On the chiral algebra of Argyres-Douglas theories and S-duality

We study the two-dimensional chiral algebra associated with the simplest Argyres-Douglas type theory with an exactly marginal coupling, i.e., the $(A_3,A_3)$ theory. Near a cusp in the space of the exactly marginal deformations (i.e., the conformal manifold), the theory is well-described by the $SU(2)$ gauge theory coupled to isolated Argyres-Douglas theories and a fundamental hypermultiplet. In this sense, the $(A_3,A_3)$ theory is an Argyres-Douglas version of the $\mathcal{N}=2$ $SU(2)$ conformal QCD. By studying its Higgs branch and Schur index, we identify the minimal possible set of chiral algebra generators for the $(A_3,A_3)$ theory, and show that there is a unique set of closed OPEs among these generators. The resulting OPEs are consistent with the Schur index, Higgs branch chiral ring relations, and the BRST cohomology conjecture. We then show that the automorphism group of the chiral algebra we constructed contains a discrete group $G$ with an $S_3$ subgroup and a homomorphism $G\to S_4 \times {\bf Z}_2$. This result is consistent with the S-duality of the $(A_3,A_3)$ theory.


Introduction
It has recently been shown that every four-dimensional N = 2 superconformal field theory (SCFT) contains a set of BPS operators whose operator product expansions (OPEs) are characterized by a two-dimensional chiral algebra [1]. These BPS operators are annihilated by two Poincaré and two conformal supercharges, and called "Schur operators." They include the highest weight component of the SU (2) R current and the Higgs branch operators, but their complete spectrum is generally highly non-trivial. The associated chiral algebra, however, determines the complete spectrum of the Schur operators, and moreover characterizes their OPEs. Indeed, some general properties of 4d N = 2 SCFTs have been uncovered with the help of the associated chiral algebras [2][3][4].
The 2d chiral algebra is especially a powerful tool to study the OPEs of strongly coupled SCFTs. Among other SCFTs, Arygres-Douglas type SCFTs are of particular importance. These theories are a series of N = 2 SCFTs with Coulomb branch operators of fractional scaling dimensions, whose first examples were found as IR SCFTs at singular points on the Coulomb branch of asymptotically free gauge theories [5,6]. Their generalizations are then constructed by wrapping M5-branes on a Riemann surface [7][8][9] as well as by considering type II string theory on a Calabi-Yau singularity [10,11]. While their behavior at generic points on the Coulomb branch is well-studied, 1 it is very recent that their physics at the conformal point symmetry of the theory is U (1) 3 , and the conformal anomalies are given by a = 15 8 and c = 2 [9,31]. 8 Moreover, it has recently been found in [31,34] that the Schur index of the (A 3 , A 3 ) theory is invariant under an action of S 4 × Z 2 , an S 3 subgroup of which is associated with the S-duality.
To identify the chiral algebra associated with the (A 3 , A 3 ) theory, we first study the Higgs branch chiral ring of the theory. According to [1], the generators of the Higgs branch chiral ring map to generators of the associated chiral algebra. In addition to them, the highest weight component of the SU (2) R current maps to the 2d stress tensor, which is another independent generator of the chiral algebra. While there could, in principle, be more generators of the chiral algebra, the minimal conjecture is that these are the complete set of generators of the chiral algebra. With this conjecture, we will bootstrap the OPEs among the generators by solving the constraint of the OPE associativity, or equivalently, Jacobi identities. We find that there exists a unique set of OPEs consistent with the Jacobi identities. We will then show that our result is consistent with the Schur index, Higgs branch operator relations, and the BRST cohomology conjecture made in [1].
Since the Schur index of the theory is identical to the character of the associated chiral algebra [1], the S 4 × Z 2 symmetry of the Schur index is expected to have a counterpart acting on the chiral algebra.
Indeed, we find that the automorphism group of the chiral algebra we constructed contains a discrete group G with a group homomorphism G → S 4 × Z 2 . The kernel of this homomorphism is generated by flipping the signs of operators of half-integer holomorphic dimensions, to which the Schur index/character is not sensitive. Moreover, we show that G contains an S 3 subgroup, which is naturally interpreted as the one associated with the S-duality. This identification is perfectly consistent with the S-duality action on the Schur index.
The organization of this paper is the following. In section 2, we briefly review known features of the (A 3 , A 3 ) theory, and then study its Higgs branch chiral ring. In section 3, after giving a very quick review of [1], we bootstrap the chiral algebra associated with the (A 3 , A 3 ) theory, with a minimal conjecture on the set of generators, and then give various consistency checks of our result. In sub-section 4, we study the automorphism group of the chiral algebra we constructed. We give a conclusion and discuss future directions in section 5. 8 We use the normalization with which a free hypermultiplet has a = 1 24 and c = 1 12 .

(A 3 , A 3 ) theory
In this section, we briefly review the S-duality and the Schur index of the (A 3 , A 3 ) Argyres-Douglas theory, and then study its Higgs branch chiral ring. The Schur index and the Higgs branch chiral ring will be important for the identification of the associated chiral algebra in the next section.

Quiver description and S-duality
As shown in [31] and further studied in [32], the conformal manifold of the (A 3 , A 3 ) theory has a weak coupling point at which the theory is well-described by the quiver diagram shown in Fig. 1. Note that each (A 1 , D 4 ) theory has SU (3) flavor symmetry, an SU (2) subgroup of which is now gauged. Therefore, there is a residual U (1) flavor symmetry acting on each (A 1 , D 4 ) sector. Similarly, there is a U (1) flavor symmetry acting on the fundamental hypermultiplet. Therefore, in total, the flavor symmetry of the (A 3 , A 3 ) theory is U (1) 3 . Since the beta function for the SU (2) gauging vanishes, the gauge coupling τ = θ π + 8πi g 2 is an exactly marginal coupling of the (A 3 , A 3 ) theory.
It was shown in [31] that the Seiberg-Witten curve of the (A 3 , A 3 ) theory is invariant under two duality transformations T and S, which act on the gauge coupling τ as 9 Therefore the duality group acts on the space of τ as P SL(2, Z). Since it relates the weak coupling limit τ = i∞ with various strong coupling limits, the duality is referred to as the S-duality. In addition to changing the value of τ , the S-duality also permutes operators of the same scaling dimensions. In particular, it permutes the flavor U (1) 3 currents through the symmetric group S 3 . The permutation of the flavor currents is uniquely determined by the change of the coupling τ through the group homomorphism This is the (A 3 , A 3 ) counterpart of the SO(8) triality associated with the S-duality of the N = 2 SU (2) gauge theory with four flavors studied in [29]. 10

Schur index
Since it will be useful in identifying the chiral algebra in Sec. 3, we next review the Schur index of the (A 3 , A 3 ) theory. The Schur index is generally defined by where H is the Hilbert space of local operators, G F is the flavor symmetry of the theory, and E, R and f i are the scaling dimension, SU (2) R charge and the flavor charges of the operator. In the case of (A 3 , A 3 ) theory, we have G F = U (1) 3 .
9 These transformations are denoted by T and S in [31]. 10 See also [35] for more on the S-duality action.
The quiver description reviewed above implies that the Schur index of the (A 3 , A 3 ) theory is evaluated as follows [15] : where a, b and c are the flavor fugacities for the U (1) 3 flavor symmetry. 11 The factor ∆(w) ≡ 1 2 (1−w 2 )(1− w −2 ) comes from the Haar measure of su(2), and are the index contributions from the su(2) vector multiplet and the fundamental hypermultiplet, respec- n g(q n ; (x 1 ) n , · · · , (x k ) n ) for any function, g, of fugacities, and χ (2). The other factor, I (A1,D4) (w, a), is the Schur index of the (A 1 , D 4 ) theory, which was first computed in [15,16] inspired by [1] and turns out to be equivalent to the vacuum character of su(3) − 3 2 . Here, we write the simplest expression for the index which was conjectured in [21] and proven in [36] (See also [22,24]): where our convention for the su(3) character is such that the character of the adjoint representation is written as χ In terms of the relabeled flavor fugacities, x = a, y = √ bc and z = b/c, the Schur index of the (A 3 , A 3 ) theory is expanded as Note that the above Schur index is invariant under the action of S 3 generated by σ 1 : x ←→ y , z : fixed , (2.8) Since this S 3 is identical to the S 3 in the homomorphism (2.2), this symmetry of the index is interpreted as the S-duality invariance of the Schur index of the (A 3 , A 3 ) theory [15]. In other words, the S-duality group acts on the Schur index through this S 3 . 12 In addition to this S 3 , it was shown in [34] that the Schur index is also invariant under (2.11) 11 To be more precise, a is the fugacity for the U (1) flavor symmetry acting on the fundamental hypermultiplet, and b and c are fugacities for the U (1) flavor symmetries acting on the two (A 1 , D 4 ) theories. 12 Note here that, since the Schur index is independent of the gauge coupling, the action of the full P SL(2, Z) is not visible The quiver diagram for the 3d reduction of the (A 3 , A 3 ) theory. Each circle stands for a U (1) vector multiplet, and each edge stands for a hypermultiplet charged under the U (1) gauge symmetries at its ends. Note that this quiver is self-mirror, namely its 3d mirror pair is itself. Therefore, its Coulomb and Higgs branches are isomorphic to each other.
Here, ζ corresponds to the charge conjugation of the theory. 13 On the other hand, σ 3 is such that which is the charge conjugation of the two (A 1 , D 4 ) theories in the quiver in Fig. 1. Hence, the Schur index of the (A 3 , A 3 ) theory is invariant under the action of the group generated by σ 1 , σ 2 , σ 3 and ζ, which turns out to be S 4 × Z 2 .
According to the discussion in [1], the above Schur index is equivalent to the character of the associated chiral algebra. This will be important when we identify the chiral algebra for the (A 3 , A 3 ) theory in Section 3.

Higgs branch from 3d reduction
Having given a quick review of the known features of the (A 3 , A 3 ) theory, we here study the Higgs branch chiral ring of the theory. When one space direction is compactified, the theory reduces to a 3d N = 4 theory. While the Coulomb branch is modified in this reduction, the Higgs branch is expected to be invariant. Therefore, the Higgs branch chiral ring of the (A 3 , A 3 ) theory can be read off from its 3d reduction.
The 3d reduction of the (A 3 , A 3 ) theory is known to be the IR limit of the gauge theory described by the quiver diagram in figure 2 [9]. 14 Note that we here remove the decoupled diagonal U (1) of the Let us denote by q ab and q ba the two scalar fields in the 13 To be more precise, ζ corresponds to the CP transformation, namely the combination of the charge conjugation and the parity transformation. Here, since our 4d theory is a strongly coupled CFT without a conventional Lagrangian description, the two transformations, C and CP , are purely defined as linear maps from the Hilbert space of operators to itself. In our case, ζ exchanges the chiral supercharge Q I ± and the anti-chiral one Q I± since the two supercharges have opposite U (1)r and SU (2) R charges (See, for example, appendix A of [1]). For this to be consistent with the supersymmetry algebra {Q I α, Q Jα } = δ I J P αα , the ζ must also involve the parity transformation. Therefore, the ζ corresponds to CP instead of C. This point, however, does not play any crucial role in this paper. 14 To be more precise, [9] shows that the mirror of the 3d reduction of the (A 3 , A 3 ) theory is described by theory associated with the quiver diagram in figure 2. Since the quiver gauge theory is self-mirror, this mirror transformation just exchanges the Coulomb and Higgs branches. The fact that this 3d quiver gauge theory is self-mirror can be seen from the general rule of abelian mirror symmetry found in [37] (See Sec. 4 in particular). q ab q bc q ac q ad q db q cd while q ba has (−1, +1). 15 The superpotential of the 3d theory is given by Note that the coefficients can be set to any non-vanishing value by rescaling the hypermultiplet scalars. Now, the Higgs branch chiral ring is the ring of gauge invariant operators composed of the hypermultiplet scalars subject to the F-term conditions. The theory has a flavor U (1) 3 = U (1) 1 × U (1) 2 × U (1) 3 symmetry acting on the hypermultiplet scalars; the U (1) 3 shifts the phases of q ij and q ji to the opposite directions by the same amount. We take the three flavor charges as in Table 1 . There are Higgs branch operators in the same supermultiplet as the flavor currents. They are called the "flavor moment maps," and written as 14) Since the flavor symmetry is abelian, these moment maps are neutral under the flavor symmetry.
The Higgs branch chiral ring is generated by the above flavor moment maps and the following gauge invariant operators: and Note that Ws and X s correspond respectively to "triangles" and "rhombi" in the quiver diagram. The normalizations of the operators are chosen so that they are easily compared with null operators of the associated chiral algebra in the next section.
The above generators satisfy the following chiral ring relations: which follow from the F-term conditions associated with the superpotential (2.12). Here we used the antisymmetric tensor such that ǫ +− = 1. In the next section, we will construct the chiral algebra for the (A 3 , A 3 ) theory which is consistent with these Higgs branch operator relations.

Chiral algebra conjecture
In this section, we will construct the chiral algebra associated, in the sense of [1], with the (A 3 , A 3 ) theory.
To that end, we first review some results of [1] in Sec. 3.1. We then focus on the (A 3 , A 3 ) theory, and identify the minimal possible set of chiral algebra generators in Sec. 3.2. In Sec. 3.3, we bootstrap the OPEs among them by assuming the absence of extra generators. We finally give various consistency checks of our result in Sec. 3.4 -3.6. We follow the convention of [1] in this and later sections.

Review of the chiral algebra construction
Let us briefly review the chiral algebra construction of [1]. In this sub-section, we consider a general 4d N = 2 SCFT. The central objects in the analysis are BPS local operators called "Schur operators," which are defined by local operators annihilated by the four supercharges Q 1 − , S 1 − , Q 2− and S 2− . They satisfy where E is the scaling dimension, (j 1 , j 2 ) is the SO(4) spin, and R is the SU (2) R charge of the operators. The 2d chiral algebra is obtained by putting the Schur operators on R 2 in the four-dimensional spacetime. 16 We then consider the "twisted translation" on R 2 as where O Schur (0) is a Schur operator located at the origin of R 2 , L −1 and L −1 are the holomorphic and anti-holomorphic translations on R 2 , and R − is the lowering operator of su(2) R . An important feature of this twisted translation is that, O twisted (z,z) is closed with respect to a nilpotent linear combination of 16 We here work in the Euclidean four-dimensional spacetime.
supercharges, Q ≡ Q 1 − + S 2− . Moreover, L −1 + R − turns out to be Q-exact. Therefore, the 2d OPEs of The OPEs of O 2d (z) form a chiral algebra, or vertex operator algebra, on R 2 . Since the structure of the chiral algebra is completely determined by the four-dimensional OPEs of the Schur operators, identifying the chiral algebra for a given N = 2 SCFT will reveal much information of the original four-dimensional OPEs.
Let us briefly review typical examples of Schur operator. In terms of the classification of irreducible N = 2 superconformal multiplets in [38], the only multiplets containing a Schur operator are B R , D R(j1,0) , D R(0,j2) and C R(j1,j2) multiplets. In particular, the highest weight component of the SU (2) R current, (J SU(2)R ) ij αα , in the stress tensor multiplet, C 0(0,0) , is a Schur operator present in any 4d N = 2 SCFT. 17 Indeed, it has E = 3, j 1 = j 2 = 1 2 and R = 1, and satisfies the Schur condition (3.1). If the theory has a flavor symmetry, the flavor current is in a B 1 multiplet. While the flavor current itself is not Schur, the bottom component of the B 1 multiplet is a Schur operator called "flavor moment map." More generally, the bottom component of B R multiplets are Schur operators called "Higgs branch operators," whose vacuum expectation values parameterize the Higgs branch moduli space of vacua. 18 It is generally shown in [1] that, by the map (3.3), the (J SU(2)R ) 11 ++ is mapped to the Virasoro stress tensor with central charge where c 4d is the four-dimensional "c central charge." 19 Similarly, a flavor moment map is mapped to an affine current with level where k 4d is the flavor central charge of the corresponding flavor current in four dimensions. 20 More generally, Higgs branch operators are mapped to Virasoro primaries.
Let us finally review the relation to the Schur index (2.3). Since E − R in (2.3) commutes with the four supercharges annihilating Schur operators, the index gets contributions only from the Schur operators. Therefore, the same quantity is calculated in terms of the chiral algebra. Indeed, it agrees with the character of the vacuum representation of the chiral algebra where the trace is taken over the Hilbert space of chiral algebra operators, and L 0 and J i 0 are respectively the zero modes of the stress tensor and the affine current associated with the i-th Cartan generator of G F . 17 Here, i, j = 1, 2 are the symmetric SU (2) R indices for the adjoint representation of su(2) R , while α = ± andα =± are spinor indices for the vector representation of so(4). 18 Higgs branch operators are local operators annihilated by Q 1 α, Q 2α and their conjugates. They saturate four unitarity bounds so that j 1 = j 2 = 0 and E = 2R. 19 Here, the normalization of c 4d is such that a free hypermultiplet contributes c 4d = 1 12 . 20 Our normalization of k 4d is such that a fundamental hypermultiplet of the flavor symmetry contributes k 4d = 2.

Generators
Let us now focus on the (A 3 , A 3 ) theory. According to our discussion in Sec. 2, the (A 3 , A 3 ) theory has at least the following Schur operators.
• U (1) 3 flavor moment maps J a . As explained in appendix A, the corresponding flavor U (1) 3 current has the flavor central charge k 4d = 8.
In particular, the last two bullets correspond to the generators of the Higgs branch chiral ring. Here, we use the same notation for the four-dimensional Higgs branch operators as for their three-dimensional reductions. According to the dictionary reviewed above, these Schur operators map to the following operators of the chiral algebra.
• The stress tensor, T , with the central charge c 2d such that The central charge is given by c 2d = −12c 4d = −24.
• A primary operator W q1q2q3 for q i = ±1. It is of dimension 3 2 and has U (1) 3 charge (q 1 , q 2 , q 3 ). Therefore • A primary operator X i q of dimension 2 for i = 1, 2, 3 and q = ±1. It has charge 2q under U (1) i . Therefore (3.12) 21 In this paper, we call flavor-charged Higgs branch operators which are not flavor moment maps "baryons." Note that these operators are independent generators of the chiral algebra. In other words, there is no smaller set of operators whose OPEs generate all of T, J i , W q1q2q3 and X i q . Indeed, it was shown in [1] that the generators of the Higgs branch chiral ring are mapped to independent generators of the associated chiral algebra. In addition, the stress tensor is also an independent generator since the Sugawara stress tensor of affine U (1) 3 currents cannot reproduce c 2d = −24. 22 The chiral algebra could, in principle, contain more generators. The minimal possible assumption is, however, that those associated with the SU (2) R current and the Higgs branch generators are the full set of generators of the chiral algebra. This minimal assumption is known to be applicable to various N ≥ 2 SCFTs such as a series of 4d rank-one N = 2 SCFTs [1], a certain class of isolated Argyres-Douglas theories [1,21,22,24], and several 4d N > 2 SCFTs [1,39]. 23 We therefore conjecture that the chiral algebra for the (A 3 , A 3 ) theory is generated by J i (z), T (z), W q1q2q3 (z) and X i q (z). Below, we show that this conjecture uniquely fixes the OPEs among the generators (up to the normalizations). We also give various consistency checks of the resulting OPEs.

OPEs among the baryonic generators
Every chiral algebra satisfies the basic consistency condition of associativity, or equivalently, the Jacobi identities among the generators. 24 This Jacobi identity condition puts a strong constraint on the possible form of the OPEs of the generators. For our conjecture to make sense, there has to be at least one set of OPEs among W q1q2q3 and X i q which are consistent with the Jacobi identities. To see that this is indeed the case, we concretely solve this Jacobi identity constraint. Our strategy is the following. We first write down the most general ansatz for the OPEs among the generators which are consistent with the charge conservation. Then we impose the constraint that all the Jacobi identities are satisfied, which gives rise to various relations among the OPE coefficients in the ansatz. Interestingly, we find that there is one and only one set of OPE coefficients consistent with these relations. For this calculation, we used the Mathematica package OPEdefs developed by [40].
Moving the detailed description of this calculation to appendix B, we only give the resulting OPEs here. Since the OPEs of the forms J i (z)O(0) and T (z)O(0) are fixed by the symmetry, we only have to describe W q1q2q3 (z)W p1p2p3 (0), W q1q2q3 (z)X i q (0) and X i q (z)X j p (0). To write down a simple expression for them, let us define J A , W A ± and X [AB] = − X [BA] for A, B = 1, · · · , 4 by where ǫ ABCD is the anti-symmetric tensor such that ǫ 1234 = 1, and raising/lowering the indices is realized 22 The Virasoro central charge c 2d of the Sugawara stress tensor of an affine U (1) n current is given by c 2d = n regardless of the level of the current. In particular, it cannot be negative. 23 See [3] for more general discussion on the chiral algebras of 4d N > 2 SCFTs. 24 In terms of the OPEs, the Jacobi identity is expressed as for |z 2 − z 3 | < |z 1 − z 3 |, where [· · · ] stands for the singular part of the OPE of the operators.
by δ AB and δ AB . In terms of these variables, the only set of OPEs consistent with the Jacobi identities is written as where the indices A, B, C, D are not summed while E and F are summed over 1, · · · , 4. For the detailed derivation of this result, see appendix B. In the rest of this section, we will give several consistency checks of this result.

Consistency with the Schur index
Let us first check that the above OPEs are consistent with the Schur index of the (A 3 , A 3 ) theory. To that end, we compute the character of the chiral algebra we constructed. Since our chiral algebra is generated by J i , T, W q1q2q3 and X i q , if there are no null operators then the character of the chiral algebra is given by where the first two terms are the contributions of J i and T , the third is that of W q1q2q3 , and the fourth is that of X i q . However, there are indeed various null operators in the chiral algebra. Therefore, the correct character is obtained by subtracting the contributions of the null operators from (3.23). Note here that, while the chiral algebra is non-unitary, every null operator has to be removed since every chiral algebra operator corresponds to a Schur operator in a unitary four-dimensional SCFT. Moreover, as explained in Appendix B, removing the null operators is necessary for the Jacobi identities to be satisfied. Below, we identify null operators of dimension less than or equal to 4, and then show that the character obtained this way coincides with the Schur index of the (A 3 , A 3 ) theory evaluated in (2.7).
The first non-trivial null operators appear at dimension 3. There are indeed 12 null operators charged under the flavor symmetry, and 3 neutral null operators. Their index contribution is given by At dimension 7/2, there are 3 null operators with charge (±1, ±1, ±1). Their index contribution is At dimension 4, there are 18 neutral null operators. In addition to them, there is one null operator for all charge combinations of the form (q 1 , q 2 , 0), (0, q 2 , q 3 ) and (q 1 , 0, q 3 ) with q k = ±2. The last set of null operators of dimension four are 9 null operators for each of charge (±2, 0, 0), (0, ±2, 0) and (0, 0, ±2). In total, the index contribution of the null operators of dimension four is given by

Consistency with the Higgs branch chiral ring
The OPEs (3.17) - (3.22) imply that there are the following null operators of dimension 3: this is that the former is not associated with a Higgs branch operator in four-dimensions, and the latter is descendants of some primary operators and therefore identified with zero in the chiral ring. Similar phenomena happen in many 4d rank-one N = 2 SCFTs [1] and 4d rank-one N > 2 SCFTs [1,39]. 25 Let us stress again that every null operator in the chiral algebra has to be removed even though the chiral algebra is non-unitary. The reason is that every chiral algebra operator corresponds to a Schur operator in four dimensions, in which no local operator is null. Moreover, as explained in Appendix B, removing the null operators is necessary for the Jacobi identities.

Consistency with BRST cohomology
Recall that the (A 3 , A 3 ) theory is obtained by an exactly marginal gauging of isolated theories, as described by the quiver diagram in Fig. 1. In such a case, the chiral algebra is conjectured, in [1], to be obtained by a certain type of BRST cohomology of the direct sum of chiral algebras for the constituents of the quiver. When the gauge coupling is turned on, some of the Schur operators in the constituents are lifted to non-Schur. The BRST cohomology is then expected to remove all such lifted operators as those that are not BRST-closed or those that are BRST-exact. Below, we will check that our conjecture is consistent with this BRST cohomology conjecture. Let us illustrate how the BRST cohomology is constructed in our case. The discussion below directly follows from the general analysis in [1]. We first write down the direct sum of the chiral algebras of the constituents. Since the quiver diagram is composed of a fundamental hypermultiplet of SU (2), two (A 1 , D 4 ) theories, and an SU (2) vector multiplet, the direct sum of the corresponding chiral algebras is generated by symplectic bosons q k s , two copies of su(3) − 3 2 current, j P n , and the b I , ∂c J ghosts in the adjoint representation of su (2). Their OPEs are written as where n, m = 1, 2 specifies in which (A 1 , D 4 ) theory the current j P n is included, the P, Q, R = 1, · · · , 8 are the adjoint indices of su (3), and the f P Q R is the structure constant of su(3). The k, ℓ = 1, 2 and I, J = 1, 2, 3 are respectively the indices for the fundamental and adjoint representations of su(2), and the s and t run over 1, 2. We raise and lower the su(3) and su(2) indices respectively by δ P Q , δ P Q and δ IJ , δ IJ .
Note that each su(3) current is decomposed into an su(2) × u(1) current and two doublets of su(2), the latter of which are charged oppositely under the u(1). We take the basis of the su(3) current so that f P QR = ǫ P QR and f P Q8 = 0 for P, Q, R = 1, 2, 3. Therefore J I n for I = 1, 2, 3 and J 8 n form an su(2) × u(1) sub-algebra of su(3) − 3

. 26
In addition to the two su(2) currents J I 1 and J I 2 arising from the (A 1 , D 4 ) sectors, there are also su(2) currents built out of q k s and b, c: where I = 1, 2, 3. In terms of these four su(2) currents, the BRST current is defined by Then the BRST charge Q BRST ≡ dz J BRST turns out to be nilpotent, which reflects the fact that the β-function of the SU (2) gauge coupling vanishes in four dimensions. With respect to Q BRST , we consider the BRST cohomology of the space of operators composed of j I n , q k s , b and ∂c. To identify the basis of the cohomology, we have used the Mathematica package OPEdefs developed in [40]. The result is the following. At dimension one, there are three independent cohomology classes represented by The OPEs among these operators coincide with (3.8). At dimension 3 2 , there are eight independent cohomology classes represented by where u i n and v i n are doublets of su(2) defined as (3.41) With the above normalizations of W q1q2q3 , the OPEs among W q1q2q3 and J i for i = 1, 2, 3 coincide with (3.17) and (3.10). At dimension two, there are eighteen independent cohomology classes, two of which are null operators.
The two null operators actually result from a null operator in su(3) − 3 2 [1,41]. In terms of the rank-three invariant tensor of su(3), d P QR , these two null operators are expressed as Therefore, there are sixteen independent cohomology classes up to null operators. Among them, those charged under the u(1) 3 symmetry are represented by With this normalization, the OPEs among X a q and J a for a = 1, 2, 3 coincide with (3.12) and (3.22). On the other hand, those neutral under the u(1) symmetry are represented by J i J j , J i′ for i, j = 1, 2, 3 and Note that (3.46) is the sum of the stress tensors in the chiral algebras for the constituents. We see that (3.46) satisfies (3.7) with c 2d = −24.
Hence, we have checked that the BRST cohomology conjecture precisely reproduces the operator spectrum of the chiral algebra we constructed, up to dimension two. In particular, we found that there are no fermionic BRST cohomology class up to dimension two. 27 Moreover, we have checked that the OPE coefficients fixed by the Jacobi identities are precisely those obtained by the BRST cohomology analysis. This is a highly non-trivial consistency check of our conjecture on the chiral algebra.

Automorphisms and S-duality
Having given various consistency checks of our conjecture, we here study the automorphisms of the chiral algebra. In particular, we study how the S-duality of the (A 3 , A 3 ) theory acts on the chiral algebra through its automorphism group.
In addition to the above transformations, the chiral algebra is also invariant under This ζ is related to the charge conjugation invariance of the four-dimensional theory. While the charge conjugate O † of a Schur operator O is not Schur unless O = 1, there exists a Schur operator O which sits in the same SU (2) R multiplet as O † . 30 The above ζ is the two-dimensional version of the map O → O. 27 Even though the Schur index has no term with negative sign, in principle, there could be cancellations between bosonic and fermionic contributions. However, as far as the BRST cohomology conjecture is correct, there are no such cancellations at least up to O(q 2 ), and therefore there are only bosonic Schur operators up to dimension two. 28 According to (3.15), the σ i are equivalent to In other words, ζ arises from the combination of the four-dimensional charge conjugation and a multiple action of the SU (2) R raising operator.
Thus, the chiral algebra we have constructed is invariant under the action of the group generated by σ 1 , σ 2 , σ 3 and ζ, which we denote by G. The generators of the group G satisfy The group G has a normal Z 2 subgroup generated by which is the map that flips the signs of operators of half-integer holomorphic dimensions. Such a map is an automorphism of any chiral algebra associated with a 4d N = 2 SCFT, since the OPEs of such a chiral algebra are always single-valued [1]. 31 The quotient of G by this Z 2 is isomorphic to S 4 × Z 2 , namely The kernel of (4.11) is generated by ζ 2 , to which the Schur index is not sensitive. 33 Another important feature of the group G is that it contains an S 3 subgroup. Indeed, σ 1 and σ 2 ≡ σ 2 ζ 2 generate an S 3 subgroup of G. Then there is a homomorphism P SL(2, Z) → S 3 ֒→ G . (4.12) We interpret this result to mean that the S-duality of the (A 3 , A 3 ) theory acts on its chiral algebra through the homomorphism (4.12). This interpretation is consistent with the action of the S-duality on the Schur index. Indeed, as reviewed in Sec. 2.2, the S-duality acts on the Schur index through S 3 generated by σ 1 and σ 2 . Since [σ 1 ] and [ σ 2 ] = [ σ 2 ] are identified with σ 1 and σ 2 , this S 3 is precisely the image of the S 3 generated by σ 1 and σ 2 under the homomorphism (4.11). Therefore, the action of the S-duality on the chiral algebra and the Schur index is characterized by the chain of homomorphisms (4.13) 31 For chiral algebras associated with 4d N = 2 SCFTs, the holomorphic dimension of an operator is an integer or half-integer.
Moreover the 2d OPEs are always single-valued. Therefore, in the OPE is half-integer, one and only one of h 2 and h 3 is half-integer. If h 3 is half-integer, one and only one of h 1 and h 2 is half-integer. Therefore, flipping the signs of operators of half-integer holomorphic dimensions is always an automorphism of the chiral algebra. In four dimensions, ζ 2 corresponds to flipping the signs of operators of half-integer SU (2) R charges, which is also a symmetry of every N = 2 SCFT due to the nature of the tensor products of su (2) representations. 32 Here, as usual, [g] ∈ G/Z 2 is the equivalence class represented by g ∈ G. 33 Indeed, the index just captures the spectrum of operators and is not sensitive to the signs of operators.

Discussions
In this paper, we have studied the chiral algebra of the simplest Argyres-Douglas theory with an exactly marginal coupling, i.e., the (A 3 , A 3 ) theory. We have conjectured that the complete set of chiral algebra generators is the minimal one, i.e., they are associated with the generators of the Higgs branch chiral ring and the highest weight component of the SU (2) R current. Then we have shown that there exists a unique set of OPEs of the generators which are consistent with the Jacobi identities. Our result is consistent with the Schur index, Higgs branch operator relations, and the BRST cohomology conjecture made in [1]. After these consistency checks, we have studied the automorphism group of the chiral algebra we constructed, and shown that the algebra is invariant under the action of G generated by (4.4) -(4.6) as well as (4.7).
This group G is equipped with a group homomorphism G → S 4 × Z 2 , which relates it to the symmetry of the Schur index. We have also shown that G contains an S 3 subgroup, through which P SL(2, Z) can act on the chiral algebra. Our work is the first non-trivial step to understanding the action of the S-duality on the chiral algebras of Argyres-Douglas type theories with exactly marginal couplings.
There are many open problems to be studied, including the following: • Is there a Hamiltonian reduction of an affine current algebra which reproduces the chiral algebra we have constructed in section 3? Such a reduction was found in [22] for (A 1 , A 2n−1 ) and (A 1 , D 2n ) Argyres-Douglas theories.
• In the BRST cohomology analysis given in Sec. 3.6, is there a systematic way of identifying the cohomology classes of any given holomorphic dimension? Finding such a systematic way will be useful in proving our chiral algebra conjecture for the (A 3 , A 3 ) theory.
• A UV N = 1 "Lagrangian" for the (A 3 , A 3 ) theory was conjectured in [42] by generalizing earlier works [43]. It would be interesting to study if the discrete symmetry G of the chiral algebra is visible in this UV "Lagrangian." • Is the chiral algebra we constructed related to a quantum integrable model? The chiral algebras for the (A N , A M ) theories without flavor symmetry have been shown to be related to the quantum integrable models obtained by the ODE/IM correspondence [44]. It would be interesting to study how our chiral algebra for the (A 3 , A 3 ) theory is related to this work.
• It would be interesting to study how the chiral algebra we constructed for the (A 3 , A 3 ) theory is related to the recent discussions on the orbifold indices of 4d N = 2 SCFTs [45].
• What is the general relation between the symmetry of the chiral algebra and that of the superconformal index for general N = 2 SCFTs? Are the two symmetries identical whenever the chiral algebra contains no operator of half-integer holomorphic dimensions?
• What is the generalization to the (A N , A N ) theory for N > 3? The (A 1 , A 1 ) theory is the theory of a single free hypermultiplet, whose chiral algebra is that of a symplectic boson. The chiral algebra of the (A 2 , A 2 ) theory is conjectured to be su ( In the rest of this section, we give a brief comment on the last bullet. According to the four-dimensional analysis in [9], the (A N , A N ) theory has c 4d = 1 24 N (N 2 + 3N − 2), U (1) N flavor symmetry, and various baryonic Higgs branch operators. Then a natural conjecture on the chiral algebra of the (A N , A N ) theory is that it is generated by the stress tensor with c 2d = − 1 2 N (N 2 + 3N − 2), the affine U (1) N current, and Virasoro primaries corresponding to the baryonic Higgs branch generators. 34 It would be interesting to study whether the constraint of the Jacobi identities fixes the OPEs among these generators as in the case of the (A 3 , A 3 ) theory.
While a detailed analysis of the (A N , A N ) chiral algebra is beyond the scope of this paper, we briefly comment on its symmetry. A crucial point is that the Schur index of the (A N , A N ) theory is invariant under the action of S N +1 ×Z 2 [34]. Here S N +1 is related to the S-duality of the theory while Z 2 corresponds to the charge conjugation (together with an action of the SU (2) R raising operator). This symmetry of the Schur index suggests that the automorphism group of the chiral algebra contains a group G equipped with a group homomorphism For the (A 3 , A 3 ) theory, such a homomorphism is identified in (4.10). Let us denote the generators of S N +1 by σ 1 , · · · , σ N and that of Z 2 by ζ so that Since ϕ is a homomorphism, for each generator g of S N +1 × Z 2 , there exists g ∈ G such that ϕ( g) = g. Then, identifying the actions of σ 1 , · · · , σ N and ζ on the chiral algebra generators will strongly constrain the possible OPEs of them, since the set of the OPEs has to be invariant under the action of G. This would be useful in constructing the chiral algebras of the (A N , A N ) theory for N > 3.
for O = J b , W q1q2q3 , X a q are complete fixed by the conformal and flavor central charges. For the other OPEs, we here write down the most general expressions and then impose the constraint that all the Jacobi identities among the generators are satisfied. We will see that this constraint uniquely fixes those OPEs among the baryonic generators.
Let us first write down the most general OPEs. The most general OPE of W q1q2q3 (z) and W q ′ we set the functions f (1) so that (3.18) and (3.19) are realized. Note that our original ansatz for the OPEs are invariant under this rescaling. In other words, we fix this rescaling degrees of freedom by fixing f (1) so that (3.18) and (3.19) are realized. Now, the last step is to study the other Jacobi identities. One subtlety here is that such a computation involves operators of dimension three, some of which could potentially be null. Since the Jacobi identities have to be satisfied up to null operators, we have to identify all such null operators involved in (the LHS of) the Jacobi identities. Since operators of dimension smaller than three turn out to be not null, our strategy is that we first impose the constraint that all operators of dimension smaller than three involved in (the LHS of) the Jacobi identities vanish, and then identify the null operators of dimension three under this constraint. With this strategy, we find the null operators of dimension three shown in (3.27). Since the OPEs of these operators with every chiral algebra generator contains operators of dimension larger than or equal to their own dimensions, these operators are all null. Note that this null property just follows from the constraints that the LHS of the Jacobi identities do not contain operators of dimension smaller than three.  (3.27). Therefore, up to null operators, the OPEs among the generators are completely fixed by the constraint that the Jacobi identities among the generators are satisfied. The resulting OPEs are summarized in (3.17) - (3.22), in terms of (3.15) and (3.16).
It would also be useful to stress here that, if we do not fix the value of c 2d while fixing k 2d = −4, the constraint that the Jacobi identities are satisfied implies c 2d = −24. Therefore, with the flavor central charge k 2d = −4 fixed, the chiral algebra generated by T, J i , W q1q2q3 and X i q exists as a consistent Walgebra if and only if c 2d = −24.