisting using the U (1) Rν symmetry. Unlike the radially quantized Hilbert-space H rad (S 2 ), the Hilbert-space depends on the choice of the R-symmetry mixing parameter ν. Due to the topological twisting, the index is well-defined only when following Dirac quantization condition is satisfied Note that the condition is always satisfied in the degenerate limit ν = ±1 since R ν=±1 ∈ Z which obvious from the fact that R ν=1 = 2R ∈ Z and R ν=−1 = 2R ∈ Z. For g = 0 case, the quantization condition is satisfied for all ν and the index is independent on the continuous parameter ν. Generally, the twisted indices can be written as follows Here H α is called the handle gluing operator at the α-th Bethe-vacuum. For s = −1 case, the operator is simply given as Here e iϕ is a α-independent overall phase factor, affected by the local counter term (A.36), which can be fixed by requiring I Σg ∈ Z for all g up to a sign. For rank 0 SCFT, the phase factor is uniquely determined by requiring I Σ g=0 = 1 in the degenerate limits, η → 1 and ν → ±1. Upon the proper choice of the phase factor, furthermore, the handle gluing operators become all positive real number in the degenerate limits, This is compatible with the dictionary for the handle gluing operators in Table 1. More generally, the twisted partition function is given in the following form Here F α is called fibering operator at the α-th Bethe-vacuum. For s = −1 case, the operator is simply given as The modular matrices contain a lot of information of the topological field theory. According to the Verlinde formula, the fusion coefficients N γ αβ can be given as The S-matrix determines how the basic operators O A α and O B α act on the Hilbert-space T-matrix is a diagonal unitary matrix The topological spin is defined only modulo 1. S and T matrices satisfies The matrix C αβ is called charge conjugation. S 0α are real and they have following pathintegral interpretation The partition function at p = 0 without insertion of loop operator, i.e. β = 0, counts ground state degeneracy GSD g on a genus g Riemann surface, Since this counts actual numbers, the partition function at p = 0 can be defined without any phase factor ambiguity. By contrast the partition function at non-zero p depends on the framing choice of the 3-manifold M g,p as well as of the knot along the fiber [S 1 ], and consequently there is no canonical framing choice. The formula above is only valid for a certain choice of the framing. The framing change affects the partition function by an overall phase factor of the form exp 2πic 2d 24 Z + 2πih β Z . For example, when M g=0,p=1 = S 3 which is different from the Z In topological field theory, Z 2 one-form symmetry is generated by an anyon O α Z 2 satisfying the fusion rule The anyon generating the one-form Z 2 symmetry . (A.59) The topological spin for the anyon α Z 2 can take only following values [83] h α Z 2 ∈ 0, 1 2 , ± 1 4 (mod 1) . (A.60) When h α Z 2 = ± 1 4 , the Z 2 symmetry is anomalous and cannot be gauged. When h α Z 2 = 0 (resp. h α Z 2 = 1/2), on the other hand, Z 2 is non-anomalous and is called a bosonic Z 2 (resp. fermionic Z 2 ) symmetry and is sometimes denoted as Z b 2 (resp. Z f 2 ).
Starting from a bosonic topological field theory TFT with bosonic or fermionic Z 2 one-form symmetry, one can gauge the one-form symmetry to obtain another topological field theory, TFT/Z 2 or TFT/Z f 2 . The gauging procedure is sometimes called the anyon condensation. The resulting theory after gauging is a non-spin TQFT for bosonic Z 2 case while the resulting theory is spin TQFT for fermionic Z f 2 case.
TFT/Z b 2 : Bosonic (non-spin) TQFT , TFT/Z f 2 : Fermionic (spin) TQFT . (A.62) The anomalous symmetry can be gauged only after tensoring with another TQFT, such as U (1) ±2 , with anomalous Z 2 one-form symmetry. At the level of modular data, the anyon condensation process can be summarized as follows. First consider the bosonic Z 2 one-form symmetry gauging. After the gauging, the Hilbert-space on H(T 2 ) is spanned by following basis acts as a diagonal matrix whose entries are all +1 or −1 acts as a permutation matrix whose square is identity. In the above, we define , the basis will be doubled to {|α; ± }. The modular data of the gauged theory is For other S-matrix elements, S (α;±)(β;±) and S [α](β;±) , of TFT/Z 2 , we need to know more information on the mother TFT beyond modular data [84].
In the fermionic Z f 2 gauging, the Hilbert-space H −− (T 2 ) of the resulting spin TQFT is Unlike the bosonic Z 2 gauging, there is no twisted sector in the Z f 2 gauging since |α Z 2 · α = |α . The modular data of the gauged theory is From the computation of Z M g,p=0 one can determines the set {S 2 0α } N −1 α=0 , while from the |Z S 3 | one can determine S 00 . So, from the two computations, one can determine whether the TQFT is unitary satisfying (A.56) or not.
Example : U (1) k theory The action for the theory is given as The action depends on the choice of a 4-manifold X 4 whose boundary is M 3 . Two different choices of the 4-manifolds, say X 4 and Y 4 , give the following difference in the action Here M 4 is a closed orientable 4-manifold obtained by gluing X 4 and Y 4 , an orientation reversal of Y 4 , along the common boundary M 3 . Since Actually, the U (1) k theory is a spin or non-spin TQFT depending on evenness/oddness of k spin (fermionic) TQFT if k ∈ 2Z + 1 . Modular data (S and T matrices) of U (1) k with even k is where α, β = 0, 1, · · · , k − 1 . (A.76) On the other hand, modular data (S and T 2 matrices) of U (1) k with odd k is where α, β = 0, 1, · · · , k − 1 .
The U (1) k theory has one-form Z k symmetry generated by For even k, the theory has Z 2 ⊂ Z k one-form symmetry generated by The topological spin of the symmetry generating anyon is Thus, according to A.61 non-anomalous and bosonic if k ∈ 8Z , non-anomalous and fermionic if k ∈ 8Z + 4 .
The Z 2 one-form symmetry is generated by According to (A.61) The Z 2 is non-anomalous and bosonic if k ∈ 4Z , non-anomalous and fermionic if k ∈ 4Z + 2 . (A.87) The non-anomalous Z 2 symmetry is generated by O α=2k which is bosonic (resp. fermionic) for even k (resp. odd k). After the one-form Z 2 gauging, the Hilbert-space on the two-torus is The modular data of the U (1) 4k /Z 2 theory is (α, β = 0, 1, · · · , k − 1) It implies that U (1) 4k /Z 2 is actually the U (1) k theory. From the Z 2 gauging, one can also confirm that the U (1) k is a non-spin (resp. spin) TQFT for even k (resp. odd k) since the Z 2 theory is bosonic (resp. fermionic).
Example : SU (2) 2k /Z 2 For odd k, the Z 2 one-form symmetry is fermionic and we have a spin topological theory after the Z 2 gauging. The Hilbert-space on the two-torus in the NS-NS sector is On the basis, the modular S, T 2 matrices are = 2 2 2k + 2 sin π(2α + 1)(2β + 1) 2k + 2 , = δ αβ e 2πih [α] with h [α] = α(α + 1) 2(k + 1) (mod 1 2 ) . (A.91) For even k, on the other hand, the Z 2 one-form symmetry is bosonic and the resulting theory after the gauging is a bosonic topological field theory. The Hilbert space on the two-torus is The modular data of the topological theory is and The Hilbert-space of SU (2) 10 × SU (2) 2 theory on the two torus is The theory has Z The modular data of the bosonic topological field theory is The non-unitary Lee-Yang TQFT, whose modular data is given in (3.11), is a Galois conjugate of the Fibonacci TQFT. (2)]

B Dual description for T [SU
In term of the 3D-3D correspondence, the theory T [SU (2) The once-punctured torus bundle (Σ 1,1 × S 1 ) ϕ with ϕ ∈ SL(2, Z) is defined as Here Σ g=1,h=1 is the once-punctured torus and ϕ ∈ SL(2, Z) is an element of mapping class group of the Riemann surface. The mapping torus actually depends only on the conjugacy class of ϕ in SL(2, Z), i.e.
The mapping torus has a torus boundary. Generally, for 3-manifold N with a torus boundary, we need to choose primitive boundary 1-cycle A ∈ H 1 (∂N, Z) to specify its associated 3D gauge theory T [N ; A] [42,61]. In the once-puncture torus bundle, there is a natural choice of the boundary 1-cycle A = S 1 punct , which is the cycle encircling the puncture in Σ 1,1 .
In the view point of 3D-3D correspondence, the conformal window in (3.103) can be geometrically understood from the following topological fact: hyperbolic if |k| > 2 .

(B.4)
We will focus on the case when the mapping torus (Σ 1,1 × S 1 ) ϕ is hyperbolic. For the case, the conjugacy class of ϕ can be decomposed into positive powers of L and R (up to sign) ϕ = ±g (L n 1 R n 2 L n 3 . . . L n L (or R n L )) g −1 n i ∈ Z >0 , The mapping torus (Σ 1,1 × S 1 ) ϕ has an alternative topological description based on an ideal triangulation. Using an ideal triangulation of (Σ 1,1 × S 1 ) ϕ , one can give an alternative description for T [(Σ 1,1 × S 1 ) ϕ ; A = S 1 punct ] following the algorithm proposed in [42]. Interestingly, the 3D gauge theory based on an ideal triangulation has only manifest N = 2 supersymmetry. We expect that the N = 2 gauge theories have enhanced N = 4 supersymmetry at IR.
The corresponding mapping torus can be decomposed into three ideal tetrahedrons [66]. According to the algorithm in [42], the corresponding 3D N = 2 field theory is The mixed CS level K for U (1) × U (1) gauge group is Gauge charges Q for 3 chirals are assigned as follows coupled to the two gauge U (1)s. The bare monopole operator is a gauge-invariant 1/2 BPS chiral primary when m 1 + m 2 = 0. The superpotential breaks the U (1) 3 flavor symmetry to U (1). From the F-maximization, the IR superconformal R-charge (ν = 0) is determined as The superconformal index at the IR conformal fixed point is punct ] (q, η, ν = 0; s = 1) Surprisingly, the index show N = 5 supersymmetry instead of N = 4 [54]. Actually, from the superconformal index computation, one can confirm that the theory is dual to the following N = 5 gauge theory

C Contour integrals
We explicitly evaluate the contour integrals that appear in this paper.

(C.2)
Here, the first and second term of the first line of (C.2) are for the path C 1 and C 2 in the 2 sin X 1 X 2 π sinh(X 1 ) sinh(X 2 ) , for Im[X 1 ] ≤ π . (C. 3) The computations for gauging SU (2) H or SU (2) C of this T [SU (2)] theory at the conformal limit (ν = 0) are straightforward since they are always reduced to the simple Gaussian integration at the three-sphere partition function level.
For the degenerate limit, say ν = 1, the three-sphere partition function Z This means that there is no divergence problem even for ν = 1 if we are considering, say, the SU (2) k gauged T [SU (2)] theory, since (C.4) always starts with linear (ν − 1) term which cancels the diverging simple pole in (C.5) from the adjoint matter. .

C.2 Z
(C.9) The third and sixth terms of (C.9) can be evaluated from the following contour integrals: • ReA ImA l πi ∞ −∞  Combining the two results (C.12) and (C.13), and restoring the overall factor in (C.9), we have (C.14) For ν = −1, thanks to (3.85), the only difference is nothing but an exchange of the role of k 1 and k 2 :  where u ± = ±πie − 2πi k+2 are the residues of A cosh(A) sinh(A) e − 2A 2 πi(k+2) at A = ±πi. Again, the integral at the last term of (C.19) can be evaluated by considering the following path:  Here B n is the n-th Bernoulli number with B 1 = 1/2. For several computations in the main text, one needs to utilize the identity