$(0, 4)$ dualities

We study a class of two-dimensional ${\cal N}=(0, 4)$ quiver gauge theories that flow to superconformal field theories. We find dualities for the superconformal field theories similar to the 4d ${\cal N}=2$ theories of class ${\cal S}$, labelled by a Riemann surface ${\cal C}$. The dual descriptions arise from various pair-of-pants decompositions, that involves an analog of the $T_N$ theory. Especially, we find the superconformal index of such theories can be written in terms of a topological field theory on ${\cal C}$. We interpret this class of SCFTs as the ones coming from compactifying 6d ${\cal N}=(2, 0)$ theory on $\mathbb{CP}^1 \times {\cal C}$

because in this case the product of curves can be understood just as a particular choice of M 4 . We would like to conjecture that "class S 2d N = (0, 4) gauge theories" that we consider in the paper can be realized as T g [CP 1 × C] where C is a Riemann surface with possible punctures. In this way the relation to N = 2 4d theories of class S becomes transparent. The dualities among different 2d theories from class S then can be understood as corresponding to different decompositions of C into pairs of pants. From this conjecture it also follows that the 2d TQFT describing the index is a reduction of Vafa-Witten 4d TQFT [13] on CP 1 . This relation may shed a light on better understanding of Vafa-Witten (VW) TQFT from categorical point of view, i.e. as functor from the category of 3-cobordisms to the category of vector spaces. So far in most of the literature the VW partition function was studied on a particular, usually closed 4-manifold. Some of the progress in understanding of VW TQFT as a functor was made in [10], where the gluing procedure of certain 4-manifolds was considered.
This interpretation is in agreement with recent calculations of the S 2 ×T 2 index of general N = 1 4d gauge theories [14,15] with topological twist along S 2 . The result has an expression that can be interpreted as the index of a (0, 2) 2d theory. In particular, in the case when C is a three-punctured sphere and g = su(3), by solving an integral equation we find index which agrees with the result from [16]. In that paper the authors propose a N = 1 4d gauge theory that flows in the IR to a strongly coupled 4d N = 2 T 3 theory with E 6 flavor symmetry and calculate its S 2 × T 2 twisted index.
However the aim of this paper is not to focus on the 4-manifold realization of twodimensional theories or their 4d gauge theory origin, but to study them purely from twodimensional point of view. The relation to 4-manifolds will be explored in detail elsewhere. Let us note that currently there are almost no non-trivial results about gauge theories with N = (0, 4) supersymmetry in the literature. Our work can be considered as a step towards improving this situation. This paper is organized as follows. In section 2 we introduce N = (0, 4) (and (4, 4)) class S theories with gauge group being a product of several copies of SU (2) and study their properties. In section 3 we consider generalization to SU (N ). In section 4 we show that N = (0, 2) (and N = (2, 2)) SQCDs with SU (N ) gauge group and 2N flavors share certain similarities.
2 Dualities of SU (2) generalized quiver 2.1 SU (2) with 4 flavors and its crossing symmetry Let us consider the simplest possible two-dimensional SQCD with N = (0, 4) supersymmetry and SU (2) gauge group. Such a theory contains at least (0, 4) vector multiplet (U, Θ) consisting of a (0, 2) Vector multiplet U and (0, 2) Fermi multiplet in adjoint representation (see appendix A for a brief review of 2d (0, 2) and (0, 4) theories). The vector multiplet contributes in total −4 to the 't Hooft anomaly coefficient 1 of SU (2) gauge group. If we want to add matter fields in the fundamental representation, the minimal choice that cancels the gauge anomaly from the vector multiplet is four fundamental (0, 4) hypermultiplets (Φ,Φ). In order for the theory to be (0, 4) supersymmetric we also have to choose the following superpotential: (2.1) The constructed theory has SU (4) flavor symmetry as well U (1) B baryonic global symmetry. The hypermultiplets form the following representation 2 w.r.t. SU (2)×SU (4)×U (1) B : (2, 4) +1 + (2,4) −1 . (2.2) As we will show later in the paper, this theory shares a lot of properties with the analogous 4d N = 2 theory, which was studied in great detail already in [18]. In particular, the flavor symmetry is enhanced to SO(8) at the classical level. This can be easily seen from the fact that for SU (2) we have 2 =2 and 4 +1 +4 −1 = 8 v of SO(8) ⊃ SU (4) × U (1). Since the (0, 4) vector multiplet does not have any scalar fields, the theory has no Coulomb branch. The Higgs branch is defined by the triplet of D-term conditions and can be represented as the H 8 //// SU (2) hyper-Kähler quotient. It is the same as the Higgs branch of 4d N = 2 theory and does not acquire any quantum corrections. The scalar fields of (Φ,Φ) transform in representation (2, 1) of SU (2) − R ×SU (2) + R of UV R-symmetry group. Following the arguments of [19] one then expects SU (2) + R , under which the scalars parametrizing the Higgs branch transform trivially, to be the SU (2) R R-symmetry of the small N = 4 superconformal algebra (SCA) in the right-moving sector of the IR SCFT.
The hyper-Kähler dimension of the Higgs branch is 8 − 3 = 5 which is the same as twice the 't Hooft anomaly coefficient of SU (2) + R or, equivalently, the level of the affine SU (2) R-symmetry algebra in the IR SCFT. It follows that the central charges of the theory are c R = 6 · 5 = 30, c L = 20 (2.3) where we also used the fact that c L − c R equals to the gravitational anomaly which is easily calculated in the UV as the difference between the numbers of left and right moving complex fermions.
We would like to conjecture that the spectrum of the (0,4) SCFT at the IR fixed point is also invariant under the action of SO(8) triality which permutes vector representation 8 v and two spinor representations 8 s and 8 c . Unlike in the N = 2 4d case, we do not need to accompany the triality action with a transformation of the the gauge coupling because it is not marginal in 2d. There are also no other apparent exactly marginal deformations of the (0,4) SU (2) gauge theory in the UV, since there is no FI parameter for SU (2) gauge group and the superpotential is completely fixed by (0, 4) supersymmetry.
As in the 4d N = 2 case [2], the symmetry under triality can be reformulated in a different way, which will be useful later in the paper when we consider more general quiver theories.

a) b)
SU (2) SU (2) SU (2) SU(2)  The flavor symmetry of the resulting theory is SU (2) 4 which is enhanced to SO (8). The chiral fields in the hypermultiplets form the following representation of the flavor group: Therefore the invariance of the spectrum under SO(8) triality is equivalent to the symmetry under permutations of SU (2) factors in SU (2) 4 flavor symmetry, or crossing symmetry of the quiver diagram (see Fig. 3). Figure 3: The symmetry under exchange of SU (2) factors in the flavor symmetry of the theory can be interpreted as the crossing symmetry of the quiver diagram. The letters x, y, z, w used to distinguish various SU (2) factors and later in the text denote the corresponding SU (2) flavor fugacities in the elliptic genus.
The statement can be checked by calculating the 2d superconformal index (also known as flavored elliptic genus 3 ) of the theory [20][21][22][23]. The NS-NS index of the theory at hand can be calculated as the following integral (see appendix B for a review of the superconformal index in 2d): (2.6) taken over a certain contour "JK" which corresponds to taking a sum of Jeffrey-Kirwan residues. For example, in the case of rank one gauge group the contour encircles only the poles coming from scalar fields with positive (or, equivalently, negative) charges w.r.t. the Cartan U (1). The factors entering the integrand are (tri-fundamental half-hyper) where x, y and z denote the fugacities corresponding to SU (2) 3 flavor symmetries, and the index of (0, 4) SU (2) vector multiplet. Here and throughout the paper we use the common notation: The fugacity v corresponds to U (1) v global symmetry -anti-diagonal Cartan of SU (2) − R × SU (2) + R R-symmetry which commutes with the supercharges used to calculate the index. The index can be understood as the (0, 2) index where the IR U (1) R R-symmetry is chosen as the Cartan of SU (2) + R and U (1) v plays the role of a flavor symmetry. See appendix B for details. Since the theory has only the Higgs branch, we expect the elliptic genus to coincide with geometrically defined (0, 2) equivariant elliptic genus [24] of the Higgs branch manifold X = H 8 //// SU (2) with empty vector bundle of left-moving fermions: .
where F T is the curvature on the tangent bundle T X.
The integral (2.6) can be calculated explicitly by residues. The result contains 8 terms, each of which has the form of ratio of products of theta functions. To make the formula simpler let us denote the collection of SU (2) 4 fugacities (x, y, z, w) as x which can be understood as the element of the maximal torus of SO(8). In the limit q → 0 the index becomes the same as Hilbert series of X calculated in [25,26], which can be written as (2.11) where θ denotes the highest root of SO (8), and χ kθ is the character for the Dynkin label given by kθ. For the sake of simplicity we later denote characters by the dimension of the corresponding representations. When k = 1, this is the character of the adjoint representation. This is the same as the Hilbert series of the (centered) one SO(8) instanton moduli space [27,28], where the first equality also holds for arbitrary simple gauge group G. The Hilbert series of (centered) 1-instanton moduli space can also be written as a sum over root vectors [29,30] as where h ∨ is the dual Coxeter number of G, and ∆ l is the set of long roots and φ is an element in the Cartan. We identify v = e µ/2 , x = e φ . There are poles at v 2 x γ = 1 for γ ∈ ∆ l . One can show that the index has the following structure: where 28 denotes adjoint representation of SO(8) and the functionĨ The denominator of (2.13) can be understood as the contribution of gauge invariant mesons constructed from bilinear combinations of the chiral fields because where two numbers in each pair denote the representations w.r.t. SU (2) gauge and SO (8) flavor group respectively. The complex dimension of the Higgs branch is 10 and the numerator of (2.13) formally corresponds to additional conditions on these 28 mesons from D-term constraints (cf. [25,26]). The index has the following expansion w.r.t. q and v written in terms of SO(8) characters: One can see that only SO(8) triality invariant representations appear in the index. The crossing symmetry of the index (2.13) can be proven explicitly, not just term by term in q and v expansion. To do this let us consider the difference between indices that differ by a non-trivial transposition of two SU (2) flavor fugacities: (2.16) Using the explicit expression for the index it is easy to show that I (0,4) ∆ (x, y, z, w; v; q) has no poles in variables (x, y, z, w) (i.e. the residues from two terms in (2.16) cancel each other). The theory has anomaly coefficient 2 w.r.t. each SU (2) flavor symmetry factor. Therefore if we further definẽ ∆ (x, y, z, w; v; q) · θ(x ± )θ(y ± )θ(z ± )θ(w ± ) 4 (2.17) it will be a function elliptic in (x, y, z, w) (i.e. invariant under the shifts x → qx, y → qy, etc.) and with no poles. It follows thatĨ − (x, z, y, w; v; q) = 0 . (2.18) The triality outer-automorphism of SO(8) can be understood as the Weyl group action of F 4 if we embed SO(8) ⊂ F 4 . This means that the series (2.15) can be formally rewritten in terms of characters of F 4 representations: The index of the analogous N = 2 4d theory has similar property [4]. As in the 4d case, it does not follow that the global symmetry actually enhances from SO (8) to F 4 in the IR SCFT because there is no conserved current of F 4 .
2.2 Dualities of quiver theories and the TQFT structure of the index

Elliptic genus and 2d TQFT
Similarly to the 4d N = 2 case [4], the crossing symmetry of the index (2.6) indicates that (2.7) and (2.8) can be used to define a 2d TQFT. Namely, let us define the Hilbert space of the 2d TQFT associated to a circle as the following space of meromorphic functions 4 : Then define the basic building blocks of 2d TQFT: Note that the last property in (2.20) is required for the integrand in the definition of η to be elliptic. Using η and C one can define a commutative product µ on H (0,4) is the identity map. The crossing symmetry property (2.18) of the index is then equivalent to the associativity of µ which can be formulated in the following 4 This space can be understood as the space of meromorphic sections of L −4 → M flat (T 2 τ , SU (2)), see appendix C for details. It would be interesting to check explicitly if this is the Hilbert space of VW TQFT associated to CP 1 × S 1 , or, equivalently, the BPS sector of the Hilbert space of T su (2) [CP 1 × S 1 ] quantized on T 2 τ . way:

Dualities between generalized quiver theories
As in [2], the crossing symmetry property of the IR spectrum of the theory depicted in Fig.  2 can be used to deduce IR dualities between various theories constructed from the basic building blocks in Fig. 1.
For example, consider a theory defined by the quiver in the l.h.s. of Fig. 4. Applying the crossing symmetry transformation in Fig. 3 to the middle part we get a different theory corresponding to the quiver in the r.h.s. of Fig. 4. From the point of view of 2d TQFT defined above the index of the theory is the partition function (which can be understood as an element of ∈ (H (0,4) S 1 ) ⊗6 ) of the sphere with 6 punctures. The first theory is a linear quiver ' Figure 4: Duality between two different (0, 4) theories with SU (2) 3 gauge group and SU (2) 6 flavor symmetry. For the sake of simplicity we suppress SU (2) inscribed inside squares and circles of the quivers. gauge theory, and the second one contains trifundamental hypermultiplet coupled to three SU (2) gauge groups.
One can consider another example of duality between two distinct 2d (0,4) theories that follows from the crossing symmetry as depicted in Fig. 5. The index of such theory can be understood as the 2d TQFT partition function of a genus two Riemann surface.
However, in the case when quiver has loops the physics is a little more complicated because the gauge group is not completely broken. Consider a theory corrsponding to a quiver with g loops and n external legs. In terms of 2d TQFT the index is the partition function of a genus ' Figure 5: Duality between two different (0, 4) theories with SU (2) 3 gauge group.
g Riemann surface with n punctures C g,n . The theory has 3g − 3 + n copies of SU (2) vector multiplet and 2g − 2 + n copies of SU (2) trifundamental chiral multiplet T . The resulting theory has SU (2) n flavor symmetry. When g > 0 a U (1) g part of the gauge symmetry remains unbroken for general expectation values of hyper-multiplets. Each unbroken U (1) factor is the the diagonal maximal torus of the gauge group i∈loop SU (2) i associated to the loop in the quiver. Following the authors of [26] in this case we will refer to the moduli space X parametrized by massless gauge-invariant combinations of hypermultiplets as Kibble branch. The naive counting of its dimensions -as n h − n v where n h,v are the numbers of hyper-and vector multiplets of the theory respectively -does not work in this case. The reason is that SU (2) 3g−3+n does not act freely on H 4(2g−2+n) space of hyper-multiplets. The mismatch of the quaternionic dimension is given by g, the rank of the unbroken part of the gauge group. It follows that the Kibble branch CFT should have the following central charges: where we calculated c L from the gravitational anomaly. Let us note that c L > 2c R /3 when g > 0. This is beacuse, unlike in the case when quiver has no loops, unbroken directions of the gauge group give rise to a non-empty complex rank 2g bundle E of left-moving Fermions, the only remnant of the usual Coulomb branch that would appear for (4, 4) theories. Again, as for the basic theory in section 2.1, at least for the large values of scalar fields, we expect the IR SCFT to have a sigma-model description in terms of target space X ∼ = H 4(2g−2+n) //// SU (2) 3g−3+n , where (0, 2) chiral multiplets play the role of complex coordinates, and a holomorphic vector bundle 5 of (0,2) Fermi multiplets E → X. The index then has the meaning of the following equivariant characteristic class [24]: where F E and F T are the curvatures on E and T X respectively. In the next section we consider example with g = 1 and n = 1 in detail. Let us note that the relation c R = 6 · (2k SU (2) + R ) ≡ 6(n h − n v ) between the right-moving central charge and the anomaly of SU (2) + R UV R-symmetry does not work when g > 0 for the following reason. In the sigma-model description SU (2) + R now acts not only on the rightmoving fermions living in the tangent bundle of the Kibble branch, but also on the left-moving fermions in the complex rank 2g vector bundle E. Therefore, similarly to what happens on the Coulomb branch of (4, 4) theories [19], we expect that in IR SCFT SU (2) + R splits into two symmetries, one is left-moving global symmetry SU (2) affine symmetry with level g, and the other is right-moving SU (2) affine R-symmetry with level (n h − n v + g), which is in agreement with the value of c R . In the UV we only see the diagonal of these two symmetries, SU (2) + R , with anomaly coefficient being half the difference of affine algebras levels, (n h − n v )/2. Figure 6: The quiver of (0, 4) theory with SU (2) vector multiplet (U, Θ) and a hyper multiplet (Φ,Φ) in adjoint representation.

Duality to a Landau-Ginzburg model
Consider the theory associated to the quiver in Fig. 6. One can show that the index of this theory satisfies the following identity: where we explicitly factored out the contribution from decoupled chiral fields (Tr Φ, TrΦ) spanning C 2 . The second factor in right hand side can be understood as the index of the (0, 2) Landau-Ginzburg model with three (0, 2) chiral multiplets Φ 1,2,3 , one Fermi multiplet Γ and the superpotential The superpotential (2.27) implies the condition which is the equation describing an embedding of C 2 /Z 2 into C 3 . The chiral fields Φ i can be mapped to the following gauge invariant operators in the chiral ring of the original gauge theory: (2.29) Then the condition (2.28) follows from the condition [Φ,Φ] = 0 imposed by the superpotential associated to Θ. The first two factors in the right hand side of (2.26) describe (0, 2) chiral fields spanning the Kibble branch of the theory, X = C 8 //// SU (2) ∼ = C 2 ×C 2 /Z 2 , and in the limit q → 0 they reproduce its Hilbert series [26]. The last factor in (2.26) is the contribution of a complex rank two holomorphic vector bundle E → X of left-moving fermions. It appears in this case because the gauge group is not completely broken (contrary to the case when a quiver does not have any loops, the gauge group is completely broken and E is empty). In terms of the original gauge theory the fibers of the bundle E are generated by massless gauge invariant Fermi multiplets TrΛΦ and TrΛΦ, where is Λ is the (0, 2) field strength Fermi multiplet constructed from the vector multiplet U . From the dimensions of the target space and the bundle E we conclude that Let us note that in this particular case (g = 1, n = 1) if we throw away the decoupled hypermultiplet (Tr Φ, TrΦ), the supersymmetry actually enhances to (4,4) and we expect to have a (4, 4) sigma model withX = C 2 /Z 2 target space. It follows that E is isomorphic to the tangent bundle TX. The resulting (4, 4) SCFT has central chargesc L =c R = 6.

N = (4, 4) theories
Most of the statements about (0, 4) theories made in previous sections also hold for their (4,4) counterparts. The main difference is that now the theory also has a Coulomb branch (and in the case of SU (2) gauge group there is no FI parameter to switch it off) that receives quantum corrections.
In particular, the index of the (4, 4) theory corresponding to the quiver in Fig. 2, (2.33) also satisfies the crossing symmetry property which means that similarly to the (0, 4) case one can use (2.31) and (2.32) to define a 2d TQFT.
We find a crossing-symmetry of the elliptic genus for this case as well. In section 3.2, we argue for the existence of 2d analog of the T N theory.
The theory has the following superpotential necessary to ensure N = (0, 4) supersymmetry.
The gauge anomaly coefficient is given by (see appendix C): which implies that we should take N f = 2N c ≡ 2N . The anomaly coefficients for the flavor SU (N f ) symmetry and U (1) B are Also, the theory has non-vanishing 't Hooft anomalies involving U (1) v : Similarly to the case with SU (2) gauge group considered in the previous section, the theory has only Higgs branch and we expect SU (2) + R to be the R-symmetry of the SCFT at the IR fixed point. By counting its anomaly coefficient in the UV theory we obtain Again, c R /6 agrees with the quaternionic dimension of the Higgs branch as expected. As in section 2.1 we find that the index of the theory has a similar crossing-symmetry property. Consider a trinion U (0,4) N describing a hypermultiplet in the bifundamental representation of SU (N ) × SU (N ) (see Fig. 7). It also has a baryonic symmetry U (1). The index is given by where we dropped v, q dependence in the expression for brevity. The vector multiplet index is given by Here we have used the flavor fugacities with We find that the index is invariant under the exchange of a ↔ b or equivalently x ↔ y: On the level of quiver diagrams this can be understood as a crossing symmetry between s-channel and u-channel (see Fig. 8). This duality or crossing-symmetry implies that the spectrum of the operators in the CFT should obey such property. It is not automatic from the global symmetry of the theory. . The equivalence to the diagram on the right represents crossingsymmetry of the index.
The crossing-symmetry can be understood as a duality. Even though the matter content on both side of the dual theories are the same, the operator contents on one side are mapped to another operators on the other side. For example, we have gauge-invariant operators of the form as in the following table (here we decomposed where Λ k is k−th antisymmetric representation and is completely antisymmetric tensor to contract the gauge indices. The first two lines are baryonic operators where as the latter four are mesonic operators. Under the exchange of U (1) x and U (1) y , the mesonic operators remain unchanged, but the baryonic operators are mapped via Let us now consider the N = (4, 4) version of the theory. The matter contents are essentially the same except that we replaced (0, 4) multiplets to (4, 4) multiplets. We can write it more explicitly in terms of N = (0, 2) superfields as in the following table: R-symmetry which an extra SU (2) I factor compared to the N = (0, 4) case. As discussed in appendix A, this R-symmetry can be understood from the dimensional reduction of 6d N = (1, 0) multiplets. The theory have the following J-type superpotential and E-terms: The N = (4, 4) gauge theory is expected to flow to two distinct CFTs on the Higgs branch and on the Coulomb branch [19,31]. We can also compute the index for this theory. The index for the trinion theory U (4,4) N consists of the free bifundamental (4, 4) hypermultiplets can be written as where u is the fugacity for the U (1) I ⊂ SU (2) I symmetry. The vector multiplet index reads . (3.17) Now we can write the index for the SQCD as where we suppressed the dependence on u, v and q. It also satisfies the crossing symmetry which implies constraints on the operator spectrum and IR duality as in the N = (0, 4) case.

Dualities of quiver theories and T
(0,4) N theory In this section, we discuss quiver gauge theories and dualities.

Quiver gauge theories
Linear quiver Let us consider linear quiver theories composed of connecting m copies of U N blocks. This will yield SU (N ) m−1 gauge theory with bifundamentals in SU (N ) i × SU (N ) i+1 where we identify SU (N ) 0 and SU (N ) m as the global symmetry groups, see Fig. 9.
... (3.20) The (quaternionic) dimension of the Higgs branch is given by c R /6. As we have discussed in section 3.1, the index of the quiver theory also enjoys crossingsymmetry. It can be also applied to the linear quiver theory, which has the global symmetry The crossing-symmetry now extends to the permutation of all the U (1) i symmetries. Therefore we have a duality map analogous to (3.12), by applying the duality repeatedly. The single-trace gauge invariant operators contains the bayonic operators detΦ i and detΦ i with i = 0, · · · , m and mesonic operators Φ 0Φ0 and Φ mΦm . Under the permutation, Circular quiver We can also consider a circular quiver theory by gauging the diagonal subgroup of SU (N ) 0 × SU (N ) m of the linear quiver. As in the case of SU (2) theories, we get a CFT on the Kibble branch with dimension m + 1, see Fig. 9. The central charge of this theory is given by c R = 6(n h − n v + 1) = 6(m + 1) , c L = 4(m + 1) + 2 . ... and SU (2) flavor symmetry acting on two hypermultiplets (see Fig.  11). On the level of indices the duality reads describes a certain Higgs branch CFT its central charges can be easily determined from the relation depicted in Fig. 11: Since at each step the one can calculate contour integrals explicitly by residues, this provides us with explicit (although quite long) expression for the index of T  Let us note that q 0 order coincides with the Hilbert series of the Higgs branch moduli space, conjectured to be the same as the moduli space of one E 6 instanton [25,29,30]. The leading terms also agree with the S 2 × T 2 partition function computed in [16].
The T (0,4) 3 is a 2d version of the celebrated E 6 SCFT of Minahan-Nemeschansky [9]. One important difference here is that our theory does not have any Coulomb branch. We can also come up with a "Lagrangian" for the "non-Lagrangian" E 6 SCFT as done in [16]. The N = (0, 2) field content can be straightforwardly read off the integral representation of the index of T (0,4) 3 . Namely, (3.24) represents combining the theory associated to the quiver in the left part of Fig. 11 together with two chiral multiplets in representations As we have discussed in section 2.2, crossing-symmetry implies the TQFT structure of the elliptic genus. But unlike the case of SU (2) theories, we have two distinct type of punctures: SU (3) (maximal) puncture and U (1) (minimal) puncture. We have already shown in section 3.1 that the index remains unchanged upon exchanging two U (1) punctures or two SU (N ) punctures in the second frame of figure 11. With the expansion 3.25 we can further show that crossing-symmetry exists in the theory with four maximal punctures up to certain order of q and v. Therefore the TQFT structure holds for the SU (3) theories as well.

T
(0,4) N theory and duality So far we have discussed 2d N = (0, 4) gauge theories without referring to its higherdimensional origin. Let us point out that theories we studied so far can be realized from M5-branes on a product Riemann surfaces. Consider 4d N = 2 class S theory of type A N −1 with the UV curve given by C with genus g and n punctures. Now, let us compactify this 4d theory on CP 1 with a partial topological twist. Since we have two independent R-symmetries SU (2) R × U (1) r , we have to choose one. Twisting with respect to SU (2) R and U (1) r gives us N = (2, 2) or N = (0, 4) supersymmetry in 2d respectively. We are interested in the N = (0, 4) twisting. In this case, for each free vector multiplets in 4d, we get one (0, 4) vector, and for each free hypermultiplets in 4d, we get one N = (0, 4) hypermultiplet. See appendix F for the detail.
Upon taking small volume limit of CP 1 , we also take the 4d gauge coupling to be small to get a 2d gauge theory, since 1/g 2 2d = vol(CP 1 )/g 2 4d . There can be also S-dual descriptions for the 4d theory, which we also dimensionally reduce to another 2d gauge theory. Note that for this case, we need to take the dual gauge couplings to zero while shrinking the volume of the sphere. In principle, dimensional reduction of these two different limits do not necessarily give the same CFT in 2d. When taking the 2d limit, we have to decouple 4d building blocks in a different way for each S-dual frames. From there we are turning on gauge couplings to RG flow to 2d CFT, which we call as T su(N ) [CP 1 × C g,n ]. Nevertheless, we find evidences that different 2d 'gauge theories' (which can also involve 'non-Lagrangian' T (0,4) N block) obtained from dual descriptions flow to the same 2d N = (0, 4) SCFT. 6 Note that since the gauge couplings undergo RG flows, the dependence on the complex structure of C g,n disappears in the IR. Crossing-symmetry (or TQFT structure) of elliptic-genus is a check of this conjecture.
As a corollary, the effective number of vector and hypermultiplets remain the same in the 2d N = (0, 4) theory as the 4d N = 2 theory. Given this assumption, we can compute the central charges of the 2d theory T su(N ) [CP 1 × C g,n ]. The number of effective vector and hypermultiplets can be decomposed in terms of a contribution from the background Riemann surface, and local contributions from the punctures [35]. For the SU (N ) theory, we get for a genus g curve, and for the maximal puncture and for the minimal puncture. We define n As we have discussed, for g = 0, we have the Higgs branch, and for g ≥ 1, we have the Kibble branch. We get for g = 0 and  We can also compute the central charges from the dual Lagrangian description. When T N theory is coupled to a quiver tail, of the form SU (N ) c ⊃ SU (N −1)×SU (N −2)×· · ·×SU (2) with bifundamentals and fundamentals attached as in the quiver diagram in the bottom of Fig. 12. This theory is dual to a linear quiver with gauge group SU (N ) N −2 , and fundamental attached to the end as in the top of Fig. 12. The SU (N ) flavor symmetry anomaly coefficient can be computed in the dual frame: a, b, c) . and x is U (1) x fugacity. In the N = (2, 2) case we have an extra left-moving U (1) Rsymmetry fugacity t. Now let us consider N = (0, 2) or N = (2, 2) SU (N ) SQCD with N fundamental and N anti-fundamental flavors, which can be obtained by coupling two copies of U N N to SU (N ) vector multiplet. In the N = (0, 2) case, similarly to the (0, 4) case, gauge anomaly contributions from chiral and vector multiplets cancel each other. The theory has the following index: .

(4.6)
One can show that the index (4.4) is invariant under the exchange of fugacities a ↔ b or, equivalently, x ↔ y. Therefore we would like to conjecture that, as in the (0, 4) and (4,4) cases, the spectrum of the SCFT at the IR fixed point is invariant under the exchange of flavor symmetries U (1) x ↔ U (1) y 4.2 Duality to a N = (0, 2) Landau-Ginzburg theory In the case of N = (0, 2) one can check that the index (4.4) satisfies the following identity: from which the symmetry under the exchange x ↔ y becomes obvious. This result can be reformulated in the following way. Let us define .
is the Calabi-Yau threefold usually referenced to as just "conifold" in the literature. Then the equation (4.7) can be written as Physically (4.12) means that gauging a diagonal subgroup of SU (N )×SU (N ) flavor symmetry from two copies of K (0,2) N is dual to just one copy of K ) j β be chiral fields from two copies of K (0,2) N in the l.h.s. of duality. The conditions detΦ (1,2) = 0 kill baryons of the theory in the chiral ring. This means that we are only left with mesons Φ i j ≡ (Φ (1) ) α j (Φ (2) ) i α which play the roles of chiral fields of the dual Landau-Ginzburg model. The condition detΦ = 0 is obviously satisfied and one can also show there are no additional conditions on Φ. Geometrically the statement can be understood as the following relation: (4.13) Also, this duality is similar to a N = (0, 2) Seiberg-like duality found in [1] in the case when there are no Fermi multiplets in fundamental representation of the gauge group. There is an important difference however, theories considered in the aforementioned paper had U (N ) gauge symmetry, not SU (N ). As we show in appendix D, the identity (4.12) can be used to derive an iversion formula for a certain integral operator with kernel constructed from theta-functions. It is analogous to the inversion formula in [36] for an operator with kernel constructed in a similar way from elliptic Gamma functions and allows us to find an explicit expression for the index of T A Review on N = (0, 2) and N = (0, 4) theory Let us summarize some basic facts about N = (0, 2) and N = (0, 4) gauge theories [37]. See also [38,39]. N = (0, 2) multiplets A general N = (0, 2) gauge theory can have the following supersymmetry multiplets: Here, the subscript ± stands for right/left-moving complex Weyl spinors respectively. An N = (0, 2) theory allows formulation in (x ± , θ + ,θ + ) superspace. A chiral superfield satisfies and has the following expansion: A Fermi superfield satisfiesD where E(Φ i ) is a holomorphic function of the chiral superfields Φ i which transforms in the same way as Ψ. This condition leads to the following expansion: where G is an auxillary superfield. Finally, the vector superfield has the following form: The corresponding field strength forms a Fermi superfield Λ, which is consistent with the fact that (bosonic) vector field in 2d is non-dynamical. There are two different types of 'superpotential' in N = (0, 2) theory. To each Fermi multiplets Ψ a , introduce a holomorphic function J a (Φ i ). Then we write the SUSY action We can write 'superpotential' as W = Ψ a J a (Φ), and integrate over the half-superspace. There is also E-type superpotential, which appears in the right-hand side of the (A.4). There is one condition we need to impose to ensure supersymmetry: Here a, b = 1, 2. We remark that N = (0, 4) supersymmetry in principle does not require N = (0, 4) Fermi multiplets to have two copies of N = (0, 2) Fermi multiplets (see e.g. [40]). In our case, as in [39], we define a (0,4) Fermi multiplet as a pair of Fermi multiplets in the conjugate representations. When a hypermultiplet couples to a vector multiplet, we have a superpotential coupling between the hypermultiplet and Fermi multiplet Θ in the vector given as This is analogous to the superpotential coupling in 4d N = 2 theory between chiral adjoint in a vector multiplet and a hypermultiplet. For a twisted hypermultiplet, the coupling is done through the E-term, instead of the superpotential (or J-term). It is given by where the right-hand side of the equation transform as the adjoint of the gauge group. For the case of Fermi multiplet, there is no coupling between Θ and Γ,Γ. But, it is possible to include a quadratic E or J term while preserving the SO(4) R symmetry. There is also a coupling between N = (0, 4) Fermi, hyper and a twisted hypermultiplet. It involves J-term given as and also the E-term These terms satisfy the constraint E · J = 0.
Note that the scalar in the hypermultiplet is uncharged under SU (2) l × SU (2) r but charged under SU (2) R , whereas the scalar in the vector multiplet is charged under the SU (2) R but uncharged under SU (2) l × SU (2) r . It has been argued that N = (4, 4) gauge theory flows to two decoupled SCFTs on the Higgs branch and the Coulomb branch [19,31]. For a large value of these scalar fields, we can trust the semi-classical description, which is given by the Higgs/Coulomb branch. For the Higgs branch theories, the R-symmetry should be given by SU (2) l × SU (2) r since the scalars are charged under SU (2) R . It is the other way around for the Coulomb branch theories. (Here the extra SU (2) R-symmetry is not visible in the UV.) Since R-symmetries on the Coulomb branch and Higgs branch are distinct, they cannot be the same SCFT.

B Review on elliptic genus
Elliptic genus for (0, 2) gauge theories The elliptic genus of N = (0, 2) supersymmetric theories was discussed in [20,21,23]. We will summarize the prescription for computing the elliptic genus of N = (0, 2) theories in this section.
Consider a two-dimensional theory with N = (0, 2) supersymmetry and a flavor symmetry group F. The elliptic genus on Ramond (R) sector is defined as while the elliptic genus on Neveu-Schwarz (NS) sector is defined as where Tr R or Tr N S are taken over the Hilbert space of SCFT on a circle, with fermions satisfying periodic or anti-periodic boundary conditions respectively. F is the fermion number, and the parameter q = e 2πiτ (B.3) specifies the complex structure of a torus. H L is the left-moving Hamiltonian, H R and J R are the right-moving Hamiltonian and U (1) R charge operator, f i 's are the Cartan generators of F, and a i are corresponding fugacities. The collection of fugacities a ≡ {a i } can be understood as the element of the maximal torus of F. By the usual argument both elliptic genera are independent ofq. The contribution of a chiral multiplet Φ transforming in a representation R is . (B.4) Where whe product is over the weights of ρ of the representation R, and x ρ ≡ i x f i ,ρ i denotes the standard pairing between an element of the maximal torus and a weight. The contribution of a Fermi multiplet Ψ in a representation R is The theta function is defined as Notice that the NS-NS elliptical genera for chiral and Fermi multiplet depend on the rightmoving J R -charge r of the multiplet. The contribution of a vector multiplet Λ with gauge group G is (−θ(z α ; q)), θ(z α ; q).

(B.8)
Here rk G is the rank of gauge group G and z is the element of the maximal torus of the gauge group G. The elliptic genus does not depend on the coupling of the theory, therefore it is always possible to compute it in the free theory limit. For a (0, 2) gauge theory with gauge group G, chiral multiplets {Φ} and Fermi multiplets {Ψ}, the elliptic genus of the theory is [20][21][22][23]: where W (G) is the order of Weyl group of G. The integral is performed over a certain contour "JK" in the moduli space of flat connections on the two-torus M flat (T 2 τ , G) which corresponds to taking a sum of Jeffrey-Kirwan residues. The absence of gauge anomaly is equivalent to the condition that the integrand is elliptic in z.
where the fugacity v labels the anti-diagonal Cartan F of SU (2) − R ×SU (2) + R mentioned above. And the vector multiplet, , .

C 't Hooft anomalies
In theories with chiral supersymmetry left-and right-moving fermions are not necessarily paired together, which in general results in non-trivial 't Hooft anomalies. Suppose the theory under consideration has a global symmetry with corresponding simple Lie group F . Then its anomaly coefficient k F is given by the following formula: where F a are the generators of F , γ 3 is the gamma matrix measuring chirality and the trace is performed over the space of Weyl Fermi fields of the theory. It follows that the anomaly coefficient k F can be calculated as the following difference between sums over the sets of (0,2) chiral and Fermi multiplets of the theory: where T (R · F ) denotes the index of representation R · F of F . For example, T [ SU (N ) ] = 1/2 and T [adj SU (N ) ] = N . In the case when the theory has two U (1) symmetries U (1) F 1,2 with corresponding charges F 1,2 , there can be a mixed 't Hooft anomaly: However, unlike in 4d there cannot be a mixed anomaly between SU (N ) and other global symmetry.
In the IR one usually expects the current corresponding to the global symmetry to become holomorphic or anti-holomorphic (i.e. left-or right-moving). In this case F enhances to the corresponding affine algebra F |2k F | acting in the holomorphic or anti-holomorphic sector of the CFT depending on the sign of k F . However, holomorphicity of the current in the IR may fail if the flavor symmetry rotates non-compact directions of the moduli space, the simplest example being U (1) symmetry acting on a free chiral multiplet.
The anomaly coefficient determines transformation properties of the index w.r.t. to corresponding fugacities. The index can be considered as a meromorphic section of L −2k F where L is a prequantum line bundle over M flat (T 2 τ , F ), the moduli space of flat connections of F -bundle over the two-torus with complex structure τ . Consider for example the case F = SU (n). Let us denote the corresponding fugacities by a = {a i } N i=1 , i a i = 1. Then the index has the following properties: Since N = 2 or small N = 4 SCA algebra of the IR SCFT has only one central element, the anomaly of the R-symmetry can be related to the the right-moving central charge. Namely, in the case of N = 2 SCA: where R is the generator of U (1) R-symmetry and k is the level of affine U (1) R-symmetry.
In the case of small N = 4 SCA: where k is the level of affine SU (2) R-symmetry and k R is the corresponding anomaly coefficient which usually can be easily computed in the UV. Once c R is known the left-moving central charge can be easily determined from the gravitational anomaly: D Proof of the elliptic inversion formula SU (2) , m > 0 has no poles, it is zero.
It follows that in order to prove the equality of two functions with positive anomaly coefficients and simple poles it is sufficient to check that they have the same poles and residues. In particular, it is easy to show that Proof. By definition the integral on left hand side is given by a residues at ξ = xa ±1 and ξ = yb ±1 : 2θ (a 2 ) θ ay bx θ aby x θ xy ab θ bxy a + θ y 2 θ 1 . (D.4) 7 We make this assumption for technical simplicity. The case with higher order poles can always be considered as a limit when simple poles collide. 8 cf appendix C 9 In other words, f is a section of a line bundle over M flat (T 2 τ , SU (2)) with divisor −m · pt and therefore it must have at least m poles It is easy to show that, as a function of a which belongs to H (2) SU (2) , it has the same poles and residues as the right hand side of (D.3). By Prop. 1 the difference between (D.4) and the right hand side of (D.3) is zero.
The formula (D.3) is a particular case of (4.12) for N = 2. Now it is easy to prove the following statement: Proof. Let us pick some a ∈ C * and consider Then from Prop. 2 it follows that we can always represent f in the following way 10 : Plugging it in the left hand side of (D.5) and applying (D.3) twice for each term in the sum we get the desired result.
Let us note that one can easily generalize the above statements for SU (N ) case, considering the following space: Then define the following basic building blocks of 1d TQFT: 10 Let us note that the Jeffrey-Kirwan contour integral prescription in (D.5) requires the choice of SU (2) charges at poles. This choice is made in the formula below by picking particular (Ai, ti) in Z2 orbit when using representation (D.2). However, the final result obviously does not depend on it. Again, the last condition in (E.1) is needed for the integrand above to be elliptic. Then (4.12) can be formulated as the following property: which is equivalent to idempotency of the operator π ≡ (id ⊗ η) • (K ⊗ id) : H It follows that π is a projector and acts as the identity map when restricted onH F Partial topological twisting of N = 2 d = 4 theory Let us compactify 4d N = 2 theory on a Riemann surface C g of genus g without punctures and take the zero-volume limit to get a 2d theory. In order to preserve supersymmetry, we perform topological twisting along C g [41]. The symmetry group of the 4d N = 2 superconformal theory includes SU (2) L × SU (2) R × SU (2) I × U (1) r , where SU (2) L × SU (2) R = SO(4) is the Lorentz group and SU (2) I × U (1) r is the R-symmetry group. Upon dimensional reduction, the symmetry group becomes SO(2) E × SO(2) C × SU (2) I × U (1) r , where SO(2) E and SO(2) C are the Lorentz group along the R 2 and C g respectively. Now, we perform topological twist along the C g direction. This type of twisting is studied in [42]. There are two independent choices of twisting. We can twist with either U (1) r or SU (2) I . If we twist by U (1) r , we get N = (0, 4) SUSY in two-dimension since Q 1 − , Q 2 − ,Q 1 − ,Q 2 − are preserved in 2d. Note that they all have charge − 1 2 under SO(2) E . If we twist with SU (2) I , Q SU (2) L SU (2) R SU (2) I U (1) r SO(2) E SO(2) C SO(2) C SO(2) C Q 1 U (1) r twisting By looking at the table 3, we see that for the U (1) r twisting, 4 components ψ + ,ψ + , q,q (and its complex conjugate) form a (0, 4) hypermultiplet in 2d spacetime, and also become scalar on C. The other two components ψ − ,ψ − (along with their complex conjugates) form a (0, 4) Fermi multiplet in 2d spacetime since they all become left-handed spinors. They become one-forms on C. Since dimH 1 (C g ) = 2g, we get g (complex) Fermi multiplets in 2d. The vector multiplets, twisting with U (1) r , give us 1 (0, 4) vector multiplet from A ++ , λ − ,λ − and g (0, 4) twisted hypermultiplets from A +− , λ + ,λ + , φ (and its complex conjugates).
SU (2) I twisting Let us consider the case of SU (2) I twisting. For this case, we get N = (2, 2) supersymmetry in 2d. Now all the components of the hypermultiplets become spinors on C. We get a pair of chiral multiplets Q = (q, ψ + , ψ † − ),Q = (q,ψ + ,ψ †− ) in 2d, that transform as spinors on C.
(q,ψ + ,ψ †− ) Table 6: The matter content of the SU (2) I twisted free vector/hypermultiplets in terms of N = (2, 2) superfields. Here R-charges of the superfield and components are written simultaneously.
The number of chiral multiplets of the N = (2, 2) twist (or SU (2) I twist) is given by the number of harmonic spinors on the curve C g or h 0 (C g , K 1 2 ). This number depends on the choice of spin structure on C g [43].