The Chiral Algebra of Genus Two Class $\mathcal{S}$ Theory

We construct the chiral algebra associated with the $A_{1}$-type class $\mathcal{S}$ theory for genus two Riemann surface without punctures. By solving the BRST cohomology problem corresponding to a marginal gauging in four dimensions, we find a set of chiral algebra generators that form closed OPEs. Given the fact that they reproduce the spectrum of chiral algebra operators up to large dimensions, we conjecture that they are the complete set of generators. Remarkably, their OPEs are invariant under an action of $SU(2)$ which is not associated with any conserved one-form current in four dimensions. We find that this novel $SU(2)$ strongly constrains the OPEs of non-scalar Schur operators. For completeness, we also check the equivalence of Schur indices computed in two S-dual descriptions with a non-vanishing flavor fugacity turned on.


Introduction
One of the most interesting recent developments in the study of four-dimensional N = 2 superconformal field theories (SCFTs) is the new duality proposed by [1], which associates a 2d chiral algebra to every N = 2 SCFT in four dimensions. 1 This map can be applied to arbitrary four-dimensional N = 2 SCFT, regardless of the existence of a weakly-coupled Lagrangian description. The associated chiral algebra constructed in this way captures the OPEs of protected BPS operators called "Schur operators", which are operators contributing to the Schur limit of superconformal index [2,3]. While the scalar Schur operators are known as Higgs branch operators and well-studied, the nature of their non-scalar cousins is not generally understood.
Associated chiral algebras have revealed various general properties of 4d N = 2 SCFTs.
It was also shown in [8] that the Higgs branch of the 4d theory is reconstructed as the associated variety [9] of the corresponding chiral algebra, which was further studied in [10][11][12][13][14]. Since the map (1.1) preserves the non-perturbative structure of OPEs among the Schur operators, the associated chiral algebra is also a powerful tool to analyze strongly coupled SCFTs, such as Argyres-Douglas type theories [15][16][17][18] and T N theories [19]. See  for recent works in this and other directions.
After these works on the associated chiral algebras, one important thing that is still to be understood is the chiral algebra nature of non-scalar Schur operators. In particular, chiral algebra generators corresponding to non-scalar Schur operators in the semi-short multiplet, C R(j,) , are not well-understood, except for the stress tensor multiplet C 0(0,0) . 2 Most of the earlier works focused on the case in which the chiral algebra is generated by the stress tensor and those corresponding to Higgs branch operators (i.e., scalar Schur operators), possibly with their partners created by extra supersymmetries. 3 In such cases, the OPEs of associated chiral algebras are often fixed by the 4d Higgs branch chiral ring and 2d Jacobi identities.
However, there are SCFTs whose chiral algebra contains a generator corresponding to generic C R(j,) . 4 In this case, 2d OPEs are not simply reconstructed from the 4d Higgs branch.
One simple such SCFT is the A 1 -type class S theory for a genus two Riemann surface without punctures. Here, A 1 -type class S theories are defined as the IR limit of the six- 1 In mathematical literature, chiral algebras are also called vertex operator algebras (VOAs). 2 Here, we use the convention of [59]. 3 In some cases, the 2d stress tensor is a composite operator and therefore not an independent generator. 4 See [21,29] for examples of such chiral algebras.
dimensional N = (2, 0) A 1 SCFT compactified on a punctured Riemann surface, C g,s , where g is genus and s is the number of punctures. 5 We denote this theory by T Cg,s , especially focusing on T C 2,0 . Unlike the other class S theories, the flavor U (1) f symmetry of T C 2,0 is emergent and not associated with any puncture on the Riemann surface. While T C 2,0 has a Lagrangian description, its associated chiral algebra has not been identified. 6 Moreover, it was proven in [1] that its chiral algebra must contain generators corresponding to non-scalar Schur operators in C 1(0,0) . These operators are neither related to the stress tensor multiplet nor to Higgs branch operators, and still to be understood. For example, it is not known whether these generators are Virasoro primary operators.
In this paper, we identify the chiral algebra χ[T C 2,0 ] associated with T C 2,0 . Our strategy is to start with a weakly-coupled description of the theory and apply to it the 2d interpretation of marginal gaugings proposed in [1]. This 2d interpretation involves a BRST reduction associated with a 4d marginal gauging. We evaluate this BRST reduction and find a set of chiral algebra generators equipped with closed OPEs. These generators are of dimension less than or equal to three, and reproduce the correct operator spectrum by normal-ordered products at least up to dimension six. Given this fact and their closed OPEs, we conjecture that they are the complete set of generators. As a further consistency check, we compute the character of χ[T C 2,0 ] based on our conjecture, and find the result in perfect agreement with the Schur index of T C 2,0 up to O(q 9 ). The 4d superconformal multiplets corresponding to these generators are also identified. We particularly show that the generators corresponding to C 1(0,0) are all Virasoro primaries.
One remarkable consequence of our OPEs is that there exists an unexpected SU (2) acting on χ[T C 2,0 ] as an automorphism. Since χ[T C 2,0 ] has no su(2) current, this SU (2) symmetry is not associated with any 2d conserved current. Moreover, one can show that there is no 4d conserved one-form current corresponding to this SU (2). 7 Therefore, this either corresponds to a 4d symmetry without conserved current, or is an accidental symmetry in two dimensions.
We find that the action of this SU (2) is trivial on the space of 2d operators corresponding to scalar Schur operators. Therefore, it is a symmetry characterizing the OPEs of non-scalar 5 When the six-dimensional theory is N = (2, 0) g SCFT for general g ∈ {An, Dn, En}, the class S theory is also characterized by embeddings su(2) ֒→ g at each puncture on Cg,s. In the case of g = A1, there is only one such embedding in addition to the trivial one. 6 Note that, while chiral algebras of class S at genus zero are well-studied in [13] even for a general Lie algebra of the 6d (2,0) theory, its generalization to higher genus cases is still to be understood. See [60] for the derived extension of [13], which will be helpful for understanding the chiral algebra interpretation of gluings that increase the genus of the Riemann surface. 7 By "conserved one-form current", we mean j = jµdx µ such that ∂µj µ = 0, as in [61]. The corresponding Noether charge is given by * j.

3
Schur operators. Indeed, the invariance under this SU (2) forbids various otherwise-possible terms in the OPEs of these operators. 8 It would be an interesting open problem to see if this new symmetry corresponds to a 4d global symmetry without conserved current.
The organization of this paper is as follows. In section 2, we will briefly review the 4d/2d duality and the BRST interpretation of 4d marginal gaugings, following [1]. In section 3, we will collect aspects of the genus two theory T C 2,0 , and describe our strategy of identifying the corresponding chiral algebra χ[T C 2,0 ]. Sections 4 and 5 include our main results. In section 4, we identify the chiral algebra generators and their OPEs, and in section 5 we discuss automorphisms of χ[T C 2,0 ] including a new SU (2) symmetry that is not associated with any 4d conserved one-form current. The final section 6 is devoted to the conclusion and discussions about future works. While the genus two theory has two independent weak coupling descriptions, we focus on one of them. The S-dual equivalence between them is checked via the superconformal index with its flavor fugacity turned off [2]. In appendix A, we provide its extension to the case of non-vanishing flavor fugacity. In appendix B, we list the null operators in χ[T C 2,0 ] up to dimension six.
2 Brief review of the chiral algebra conjecture In this section, we will review the duality between four-dimensional N = 2 SCFTs and 2d chiral algebras, following [1] (see also [62] for a lecture note). Readers familiar with this 2d/4d correspondence can skip this section. We follow the convention of [1] unless otherwise stated.

General properties of associated chiral algebras
The map (1.1) from 4d N = 2 SCFTs to chiral algebras was defined in [1] by considering cohomology classes of local operators with respect to a particular linear combination of Poincaré and conformal supercharges. At the origin, each cohomology classes is represented by a local operator annihilated by Q 1 − ,Q 2− , S − 1 andS 2− , which is called a Schur operator. The 4d unitarity implies that the quantum numbers of Schur operators satisfy where E is the conformal dimension, (j,) is the Lorentz spin, R is the SU (2) R charge, and r is the U (1) r charge.
Schur operators are classified by superconformal multiplets containing them. We call an N = 2 superconformal multiplet containing a Schur operator, a "Schur multiplet". In the 8 Such a symmetry is also present in the chiral algebra studied in [29]. The associated chiral algebra χ[T ] contains special sub-algebras uniquely fixed by the 4d symmetry of T . In particular, the Virasoro sub-algebra of χ[T ] arises from the self-OPEs of the SU (2) R current in the stress tensor multiplet,Ĉ 0(0,0) . It is for this reason that the 2d Virasoro central charge, c 2d , is fixed by a 4d Weyl anomaly coefficient, c 4d , as 9 When T has a flavor symmetry G, the flavor current multipletB 1 gives rise to the following affine Kac-Moody G sub-algebra in χ[T ]: where f AB C is the structure constant of G, and A, B, C = 1, · · · , dim G. The level, k 2d , of the affine Kac-Moody algebra is fixed by the 4d flavor central charge, k 4d , as 10 Here, we normalized c 4d such that a free hypermultiplet has c 4d = 1 12 . 10 We use the convention that the k 4d = 2 for a hypermultiplet in the fundamental representation of G.

5
This implies that every 4d global symmetry that commutes with the N = 2 superconformal symmetry gives rise to a corresponding affine current in χ[T ] unless the 4d symmetry has no conserved current.
The OPEs of the other 2d operators are more non-trivial, reflecting the 4d OPEs of Schur operators that are not fixed by the global symmetry of T . Among these operators, those corresponding toB R are relatively well-understood, since the corresponding Schur operators are Higgs branch operators whose vacuum expectation values parameterize the Higgs branch of T . Indeed, the OPEs of 2d operators corresponding toB R are often reconstructed from the Higgs branch chiral ring of T . The Hall-Littlewood (HL) chiral ring is an extension of the Higgs branch chiral ring by Schur operators inD R(j,0) . 11 Similarly, HL anti-chiral ring is defined for D R(0,) andB R . A common feature of these multiplets,D R(j,0) , D R(0,) andB R , is that the corresponding 2d operators are guaranteed to be Virasoro primary operators.
When the 4d theory T has a Lagrangian description, Schur operators in the D R(0,) and D R(j,0) multiplets are gauge invariant composite operators involving at least one gaugino in a vector multiplet. Therefore, for Lagrangian theories, 2d operators corresponding toD R(j,0) or D R(0,) are always nilpotent with respect to the normal ordered product. The simplest example of D R(0,) ,D R(j,0) type multiplets is an N = 2 free vector multiplet, D 0(0,0) ⊕D 0(0,0) , whose associated chiral algebra is the small (b, c)-ghost system. 12 Its OPE is expressed in terms of λ := b,λ := ∂c as which we will to review the 2d interpretation of a 4d marginal gauging below. It is also known that, in A 1 type class S theories,D R(j,0) andD R(0,) exist only if the genus of the Riemann surface is non-zero.
The last type of Schur multiplet,Ĉ R(j,) , is not well-understood, except for the stress tensor multipletĈ 0(0,0) . As mentioned above, the Schur operator inĈ 0(0,0) is the highest weight component of the SU (2) R conserved current, and corresponds to the 2d stress tensor in the associated chiral algebra. The chiral algebra nature of the otherĈ-type multiplets are, however, not very clear. Indeed, unlike the other Schur multiplets, a chiral algebra generator corresponding toĈ R(j,) is not guaranteed to be a Virasoro primary operator.

BRST reduction corresponding to gauging
Let T be a 4d N = 2 SCFT with a flavor symmetry G whose flavor central charge is where h ∨ is the dual Coxeter number of G. In this case, we can construct a new N = 2 SCFT, T G , by marginally gauging the flavor G symmetry of T . Indeed, k 4d = 4h ∨ precisely coincides with the condition for a vanishing β-function.
On the associated chiral algebra side, this procedure corresponds to considering a BRST cohomology of the tensor product of χ[T ] and the small (b, c)-ghost system in the adjoint representation of G. To describe this, let us consider the following BRST charge where J A for A = 1, · · · , dim G is the affine G currents in χ[T ], and is the affine G currents in the (b, c)-ghost sector. Note that k 4d = 4h ∨ implies the affine currents J A have level k 2d = −2h ∨ , which is precisely the condition that Q 2 BRST = 0. When the gauge coupling is turned off, the chiral algebra of the T G is given by where J A tot,0 is the zero-mode of J A tot := J A + J A gh , and (b A , ∂c A ) stands for the small (b, c)ghost system associated with G. The constraint J A tot,0 |ψ = 0 corresponds to the Gauss law constraint that is present even for the zero gauge coupling. When the gauge coupling is turned on, some Schur operators are lifted to non-Schur operators, giving rise to a reduced chiral algebra. This reduced algebra was conjectured in [1] to be given by the Q BRST -cohomology As shown in [1], the associated chiral algebra is independent of marginal couplings of the fourdimensional theory. Therefore (2.9) is identified as the chiral algebra of T G at any generic value of the gauge coupling.
In Sec. 4, we will use the above prescription to identify the chiral algebra χ[T C 2,0 ] associated with genus two class S theory.

Genus two theory
The main purpose of this paper is to identify the chiral algebra of the genus two class S theory, T C 2,0 , obtained by compactifying the six-dimensional N = (2, 0) , A 1 theory on a genus two Riemann surface C 2,0 . Here we collect known facts about this theory, mainly following Sec. 5.4 of [1], so that we can use them in identifying its chiral algebra in the next section. Each circle node stands for an SU (2) gauge group, while each triangle node stands for the T 2 theory, i.e., the theory of four free hypermultiplets. Each corner of the triangles stands for an SU (2) flavor subgroup of T 2 . When a corner of a triangle is connected to a circle, the corresponding SU (2) flavor subgroup of T 2 is coupled to the gauge group associated with the circle.

Weak coupling descriptions
The T C 2,0 theory has two different weak coupling descriptions, as shown in quiver gauge theories in Fig. 1, corresponding to two different pants decompositions of C 2,0 . These two descriptions are expected to be S-dual to each other. In Fig. 1, each circle node stands for an SU (2) gauge group, while each triangle node stands for the T 2 theory, i.e., the theory of four free hypermultiplets. The T 2 theory has Sp(4) flavor symmetry, whose SU (2) 3 subgroup is manifest in Fig. 1; each corner of the triangles stands for one SU (2) flavor subgroup. When a corner of a triangle attaches to a circle, the corresponding SU (2) flavor subgroup is gauged by the gauge group associated with the circle.
The above SU (2) 3 flavor subgroup of the T 2 theory can be explicitly seen as follows. In the N = 1 language, the T 2 theory is composed of eight chiral multiplets, which we denote by Q ab and Q ab for a, b = 1, 2. There is an obvious N = 2 flavor SU (2) 2 symmetry under which Q ab andQ ab transform respectively as 2 ⊗ 2 and 2 ⊗ 2. 13 In addition, there exists an extra flavor SU (2) symmetry under which (Q 1bc , Q 2bc ) defined by transforms as a fundamental representation, where ǫ ab is the anti-symmetric tensor such that ǫ 12 = −ǫ 12 = 1. Therefore, Q abc can be regarded as a "half-hypermultiplet" in the trifundamental representation, 2 ⊗ 2 ⊗ 2, of SU (2) 3 [19].
As mentioned in section 1, the genus two theory T C 2,0 has an accidental flavor U (1) f symmetry that is not visible in its class S construction (see [63] for similar accidental en-hancements of flavor symmetries of A 1 -type class S theories). As a result, the most general expression for its superconformal index contains a fugacity for this flavor U (1) f symmetry.
The explicit action of this U (1) f will be shown in sub-section 3.2.
The S-duality equivalence between two quiver descriptions in Fig. 1 has been checked in terms of the Schur limit of their superconformal indices with the flavor fugacity turned off [2].
The extension of their proof to the case with a non-vanishing flavor fugacity is discussed in Appendix A of this paper. Given these results, we assume the S-duality equivalence between the two quiver descriptions. In the rest of this paper, we mainly focus on the left description in Fig. 1, which we call the "dumbbell quiver".

Higgs branch and flavor symmetry
In this sub-section, we describe the U (1) f flavor symmetry and a chiral ring relation of the genus two theory T C 2,0 . We focus on the dumbbell quiver description of the theory (i.e., the left quiver in Fig. 1), and denote by Q abc and S def two trifundamental half-hypermultiplets of SU (2) 3 . Let SU (2) 1 , SU (2) 2 and SU (2) 3 be the gauge groups corresponding to the left, middle and right circles in the quiver, respectively. Without loss of generality, we say SU (2) 1 is diagonally gauging the SU (2) × SU (2) symmetry corresponding to b and c of Q abc , and SU (2) 3 is diagonally gauging the SU (2) × SU (2) symmetry corresponding to b and c of S abc .
The remaining gauge group SU (2) 2 is diagonally gauging SU (2) corresponding to a of Q abc and S abc . Then we see that, under SU (2) 1 , the left half-hypermultiplet Q abc is decomposed Similarly, S abc also decomposes into 1 ⊕ 3 under SU (2) 3 as Note that these are all in the fundamental representation of SU (2) 2 . It is also useful for us below to define The superpotential of the dumbbell quiver is then written as F ǫ de 14 The σ matrices and antisymmetric tensor, ǫ, are in the same convention with [64] and (σ A ) ab := A is the chiral superfield arising from the vector multiplet associated with SU (2) i . From the above superpotential, we see that there is a flavor U (1) f symmetry under which φ a andφ a have charge +1 and −1 while all the other fields are neutral. The generators of the Higgs branch chiral ring can also be read off from the above superpotential. Indeed, it is generated by the moment map of U (1) f and the following Schur operators in the B 2 multiplet: whereφ a := ǫ abφ b and Q a A := ǫ ab Q Ab . These Higgs branch operators satisfy the chiral ring relation as seen from the superpotential [65].
Note that, when taking the decoupling limit of the middle gauge group SU (2) 2 , the second line of (3.5) vanishes. Then the theory splits into the following three sectors: C ).
This means that the genus two theory T C 2,0 can be regarded as a theory obtained by marginally gauging a diagonal SU (2) 2 global symmetry of the above three sectors. 16 We will use this construction of T C 2,0 in identifying its associated chiral algebra in the next section.

Chiral algebra of genus two theory
In this section, we construct the chiral algebra χ[T C 2,0 ] associated with the genus two theory T C 2,0 . We regard T C 2,0 as a theory obtained by marginally gauging the three sectors discussed at the end of the previous section. Then χ[T C 2,0 ] is constructed via the BRST reduction corresponding to this marginal gauging. 15 The flavor moment map is a Schur operator in the flavor current multiplet B1. 16 Note that N = 4 SYM theory has an N = 2 flavor SU (2) symmetry.

Chiral algebras of the three sectors
The chiral algebras of the three sectors discussed at the end of sub-section 3.2 are already identified.
The chiral algebra of a fundamental hypermultiplet of SU (2) is the symplectic boson algebra generated by φ ai for a = 1, 2 and i = ± such that Here, the 2d operators φ a+ and φ a− correspond respectively to the 4d Schur operators φ a andφ a described in (3.4). 17 Recall that the 4d U (1) r symmetry is generally preserved by 2d OPEs. Since the 4d Schur operators φ a andφ a are neutral under U (1) r , so are φ a± . This chiral algebra has sp(4) currents, which contains as sub-algebras the su (2) currents and the u(1) current In the BRST reduction discussed below, J A matter will be involved in the BRST current and therefore disappears from the spectrum. On the other hand, J will give rise to a non-trivial BRST cohomology class corresponding to the flavor U (1) f symmetry of T C 2,0 .
Let us now turn to the other sectors, i.e., N = 4 SU (2) SYM theories. The chiral algebra associated with the N = 4 SU (2) SYM was conjectured in [1] to be the small N = 4 super Virasoro algebra at the Virasoro central charge c = −9. This algebra is generated by su (2) currents J A at level −3/2, and N = 4 supercurrents G a ,Ḡ a . Their non-trivial OPEs are given by 17 The 2d operators q a,1 := 1 √ 2 (φ a+ + φ a− ) and q a,2 := 1 √ 2i (φ a+ − φ a− ) correspond to the 4d Schur operators Q a and S a , respectively. Their 2d OPEs are given by q a,i (z)q b,j (0) ∼ ǫ ab δ ij /z.
where f ABC = ǫ ABC is the structure constant of SU (2), and T sug := 2J A J A is the Sugawara stress tensor. The U (1) r charges of J A , G a andḠ a are respectively 0, 1 2 and − 1 2 . 18 Note that the above OPEs preserve this U (1) r symmetry as expected.
The small N = 4 Virasoro algebra at c = −9 has various null operators, which we need to remove from the spectrum. Fortunately, there is a special free field realization of this algebra which makes all these null operators automatically vanishing [66]. As shown in [12], this free field realization is concisely expressed in terms of a βγbc ghost system. To describe it, let us change variables as Then the free field realization is given by Here the OPEs of the βγbc system are given by In the next sub-section, we use this free field realization for each of the two N = 4 SYM sectors. Note that b and c that appear in the above free field realization have nothing to do with the bc ghost system arising in the BRST reduction.

BRST reduction
Let us now consider marginally gauging the three sectors to obtain the genus two theory T C 2,0 . As reviewed in sub-section 2.2, on the chiral algebra side, this corresponds to a BRST reduction of the tensor product of chiral algebras associated with these three sectors.
Recall that two of the three sectors are described by be the generators of the small N = 4 super Virasoro algebra (at c = −9) associated with the k-th N = 4 SYM sector. The remaining sector is the theory of a fundamental hypermultiplet of SU (2), whose chiral algebra is generated by the symplectic bosons φ a± such that (4.1).
In the tensor product of these three chiral algebras, The Schur operators corresponding to G a andḠ a are in D 1 2 (0,0) andD 1 2 (0,0) , respectively. Their U (1)r charges can be seen from Table 1. is the su(2) current corresponding to the 4d SU (2) symmetry that we are gauging, where J A matter is given by (4.2). With this J A , the relevant BRST current is given by where J A gh is defined in (2.7). We see that Q BRST = dz 2πi J BRST is nilpotent, which reflects the fact that the corresponding SU (2) gauging is exactly marginal in four dimensions.
According to the gauging prescription reviewed in sub-section 2.2, the chiral algebra χ[T C 2,0 ] is identified as the following BRST cohomology: where (x, y, · · · ) stands for the chiral algebra generated by x, y, · · · , and J A tot := J A + J A gh . We evaluate this cohomology explicitly, using the Mathematica package OPEdefs developed by [67,68]. While this computation is rather involved, one can simplify it by using the free field realization of (J A (k) , G a (k) ,Ḡ a (k) ) reviewed in the previous sub-section. 19 Evaluating the cohomology classes in (4.11) up to a high order of holomorphic dimensions of operators, we find that there exists a set, S, of operators in χ[T C 2,0 ] with the following properties.
• The operators in S are of dimension less than or equal to three.
• Every Q BRST -cohomology class of dimension less than or equal to six is either an operator in S, the derivative of an operator in S, or a normal ordered product of these operators.
• The operators in S form closed OPEs.
The list of operators in S is shown in Table 2 with their quantum numbers and corresponding 4d superconformal multiplets.
Given these facts, we conjecture that the above S is the complete set of generators of the chiral algebra χ[T C 2,0 ]. Similar conjectures have been made for various theories [1,28,40], leading to consistent results. We will perform a further consistency check of our conjecture in sub-section 4.5. In the next sub-section, we describe which Q BRST -cohomology classes correspond to the operators in S.
2d generator h f r 4d multiplet current J, and r is the U (1) r charge. We conjecture that this is the complete set of generators.
Note that the indices I =↑, ↓ and A = 1, 2, 3 have nothing to do with the global U (1) symmetry.
They play an important role in the study of automorphisms of χ[T C 2,0 ] in section 5.2.
Note that, as shown in [1], the Macdonald index of T C 2,0 implies that there are at least those generators listed in Table 2. Although the index computation does not imply the absence of extra generators, we have shown here that there is no extra generators up to dimension six.
Combined with the fact that they form closed OPEs, this is a strong evidence for the above conjecture.

List of generators of χ[T C 2,0 ]
We here describe the Q BRST -cohomology classes corresponding to the generators listed in Table 2, in terms of their representatives. Note that these cohomology classes are labeled by three quantum numbers (h, r, f ), where h is the holomorphic dimension, r is U (1) r charge, and f is flavor U (1) f charge.
First, at dimension one, the only generator is the u(1) current Since the genus two theory T C 2,0 has c 4d = 13/6, the Virasoro central charge is c 2d = −26. The other two bosonic operators are written as At dimension two, there are also the following fermionic generators: , , . (4.16) The U (1) f charge of D ±I and D ± I are both ±1. Here, one could instead take ǫ ab φ a± G b (i) and ǫ ab φ a±Ḡb (i) as independent operators, but we will see below that the above linear combinations make automorphisms of χ[T C 2,0 ] more transparent. From their U (1) r charges, we see that the corresponding Schur multiplets of D ±I andD ± I are as in Table 2. At dimension three, there are five generators, which are all neutral under the global U (1) f symmetry. We split them into two parts; and (1) ) .

OPEs of generators
Here, we describe the OPEs of generators listed in Table 2. Note that these are uniquely fixed by the BRST cohomology computation. First, the OPEs of T and J are given by 20 Note that T cannot be the Sugawara-type stress tensor associated with J since c 2d = −26.
Recall here that the Virasoro central charge c 2d = −26 follows from the 4d central charge Next, we find that all the other generators in Table 2 are primary operators in the sense of the Virasoro sub-algebra generated by T and the u(1) sub-algebra generated by J. Note that, while operators corresponding to B, D andD-type multiplets are guaranteed to be Virasoro primaries [1], those corresponding toĈ-type multiplets are not. Therefore, this result is already non-trivial. The fact that they are primary operators implies that their non-vanishing OPEs with J and T are the following: (4.25) The OPEs of B ± , D ±I ,D ± I , X,X and C A are more non-trivial. We evaluate them to find that the only non-vanishing OPEs of these operators are the following. First, the OPEs of dimension-two operators are where ǫ IJ and ǫ IJ are anti-symmetric tensors such that ǫ ↑↓ = ǫ ↓↑ = 1, and (σ A ) I J is the (I, J)-element of the Pauli matrix σ A .
The OPEs of dimension-two and three operators are (4.36) Finally, the OPEs of dimension-three operators are Note that the OPEs of generators that are not listed above all vanish. In particular, we find The vanishing of these OPEs are necessary for our conjecture that those in Table 2 is the complete set of generators, since no operator composed of those in Table 2 can appear in these OPEs for dimensional and symmetry reasons.

Consistency check with Schur index
In sub-section 4.2, we conjecture that S is the complete set of generators of the chiral algebra χ[T C 2,0 ] associated with the genus two theory T C 2,0 . Operators in S are listed in Table 2. As discussed in sub-section 4.2, we have checked that every Q BRST -cohomology class of dimension less than or equal to six is generated by those in S. Here, we perform a further non-trivial where L 0 and J 0 are the zero modes of T and J respectively, and q, a ∈ C such that |q| < 1 and |a| = 1. This character can be computed order by order in q as follows. First, if χ[T C 2,0 ] has no null operator, I(q, a) is given by where P.E.[g(q, a)] := exp ∞ n=1 1 n g(q n , a n ) for any function g. Indeed, the above expression counts all composite operators built out of ∂ k O for O ∈ S and k ≥ 0. However, χ[T C 2,0 ] contains many null operators, corresponding to 4d operator relations in T C 2,0 . These null operators must be removed from the spectrum. Therefore, the vacuum character I(q, a) of χ[T C 2,0 ] is obtained by subtracting the contributions of these null operators from (4.41). This can be done by identifying the null operators order by order in q. We list all the null operators up to dimension nine in Appendix B.
On the 4d side, the vacuum character I(q, a) is identified with the Schur index defined by where H is the space of local operators in T C 2,0 , and E, R and f are the dimension, SU (2) R charge and U (1) f charge of operators. Since T C 2,0 has a Lagrangian description, one can compute it via an integral formula [69,70].
We have checked that (4.42) agrees with (4.40) up to O(q 9 ). Combined with our check of the BRST cohomology classes up to dimension six, this gives a strong evidence for our conjecture that S is the complete set of generators of χ[T C 2,0 ].

Automorphisms and new SU (2)
In this section, we discuss automorphisms of χ[T C 2,0 ], based on its OPEs we identified in the previous section. We will see that there exists an unexpected SU (2) automorphism sub-group in addition to those associated with 4d flavor and U (1) r symmetries.

Expected automorphisms
Let us first recall that 4d flavor U (1) f symmetry gives rise to an u(1) current, whose zero-mode  Table 2.
The charge conjugate for this U (1) f leads to a Z 2 automorphism under which with the other generators kept fixed. This Z 2 corresponds to φ a± → ±iφ a∓ , which preserves (4.1) and Q BRST , and therefore is an automorphism of χ[T C 2,0 ].
There is also another Z 2 automorphism corresponding to exchanging two small N = 4 super Virasoro algebras. Under this Z 2 , the generators of χ[T C 2,0 ] transform as with the other generators kept fixed. The combined action of the above two Z 2 corresponds to exchanging the left and right sides of the left quiver in Fig. 1. Note that D ±I andD ± I are eigenstates of this combined Z 2 -action, where D ±↑ andD ± ↑ have eigenvalue +1 while D ±↓ andD ± ↓ have eigenvalue −1. 19 As reviewed in sub-section 2.1, the 4d U (1) r symmetry of T C 2,0 also gives rise to a U (1) automorphism of χ[T C 2,0 ]. The U (1) r charges of the chiral algebra generators are shown in

New SU(2) symmetry
In addition to the above In contrast to U (1) f and U (1) r , the above SU (2) symmetry does not correspond to any conserved one-form current in four dimensions. This follows from the fact that every conserved one-form current in 4d N = 2 SCFT is either a flavor current or an R-symmetry current. 21 It is clear that our new SU (2) is not associated with any 4d R-current. 22 It is also clear that this SU (2) does not correspond to a 4d flavor current, since the genus two theory T C 2,0 has only one flavor current multiplet,B 1 , corresponding to U (1) f . Therefore, the above SU (2) symmetry either corresponds to a 4d global symmetry without conserved currents, or is an accidental symmetry in two dimensions. 23 It would be interesting to study each of these possibilities further.
While its four-dimensional origin is still to be understood, we see that this SU (2) symmetry strongly constrains the OPEs of does not forbid C 2 (0)/z 3 to arise in the OPE of X(z)X(0), which is however prohibited by 21 Here, the flavor symmetry of a 4d N = 2 SCFT is defined as a global symmetry that commutes with N = 2 superconformal symmetry. 22 The only SU (2) sub-group of the 4d R-symmetry group is SU (2)R, which is broken in the 4d/2d correspondence. 23 In the latter case, it can be an accidental enhancement of a smaller 4d global symmetry. It would particularly interesting to see if this SU (2) is an accidental enhancement of an abelian higher-form symmetry [61] in four dimensions. Note that the above SU (2) symmetry cannot be interpreted as coming from an "SU (2) higher form symmetry" in four dimensions since every higher form symmetry is abelian.

20
the SU (2) symmetry; an SU (2) triplet just cannot appear in the tensor product of singlets.
There are indeed various terms in the OPEs of generators that are forbidden by the SU (2) symmetry.
Note that, since 2d OPEs are determined by 4d OPEs, this SU (2) symmetry also constrains the OPEs of Schur operators in the genus two theory T C 2,0 . In particular, every 4d operator relation involving Schur operators gives rise to a null operator in χ[T C 2,0 ], and therefore must be in some representation of SU (2). Since J and B ± are neutral under it, this SU (2) acts trivially on the Higgs branch. To see the effects of this SU (2), one needs to look at Schur operators with non-vanishing spins. 24 This means that the above unexpected SU (2) symmetry of χ[T C 2,0 ] constrains the OPEs of non-scalar Schur operators in T C 2,0 . We leave a detailed study of these non-trivial constraints for future work.

Conclusions and discussions
In this paper, we studied the chiral algebra χ[T C 2,0 ] associated with the A 1 -type genus two class S theory T C 2,0 . We focus on the weak coupling description of the theory corresponding to the left quiver in Fig. 1, and apply the BRST reduction reviewed in sub-section 2.2. We found that (1)  One important and remarkable consequence of our OPEs is that there exists an unexpected SU (2) automorphism sub-group of χ[T C 2,0 ]. As discussed in section 5, this SU (2) symmetry is not related to any conserved one-form current in four dimensions, and therefore either corresponds to a 4d symmetry without conserved current or is an accidental symmetry in two dimensions. We found that this SU (2) acts trivially on 2d operators corresponding to 4d Higgs branch operators. Therefore, this SU (2) is a symmetry characterizing the OPEs of non-scalar Schur operators.
While there has been various progress about the associated chiral algebra of class S, our work is the first step to understand the chiral algebras of class S at higher genera. There are indeed many open problems related to this work in this direction: • What is the four-dimensional origin of the new SU (2) symmetry that we discussed in sub-section 5.2? As mentioned already, there is no 4d conserved one-form current corresponding to it. One possible way to understand it is to see how this SU (2) symmetry emerges in the localization computation studied in [42,52].
• What is the chiral algebra of class S associated with a Riemann surface of higher genus?
It would be particularly interesting to see if there is a non-abelian automorphism subgroup that does not correspond to a 4d conserved one-form current. It would also be interesting to consider a generalization to genus two theories arising from higher rank 6d (2,0) theories.
• Is there any free field realization of χ[T C 2,0 ]? As shown in [12,14,56], the chiral algebras of a class of 4d N = 2 SCFTs have a beautiful free field realization that makes all the null operators trivially vanishing. It would be interesting to search for a similar realization of χ[T C 2,0 ], which would also be useful for solving the problem discussed in the previous bullet.
• As shown in appendix D of [8], the (normalized) Schur index of the genus two theory T C 2,0 satisfies a sixth-order modular linear differential equation. This suggests that

A S-duality equivalence of Schur index
As mentioned in section 3, T C 2,0 has two weak coupling Lagrangian descriptions corresponding to the quiver diagrams shown in Fig. 1. These two quiver descriptions are expected to be related by S-duality, which implies the Schur indices computed via these quivers are equivalent.
This equivalence has been checked in the case of vanishing flavor fugacity in [2]. In this appendix, we will extend it to the case of non-vanishing flavor fugacity. Note that, since the flavor U (1) f symmetry of this theory is not visible in its class S construction, this extension does not immediately follow from the class S interpretation of S-duality.
First, we focus on the left quiver diagram in Fig. 1. Let a be a U (1) f flavor fugacity. The Schur index of T C 2,0 is evaluated as where the contour integrations are taken over is the factor arising from the Harr measure of SU (2) k gauge group, and are respectively the index contributions from an SU (2) vector multiplet, a fundamental hypermultiplet, and an adjoint hypermultiplet. Here, we used the plethystic exponential defined by P.E.[g(q; x 1 , · · · , x k )] := exp ∞ n=1 1 n g(q n ; x n 1 , · · · , x n k ) , (A.5) for arbitrary function g of fugacities.
On the other hands, the right quiver diagram in Fig. 1 indicates the index of T C 2,0 is evaluated as where b is a fugacity for the U (1) f flavor symmetry, and is the index contributions from trifundamental hypermultiplet. The relation between a and b will be clear below. 23 When the flavor fugacity is turned off, the equivalence of (A.1) and (A.6) was shown in [2].
Indeed, it was shown in [2] that where R runs over irreducible representations of su (2), , and χ R (x) : . Using this expression together with one can show that the equivalence in the case of a = b = 1.
Our aim in this appendix is to generalize the above proof to the case of a, b = 1. First, by comparing the first few terms of I 1 (q; a) and I 2 (q; b), we see that the collect identification of the flavor fugacity is Below, we rewrite I 1 and I 2 to show that I 1 (q; b 2 ) = I 2 (q; b).
The last line are rewritten for symmetry f (x −1 ) = f (x).
We now evaluate the residues of the contour integral. The poles of the integrand are at 26 First, we evaluate the residue at the pole at x = q 1 2 +k b 2 . The residue involves the following factor from the Pochhammer symbol and therefore we find In the last equality, we used the formula 26 There is no pole at x = 0, since f (0) takes finite value and the summation part in the integrand is zero in the limit x → 0 .
From the above calculations, we see that the residue at x = q 17) Note here that one can further rewrite the sum over m as Combining the above result with the residues at x = q 1 2 +k b −2 , we finally get the following expression for I 1 (q; b 2 ): A.2 Rewriting I 2 Let us now turn to I 2 (q; b). Using the identity I half tri-fund (q; x 1 , x 2 , x 3 , b)I half tri-fund (q; x 1 , x 2 , x 3 , b) = I half tri-fund (q; x 1 , bx 2 , x 3 , 1)I half tri-fund (q; and (A.8), we can more simplify I 2 (q; b) as (A.21) Note that the sum over R can be rewritten as . (A.23) From this and , (A. 24) we see that the integrand of (A.21) has a pole at x = ±b ±1 q k+1 2 for all k ∈ Z ≥0 . 27 27 For careful evaluation, we can see that x = 0, ±b ± are also not poles.

26
Note that the residue at x = bq k+1 2 is evaluate as (A. 25) Combining this and its cousin obtained by b → −b −1 , we find Since the sum over k and m is rewritten as 28 the expression (A.26) is further rewritten as A.3 Proof of I 1 = I 2 From (A. 19) and (A.32), we see that proving I 1 = I 2 is equivalent to proving (A. 35) In the rest of this appendix, we prove (A. 33 Therefore the expression in the most inner bracket is regular at b = sq − ℓ

A.3.1 Poles and residues on both sides
Let us focus on the poles at b = q ℓ 2 for ℓ > 0. Its generalization to the other poles is straightforward. Note that these poles are second order poles. One can see that the coefficients of the most singular terms on both sides of (A.33) are identical. Indeed, when b ∼ q ℓ 2 , the LHS behaves as .
On the other hand, the RHS behaves as .
Thus, the most singular terms on both sides agree.
Let us next evaluate the residues on both sides. First, we compute the residue on the LHS. The residue coming from h 1 (b) is evaluated as while, the residue coming from h 2 (b) is evaluated as Combining the above two, we see that the residue of the LHS of (A.33) is We next turn to the RHS of (A.33). The residue coming from (A. 42) and that arising from Therefore, the residue of the RHS of (A.33) at b = q ℓ 2 is evaluated as

A.3.2 Coincidence of the residues
We here show that the residue (A.40) of the LHS of (A.33) agrees with the residue (A.43) of the RHS.
Note first that the second line of (A.40) minus that of (A.43) is simplified as Therefore, all we need to show is the equivalence of the first lines of (A.40) and (A. 43). Note that this is equivalent to proving the identity Using the identities we see that proving (A.45) is equivalent to proving Below, we show that (A.48) indeed holds. To that end, first note that (A.50) Using these identities, the LHS of (A.48) is rewritten as which can be shown to vanish by a straightforward calculation. Therefore, (A.48) is an identity, which completes our proof of the equivalence between (A.40) and (A.43).

B Null operators of genus two chiral algebra
In this appendix, we list the null operators in the chiral algebra χ[T C 2,0 ] whose holomorphic dimension (which we denote by h) is less than or equal to nine. Interestingly, such null operators only exist at h = 4, 5 and 6, up to composite operators involving them or their derivatives. Note that the absence of independent null operators at h = 7, 8 and 9 does not mean the absence of such operators at h ≥ 10. Indeed, the modular linear differential equation studied in [8] suggests an independent null operator involving T 6 at h = 12. It would be interesting to extend our results here to higher dimensions.
Below, we list these null operators as operator relations. We also classify these operator

B.3 Dim 6
We here list independent null operators of dimension six. We again omit all such operators that trivially follow from lower-dimensional nulls. Finally, it turns out that dimension six operators with r = ±2 are all null: In general, D andD type Schur operators in the Lagrangian theory are nilpotent since they involve gauginos [1,8]. The above relations reflect this property of D andD type operators in four dimensions. 30 For completeness, we here comment that there are also operator relations of the form 0 = C [A C B] + if ABC (σ C ) J I ∂(D +I ∂D − J +D + J ∂D −I ). These are, however, relations trivially following from the OPE of C A (z)C B (0). We do not list such operator relations as mentioned at the beginning of this appendix.