Argyres-Douglas Theories, the Macdonald Index, and an RG Inequality

We conjecture closed-form expressions for the Macdonald limits of the superconformal indices of the (A_1, A_{2n-3}) and (A_1, D_{2n}) Argyres-Douglas (AD) theories in terms of certain simple deformations of Macdonald polynomials. As checks of our conjectures, we demonstrate compatibility with two S-dualities, we show symmetry enhancement for special values of n, and we argue that our expressions encode a non-trivial set of renormalization group flows. Moreover, we demonstrate that, for certain values of n, our conjectures imply simple operator relations involving composites built out of the SU(2)_R currents and flavor symmetry moment maps, and we find a consistent picture in which these relations give rise to certain null states in the corresponding chiral algebras. In addition, we show that the Hall-Littlewood limits of our indices are equivalent to the corresponding Higgs branch Hilbert series. We explain this fact by considering the S^1 reductions of our theories and showing that the equivalence follows from an inequality on monopole quantum numbers whose coefficients are fixed by data of the four-dimensional parent theories. Finally, we comment on the implications of our work for more general N=2 superconformal field theories.


Introduction
When they were first constructed, Argyres-Douglas (AD) theories were defined as singular points on the Coulomb branches of certain N = 2 gauge theories where mutually nonlocal BPS states become massless [1,2] (see also the generalizations in [3]). This definition lead to remarkable insights into AD theories (e.g., the construction of the Coulomb branch chiral ring via the Seiberg-Witten curve and much more). On the other hand, the approach of starting from a UV gauge theory makes the computation of many observables in AD theories difficult. For example, since the superconformal U(1) R ⊂ U(1) R ×SU(2) R symmetry is emergent from this perspective, computing the superconformal index of an AD theory is highly non-trivial. At a more conceptual level, this construction obscures the inherent simplicity of AD theories by adding many extraneous degrees of freedom.
In a recent pair of papers [4,5], we advocated a different approach for computing the superconformal index of AD theories in the Schur limit. 1 Our starting point was the class S realization of AD theories as compactifications of the (2, 0) theory on a sphere, C, with an irregular singularity and at most one additional regular singularity [7,8] (see [4] for a review). We then considered the topological quantum field theory living on C (in this case, two-dimensional q-deformed SU(2) Yang-Mills theory (YM) [9]) and defined a state in this theory corresponding to the irregular singularity (the states corresponding to regular singularities were already given in [9]). The resulting Schur index for the (A 1 , A 2n−3 ) theory then has a simple representation as a sum over the components of the wave function of the irregular singularity in the basis of irreducible SU(2) representations weighted by certain coefficients, while the index for the (A 1 , D 2n ) theory has a representation as a product of wave functions for the corresponding regular and irregular singularities [4].
An important aspect of our approach is that the irregular state is a simple and natural deformation of the regular state. As a result, it is possible to generalize our construction to limits of the index with more fugacities (and also, possibly, to the larger zoo of AD theories considered in [8,10], although we leave the study of such theories to future work).
In this note we propose just such a generalization to the Macdonald limit of the index.
Recall that this limit is given by [11] 2 I(q, t; x) = Tr H (−1) F e −β∆ q 2j 1 t R+r where the trace is taken over the Hilbert space of local operators, H, ∆ = Q 2− , (Q 2− ) † , j 1,2 are the SO(4) spins, R is the SU(2) R Cartan, r is the U(1) R ⊂ U(1) R × SU(2) R charge, and the f i are flavor charges. The fugacities q, t, and x i are complex numbers satisfying |q|, |t| < 1 and |x i | = 1. We arrive at the last equality in (1.1) by well-known arguments which show that only operators satisfying ∆ = E − 2j 2 − 2R + r = 0 contribute to the index. 1 See also the very interesting orthogonal approach to the problem presented in [6]. 2 In all formulas below, we follow the conventions of [12].
In our previous papers [4,5], we learned new things about AD theories by studying analytic properties of the Schur index. For example, we saw that the pole structure of the Schur limit-in particular the absence of certain poles-somewhat surprisingly encoded the spectrum of N = 2 chiral primaries 5 even though these operators do not directly contribute in this limit! 6 We took this fact as an indication of the simplicity of AD theories: the Schur sector does not consist of entirely new degrees of freedom but rather is highly constrained by the physics of the Coulomb branch. At the same time, we learned some lessons about more general N = 2 SCFTs. For example, we saw that the S 1 reductions of theories with generic N = 2 chiral ring spectra (i.e., theories with non-integer and non-half-integer dimensional N = 2 chiral primaries) should have three-dimensional flavor symmetries (acting on SU(2) L ⊂ SU(2) L × SU(2) R ≃ SO(4) R charged primaries) that mix with the four-dimensional U(1) R symmetry upon compactification (at least as long as the four-dimensional theory has a Coulomb branch).
In a similar spirit, we will study the pole structure of the more refined Macdonald index below. We will see that away from the Schur limit (and also away from the HL limit, i.e., taking generic t = q) many of the poles that were missing (and whose absence encoded aspects of the Coulomb branch physics) reappear. 7 This behavior arises because of an intricate set of operator relations in the Schur/Macdonald sector that we will only scratch the surface of in this note (but which we will return to as part of a larger study [15]).
In addition, we extend our discussion of the dimensional reductions of the (A 1 , A 2n−3 ) and (A 1 , D 2n ) theories that we initiated in [5]. However, instead of studying the resulting 5 These are primaries that are annihilated by the full set of anti-chiral Poincaré supercharges (theĒ operators in the language of [13]; see also the earlier classification in [14]). They are often referred to as "Coulomb branch operators" since, in all known examples, their vevs parameterize the Coulomb branch of a theory. 6 This surprise may be related to the fact that the authors of [6] were able to reproduce the Schur index by a BPS calculation on the Coulomb branch. 7 It would therefore be interesting to understand if the Coulomb branch BPS physics arguments of [6] carry over to the Macdonald limit.
S 3 partition functions, we will instead examine aspects of the resulting three-dimensional indices. This study gives rise to a derivation of the equivalence of the HL limit of the (A 1 , A 2n−3 ) and (A 1 , D 2n ) indices and the corresponding Hilbert series (for the (A 1 , A 2n−3 ) theories, this series was computed in [16]). Even more interestingly, we will find a sufficient condition for this equivalence that involves an inequality on the quantum numbers of three-dimensional monopole operators weighted by the mixing coefficients of the U(1) R symmetry descending from four dimensions with the topological symmetries of the dimensionally reduced theory. We will see that, as long as this mixing is sufficiently small, the HL limit of the index and the Hilbert series must agree. In generic N = 2 theories, this result suggests a new criterion for the absence of index contributions due to exotic D type HL operators (in the notation of [13]) and, possibly, a new constraint on RG flows between four and three dimensions that preserve eight supercharges.
Another important aspect of our previous work [4] involved a comparison of the Schur indices of the (A 1 , A 3 ) and (A 1 , D 4 ) theories with the torus partition function of the corresponding two-dimensional chiral algebras (in the sense of [17]). While the chiral algebras for the (A 1 , A 2n−3 ) theories with n > 3 and the (A 1 , D 2n ) theories with n > 2 are still not known (see, however, [6] for the chiral algebras of many other AD theories), there are certain universal chiral sub-algebras that must be present in these theories on symmetry grounds.
Moreover, we can use these chiral subalgebras to check the existence of certain operator equations predicted by our formulas. Note, however, that the chiral algebra only respects the superconformal quantum numbers of the Schur index and not all of those appearing in the Macdonald refinement. As a result, operator equations in the four-dimensional theory descend to null state relations in the two-dimensional chiral algebra that generally include terms whose four-dimensional pre-images violate the Macdonald quantum numbers (in this sense the Macdonald index also contains more refined information than the chiral algebra). 8 The plan of this paper is as follows. In the next section we briefly motivate our conjectures for the Macdonald indices of the (A 1 , A 2n−3 ) and (A 1 , D 2n ) theories. We then discuss certain checks applicable to the low-rank theories. In particular, we discuss checks arising from S-dualities involving these theories as building blocks [18][19][20] (see also the discussion in [21]). 9 We then move on to more general checks involving the RG flow, the emergence 8 A particularly trivial example of this phenomenon is given by those theories whose chiral algebras have a Sugawara construction. In that case, one finds an equation involving on one side a quadratic composite built out of currents (contributing at O(t 2 ) to the Macdonald index) and involving on the other side a stress tensor (contributing at O(qt) to the Macdonald index). 9 We hope that further generalizations of our work to the full class of theories considered in [8,10] will of known Higgs branch relations, the equivalence of the HL limit and the Higgs branch Hilbert series, and the matching of operator relations to null states in the chiral algebras.
In the following section we write our inequality on monopole quantum numbers and explain why the HL limits of the (A 1 , A 2n−3 ) and (A 1 , D 2n ) indices should coincide with their Higgs branch Hilbert series. Finally, we include an initial discussion of the analytic properties of the index and the resulting consequences for the operator spectrum. We close with some conclusions.

Motivating our Conjectures
While the Macdonald index of a four-dimensional N = 2 SCFT of class S does not correspond to a correlator in a two-dimensional q-deformed YM theory on the punctured compactification curve, C, general arguments suggest that it should still correspond to a correlator in some topological quantum field theory (TQFT) living on that curve [11].
Therefore, it is reasonable to assume that, just as in [4], we may find a state associated with the irregular singularities present in the class S constructions of the (A 1 , A 2n−3 ) and To give some further justification for our conjectures in (1.2), it is useful to first recall the Macdonald index of an A 1 theory corresponding to C of genus g with m regular punctures [11] where the coefficients, C λ , and the regular singularity wave function, f λ (q, t; x k ), are defined in (1.3) and (1.7) respectively. Note that the power of C λ is determined by the topology of C. Now, our formulas in (1.2) come from considering one and two punctured spheres and replacing a regular singularity wave function with that of the irregular singularity.
Therefore, in the case of the (A 1 , A 2n−3 ) theory, we have g = 0, m = 1, and so it is natural for C λ to appear to the first power. On the other hand, in the case of the (A 1 , D 2n ) theory, we have g = 0, m = 2, and so it is natural for C λ to not appear in the corresponding index.
We can further justify the form of the irregular singularity wave function by noting lim t,q→1 allow us to consider the non-self-dual S-duality discussed in [18] and its generalizations.
In other words, the irregular singularity wave function is a natural deformation of the regular singularity wave function and the corresponding Macdonald polynomials (up to a pre-factor that reflects the fact that the irregular singularity has U(1) instead of SU (2) flavor symmetry). More abstractly, we previously suggested [4] that, in analogy with the description of irregular singularities as coherent states in the generalized AGT correspondence [7,22], the irregular singularity state appearing in the index should be thought of as an analog of a coherent state in the TQFT on C. Therefore, if we think of the limit t, q → 1 as a sort of classical limit, it is natural for the regular state wavefunction to coincide with the irregular state wavefunction up to an overall normalization. 10 Another zeroth-order motivation for our conjecture is that it correctly reproduces the Schur limits of the (A 1 , A 2n−3 ) and (A 1 , D 2n ) superconformal indices [4,6]. Indeed, taking t → q in (1.2), we find that the irregular singularity wave function reduces to the expression in [4] lim t→qf (n) In particular, the second and third factors in (2.3) are just the deformed Schur polynomials we studied in [4]. Moreover, the C λ become the (normalized) coefficients of the irregular singularity wave function discussed in the expression for the (A 1 , A 2n−3 ) Schur index [4] lim 2 ) is the q-deformed dimension and N (q) ≡ 1/(q 2 ; q) ∞ . The regular singularity wave function factor in the (A 1 , D 2n ) index, f λ , behaves in the desired way under t → q by construction [11].
As a final motivation, recall that one general property of the Macdonald index is that it is finite in the limit we take q → 0 with t held fixed (this is the HL limit of the index). 11 Indeed, this statement follows from the general expression (1.1) and the fact that all Macdonald operators have j 1 ≥ 0. Note that this property is manifestly satisfied by our λ , and f λ are all finite in this limit. 10 For instance, recall that in the basic example of the quantum mechanics of a simple harmonic oscillator, coherent states are the closest analogs to classical physics: the uncertainty, ∆x∆p, is minimized in these states, and x and p are oscillatory with the classical frequencies and amplitudes. 11 A priori, it need not be the case that the Macdonald index is finite if we instead send t → 0 and keep q fixed. However, our conjectured forms of the (A 1 , A 2n−3 ) and (A 1 , D 2n ) indices are also finite in this limit. In section 6, we will see what this statement implies for the operator spectra.

Low-Rank Checks
In this section we perform checks of our conjecture (1.2) that only apply to the subset of theories in our class that have rank zero or one. Since theories of free hypermultiplets have a Lagrangian description, we can evaluate their indices by direct computation. For the (A 1 , A 1 ) theory, we have where x is a fugacity for the Sp(1) ≃ SU(2) flavor symmetry. 12 On the other hand, our conjecture implies We have checked that the two expressions (3.1) and (3.2) coincide to high perturbative order in q and t. This agreement is highly non-trivial because our conjecture does not rely on a Lagrangian description of the theory. Note also that this agreement implies that the manifest U(1) flavor symmetry in (3.2) is appropriately enhanced to SU(2).
Next, let us consider the (A 1 , D 2 ) theory. Since this SCFT is a theory of hypermultiplets, its Macdonald index is similarly evaluated as where x and y are fugacities for the Sp(2) flavor symmetry. 13 On the other hand, our conjecture implies that (3.4) 12 We use the convention such that the character of a fundamental representation of su(2) is x + x −1 . 13 We choose the basis of the Cartan subalgebra of sp (2) so that x and y can be regarded as fugacities for monomial su (2) representations monomial su(2) representations Table 1:  and (A 1 , D 4 ) theories. One simple check of our conjectures in these cases is that we find the correct flavor symmetry enhancement to SU(2) in the case of (A 1 , A 3 ) and SU(3) in the case of (A 1 , D 4 ). 15 14 As discussed below, the (A 1 , D 4 ) theory also has an enhanced flavor symmetry. 15 These symmetries can be understood as a consequence of the corresponding flavor symmetries of the RG flows from the SU (2) gauge theories with N f = 2, 3 described in [2]. To see this enhancement, first recall that our conjecture implies

The (A 3 , A 3 ) S-duality
In this subsection, we perform another check of the (A 1 , D 4 ) index. As discussed in [18] (see also the discussion in [19,21]), we can consider taking two (A 1 , D 4 ) theories along with a doublet of hypermultiplets and gauging a diagonal SU(2) flavor symmetry. The resulting coupling is exactly marginal, and the theory we obtain is identical to the (A 3 , A 3 ) theory [18]. This exactly marginal gauging implies that the Macdonald index of the (A 3 , A 3 ) theory can be written as fund (q, t; x, w) is the measure factor and I fund (w) are the vector multiplet and hyper multiplet indices, respectively.
On the conformal manifold of the (A 3 , A 3 ) theory, there are various cusps where a dual gauge coupling goes to zero. These cusps are related to each other by the S-duality group, which acts on the three mass parameters (corresponding to the U(1) 3 flavor symmetry) via S 3 [18]. In terms of the corresponding flavor fugacities, this action gives rise to [4] x Since the superconformal index is invariant under the S-duality, the index should satisfy the identity We have checked that our conjecture for I (A 1 ,D 4 ) (q, t; x, y) correctly reproduces (3.8), via (3.6), up to a high perturbative order in q and t. This result is highly non-trivial evidence for our conjecture. index. Our test again involves an S-duality in a similar spirit to the one described in the previous subsection. To that end, recall that the authors of [19] considered an SCFT built by taking an (A 1 , A 3 ) theory, an (A 1 , D 6 ) theory, and a fundamental hypermultiplet and gauging a diagonal SU(2) flavor symmetry. This gauging turns out to be exactly marginal, and the resulting SCFT is the (A 2 , A 5 ) theory. 16 Like the (A 3 , A 3 ) theory discussed above, the conformal manifold of this theory has multiple cusps where the duality group (SL (2, Z) in this case) acts on the two mass parameters of the theory (corresponding to the U(1) 2 flavor symmetry). Moreover, in [20], this action was argued to be via S 3 .
To see this symmetry at the level of the index, we first construct the (A 2 , A 5 ) index from the building blocks described in the previous paragraph It is then straightforward to check (as we have done perturbatively in q and t) that the index is invariant under the following action on the flavor fugacities i.e., that Let us now find the action of S 3 more explicitly. Denote the transformation in (3.10) as f . The index is also symmetric under the transformation, g, which takes x → 1 x and leaves y invariant. In terms of the corresponding chemical potentials, m x,y , we have Note that we have the relations which we recognize as the defining relations of S 3 (there cannot be additional relations since S 3 is the smallest non-Abelian group).

General Checks
In the following subsections, we perform checks of our conjectures that apply to all (A 1 , A 2n−3 ) and (A 1 , D 2n ) theories.

HL limit and Higgs branch relations
In this subsection, we recover the known Higgs branch relations for the (A 1 , A 2n−3 ) and (A 1 , D 2n ) theories [4,16,23] from our index. The most efficient way to find these relations is to consider the HL limit of the index [11]: we keep t fixed and set q → 0. In this limit, the index counts 3/8 BPS operators and takes the vastly simplified form where the trace is over the subset of operators contributing to the Macdonald index with A priori, there are two types of operators that can contribute in the HL limit. Using the nomenclature of [13], the first type are highest SU(2) R weight primaries ofB R multiplets, and the second type are highest SU(2) R weight first-level superconformal descendants in the more exotic D R(0,j 2 ) multiplets. TheB R primaries (with highest SU(2) R weight) are the familiar N = 1 chiral operators with scaling dimension E = 2R (they are anti-chiral with respect to the second set of Poincaré supercharges) whose vevs can parameterize the Higgs branch. 17 On the other hand, the generic D R(0,j 2 ) multiplets are less familiar. 18 In spite of the possibility of having exotic contributions to the HL index, we will find a consistent picture in which there are onlyB R type contributions. 19 In particular, we will see that the HL limit of our conjecture (1.2) coincides with the corresponding Higgs branch characters / Hilbert series (i.e., we will find a set of contributions equivalent to those coming fromB R operators modulo known Higgs branch constraints). In section 5, we will argue that this agreement follows from general principles. Note also that this picture is consistent with the chiral algebras of the (A 1 , A 3 ) and (A 1 , D 4 ) theories described in [4,6].
Indeed, from these chiral algebras, we know that there are no D R(0,j 2 ) multiplets in these theories. Finally, as we will see in section 6, simple analytic properties of our Macdonald indices rigorously forbid multiplets of the typeD R(j 1 ,0) with R = 0, 1 2 , 1 and j 1 ≥ R (along with their conjugates). 17 For example, the highest SU (2) R weight primaries ofB 1 2 are free hypers and those ofB 1 are holomorphic moment maps for N = 2 flavor symmetries. 18 However, the multiplets with R = j 2 = 0 are well-known since they contain free vectors. 19 The D 0(0,0) multiplets cannot be present since they contain free vectors. Similarly, the arguments in [18] rule out the presence of D 0(0,j2) multiplets with j 2 > 0.
To understand the above discussion more quantitatively, let us first take the HL limit of These formulas in turn imply that .

(4.3)
To see the physical meaning of the above expression, recall that the Higgs branch operators of the theory are M (with R = 1 and r = 0) and N ± (with R = n−1 2 and r = 0; the superscript is the U(1) charge for n > 3 and the SU(2) flavor weight for n = 3). These operators are subject to the Higgs branch constraint The quantum numbers of this constraint are R = n − 1 and r = 0. As a result, we see that (4.3) has a simple interpretation: it counts all products of M and N ± subject to (4.4).
Therefore the HL limit of our index agrees with the character of the Higgs branch chiral ring (which in turn agrees with the Higgs branch Hilbert series computation in [16]). In section 5 we will see that this agreement is no accident. This result also suggests that the Higgs branch chiral ring is equivalent to the HL chiral ring. 20 Let us now turn to the (A 1 , D 2n ) theory. Our conjecture (1.2) implies that the HL index of the theory is given by Note that f λ (0, t; y) = P HL λ (t; y)/ √ 1 − t(1 − ty 2 )(1 − ty −2 ) where P HL λ (t; y) is the HL polynomial: P HL λ (t; y) = χ su(2) λ (y) − tχ su (2) λ−2 (y) for λ > 0 and P HL 0 (t; x) = √ 1 + t. Using these expressions, we find that As in the case of the (A 1 , A 2n−3 ) theory, it is now straightforward to show that (4.6) agrees with the character of the Higgs branch chiral ring, char(H) (see appendix A for more detail).
We will return to discuss this agreement from first principles in section 5. This result is also strong evidence that the Higgs branch chiral ring is equivalent to the HL chiral ring. 21

Operator equations beyond the Higgs branch and null states in the chiral algebra
In this subsection, we would like to check certain operator constraints predicted by our conjectures that go beyond Higgs branch / HL chiral ring relations. In particular, we will focus on constraints involving the stress tensor multiplet (since the corresponding operators are universal) and / or derivatives of HL operators. Recall that the stress tensor multiplet is counted by the Macdonald index starting at O(qt).
In theories with flavor symmetries (for simplicity, and in order to directly connect our discussion to the (A 1 , A 2n−3 ) and (A 1 , D 2n ) SCFTs, we will assume the theories in question have either a simple global symmetry group or a single Abelian factor or both; generalizations of this ansatz are straightforward), the simplest operator constraints (that are counted by the Macdonald index) we can imagine involving the stress tensor multiplet and / or derivatives of HL operators take the form 22 where the M 11I are holomorphic moment maps for the simple global symmetry group (the first two numbers in the superscript are SU(2) R indices set to highest weight, and the third is an adjoint flavor index), M 11 U (1) is a holomorphic moment map for a U(1) flavor symmetry, J 11 ++ is the Schur component of the SU(2) R current (which sits in the stress tensor multiplet), the various κ's are theory-dependent constants, the f I JK are the structure constants of the flavor symmetry group, and d I JK is the rank-three symmetric invariant tensor. 23 There is no ordering ambiguity in (4.8) U (1) (0), and J 11 ++ (x)M 11I (0) OPEs do not contain singular terms. 24 Before continuing, let us note that the free hypermultiplet theory (the (A 1 , A 1 ) theory) satisfies (4.8). Indeed, in this theory we have J 11 Since the highest-weight component of the hypermultiplet is chiral, we can commute the hypermultiplets (and their derivatives). Therefore, it is trivial to see that these operators satisfy the first constraint in (4.8) with κ 1 = κ ′ 1 = 0 (d I JK = 0 in this case). Clearly we cannot use the HL index to study (4.8), since the operators in these relations do not contribute to this limit of the index. Moreover, in a general theory, it is non-trivial to use the Macdonald index to conclude that a constraint like (4.8) is present. Indeed, while this constraint contributes to the Macdonald index negatively starting at O(qt 2 ), there may instead be contributions of this same type coming from fermionic Macdonald operators. On the other hand, when we have a chiral algebra description of the Schur sector, we expect that (4.8) will descend to a relation in this chiral algebra. 22 By "simplest," we mean the lowest-dimensional independent operator constraints involving these operators and other universal operators. 23 In general, if the theory has additionalĈ 1(0,0) multiplets, they may appear in (4.8). 24 The absence of singularities in the OPEs of two holomorphic moment maps follows from a standard theorem regarding chiral ring operators. On the other hand, the absence of singularities in the J 11 ++ (x)M 11I (0) and J 11 ++ (x)M 11 U(1) (0) OPEs follows from SU (2) R symmetry and general Ward identities. Indeed, by SU (2) R conservation and dimensional considerations, the only potentially singular term in these OPEs would arise from a Schur operator in a multiplet of typeB 2 . If such a term were present, then its chiral algebra image would contribute to the O(z −1 ) term in the T (z)J I (0) (or T (z)J U(1) (0)) chiral algebra OPEs. However, this term is fixed by Ward identities to be a descendant. Since the chiral algebra image of aB 2 operator is a Virasoro primary [17], this is a contradiction.
However, the null state equation corresponding to (4.8) in the chiral algebra will in general include operators whose four-dimensional pre-images have Macdonald quantum numbers that are different from the corresponding quantum numbers of the operators appearing in (4.8). The reason for this discrepancy is that the mapping of the four-dimensional Schur sector to the two-dimensional chiral algebra involves an SU(2) R twist [17].
To see this violation of Macdonald quantum numbers more explicitly, first recall that a Schur operator transforms as the highest SU(2) R -weight component of an operator of SU(2) R spin R, O i 1 ···i 2R (we suppress Lorentz and flavor indices for simplicity), i.e., O Schur = O 1···1 . However, the chiral algebra states are associated with non-trivial representatives of cohomology classes of a certain nilpotent supercharge that is a linear combination of a Poincaré supercharge and a special supercharge. As a result, in order for translations in the chiral algebra plane to be compatible with this cohomology structure, we must study the cohomology classes associated with "twisted-translated" operators [17] O

The result of this process is written symbolically as O A (z) = χ [O 1···1
A ], where χ [· · · ] is the map that takes a Schur operator in four dimensions and gives a chiral algebra operator in the associated two-dimensional theory.
The mixing in (4.9) has additional implications for the images of composite operators.
In particular, where the a k are certain theory-dependent constants determined by the particular O . This fact makes finding the mapping between relations in four-dimensions and those in two dimensions highly non-trivial in general.
Let us now apply this discussion to the chiral algebra analog of (4.8). From the dictionary constructed in [17], we have the following 4d/2d maps where T is the two-dimensional holomorphic stress tensor, the J I (and J U (1) ) are currents of the two-dimensional Affine Kac-Moody algebra corresponding to the four-dimensional flavor symmetry, and ∂ is the holomorphic two-dimensional derivative (the normalization constants are determined in [17]).
Given the above discussion, we see that the relations in (4.8) descend to In writing these equations, we have assumed that there are no higher-spin symmetries (i.e., that the theory is interacting). Indeed, if there are higher-spin symmetries, then we might find contributions of Schur operators in multiplets of typeĈ 0( 1 2 , 1 2 ) in the R = 1 OPEs of the SU(2) R currents and moment maps and in the R = 1 OPEs of the moment maps with themselves. 25 The remaining operators that mix-in must be four-dimensional descendants and hence they must also be two-dimensional descendants (these are the operators multiplying γ andγ in (4.12)).
Note that in the case of the (A 1 , A 2n−3 ) and (A 1 , D 2n ) theories with n > 3 and n > 2 respectively, we do not know the full chiral algebras. However, we do know that there exist universal sub-algebras consisting of the affine Kac-Moody algebras corresponding to the U(1) and SU(2) × U(1) flavor symmetries as well as, in the case of the (A 1 , A 2n−3 ) theories, independent Virasoro sub-algebras (the (A 1 , D 2n ) theories all have Sugawara stress 25 For example, in the case of the (A 1 , A 1 ) theory, such pollution contaminates the expression in (4.12) (we have a Schur operator of the form ∂ ++ Q 1 I ∂ ++ Q 1 J sitting in aĈ 0( 1 tensors). 26 Using these sub-algebras, we will argue that the null state in (4.12) does not exist. We will see this picture is confirmed by our conjectures for the Macdonald indices.
Therefore, we arrive at an internally consistent set of results.
Let us first consider the (A 1 , A 3 ) SCFT. In this theory, we know that the only Macdonald / Schur operators consist of (derivatives of) products of the flavor symmetry moment maps and the SU(2) R current. Since there is a missing second 3 representation at O(qt 2 ) in table 1, we conclude that there must be a relation of the form (4.8) (note that d I JK = 0). We can find the corresponding two-dimensional null relation as in (4.12) by computing the following inner products for the states corresponding to the operators appearing in (4.12) In (4.13), I, J = 1, 2, 3 is an SU(2) adjoint index. We have defined (J∂J) To arrive at the last set of equations in (4.13), we have used the fact that k = − 4 3 and c = −6 (see the discussion in [4,6]). From these matrix elements it is straightforward to see that In particular, we find that in four-dimensions We can proceed similarly for the (A 1 , D 4 ) theory. Note that there is a small subtlety: at O(t 2 ) there is already a missing 8 representation due to the HL constraint (4.7) (with n = 2). In particular, it is easy to check that the corresponding chiral algebra null vector is 3 k + 1 = 0 (with A, B = 1, · · · , 8 and k = −3/2). 27 As a result, at level three, we have that d A CD ∂J C J D = 0. However, this is not an independent constraint since it follows from the level two constraint (and properties of the HL ring). Now, checking table 2, we see that there is an additional missing 8 representation at O(qt 2 ) and so we conclude there should be a relation of the form (4.8). We find the corresponding equation in the chiral algebra by computing the following matrix elements where A, B = 1, · · · , 8 are adjoint indices and we have used the fact that k = − 3 2 and c = −8 (see [4,6]). We have normalized the structure constants in accord with the conventions of the previous example. As a result, we find the following null state The corresponding four-dimensional operator equation is

Higher-rank theories
Let us first consider the (A 1 , A 2n−3 ) theories with n > 3. In this case, it is straightforward to expand our conjectured form of the index and observe that, subject to the assumptions that the only low-dimensional operators for generic n are (derivatives of) products of the 27 Mixing with ∂J A is forbidden by symmetry.
flavor moment maps and the SU(2) R current, there cannot be constraints of the type (4.8) at O(qt 2 ). In the chiral algebra we therefore expect there will not be a constraint of the form (4.12) subject to the same assumptions.
Indeed, we can compute the following matrix elements (from now on we change notation and take J U (1) → J hoping that confusion will not arise) where, without loss of generality, we have set the U(1) two point function, k 1 , equal to unity, and we have used the fact that c = −6n + 14 − 6 n [4,6]. It is straightforward to check that the corresponding matrix is not degenerate, thus confirming our intuition from the Macdonald index.
Next, let us study the (A 1 , D 2n ) theories with n > 2. These SCFTs have SU(2) × U(1) flavor symmetry. Expanding our conjectured form of the index subject to the assumptions we made in the (A 1 , A 2n−3 ) case above, we see that there should not be constraints of the type (4.8) at O(qt 2 ) either in the SU(2) adjoint channel or in the singlet channel. This calculation is straightforward but tedious and so we relegate it to appendix B.
We see that the constraint (4.8) and the corresponding chiral algebra null state (4.12) are very special. Unlike the singlet O(t 2 ) Higgs branch constraints in (4.4) (for n = 3), the first equation of (4.7), and the Sugawara stress tensor equations in the corresponding chiral algebras, the constraints we have studied in this subsection apply to a finite number of theories. This relation is particularly intriguing because it only seems to apply to theories with low-dimensional Coulomb branch (complex dimension one or zero).

The RG flow
In this section, we will study the compatibility of our conjectures for the Macdonald indices with an intricate set of RG flows described in [4]. Recall that these RG flows are triggered by giving a vev to some Higgs branch operator, O, of SU(2) R weight R O and charge f k,O under some U(1) k flavor symmetry (this generator may also be a Cartan of a non-Abelian flavor symmetry). Schematically, these RG flows are expressed as A 1 ) , (4.20) where T U V is the UV SCFT while T IR is the IR SCFT from which the decoupled axiondilaton multiplet is excluded (for these RG flows the axion-dilaton is a free hypermultiplet, i.e., the (A 1 , A 1 ) theory). Therefore, we will again use the prescription of [24] for relating the indices of the resulting IR endpoints of the RG flow to the indices of the corresponding UV endpoints where I IR is the index of the IR SCFT, and I −1 vect ≡ P.E. q+t 1−q is the index of the decoupled axion-dilaton multiplet. Note that, since the indices we study here have an additional superconformal fugacity compared to the Schur indices we analyzed in [4], we will need some more powerful mathematical tools for isolating the residues in (4.21). We will see that two particularly useful tools in our case are the q-binomial theorem and Bowman's generalization of Heine's transformation formula [25].

Rewriting the indices
Before beginning our analysis of the RG flows with endpoints in our class of theories, we would like to rewrite our conjectures for the Macdonald indices in such a way that we can easily extract the IR physics on the Higgs branch (i.e., so that we can straightforwardly apply (4.21) to an RG flow with (A 1 , A 2n−3 ) or (A 1 , D 2n ) as the short-distance fixed point).
Let us start with the (A 1 , A 2n−3 ) theory. Our conjecture (1.2) implies that where P λ andP λ are given in (1.4) and (1.6). Since the A 1 Macdonald polynomial is the ultraspherical polynomial, there is a simple expression for P λ (q, t; t 1 2 ): 23) where N λ (q, t) is the normalization factor given below (1.4). See appendix D for a derivation of this expression. By combining the above two equations and using the infinite q-binomial theorem ∞ λ=0 (a;q) λ (q;q) λ x λ = (ax;q)∞ (x;q)∞ , we obtain (4.24) We will use this expression to evaluate (4.21) for the (A 1 , A 2n−3 ) theory below.

The
Given our rewriting of the (A 1 , A 2n−3 ) index in (4.24), we will study the following RG flow In (4.24), the residue at x = t n−2 2 comes from the term proportional to (4.29) All the other terms are finite at x = t n−1 2 , and moreover the sum of all such finite terms is convergent because of the conditions |t| < 1 and |q| < 1. Therefore, the residue is evaluated as Res x=t n−1 2 Let us now use (4.26) to study the following RG flow (4.31) Recall from the discussion in [4] that we can generate this RG flow by turning on L 2 = 0 and keeping the vevs of the remaining generators of the Higgs branch set to zero (these operators were discussed around (4.7)). Therefore, from (4.21), we see that to study the flow (4.31), we should take the residue of I (A 1 ,D 2n ) (t, q; x, y) at x = t n 2 y. To calculate this residue, we have to understand the analytic structure of the basic hypergeometric series. It turns out that there is a particularly useful rewriting of the basic hypergometric series due to Bowman [25]. Indeed, he found that (4.33) This rewriting makes it manifest that, as a function of z, 4 ϕ 3 ( a; b; z) has simple poles at q k z = 1 for k = 0, 1, 2, 3, 4, · · · . This property of the basic hypergeometric series implies that, in the sum over m 1 in (4.26), the terms with m 1 = 0 have simple poles at x = t n 2 y. All the other terms (with m 1 > 0) are finite at x = t n 2 y (as long as the values of the other fugacities, q, t, and y, are generic). Moreover, the sum of all such finite terms is convergent due to the conditions |q| < 1, |t| < 1, and |y| = 1. Therefore, the residue of I (A 1 ,D 2n ) (q, t; x, y) at x = t n 2 y is evaluated as where y ≡ √ ty 2 is the fugacity for the correct IR flavor symmetry as discussed in [4]. This result is perfectly consistent with the RG flow described in (4.31), since the IR SCFT with the axion-dilaton removed is just a free hypermultiplet, i.e., the (A 1 , A 1 ) theory.

The
Finally, we study the RG flow From the discussion in [4], we know that this RG flow can be initiated by turning on This result is in perfect agreement with the RG flow in (4.35).

The HL Limit vs the Higgs Branch Hilbert Series and an RG Inequality
In section 4.1, we saw that the HL limits of the (A 1 , A 2n−3 ) and (A 1 , D 2n ) indices agreed with the corresponding Higgs branch Hilbert Series. In this section, we would like to demonstrate that this agreement must occur.
Let us outline the argument. First recall that the index can be thought of as a twisted partition function on S 1 × S 3 . We can then take the four-dimensional index and reduce it to the three-dimensional index via a Z n (n → ∞) quotient of the Hopf fiber of the S 3 [26]. Since the HL index does not count operators with angular momentum quantum numbers along this fiber, we see that it is invariant under the reduction to three dimensions.
Moreover, as we will see, the HL index reduces to the three-dimensional Higgs index. This latter index is then equivalent to the three-dimensional Higgs branch Hilbert series [27] (which is, in turn, equivalent to the four-dimensional Higgs branch Hilbert series). 28 Note, however, that the above discussion is somewhat non-trivial in our case. Indeed, as observed in [5], Therefore, we should check that this mixing does not spoil the above argument. Intuitively, we do not expect this to be the case since, as we saw in [5], the mixing with topological symmetries was associated with the absence of certain poles involving non-HL operators.
To understand the above discussion in more detail, first recall the form of the fourdimensional Lens space index (i.e., the partition function on S 3 /Z n × S 1 ) where the n = 1 case is the usual superconformal index, while the limit n → ∞ yields the three-dimensional index (here β is the circumference of the S 1 ) [26]. The quotient acts on the Hopf fiber via the phase, exp 2πi n (see [27] for a thorough review).
Let us now rewrite (5.2) using the substitution p → √ xxy, q → √ xxy −1 , and t → x Now, consider taking the HL limit. This amounts to taking p, q → 0 with t fixed. In terms of our redefined fugacities, this is equivalent tox → 0 with x fixed. Since the HL limit is independent of j 1 , we drop the dependence on y. In particular, we find reduction is not "bad"-see the discussion in [27]). In our cases of interest, the dimensional reductions are all "good." 29 There is no such mixing only in the (A 1 , D 4 ) case. and the final trace is over states with j 2 − r = 0.
We can now take the limit n → ∞ in (5.4) and map the various charges appearing in (5.4) to three dimensions via the following dictionary Let us also defineẼ HereẼ is the three-dimensional scaling dimension for short multiplets that contribute to the (three-dimensional) index (while E is the corresponding scaling dimension in fourdimensions). We then find This is just the form of the Higgs index given in [27] (modulo mixings with topological symmetries, which were vanishing in the theories considered there).
To complete the argument, note that in the limit (5.7), only operators satisfyingẼ = I R 3 + c a H C a contribute to the index. In principle, we could imagine two types of operator contributions: those from operators that are charged under the topological symmetries and those from operators that are not. Let us first consider the case of operators that are singlets under the topological symmetries. In this case, we needẼ = I R 3 . In our theories, such contributions come from three-dimensional Higgs branch scalars (since we have identified the correct symmetries of the IR theory).
Next, let us consider potential contributions from operators charged under the topological symmetries. A sufficient condition to rule out such contributions in our theories is to show that for any monopole primary, O, the following inequality holds In (5.8) we have used the fact that monopole primaries are SU(2) R -neutral. In both the (A 1 , A 2n−3 ) and (A 1 , D 2n ) cases, it is easy to check that the mirror of the above inequality holds for the matter operators in the mirror theories (see appendix C). Therefore, (5.8) holds for all short multiplets of our S 1 reductions, and the Higgs branch Hilbert series and HL index agree as promised.
Note that (5.8) is an inequality that depends on both ends of the β → 0 limit of the RG flow. Indeed, while the LHS and the H C a (O) are determined by the three-dimensional longdistance physics, the mixing coefficients, c a , are determined by the UV four-dimensional theory. It would be interesting to understand if (5.8) is an inequality that holds for all RG flows from four dimensions to three dimensions that preserve eight supercharges or if it can be violated in some theories by sufficiently large U(1) R mixing with Coulomb branch symmetries.

Comments on Analytic Properties of the Index
In this section, we would like to make some preliminary comments on the analytic structure of the (A 1 , A 2n−3 ) and (A 1 , D 2n ) Macdonald indices. It turns out that these indices have the following properties: (i) The coefficients of all the terms in the q, t expansion are positive integers.
(ii) The t → 0 with q fixed limits of our indices are finite and equal to unity.
(iii) Certain poles associated with non-HL operators that were absent in the Schur limit reappear in the Macdonald limit.
Let us first examine (i). For the (A 1 , A 2n−3 ) theories, it follows from (1.6), (4.22), and (4.23) that It is now clear that, when expanded in powers of q and t, the index has only positive integer coefficients. It is straightforward to prove the same statement for the (A 1 , D 2n ) theories. Indeed, as shown in appendix D, the (A 1 , D 2n ) index can be rewritten as where λ m q ≡ (q; q) λ /{(q; q) m (q; q) λ−m } is the q-binomial coefficient and therefore a polynomial in q with positive integer coefficients. It is manifest in (6.2) that, in the expansion in powers of q and t, all the coefficients are positive integers.
These statements are consistent with the conjecture that all the Schur operators in the (A 1 , A 2n−3 ) and (A 1 , D 2n ) theories are bosonic operators. In particular, they are consistent with the conjecture that the only Schur generators in our theories are the SU(2) R current, the flavor currents, and the baryons. 30 More generally, it is consistent with all the checks we performed above (we were not forced to include fermionic degrees of freedom).
Let us now discuss property (ii). It is straightforward to check that (6.1) and (6.2) imply lim t→0 where we have held q fixed. From this simple fact, it immediately follows that: • The (A 1 , A 2n−3 ) and (A 1 , D 2n ) theories do not have multiplets of typeD R(j 1 ,0) with R = 0, 1 2 , 1 and j 1 ≥ R or conjugate multiplets of type D R(0,j 2 ) with R = 0, 1 2 , 1 and j 2 ≥ R.
To understand this statement, first note that theD R(j 1 ,0) multiplets with R = 0, 1 2 , 1 (except forD 1(0,0) ) and their conjugate multiplets do not participate in recombination rules that give long multiplets [13] (here we are using the absence of higher-spin symmetries). Their single letter contributions to the index are Note that (6.4) is singular in the t → 0 limit of the index for j 1 > R. Moreover, it is easy to check that these are the only non-recombinant Macdonald multiplets that contribute singularly in the limit t → 0 and that these multiplets do not experience "accidental" cancelations when q is allowed to remain arbitrary. Since the limit (6.3) is non-singular, it is then impossible to haveD R(j 1 ,0) multiplets with R = 0, 1 2 , 1 and j 1 > R (similar statements hold for the corresponding conjugate D multiplets). To rule out the remaining multiplets with R = j 1 = 0, 1 2 , 1, we note that these multiplets would contribute a q-dependent piece to (6.3). The absence of higher-spin symmetries forbid any canceling contributions to the lowest-dimensional such contributions. Therefore, these multiplets are also absent.
These results are entirely consistent with the picture we have described so far. Moreover, the above conclusions extend and confirm the results of [12], which imply the absence of multiplets of type D 0(0,j 2 ) andD 0(j 1 ,0) in our theories.
Let us now turn to the property (iii). To see an example of (iii), consider the index of the (A 1 , A 2n−3 ) theory for n ≥ 3. Let us study the residue at x = q 2 t with some non-HL operator, O. From the expression (4.24), we see that the residue of Note that this residue is indeed zero for t → q.
The residue can vanish in this limit because, while some constraint in the theory has the same Schur quantum numbers as O, it has different Macdonald quantum numbers. One way in which such a constraint may arise is if we have a relation of the form where J 11 ++ is the Schur component of the SU(2) R current and O is a second derivative of an HL operator. Indeed, we saw an example of such a constraint in (4.15) for n = 3 (and k = 1). The basic point is that although J 11 ++ has the same Schur quantum numbers as two derivatives, it has different Macdonald quantum numbers. As a result, constraints like (6.6) will not cancel a pole associated with the second derivative of an HL operator in the generic Macdonald limit but may cancel that pole in the Schur limit.
Constraints of the form (6.6) are intriguing. Indeed, the SU(2) R current knows about all sectors of the AD theory, since any multiplet has operators charged under SU(2) R (in the case of "Coulomb branch" operator multiplets, these are superconformal descendants).
Moreover, in our previous work [5] we saw that the absent poles in the Schur limit were intimately connected with the fact that the quantum numbers of the "Coulomb branch" operators were secretly encoded in the index. Therefore, understanding the physics associated with constraints of the form (6.6) may point the way to constructing the full index of our AD theories.

Discussion
We have generalized our construction of the Schur indices of the (A 1 , A 2n−3 ) and (A 1 , D 2n ) theories to the Macdonald limit. In performing various checks of our conjectures, we arrived at an intriguing inequality on monopole quantum numbers in the dimensional reductions, and we found some interesting operator relations involving the SU(2) R current and various HL operators. We expect both these results to be useful in studying generic N = 2 theories.
A natural (partial) list of future directions include: • Generalize our formulas to include the final superconformal fugacity. Better understanding the operator relations involving the SU(2) R current and the HL operators might be useful.
• Find the full set of theories which satisfy (4.8). Do the corresponding chiral algebras necessarily have stress tensors given by the Sugawara construction? Is the list of these theories finite? Perhaps further understanding this equation can give additional insight into the possible set of low-rank N = 2 theories with flavor symmetries (recall that this equation only held in the rank zero and one theories we studied). 31 • Can the inequality we found on monopole operator quantum numbers, (5.8), be violated in more general theories? Are there interesting theories for which this is an equality? In this case, (5.8) might be an interesting generalization of the topological criterion for the appearance of D multiplets in the special subset of class S theories with only regular singularities [11].
• On the other hand, if (5.8) cannot be violated, then it may imply interesting constraints on UV versus IR physics of general RG flows from 4d to 3d preserving eight supercharges. In this sense it would be somewhat similar in spirit to other constraints on the RG flow that are already known (e.g., [29]) or conjectured (e.g., [30]). The main novelty would be a constraint on flows between dimensions. See appendix C for a very non-trivial check of this inequality in our theories (especially in the large central charge limit).
• Further study the mathematical meaning of our deformation of the Macdonald polynomials given in (1.6). Since the A 1 Macdonald polynomials are equivalent to ultraspherical polynomials, our deformation can also be regarded as a deformation of these latter polynomials. T. N. is also partially supported by the Yukawa Memorial Foundation.
It is again straightforward to check that the matrix of the above inner products has nonvanishing determinant. These results therefore serve as a useful consistency condition for our conjectures.

Appendix C. Explicit check of our monopole inequality
In this appendix, we verify that the inequality ( whereÕ is the (matter) primary dual to O, andH a is the flavor symmetry dual to H C a . First consider the (A 1 A 2n−3 ) theory. Recall from [8] that the three-dimensional mirror of this theory can be reached by an RG flow from N = 4 SQED with N f = n − 1. We denote the fundamental fields X I and the anti-fundamental N = 4 partners Y I (with I = 1, · · · , n − 1). To demonstrate (C.2), it suffices to show it holds for all operators built from the squarks.
To prove this latter statement, recall that c a = 1 n √ 2 (−1) n+a a(a + 1) [5]. Therefore, we have a c aH a (X I ) = 1 n √ 2 a (−1) n+a a(a + 1) · ν Ia = 1 2n (−1) n+I · (I − 1) + n−2 a=I (−1) n+a where the ν I are the weights for the fundamental representation of the SU(n − 1) flavor symmetry. Note that the same inequality holds for X †I , Y I , and Y † I . If n + I is even, then, we have that the LHS of (C.3) is I 2n ≤ n−2 2n < 1 2 . On the other hand, if n+I is odd, we have that the LHS is −I+1 2n with −I+1 2n ≤ n−3 2n < 1 2 . As a result, we see that all gauge-invariant matter operators built from the squark superfields satisfy (C.2). 32 where 4 ϕ 3 is the basic hypergeometric series given by On the other hand, in the case of m 1 ≤ m 2 , the sum over λ becomes (q; q) m 2 (t; q) m 2 −m 1 (t 2 ; q) m 2 (q; q) m 2 −m 1 (1 − tq m 2 )(q nm 1 t