Three-Dimensional Topological Field Theories and Non-Unitary Minimal Models

We find an intriguing relation between a class of 3-dimensional non-unitary topological field theories (TFTs) and Virasoro minimal models $M(2,2r+3)$ with $r \geq 1$. The TFTs are constructed by topologically twisting $3d$ ${\mathcal N}=4$ superconformal field theories (SCFTs) of rank-0, i.e. having zero-dimensional Coulomb and Higgs branches. We present ultraviolet (UV) field theory descriptions of the SCFTs with manifest ${\mathcal N}=2$ supersymmetry, which we argue is enhanced to ${\mathcal N}=4$ in the infrared. From the UV description, we compute various partition functions of the TFTs and reproduce some basic properties of the minimal models, such as their characters and modular matrices. We expect more general correspondence between topologically twisted $3d$ ${\mathcal N}=4$ rank-0 SCFTs and $2d$ non-unitary rational conformal field theories.


INTRODUCTION
Two-dimensional rational conformal field theory (RCFT) has played an essential role in many different problems in theoretical physics and mathematics.They are characterized by the property that the Hilbert space decomposes into a finite sum of the representations V α and V ᾱ of some chiral algebras A and Ā, (1) The simplest class of RCFTs is the Virasoro minimal models, which have a finite number of irreducible representations of the Virasoro algebra.They have a wide range of applications in the study of two-dimensional critical systems, even when they are non-unitary.
The partition function of an RCFT on a torus with complex structure τ can be written as a combination of a finite number of holomorphic and anti-holomorphic functions in q = e 2πiτ , Z(τ, τ ) = α, ᾱ M α, ᾱχ α (q) χᾱ (q) , (2) where the holomorphic functions χ α (q) are called the characters of the representations V α .The invariance of the partition function under the modular transformation, implies that the RCFT characters transform as vectorvalued modular functions.
In this letter, we construct a novel class of threedimensional topological field theories (TFTs) which are expected to support two-dimensional rational chiral algebras on their boundaries.We will argue that these 3d TFTs can be constructed from a certain family of 3d N = 4 superconformal field theories (SCFTs).A key characteristic of these SCFTs is that they are rank-0, i.e. their Coulomb and Higgs branches are zero-dimensional.
The first examples of such theories were discovered in [1,2].
In general, these 3d theories do not admit a Lagrangian description that preserves the full N = 4 symmetry.Instead, we will present an ultraviolet (UV) field theory description with manifest N = 2 supersymmetry which flows to an infrared (IR) fixed point with enhanced supersymmetry.Each N = 4 theory at the fixed point admits two topological twists [3] which produce two distinct 3d TFTs.These topological theories are in general non-unitary and do not have local operators.
Despite the absence of a Lagrangian description of the IR theory, the N = 2 UV description enables exact computations of various observables in the topologically twisted theories.These computations allow us to extract the data of the corresponding boundary algebra, such as its characters and modular data.
For concreteness, in this letter, we focus on a simple class of TFTs that reproduce the data of non-unitary Virasoro minimal models M (2, 2r + 3) for r ≥ 1.However, we expect that this correspondence exists for a more general class of rank-0 theories and non-unitary RCFTs.We will discuss their construction and classification in an upcoming paper [4].
Let us consider the following class of 3d N = 2 abelian Chern-Simons matter theories, which we call T r : with the superpotential deformation, The charge of the a-th chiral multiplet Φ a under the b-th U (1) gauge symmetry is δ ab .There are mixed Chern-Simons interactions among the abelian gauge fields given by the following level matrix [5]: which coincides with 2C(T r ) −1 , where C(T r ) is the Cartan matrix of the tadpole graph, obtained by folding the Cartan matrix of A r in half.The V mi 's are 1/2 BPS, gauge-invariant, bare monopole operators with fluxes [6]: After the monopole deformation, the 3d N = 2 gauge theory has an unbroken U (1) flavor symmetry which we denote by U (1) A .The charge A of this flavor symmetry is where M a is the topological charge of a-th U (1) gauge symmetry.The theory also has a U (1) R R-symmetry which can be mixed with the U (1) A flavor symmetry.We denote the R-charge at general mixing parameter ν ∈ R by R ν , i.e., We choose the reference R-charge, R 0 , to be the superconformal R-charge, which can be determined by Fmaximization [7].

Supersymmetry enhancement
Here we claim that the N = 2 gauge theory T r flows to an N = 4 rank-0 SCFT in the IR with an accidental supersymmetry (SUSY) enhancement.For the r = 1 case, SUSY enhancement was claimed in [1] by demonstrating several pieces of non-trivial evidence.We give similar evidence for general r.Under the SUSY enhancement, the manifest U (1) R ×U (1) A symmetry is expected to become an SO(4) R ≃ SU (2) C × SU (2) H R-symmetry with the following embedding Here J C/H 3 are the Cartan generators of the SU (2) C/H R-symmetries, whose charges take half-integral values.
To see the SUSY enhancement, we compute the superconformal index I sci (q, η; ν) which is defined as Here H rad (S 2 ) is the Hilbert space of radially quantized theory on S 2 and j 3 ∈ Z 2 is the spin.The index can be computed via supersymmetric localization [8,9] and we find Only q 1 2 Z ≥0 -terms appear in the index, which is the first sign of an enhancement.
Further, the terms − η + 1 η q 3/2 can only come either from extra SUSYcurrent multiplets or chiral primary multiplets with superconformal R-charge 3 [10].Performing a semiclassical analysis of H rad (S 2 ), one can verify that there are two 1/4 BPS dressed monopole operators which have R 0 = j 3 = 1 and A = ±1, which are exactly the same as that of 1/4 BPS operators in extra-SUSY multiplets.The monopole operators are . Here (φ a , ψ a ) are the (scalar, spinor) in the a-th chiral multiplet.On the other hand, we cannot find any chiral primary operator with R 0 = 3 in the semi-classical analysis.Thus, it is natural to conjecture that there exist extra SUSY-current multiplets in the IR.Another supporting fact is that there is an exact match between the central charges of U (1) R and U (1) A which can be computed using localization [11][12][13][14].This is expected if the symmetry enhancement occurs, as they would be related by an element of the Weyl group of SO(4) R .Finally, the T r theory has a dual field theory description with manifest N = 3 SUSY [15].Combining the manifest N = 3 symmetry with the superconformal index computation, one can argue that the symmetry is enhanced [16].

TWO TOPOLOGICAL TWISTS
Being 3d N = 4 theories, each of the IR SCFTs admits two nilpotent topological supercharges Q A and Q B , which we can use to perform two topological twists.They are defined by replacing the SU (2) E rotation group by the diagonally embedded SU (2) subgroup of SU (2) E × SU (2) H or SU (2) E × SU (2) C , which we call the topological A-twist or B-twist.We denote the resulting topological field theories by TFT A and TFT B respectively.
The local operators of the two topologically twisted theories are the Coulomb branch chiral rings and the Higgs branch chiral rings respectively.At the level of the superconformal index, the topological A-twist is realized by taking the limit ν → −1 with η = 1, while the topological B-twist is realized by taking the limit ν → 1 with η = 1 [17,18].For the class of theories discussed in the previous section, we find which agrees with the expectation that the Higgs and Coulomb branches are trivial for this class of theories.
In this paper, we focus on the properties of the Atwisted theories and leave a general analysis for the Btwisted theories in an upcoming paper by one of the authors [19].

FERMIONIC SUM REPRESENTATIONS OF THE MINIMAL MODEL CHARACTERS
The motivation for the UV description T r comes from the following expressions for the characters of nonunitary Virasoro minimal models M (2, 2r + 3) [20][21][22][23][24]: where the r × r matrix K r coincides with the Chernsimons level matrix (6) and Q α are rank-r vectors whose components are We also define which are the conformal dimensions and the central charge.Finally, the denominator is a product of q-Pochhammer symbols, defined by The simplest non-trivial example is the Virasoro minimal model M (2, 5), whose characters are These characters transform as a vector-valued modular function under the SL(2, Z) transformation (3).
where S and T are the generators of SL(2, Z) that satisfy the relation S 2 = (ST ) 3 = I [25].They can be explicitly written as We note that these are the simplest examples of characters that can be written in a so-called fermionic sum representation, where A is a r×r positive definite symmetric matrix, B is a r-dimensional vector and C is a real number.There exists a large class of CFTs whose characters can be written in a fermionic sum representation or its generalizations.In particular, they have been extensively studied with regard to the characters of Virasoro minimal models [20][21][22][23][24] and the characters of a certain class of logarithmic CFTs [26].

HALF-INDICES
The half-index or the supersymmetric partition function on D 2 × q S 1 , introduced in [27,28], counts the boundary operators annihilated by the supercharges that are preserved by a chosen supersymmetric boundary condition on ∂(D 2 × q S 1 ) ≃ T 2 .More precisely, it computes where the trace counts the local operators on the boundary torus.If the boundary condition is compatible with the topological supercharge Q A (or Q B ) in the IR, the half-indices in the limit ν → −1 (or ν → 1) with η = 1 calculate the characters of the boundary algebra for each topologically twisted theory.The UV description of the T r theory ( 4) is designed in a way that its half-index reproduces the characters of the Virasoro minimal model M (2, 2r + 3) in a specific limit.
Indeed, if we impose the Dirichlet boundary conditions for all the N = 2 U (1) vector multiplets and the deformed Dirichlet boundary conditions for all the chiral multiplets in the T r theory [29], the half-index reads [30][31] where we define (x; q) ∞ := ∞ n=0 (1 − q n x).We observe that this expression in the A-twist limit η = 1, ν = −1 coincides with the vacuum character of the Virasoro minimal model M (2, 2r + 3) up to an overall factor [32], The characters of other modules M α can be obtained by inserting loop operators.We consider the Wilson loops L α=1,••• ,r , whose charge under the a-th U (1) gauge group factor is given by the formula (Q α ) a in (15).The halfindex for all α = 1, • • • , r.The choice of these particular sets of loop operators will be justified in the following section.
In order to claim that these expressions are the characters of the boundary algebra of TFT A it is crucial to ensure that the boundary conditions are compatible with the topological supercharge Q A in the IR theory.In general this is a non-trivial task for theories which only have N = 2 descriptions.See [19] for the discussion of this issue in the context of deformable boundary conditions in the holomorphic-topologically twisted theory.

PARTITION FUNCTIONS ON SEIFERT MANIFOLDS
The supersymmetric partition functions of N = 2 theories on a Seifert three-manifold M 3 are completely determined by the twisted effective superpotential W and the dilaton potential Ω [13,[34][35][36].For the T r theory, we have where ζ is the real mass parameter for U (1) A symmetry.If M 3 is a degree-p circle bundle over a genus g Riemann surface, the partition functions can be written as where with x a = e 2πiua [37].These functions are then evaluated on the solutions to the so-called Bethe equations: which reads, for T r , This system of equations has exactly r + 1 solutions, which we denote by {u * α=0,••• ,r }.In the twisting limit (ν, η) = (−1, 1), the supersymmetric partition function ( 27) can be written in terms of the modular data [38]: where α labels modules of boundary algebra and (S, T ) are the modular matrices that transform the characters as in (20).By comparing (27) and (31), we can extract the modular data, more precisely the set {S 2 0α , T −1 αα }, by identifying it with {H(u * α ) −1 , F (u * α )} [39].The full modular data and the precise map between the Bethe vacua u * α and the modules M α (or equivalently, the loop operators L α ) can be constructed by requiring the relation [40] [41] L α (u * 0 ) = S α0 /S 00 = ± H(u * 0 )/H(u * α ) , where L α (u * β ) is the loop operator L α evaluated on the Bethe-vacuum u * β .For this class of theories, we consider Wilson loops L with gauge charges (Q 1 , . . ., Q r ) which contributes Then the following identity together with the SL(2, Z) relations and the nonnegativity of the fusion rule coefficients, determines the S matrix up to an overall sign and the T matrix up to an overall phase factor of the form exp( 2πiZ 3 ).This procedure determines the precise set of lines L α as stated in the previous section.The S-and T -matrices computed in this way agree with the modular matrices of the Virasoro minimal model M (2, 2r + 3), as given in (20).