The Coulomb and Higgs Branches of $\mathcal{N}=1$ Theories of Class $\mathcal{S}_k$

Even though for generic $\mathcal{N}=1$ theories it is not possible to separate distinct branches of supersymmetric vacua, in this paper we study a special class of $\mathcal{N}=1$ SCFTs, these of Class $\mathcal{S}_k$ for which it is possible to define Coulomb and Higgs branches precisely as for the $\mathcal{N}=2$ theories of Class $\mathcal{S}$ from which they descend. We study the BPS operators that parameterise these branches of vacua using the different limits of the superconformal index as well as the Coulomb and Higgs branch Hilbert Series. Finally, with the tools we have developed, we provide a check that six dimensional $(1,1)$ Little String theory can be deconstructed from a toroidal quiver in class $\mathcal{S}_k$


Introduction
Since the seminal work of Gaiotto [1], a lot of milage has been gained in the study of four dimensional superconformal field theories (4d SCFTs) when compactifications of six dimensional SCFTs (living on M5 branes) on various Riemann surfaces are considered. Focusing on the so called Gaiotto's class S of N " 2 SCFTs we have learned a lot about the 4d gauge theory thinking about simpler theories in two dimensions; a 2d CFT in the case of the partition function on S 4 [2] or a 2d TQFT in the case of the superconformal index [3,4]. For generic N " 1 SCFTs the six dimensional (M-theory) approach [5] has been proven to be far more complex. However, there is a class S k of N " 1 SCFTs, obtained by compactifications on Riemann surfaces of the 6d SCFTs living on M5 branes probing a Z k singularity [6], for which there is hope that similar results with N " 2 are attainable [7][8][9][10]. 1 In the study N " 2 SCFTs it has been proven fruitful to concentrate in one of the two distinct branches of supersymmetric vacua; the Coulomb branch or the Higgs branch. These vacua are parameterised by vacuum expectation values of certain local gauge invariant operators with very special properties. In the case of the Coulomb branch, the so called Coulomb branch operators obey chiral 1/2 BPS shortening conditions and form a closed chiral ring. On the other hand, the Higgs branch operators are annihilated by two of the supercharges Q and two of the r Q and they also form a closed ring of operators with very special mathematical structure.
The study of the Coulomb branch of N " 2 theories led to several important developments related to non-perturbative properties of the corresponding gauge theories. In the seminal works [17,18] Seiberg and Witten showed that the low-energy effective action of N " 2 gauge theory can be completely specified by an holomorphic function containing both perturbative and non-perturbative contributions. Nekrasov gave a micropic derivation of the Seiberg-Witten results computing the instanton partition function [19] which finally 1 There is a larger class of N " 1 theories which are obtained as Γ orbifolds of N " 2 theories in class S, known as SΓ [11][12][13][14][15][16] for which there is also hope to extend our results [7][8][9] to. Theories in class SΓ are obtained by compactifications on Riemann surfaces of the 6d (1,0) SCFTs living on M5 branes probing a Γ P ADE singularity. was used by Pestun on the supersymmetric localization on S 4 [20]. On the other hand, it is also long known that the Higgs branch of a N " 2 theory is endowed with a HyperKähler structure that fixes the metric [21][22][23].
Correlation functions of the Coulomb branch operators are non-trivial functions of the marginal coupling constants of the SCFT. Nonetheless, they can be determined from a simple deformation of Pestun's matrix model via [24][25][26][27] 2 . On the other hand, correlation functions of Higgs branch operators are independent of the coupling constants [29,30] and thus fixed, up to multiplet recombination, by a tree level calculation, which can be mapped to a 2d chiral algebra computation [29].
The purpose of this paper is to show that theories in class S k , precisely because they are orbifold daughters of N " 2 SCFTs in class S also have similar branches of supersymmetric vacua as their N " 2 mother theories. They enjoy, as opposed to generic N " 1 theories, a second U p1q (anomaly free [6]) global symmetry on the top of their R-symmetry, which allows for a well defined Coulomb and Higgs branch.
The content of this paper is as follows. In Section 2 we review some basic facts about the theories of class S k that we are going to study in this paper, while taking the opportunity to introduce notation. In Section 3 we introduce the Hilbert series (HS) which will be our first tool for counting operators in chiral rings. Our second tool, the Superconformal index (SCI) and some interesting limits of it are introduced in Section 4. In Section 5 we study some examples of genus zero theories 3 with a Lagrangian description. We separately compute the Coulomb branch SCI, the Coulomb branch Hilbert series and show that they are equal. What is more, we compute the Hall Littlewood limit of the superconformal index, as well as the Higgs branch Hilbert series and also find that they are equal, precisely as for the mother N " 2 SCFTs of class S [31]. Succeeding that, in Section 6, we study some examples of genus one theories and we compute their Higgs branch HS as well as their Hall Littlewood index. We take the opportunity to analyse how the 6d p1, 1q little string theory (LST) [32] can be deconstructed using a toroidal class S k theory, in the past also referred to as the "Conformal Moose". In Section 7, we consider strongly interacting class S k theories, without a Lagrangian description, arising as S-dual descriptions (pants decompositions) of the A 1 four punctured sphere in class S 2 . We compute the Coulomb and Hall Littlewood indices for these theories and show that they can be written in a simple closed form. Finally, we end with some conclusions in Section 8.
In Appendix A we provide a short review of the so called algebra-geometry correspondence, whilst in Appendix B we collect the mathematical identities that we used through the paper. Finally important facts about N " 1 superconformal representation theory, short and semi-short multiplets, and the Superconformal Index are reviewed in Appendices C and D. 2 Or can also be obtained using data from the SW curve for non-Lagrangian cases [28]. 3 Throughout the paper we will use the jargon "genus g theories" by which we mean that these are theories in class S k obtained by compactifications on Riemann surfaces of genus g of the 6d (1,0) SCFTs living on M5 branes probing a Z k singularity.  2 4d N " 1 quiver gauge theories in Class S k In this Section we review some background material for quiver gauge theories in Class S k introduced in [6]. Experts can skip reading this Section. We will mostly follow the notation of [10]. From an M-theory perspective these theories arise considering a stack of N M5-branes probing a A k´1 singularity. In the following our analysis will be based on the corresponding type IIA brane configuration reported in Table 1.
For k " 1 we are describing N " 2 theories of Class S which arise through the compactification of the 6d p2, 0q SCFT living on the stack of M5-branes over a 2d Riemann surface of genus g decorated with punctures. The R-symmetry group of these theories is U p2q » SU p2q R N "2ˆU p1q r N "2 . Looking at Table 1 we observe that the sup2q R N "2 » sop3q R N "2 subalgebra of the R-symmetry group is described by rotations along tx 7 , x 8 , x 9 u directions. This symmetry is broken to a U p1q R by the orbifold projection. The unbroken U p1q R corresponds to the Cartan generator of the SU p2q R N "2 . On the other hand, the U p1q r N "2 » SOp2q subgroup is described by rotations along tx 4 ,x 5 u directions and it is left unbroken by the orbifold.
For k " 1, the Class S theories obtained from the string theory setup in Table 1 are known as linear quiver gauge theories, with a quiver diagram like the one in Figure 1. The circular nodes of the quiver correspond to SU pN q gauge groups while the square boxes correspond to SU pN q flavour groups. We can glue together by gauging the two extremal flavour nodes of the linear quiver, this way we obtain a circular or elliptic Class S theory. An example of the latter is given in Figure 7.
The 4d Class S k theories are realised as twisted compactifications of the 6d N " p1, 0q SCFT on the worldvolume theory on N M5-branes probing the transverse A k´1 singularity which has up1q t 'supkq β 'supkq γ global symmetry. The compactifications typically preserve n " 0 n " 1 n " 2 n " 3 n " 4 SU pN q pi,n´1q SU pN q pi,nq SU pN q pi´1,nq U p1q t U p1q αn U p1q β i`1´n U p1q γ i V pi,nq 1 Adj Table 2: Symmetry transformations of the fields of a class S k theory.
only a Cartan subalgebra up1q t 'up1q 'k´1 β 'up1q 'k´1 γ which are 'intrinsic' global symmetries carried by all class S k theories. Moreover, there is also the N " 1 up1q r R-symmetry. The Cartan generators of these symmetries are given by a linear combination of the Cartans tR N "2 , r N "2 u of the sup2q R ' up1q r N " 2 R-symmetry algebra r " 2 3 p2R N "2´rN "2 q , q t " R N "2`rN "2 . (2.1) where q t denotes the Cartan of the up1q t symmetry. We report an example of such a quiver gauge theory in Figure 2. We summarize the transformations of the field content for this theory in Table 2.
For k ě 2, the orbifold breaks half of the initial supersymmetry giving rise to 4d N " 1. States of the N " 2 mother theories of Class S live in unitary representations of the sup2, 2|2q superconformal algebra. They are labelled by the pE, j 1 , j 2 , R N "2 , r N "2 q quantum numbers which denote representations under the maximal bosonic subalgebra up1q E ' sup2q 1 ' sup2q 2 ' sup2q R N "2 ' up1q r N "2 Ă sup2, 2|2q . (2. 2) The orbifold action breaks the sup2, 2|2q superalgebra down to sup2, 2|1q'up1q qt . Representations of this algebra are labelled by the Cartans pE, j 1 , j 2 , r; q t q of the maximal bosonic subalgebra From the initial eight N " 2 supercharges, only the ones with q t " 0 charge survive after the orbifold projection, to give the four supercharges of the sup2, 2|1q superalgebra Q :ˆ1 2 ,˘1 2 , 0,´1; 0˙, r Q :ˆ1 2 , 0,˘1 2 , 1; 0˙. (2.5) The pj 1 , j 2 , r; q t q quantum numbers of the field content is presented in Section 4 in Tables  4 and 5. While the complete list of supercharges is reported in Table 3. The punctures carry additional data associated to inserting a variety of defect operators in the worldvolume of the 6d N " p1, 0q SCFT, which are localised at the punctures Figure 2: Quiver for a Class S k theory. The gauge nodes are labelled by pi, nq, where i " 1, ..., k is the Z k orbifold index while n " 1, ..., ´3 is the label from the N " 2 mother theory. In this example we set ´2 " 4.
[ 6,11,33] of the Riemann surface, and spacetime filling in tx 0 , x 1 , x 2 , x 3 u. In this paper we will focus only on maximal and minimal punctures which are well understood. A complete classification of puctures is yet missing [11,33].
Maximal punctures are labelled with a "colour" c P t1, 2, . . . , ku quantum number, a sign σ "˘1 and an "orientation" o " l{r. We will label these maximal punctures with the notation s o,σ c . Maximal punctures also carry an associated supN q 'k flavour symmetry algebra. Minimal punctures carry a up1q symmetry under which the baryonic operators of the form det Q i and det r Q i are charged, while mesonic operators of the form M i,i "Q i Q i are uncharged.
The analogous of the Class S linear quiver gauge theory, in the context of Class S k , are theories associated to spheres with ´2 minimal punctures and two maximal punctures s l,c l & s r,c r with c r " pc l` ´2q mod k, and 1 2k unit of flux for U p1q t . As we will discuss these theories will be the basic building blocks for more involved theories, for this reason we refer to these as "core" theories. These theories admits a weakly coupled Lagrangian description associated to a pair of pants decomposition of the Riemann surface into a chain of n " 0, 1, . . . , ´3 spheres with one minimal puncture and two maximal punctures s l,c l`n & s r,c l`n`1 . These three punctured spheres correspond to quiver theories of bifundamental chiral multiplets called the free trinion which is pictured in Figure 3. See Table 18 for the corresponding charges under symmetries. Figure 3: On the left we depict the quiver diagram for the free trinion theory of bifundamental chiral multiplets (arising as the orbifold of a bifundamental hypermultiplet). On the right we depict the associated sphere with one minimal puncture and two maximal punctures s l,c and s r,c`1 (the fugacities z l and z r label the global symmetry groups associated with the two maximal punctures).
The maximal punctures of equal colour, opposite orientation and equal sign of the n th and n`1 th three punctured spheres are then glued with tubes associated to spheres with two maximal punctures s l,c l`n`1 & s r,c l`n`1 . This gluing corresponds to gauging the diagonal supN q 'k z Ă supN q 'k z l ' supN q 'k zr of the two maximal punctures with free N " 1 vector multiplets and k bifundamental chiral multiplets Φ i . This type of gluing is called Φ-gluing 4 and is pictured in Figure 4. The representations under the global symmetries are summarised in Table 5. From these theories it is possible to construct theories associated to tori with minimal punctures by Φ-gluing the 'open' maximal punctures s l,c l & s r,c r , these are Z kˆZ orbifold theories of N " 4 SYM. Unless c r´cl " 0 mod k (or equivilantly ´2 mod k " 0) this procedure breaks the up1q 'k´1 β ' up1q 'k´1 γ symmetry. The quiver diagram of such theories can be found in Figure 6. We will discuss theories of this type in Section 6.
Finally, a general Lagrangian theory in Class S k , made using the above ingredients, associated to a genus g Riemann surface with punctures has superpotential Figure 4: Quiver associated to Φ-gluing.

The Moduli Space of Supersymmetric Vacua
In order to appreciate how special class S k theories are, in the first part of this Section, we review the notion of moduli space of vacua valid for a 4d N " 1 theory. Theories with N " 1 supersymmetry can have a moduli space of supersymmetric vacua M which is given by the space of solutions to the F-term and D-term constraints modulo gauge transformations by gauge group G. Setting the F-terms and the D-terms to zero and modding out by G is equivalent to dropping the D-term constraints and modding out by complexified gauge transformations G C [35]. Therefore the moduli space can be described by the following quotient between the master space F [36,37] and the complexified gauge group here v i denote the scalar vevs, F i " BW {Bv i are the F-term constraints and W is the superpotential. From a geometrical point of view F is a complex algebraic variety, therefore it can be characterised in terms of a quotient ring R{I where R " Crv 1 , v 2 , . . . s is the polynomial ring in the v i and I " xF 1 , F 2 , . . . y is the the F-terms ideal 5 . On the other hand, the moduli space is given by the spectrum M " spectAu of some algebra A. In Appendix A we have spelled out a lot of useful details regarding this algebra-geometry dictionary. In generic N " 1 gauge theories, the superpotential can receive quantum corrections which can be perturbative and non-perturbative, which, however, are constrained by holomorphicity arguments [38]. The same is true for the moduli space of vacua M -M Classical . In the quantum level it is generically different from the classical one since the metric receives corrections and singularities may appear (when massive particles becomes massless). Generally, the problem of obtaining the quantum moduli space can be a difficult one. 5 Given a polynomial ring Crx1, ..., xns and some polynomials f1, ..., fs P Crx1, ..., xns we define xf1, ..., fsy :" ř s i"1 hifi with h1, ..., hs P Crx1, ..., xns , and we call xf1, ..., fny the ideal generated by f1, ..., fn.
As long as we have supersymmetry, there exist further tools which allows us to study the algebra A, the spectum of which gives the moduli space M " spectAu. These are the Hilbert Series (which we will introduce in this Section) and the superconformal index (which we will introduce in the next Section).

Higgs and Coulomb Branches for Class S k Theories and the Hilbert Series
Before plunging in Class S k a lightning review of Class S is in order. For N " 2 SCFTs the Coulomb branch is parameterised by the so called Coulomb branch operators which obey chiral 1/2 BPS shortening conditions Q I α O C " 0 @ I, α (E r in [39] notation) with E " r N "2 and form a closed chiral ring. On the other hand, the Higgs branch operators obey the shortening condition E " 2R N "2 (B R in [39] notation) which means that they are annihilated by one Q and one r Q and they also form a closed subsector of operators with interesting mathematical structure. These two shortening conditions E " r N "2 and E " 2R N "2 define subsectors of operators which are closed (do not mix with other under renormalization) and have ring structure.
The full moduli space for theories in class S k are very rich and share many similarities with N " 2 theories, in particular they possess Coulomb, Higgs and mixed phases. The Coulomb moduli spaces have been studied in [7,10,40]. Here we will mainly focus on two truncations of the full moduli space which, in N " 2 nomenclature, we will refer to as Higgs and Coulomb branches. As we will see these truncations are well defined and will be useful.

The Higgs Branch
Let us first present the definition for the Higgs branch for theories of Class S k . This definition is valid for theories either with a Lagrangian description or those related to theories with a Lagrangian description by dualities and can be made by restricting to scalar operators that have where q t and r denote the charges of the up1q t and N " 1 up1q r R-symmetry respectively. Provided that the R-symmetry is not broken (which is true for any N " 1 SCFT) and that up1q t is also anomaly free (shown in [6]) we can always decompose a ring R under the up1q r ' up1q t grading. The two conditions on (3.2) combine to with the first equality being the BPS condition B r,pj 1 ,0q in the notation [39] (or LB 1 rj 1 , 0s prq E in the the notation of [41]) as summarised in Table 14. These type multiplets are isolated short multiplets from all other types of multiplets (both the continuum of long and other short multiplets) by a gap. What is special about the Higgs branch for theories of Class S k , as opposed to general N " 1 SCFTs, is that the R-symmetry of the N " 1 superconformal algebra is further related to the charge q t of the extra flavour U p1q t as in (3.3). For our basic "core" theories this definition coincides with turning on generic diagonal vevs for scalars in Q and r Q chiral multiplets while setting to zero vevs for the scalars in the Φ chiral multiplets. Those choices of vevs completely breaks the SU pN q gauge symmetry at each node. Hence, we refer to this sub-branch of M as the Higgs branch. From the point of view of the pi, nq th gauge node the theory is SQCD with N f " 3N c . It is known that the quantum moduli space of SQCD with N f ě N c`1 coincides with the classical one [42]. Therefore we have where Q pi,nq , r Q pi,nq are the scalar vevs of the theory which have r N "2 " 0. The only nontrivial F-terms in a Lagrangian theory using the building blocks we described in Section 2 associated to a Riemann surface of genus g with punctures on this branch are therefore just F pi,nq " BW S k {BΦ pi,nq where W S k is given in (2.6). F H can be characterised by the quotient ring R H {I H where R H " CrQ pi,nq , r Q pi,nq s , I H " xF p1,1q , . . . , F pk,1q , . . . , F p1,3g´3` q , . . . , F pk,3g´3` q y . (3.5) We are now in a position to define the Higgs branch Hilbert series for our class S k theories. The total independent grading on the ring R H can be paramterised by a fugacity τ for the generator E " 2q t " 3 2 r, from the conditions (3.2), as well as the fugacities for U p1q k´1 βˆU p1q k´1 γ and any other global symmetries. The Higgs branch Hilbert series is then defined to be [43,44] HSpτ ; HBq :" Tr HB τ E e´J , (3.6) where Tr HB denotes the trace over the space of operators parametrising the Higgs branch (3.4) and e´J collectively denotes the fugacities for the remaining intrinsic U p1q k´1 γÛ p1q k´1 β and any other global symmetries. See also [45,46] for a detailed analysis of the Hilbert series for N " 1 SQCD. For gauge theories (3.6) takes the general form of a integral over the gauge group G of the Hilbert series of the master space F H which, by abuse of notation, we often simply denote as F H HSpτ ; HBq " where dµ G pzq denotes the Haar measure of G. Examining Table 14, where short representations of the N " 1 superconformal algebra are listed, it is clear that the Hilbert series (3.6) counts the top components of D p0,0q and B r,p0,0q multiplets of the N " 1 superconformal algebra. These multiplets have E " 3 2 r and j 1 " j 2 " 0. We can therefore see that, for k " 1, this definition does indeed coincide with the usual Higgs branch definition for N " 2 theories.

The Coulomb Branch
We can also make a similar definition for the Coulomb branch. Namely, analogously to N " 2 theories, we may define a consistent truncation of the moduli space by restricting those operators which have The two conditions in (3.8) combine to with the first equality being the BPS condition of isolated short multiplets; B r,pj 1 ,0q in the notation [39] (or LB 1 rj 1 , 0s prq E in the the notation of [41]), summarised in Table 14. What is special about the Coulomb branch for theories of Class S k , as opposed to general N " 1 SCFTs, is that the R-symmetry of the N " 1 superconformal algebra is further related to the charge q t of the extra flavour U p1q t as in (3.9).
For Lagrangian theories this coincides with setting the scalars arising from the free trinions to zero while giving the scalar in the Φ chiral multiplets (defined in Table 2) generic diagonal vevs. On this branch the gauge symmetry is broken down to the stabiliser subgroup of SU pN q kp3g´3` q with respect to the vevs Φ pi,nq which is given by U p1q pk´1`δ k,1 qp3g´3` q where is the number of punctures. We have that where Φ pi,nq are the scalar vevs of the theory that have R N "2 " 0. On this branch all of the F-terms in a Lagrangian theory are trivial because they are all proportional to either a Q or r Q. Therefore F C is simply associated to the freely generated ring R C " CrΦ pi,nq s. Similarly, we may also define the Hilbert series for the Coulomb branch HSpT ; CBq :" Tr CB T E e´J , (3.11) where Tr CB denotes the trace over the space of operators parametrising (3.10). In other words, the space of scalar operators of the theory satisfying (3.8) pE " 3 2 r "´q t q. Because the F-terms are trivial, for Lagrangian theories the F-flat Hilbert series (3.11) may be computed by multiplying the contribution coming from each Φ-multiplet and integrating over the gauge group. One extra simplification that arises is the fact that, because Q " r Q " 0, the pi, nq th node in the quiver is not coupled to the pj, m ‰ nq th . Therefore (3.11) reduces to a product of factors associated to the Φ-gluing of colour c and positive sign σ "H (3.12) the z's denote fugacities for the product gauge group, which are integrated over using the invariant measure dµ. The integrals h Φc were computed in [10] h Φc " PE where the symbol PE denotes the Pleythistic Exponetial defined in (B.1). As we will discuss in Section 4 the factor F Φc , appearing in equation (3.12), coincides with the Coulomb limit of the Superconformal index of the Φ-gluing factor (4.19). Therefore we reach the conclusion that for these Lagrangian theories HSpT ; CBq " I C , where I C denotes the Coulomb branch Superconformal index (defined in (4.18)). Table 3: Supercharges of the sup2, 2|2q superalgebra and its sup2, 2|1q subsuperalgebra. The orbifold breaks sup2, 2|2q Ý Ñ sup2, 2|1q ' up1q t and projects out any supercharge with q t ‰ 0. In radial quantisation S " Q : .

The Superconformal Index and some Interesting Limits
The right-handed N " 1 Superconformal index computed with respect to r Q 9 is given by [47,48] (see Table 3 for the quantum numbers of the supercharges and [49] for a review).
where Tr denotes the trace over the Hilbert space on S 3 in the radial quantisation, pE, j 1 , j 2 , rq denote the Cartans of the maximal compact bosonic subalgebra up1q E ' sup2q 1 ' sup2q 2 ' up1q r Ă sup2, 2|1q and q t denotes the generator for the 'intrinsic' global up1q t symmetry. What is more, e´J collectively denotes the fugacities for the remaining intrinsic U p1q k´1 γˆU p1q k´1 β and any other global symmetries. Finally, the different δ 1˘, δ 2˘, r δ 1 9 , r δ 2 9 are given by p " τ σ , q " τ ρ , t " τ 2 . The superconformal index (4.1) receives contributions only from those states satisfying r δ 9 " r δ 1 9 " 2t r Q 9 , r S 9 u " E´2j 2´3 2 r " 0 . Special attention should be paid to the fugactity t. When k " 1 the combinations and (4.2) are elements of the enhanced sup2, 2|2q superconformal algebra, see Table 3. When k ě 2 there is generically no N " 2 enhancement and q t generates a global U p1q t symmetry of the corresponding theory. Note that if the lowest component of a chiral superfield is given by f then the fermion which also contributes to the index has δ 1˘r r Q 9 f s " 2´δ 1˘r f s , r δ 2 9 r r Q 9 f s " 4´r δ 2 9 rf s . (4.6) Table 4: Letters satisfying the BPS condition (4.4) for the free trinion theory associated to a sphere with one minimal puncture (with associated U p1q valued fugacity α) and two maximal punctures s l,c and s r,c`1 . Here χ o,i " χ p1,0,...,0q pz o,i q, χ o,i " χ p0,0,...,1q pz o,i q are shorthand for the characters of the fundamental and anti-fundamental representations of SU pN q, defined in (B.9). We importantly note that δ 1˘, r δ 2 9 ě 0.
The importance of the above relations is due to the fact that, for any chiral superfield f , the condition that each state contributing to the index has δ 1˘, r δ 2 9 ě 0 is equivalent to We will employ the above inequalities in Section 7 in order to ensure that the limits of the SCI are well defined for theories without a Lagrangian description. We report in the last columns of Table 4 and Table 5 the values of the different δ, r δ for the field content of the free trinion and Φ-gluing.
We reviewed the construction of the basic Lagrangian theories in class S k in Section 2. The letters of the free trinion of Figure 3 that contribute to the index are listed in Table  4. The free trinion, that corresponds to an orbifolded N " 2 hypermultiplet, contributes to the index a factor I s l,c ,s r,c`1 " where Γ e pzq denotes the Elliptic Gamma function, defined in (B.3) and with ś k i"1 γ i " ś k i"1 β i " 1. Moreover, as discussed in Section 2, we observe that the colour label of the two maximal punctures is sfhited by one.
We will also sometimes adopt the notation I s l,c ,s r,c`1 " I zr z l α , leaving implicit the colour of the punctures. The Φ-gluing of two maximal punctures of the same sign σ "`, of colour c and opposite orientation contributes (4.9) Physically the above corresponds to the contribution of an orbifolded N " 2 vector mul- Table 5: Letters satisfying the BPS condition (4.4) of the free N " 1 theory corresponding to a tube which implements the Φ-gluing of two punctures of equal colour, opposite orientation and sign σ "`. For the vector multiplet piece we must take into account the equation of motion Bλ " B`9λ´`B´9λ`" 0. Here χ i " χ p1,0,...,0q pz i q, χ i " χ p0,0,...,1q pz i q and χ adj.
tiplet. It contains an N " 1 bifundamental chiral field Φ and N " 1 vector V in the adjoint, as can be seen in Table 5. The factors κ and ∆ are defined in (D.6). 6 Note that in the above, and throughout, products and sums over i shall always be taken modulo k, i.e. i`k " i unless otherwise stated. For instance, we denote the Φ-gluing of two three punctured spheres to obtain the theory associated to a sphere with two minimal and two maximal punctures at the level of the index by 7 we observe that the colour label of the second maximal puncture has been shifted by two. Let's now consider more in details the expression (4.10), starting from it we can understand some general properties of the theories taken into consideration. In order to do this let's consider the simplest case: k " N " c`1 " 2 and let's expand (4.10) in terms of N " 1 6 The factor δ k,1 is to account for the fact that for k " 1 Φ sits in the adjoint of supN q adj.
while for k ą 1 Φ sits in bifundamenal representations. 7 The theory obtained in this way corresponds to SQCDn, that is to say the orbifold of N " 2 SQCD.
index equivalence classes 8 I rr r,j 1 s˘[ 50-52] as (4.11) where β " β 1 " β´1 2 and γ " γ 1 " γ´1 2 and we defined t 1 " t{ppqq 2{3 so that the expansion is made using the free R-symmetry. The equivalence class r 1 3 , 1 2 s´, which has only a single representativeĈ p 1 2 ,0q containing a spin 3{2 current and contributes to the index a factor proportional to`ppqq 2{3 pp`qq{p1´pqp1´qq, is absent. There are only two possible explantions of this fact. The first is that the theory has no enhancement to N " 2 supersymmetry. The second is that the theory contains a number of B 7 3 p 1 2 ,0q multiplets (which is the single representative of the r 1 3 , 1 2 s`equivalence class), that cancel the contribution arising from the equivalence class r 1 3 , 1 2 s´in such a way that we do not see it in the index expansion. Note that all of the equivalence classes in the above contain only a single representative. In particular one can replace I r´2 3 ,0s´" I B 4 3 ,p0,0q , I r0,0s`" IĈ p0,0q and I r0,0s´" I B 2,p0,0q . The net degeneracy [51], defined in (C.12), of the r0, 0s˘equivalence classes counts NDr0, 0s "#r0, 0s`´#r0, 0s´" #B 2,p0,0q´#Ĉp0,0q "#marginal operators´#conserved currents "30´36 . We observe that, when k " 1, the index (4.1) admits various interesting limits involving the three fugacities p, q, t (or ρ, σ, τ ) in which the index receives contribution only from states annihilated by two or more N " 2 Poincaré supercharges (one of them, of course, always being r Q 9 ) [31]. On the other hand when k ě 2 the situation is different. We notice that the index of a generic N " 1 SCFT admits no non-trivial limits in which the states contributing to it are annihilated by more than one supercharge since r δ 9 rQ α s ‰ 0 and r δ 9 r r Q 9 s ‰ 0. However, when the N " 1 SCFT has flavour symmetry, we may consider taking limits also involving the flavour fugacities. For generic theories there is no guarantee that such limits are well defined. Moreover, the index in certain limits; although not leading to extra superconformal shortening, can often admit drastic simplifications. Similar ideas have also been deployed in studying 'non-generic' N " 1 SCFTs in e.g. [6,49,51].

The Hall-Littlewood Limit of the Index
We can study the limit which, for N " 2 theories, is equivalent to the so-called Hall-Littlewood limit of the index [31]. 9 For N " 1 Class S k theories it was first considered in [6], and it reads σ Ñ 0 , ρ Ñ 0 , τ fixed, (4.13) or, equivalently, p, q Ñ 0 with t held fixed. From (4.8) and (4.9) we see that the Hall-Littlewood limit of the indices for the Lagrangian building blocks is well defined. The existence of this limit is equivalent to the fact that each letter contributing to the basic building building blocks have δ 1˘ě 0 (see Table 4 and Table 5), as is the case for all N " 2 theories. We can therefore write HLpτ, . . . q :" lim σ,ρÑ0 I " Tr HL p´1q F τ 2qt e´β r δ 9 e´J , (4.14) here Tr HL denotes the restriction of Tr to the states satisfying δ 1˘" 0, i.e. 2q t " 3 2 r`3j 2 " E`j 2 and j 1 " 0. The indices for the building blocks (4.8) and (4.9) become As pointed out in [31,54], for Class S theories at genus g " 0, the Hall-Littlewood limit of the index coincides with the Hilbert series of the Higgs branch. For Lagrangian theories this can be explicitly proved and can be argued to extend to theories related to Lagrangian theories by S-duality [55]. To the best of the author's knowledge a full proof that extends to all class S theories is currently lacking. In Section 5 we will demonstrate that the same property also holds for Lagrangian theories made using Φ-gluing in class S k at genus g " 0.

The Coulomb Limit of the Index
The Coulomb limit of the index for N " 2 theories is given by (4.17) or, equivalently, t, p, q Ñ 0 with T :" pq{t " σρ and V :" p{q " σ{ρ held fixed. An extensive study of this limit of the index was given in [10]. For generic N " 1 theories we would have no reason to believe that this limit exists since r δ 2 9 ě 0 is no longer guaranteed. However, let us assume that it does. In this limit the index would take the form Here Tr C denotes the restriction of Tr to states with r δ 2 9 " 0. Indeed, we can see at the level of the Lagrangian building blocks that the limit does exist and, moreover, is conjectured to exist for all theories in class S k of type A N´1 [10]. The indices for the Lagrangian building blocks (4.8) and (4.9) become For Lagrangian theories made with Φ gluing the interpretation of this limit of the index is clear. The Coulomb limit of the index is simply counting the possible gauge invariants that can be made from the bifundamental scalar fields in the chiral multiplets Φ. These operators are the top components of the 1{2-BPS multiplets D p´1q p0,0q and B p´2 r 3 q r,p0,0q which simultaneously have E " 3 2 r "´q t and j 1 " j 2 " 0 (see Appendix C). Indeed we demonstrated in Section 3.1.2 that, for those theories, I C can be given the interpretation of a Hilbert series constructed to count the above 1{2-BPS multiplets on the Coulomb branch.

The Schur & Madonald Limits of the Index
For completeness of our discussion we can also define analogues of the Schur and Macdonald limits of the index of [31]. The analogue of the Macdonald index for the specific case of class S k theories, was originaly considered in Appendix B of [6]. It is obtained by taking σ Ñ 0, which is well defined for Lagrangian theories because each letter has δ 1`ě 0, while holding ρ, τ fixed.
The stress tensor of the associated chiral algebra is identified with the top component of the sup2q R N "2 current, namely j µ 11 . This current lives in the stress tensor multipletĈ 0p0,0q which contains conserved sup2q R N "2 and up1q r N "2 currents j µ pIJq and j µ with I, J " 1, 2 sup2q R N "2 indices. This current enters the Schur index I S with a factor I Ŝ C 0p0,0q " q 2 {p1´qq. Under the decomposition sup2, 2|2q Ñ sup2, 2|1q ' up1q t the N " 2 stress tensor multiplet decomposes aŝ ; (4.27) see Figure 5.
Since, the stress tensor multiplet of the mother SCFT sits in trivial representations of any flavour symmetries (e.g. it sits in a trivial SU pN f q representation) under the Z k orbifold the projection of the stress tensor should simply beĈ 0p0,0q Z k ÝÑĈ p0,0q 'Ĉ p 1 2 , 1 2 q . Indeed the multipletsĈ p 1 2 ,0q andĈ p0, 1 2 q contain additional supersymmetry currents which would lead to enhanced N ě 2 supersymmetry if present.
Underlined are those states, with given j 2 , which have r δ 9 " E´2j 2´3 2 r " 0 and thus can contribute to the right-handed index (4.1).
We can identify in the decomposition theĈ p0,0q as the up1q t flavour current multiplet whileĈ p 1 2 , 1 2 q is of course the N " 1 stress tensor multiplet whose lowest component is the up1q r current. They are built from linear combinations of j µ 12 " j µ 21 and j µ in accordance with (4.5).

Genus Zero Theories
In this Section we consider Class S k theories which arise from compactifications of 6D SCFTs on Riemann surfaces of genus g " 0. We show that, for this subclass of theories, the Hall-Littlewood limit of the index coincides with the corresponding Higgs branch Hilbert series, as is the case for N " 2 Class S theories. In this Section, we provide the explicit expression of the Higgs branch Hilbert series for some Lagrangian genus zero theories, i.e. the free trinion and the interacting SCFT associated to a sphere with two maximal and two minimal punctures. The study of genus g " 0 theories without a Lagrangian description is presented in Section 7.

Hilbert Series and the Hall-Littlewood Limit of the Index
We are now in a position to show that the Hall-Littlewood limit of the index coincides with the Higgs branch Hilbert series at genus g " 0 for Lagrangian theories made using Φ-gluing.
For the theory corresponding to a sphere with ´2 minimal punctures and two maximal punctures the relevant F-terms for the Higgs branch are for n " 1, . . . ,´3` and i`k " i " 1, . . . , k. For genus g " 0 these constitute kp´3` q independent constraints on the Q pi,nq and r Q pi,nq . More precisely the ideal I H comprised of the list of the F pi,nq forms a regular sequence in R H " CrQ pi,nq , r Q pi,nq s, see Appendix A. This means that the variety whose coordinate ring is given by the quotient ring R H {I H is a complete intersection and we may apply letter counting techniques to compute the Hilbert series for the master space F H .
The Hilbert series of the Higgs branch of the theory associated to the a sphere with ´2 minimal punctures and two maximal punctures s l,1 , s r,` ´1 is precisely given by and we see that F H " ś ´2 n"1 HL s l,ǹ ,s r,ǹ`1 ś ´2 n"2 HL Φǹ and therefore HSpτ q " HLpτ q for this class of genus zero theories. The contribution of the λ 9 pi,nq to the Hall-Littlewood index coming from Φ-gluing precisely plays the role of the F-term constraints (5.1) in the Higgs-branch Hilbert series.
In other words, we see that for this class of theories Tr HL " Tr HB . Indeed, one can see that if one further introduces the condition j 2 " 0 into those defining the Hall-Littlewood limit of the index r δ 9 " δ 1˘" 0 we have E " 2q t " 3 2 r, j 1 " j 2 " 0 which are precisely the conditions defining the Higgs branch (3.4).

The Free Trinion
Let us consider the Hall-Littlewood index/ Hilbert series of the g " A N´1 theory associated to a sphere with one minimal puncture with fugacity α and two maximal punctures s l,1 and s r,2 a.k.a. the free trinion. The expression for the Higgs branch Hilbert series was given in (4.15) and it reads

5.4)
note that we set z l " u and z r " v with respect to (4.15). We checked for various low values of N in expansion around τ " 0 that the identity A"1 Apλ A´λA`1 q s λ puqs λ pvq and where the SU pN q characters are defined in (B.9). In the second line we rewrote the expression in terms of Schur polynomials, the relevant definitions and identities can be found in Appendix B. Therefore, we can write (5.4) as Specialising to the case g " A 1 , where we explicitly know the Littlewood-Richardson coefficients c ν λµ , we have

Core Interacting Theories
Let us first consider the Hall-Littlewood index/ Higgs-branch Hilbert series for the interacting SCFT associated to a sphere with two minimal punctures and two maximal punctures s l,1 , s r,3 . It is given by (5.10) An important observation that will allow us to write down the Highest Weight Generating (HWG) function [56] for the Hilbert series for this theory is the fact that there is, at the level of the Lagrangian, a symmetry enhancement which we can make manifest in (5.10) by writing under SU p2N q Ñ SU pN qˆSU pN qˆU p1q and Q L i , r Q L i are the bifundamental chiral multiplets of the first free trinion and Q R i , r Q R i the second. Then the Hall-Littlewood index / Hilbert series can be written as (5.13) The symmetry enhancement (5.11) allows us to conjecture the following expression for the Highest Weight Generating (HWG) function for the Hilbert series as 14) where tµ piq A u denotes a set of highest weights for the i-th flavour node. The corresponding Hilbert series is given by (5.15) We checked in an expansion around τ " 0 that (5.15) agrees with (5.13) for pN, kq " tp2, 2q, p2, 3q, p3, 2qu. For theories associated to -punctured spheres with two maximal punctures and ´2 ą 2 minimal punctures there is no symmetry enhancement (5.11) and the number of monomials in the PLog of the HWG is not finite.
We can however perform the expansion of the integral (5.2) around τ " 0. For example, for k " N " 2, ´2 " 3 we have q denotes the characters of the fundamental representation of the corresponding SU p2q's.

Genus One Theories
In this Section we consider genus one theories of Class S k . As is the case for N " 2 Class S theories, for g " 1 theories the Hall-Littlewood limit of the index does not coincide with the p1, 1q p3, 1q p3, 2q p3, 3q p3, 4q p3, 5q Figure 6: Section of the quiver diagram of the Z kˆZ orbifold theory of N " 4 SYM. Circular nodes denote U pN q vector multiplets and directed arrows denote chiral multiplets. Horizontal lines between node pi, nq and pi, n`1q denote Q pi,nq fields. Vertical lines between node pi, nq and pi´1, nq denote Φ pi,nq fields. Diagonal lines between pi´1, n`1q and pi, nq denote r Q pi,nq fields. The quiver should be periodically identified in both directions, such that it has the topology of a tessellation of the torus.
corresponding Higgs branch Hilbert series (which, as we discussed in Section 3.1.1, counts only B and D p 1 2 q p0,0q multiplets), but it differs by C,Ĉ and D pj`1q type multiplets. The theories that we study are the N " 1 upN q ' k toroidal quiver gauge theories realised as the string length l s Ñ 0 limit of the worldvolume theory on a stack of N D3branes probing a transverse C 3 {pZ ˆZ k q singularity where the quotient acts by pz 1 , z 2 , z 3 q Þ Ñ pω k z 1 , ω z 2 , ω´1 ω´1 k z 3 q , ω k k " ω " 1 . These are orbifolds of N " 4 SYM. In the case where the up1q ' k Ă upN q ' k is not gauged, these are precisely the theories in class S k associated to Riemann surfaces of genus one with punctures. For k " Ñ 8 we will provide a check that the six dimensional p1, 1q little string theory can be deconstructed from a toroidal quiver in class S k .
The action (6.1) is an element of the SOp6q R R-symmetry group of N " 4 SYM. If we denote the N " 4 supercharges by Q q 1 q 2 q 3 α with 8q 1 q 2 q 3 "´1 and Q q 1 q 2 q 3 9 α with 8q 1 q 2 q 3 "`1 where the q i "˘1 2 . The orbifold acts by (6. 2) The surviving supercharges are those with q 1 " q 2 " q 3 Q´´ά " Q α , Q``9 α " Q 9 α . (6. 3) The Cartans q 1 , q 2 , q 3 of sop6q R are related to the more natural N " 1 symmetries by there is also the overall U p1q generated by b.
It will later be useful to have the Hilbert series for this quotient space. This is given by the Molien series [43,44] 5) where τ 1 , τ 2 , τ 3 are related to the SOp6q R Ą U p1q 3 ü C 3 toric action generated by q 1 , q 2 , q 3 . Moreover LCM p , kq denotes the Lowest Common Multiple of and k. For " k these varieties are complete intersections since This space can be realised as C 3 {pZ kˆZk q ãÑ C 4 defined by the equation with w i " z k i , w 4 " z 1 z 2 z 3 . As a warm up we will now review the computation for k " 1.

Class
The quiver diagram for this theory is given in Figure 7. In the following subsection we compute the Hilbert series of the mesonic moduli space M mes [57].

Hilbert Series
. Let us begin with the case of N " 1. The master space for M is then associated to R{I with R " rQ 1 , . . . , Q , r Q 1 , . . . , r Q , Φ 1 , . . . , Φ s and I " xF Q 1 , . . . , F Q , F r Q 1 , . . . , F r Q , F Φ 1 , . . . , F Φ y with F Qn " r Q n pΦ n`1´Φn q , F r Qn " Q n pΦ n`1´Φn q , F Φn " Q n´1 r Q n´1´Qn r Q n , (6.8) where n " n` . We perform a primary decomposition of the above ideal and we select the prime ideal corresponding to a mesonic branch, the corresponding Hilbert series is The τ i are the same as those in (6.5). Explicitly, in terms of the parameters appearing in (4.1) where b " ś n"1 α n is the product of all of the fugacities for minimal punctures. So, the Hilbert series for M mes is given by Figure 7: The circular quiver with gauge group U pN q .
We observe that HSpτ 1 , τ 2 , τ 3 ; M mes q " M pτ 1 , τ 2 , τ 3 ; CˆC 2 {Z q. The Hilbert series for the Higgs branch can easily obtained by considering the τ 1 Ñ 0 limit As we can see taking the PLog of (6.12) the Higgs branch is generated by one generator M " Q 1 r Q 1 "¨¨¨" Q r Q of dimension 2 and two generators B " ś n"1 Q n , r B " ś n"1 r Q n of dimension . They satisfy the following relation at order 2 M " B r B , (6.13) this corresponds to the variety C 2 {Z . The Coulomb branch is simply a copy of C parametrised by tΦ 1 , . . . , Φ u 14) The corresponding expressions for the mesonic branch for general N ,can be obtained as the N th symmetric product of the N " 1 case [43,44,[57][58][59] and 6.15) and the Hilbert series is given by Figure 8: Quiver diagram of the k " 2 theory associated to a torus with " 2 minimal punctures.

Hall-Littlewood Index
Let's now move to the computation of the Hall-Littlewood index of the theory in Figure 7.
Let's begin the case with N " 1 and generic the computation can be explicitly done and we get 17) therefore the ratio between the Higgs branch Hilbert series (6.12) and the above index reads HLpτ 2 , τ 3 q HSpτ 2 , τ 3 ; HBq " PEr´τ 2 τ 3 s " PEr´τ 2 s " PErHL D 0,p0,0q s , here HL D 0,p0,0q denotes the Hall-Littlewood index of the free N " 2 vector multiplet [31,39]. We now want to compute the expression of the Hall-Littlewood index for a generic value of N . Since D 0,p0,0q is a free field multiplet it naturally decouples from the theory and we would like to conjecture that the Hall-Littlewood index for general N can still be obtained as the coefficient HL N of the following expansion PErνHL N "1 pτ 2 , τ 3 qs " We verified this conjecture for various low values N and , that is to say for p , N q " tp1, 2q, p2, 2q, p3, 2q, p1, 3q, p2, 3q, p1, 4qu.
6.2 Class S k k ě 2 Let us move to the case of general values of k, , while again focusing on N " 1. Let's firstly consider k " " 2. The quiver diagram is given in Figure 8.
After performing the integrals, we have 3`1˘( 6.22) Note that the final expression is independent of α δ and both γ and β. This is because for the quivers with U pN q gauge groups the U p1q transformations generated by q αn , q γ i , q β i are isomorphic to gauge transformations, while for SU pN q gauge groups they are global symmetries. As before the Higgs branch is reached by considering the τ 1 Ñ 0 limit We notice that the Hilbert series splits into that for C 2 {pZ 2ˆZ2 q (the mesonic Higgs branch moduli space), and two copies of pC{Z 2 q 2 , minus the two common C{Z 2 -line intersections. From the Plethystic Logarithm we recognise the generators as , with fugacity τ 2 2 and τ 2 3 , respectively. At the next order we have the relations B 1 r The higher order terms are Hilbert syzygies (a.k.a. relations between relations).
We can also consider the Coulomb limit corresponding to the operators For higher values of k, the computations of the Hilbert series become increasingly complex, due to the requirement of making primary decomposition in a larger number of variables using Macaulay2. We were able to compute the Higgs-branch Hilbert series for k " " 3. Again setting α 1 α 2 α 3 " 1 since that can be reintroduced by rescaling τ 2 , τ 3 , is HSpτ 2 , τ 3 ; HBq " PErτ 3 2`τ 3 3 s`PEr3τ 3 2 s`PEr3τ 3 3 s´PErτ 3 2 s´PErτ 3 3 s (6. 26) we note that γ i , β i do not appear and α n enters only via α 1 α 2 α 3 in the Hilbert series for the Higgs Branch. Moreover, we conjecture that the form of the Hilbert series for arbitrary k " reads HSpτ 2 , τ 3 ; HBq " PErτ 2`τ 3 s`PEr τ 2 s`PEr τ 3 s´PErτ 2 s´PErτ 3 s . (6.27) This form for the Hilbert series implies that the moduli space is made up of a copy C 2 {pZ Ẑ q, two copies of pC{Z q minus C{Z common intersections. The Plethystic Logarithm of the Hilbert series (6.27) reads PLogrHSpτ 2 , τ 3 ; HBqs " pτ 2`τ 3 q´p 2´1 qτ 2 τ 3`¨¨¨, (6.28) where, at the first order of the expansion we recognise generators with fugacity τ 2 and generators with fugacity τ 3 . While, at the next order of the expansion, we observe the presence of 2´1 relations between the generators. Finally the higher terms in the PLog expansion are Hilbert syzygies.

Hall-Littlewood Index
We can also compute the corresponding Hall-Littlewood index for our theories. The general expression is given by  For general " k with N " 1 we were propose the following conjecture, which we checked for various low values of " k, HLpτ 2 , τ 3 q " PEr τ 2 s`PEr τ 3 s´1 . (6.31) The ratio HL{HS is then counting, with signs, the protected operators of the theory which have j 1`j2 ě 1 2 and r`2j 2 " 4 3 q t and are given in (C.17). It is possible compute the Hall-Littlewood index for the class S k theories with SU pN q gauge groups. For k " " 2 with SU p2q k gauge group this reads HL α 1 α 2 " where α 1 and α 2 denote the global symmetry fugacities associated to the minimal punctures.

Deconstruction Limit
In this subsection we digress a bit from the main objective of this paper to obtain an exact check for the dimensional-deconstruction prescription of Arkani-Hamed, Cohen, Kaplan, Karch and Motl [32]. The last few years there have been a few precision tests [61][62][63][64][65] of the deconstruction proposal [32]. Of most interest to us in this article is the fact that [61,65] were able to show that the 1 2 -BPS partition function of the N " p2, 0q theory is equal to the Higgs branch Hilbert series of the corresponding 4d N " 2 theory in the deconstruction limit. This naturally leads one to expect that a similar story should also exist for the N " p1, 1q Little String Theory (LST). The N " p1, 1q LST arises as the worldvolume theory on a stack of N parallel NS5-branes in type IIB string theory [66][67][68][69][70]. The basic deconstruction proposal is that the 6d theory may be effectively described by considering the N " 1 upN q ' k toroidal quiver gauge theory realised as the l s Ñ 0 limit of the worldvolume theory on a stack of N D3-branes probing a transverse C 3 {pZ ˆZ k q singularity where the quotient acts as in (6.1). We then go to the point in parameter space where the vevs v 5 " xQ pi,nq y, v 6 " xΦ pi,nq y and couplings G " g YM pi,nq are equal for all nodes in the quiver and then take the limit 10 while holding 2πR 5 Gv 5 " and 2πR 6 Gv 6 " k fixed. The main point is that, in this limit, the transverse C 3 {pZ ˆZ k q can be approximated by T 2ˆR4 where the radii of the torus are r 5 " v 5 {k and r 6 " v 6 { . Performing T-duality along the two circles and then S-duality gives the rank N N " p1, 1q LST on a torus with radii R 5 , R 6 .
Representations of the N " p1, 1q supersymmetry algebra may be decomposed into a finite sum of representations of the bosonic subalgebra given by the sum of Lorentz algebra sop6q and R-symmetry algebra sop4q -sup2q R 1 ' sup2q R 2 . The supercharges sit in the representations Q P r0, 1, 0s p 1 2 ,0q 1 2 and r Q P r0, 0, 1s p0, 1 2 q where η µ , r η µ , µ " 1, 2, . . . , 6 denote the 't Hooft symbols which intertwine between sup4q and sop6q. The 4d N " 1 supersymmetry algebra plus the residual global up1q t ' up1q b symmetry algebras can be embedded into the 6d N " p1, 1q algebra with the relations 11 The relationship between the 4d and 6d supercharges is given in Table 6.
Let us now move to our candidate set of operators that will reproduce the 1 2 -BPS scalar operators in the p1, 1q LST. Let's assume that, after primary decomposition, we can always identify a irreducible mesonic branch M mes inside the full moduli space M. M mes will be our candidate for reproducing the 6d 1 2 -BPS ring. For any N " 1 theory, the operators parametrising M are themselves 1 2 -BPS with respect to the N " 1 supersymmetry algebra, namely they are annihilated by r Q 9 α . 11 These can be obtained in the following way: the first two relations are simply the identifications one would make between the Cartans for sup2q1 ' sup2q2 -sop4q Ă sop6q. When compactifying the 6d theory on T 2 the spinor label h1 h1 " H " HR`HL on T 2 and 2HL " q1, 2HR " q2. Finally 2R2 " q1´q2 is fixed by demanding that R2 evaluates to zero on the N " 1 subsector. This then fixes q3 " R1. The identifications (6.4) then give (6.38).
" Figure 9: Visualisation for the example of p " 3. On the left Q pi,nq Q pi,n`1q Q pi,n`2q Φ pi,n`3q Φ pi´1,n`3q r Q pi´1,n`2q Φ pi´1,n`2q r Q pi´1,n`1q r Q pi,nq reduced to ś 2 m"0 Q pi,n`mq Φ pi,n`m`1q r Q pi,n`mq . Again further applying the F-terms means this operator can be completely written in terms of M .
One may wonder why we have picked the subvariety M mes as opposed to the full moduli space M. Shortly we will prove that, for the theory with U p1q gauge groups, in the k, Ñ 8 limit that M mes coincides with M. Setting, without loss of generality in this limit k " , the possible gauge invariant operators parametrising M for the theory with U p1q gauge groups are in correspondence with the number of closed, directed paths that one can draw on the quiver diagram. There are four main types of operators. There are those which involve an equal number p ď k " of Q, r Q and Φ fields. Such an operator enters the Hilbert series with fugacity τ p 1 τ p 2 τ p 3 . Operators of this form correspond to picking a base node, say pi " 1, n " 1q, and drawing a closed loop involving p vertical steps, p horizontal steps and p diagonal steps. However, recall the F-terms set F Φ pi,nq " r Q pi,n´1q Q pi,n´1q´Qpi´1,nq r Q pi,nq " 0 , (6.39) F Q pi,nq " Φ pi,n`1q r Q pi,nq´r Q pi`1,nq Φ pi`1,nq " 0 , (6.40) F r Q pi,nq " Q pi,nq Φ pi,n`1q´Φpi,nq Q pi´1,nq " 0 . (6.41) In terms of the quiver diagram Figure 6, this means that the operations of moving right, up or diagonally all commute. In other words this means that the associated operator is independent of the choice of base node and of the specific path chosen, it depends only on the length 3p. See Figure 9 for a diagrammatic example. We can therefore write these operators as M p with, say, M " r Q p1, q Q p1, q Φ p1,1q . The other types of operators involve only Q's, r Q's or Φ's. They can be written as powers of they enter the Hilbert series with fugacity τ 1 , τ 2 and τ 3 , respectively. In degree ě minpk, q there can be complicated relations and higher syzgies between M , B Q i , B r Q i and B Φn , leading to a complicated structure for M. However, the main point is that in the " k Ñ 8 limit the B Q i , B r Q i and B Φn operators all become infinity heavy and their dimensions E " k " tend to infinity, in particular lim Ñ8 τ 1,2,3 Ñ 0 and their contribution to the Hilbert series vanishes. This also implies that the corresponding ring becomes freely generated, since there are no relations between the operators in degree smaller than minpk, q Ñ 8. Only the dimension of M remains finite in the limit. M is a purely mesonic operator and therefore, in this limit, one indeed expects M mes to coincide with M.
The above arguments can be simply extended to the case N ě 2 by taking traces. Each path with an equal number p of Q, r Q and Φ's now corresponds to A " 1, . . . , N operators of dimension 3pA corresponding to operators schematically of the form trpQ p r Q p Φ p q A . As long as we consider the traces we can apply the same rules that we did for N " 1, namely that the operations of moving right, up or diagonally commute. This again means any path of length 3p can be written as the loop given by M p with, say, M " r Q p1, q Q p1, q Φ p1,1q which is now a NˆN matrix. The N operators corresponding to one of these closed paths can then be written as M p pAq with As before the M pAq can have complicated relations with the operators which are of the form and tr B A Φ i but the dimensions of the latter are Ak " A Ñ 8. We therefore arrive at the conclusion that, in the deconstruction k " Ñ 8 limit M coincides with M mes for all N lim ,kÑ8 HSpτ 1 , τ 2 , τ 3 ; M mes q . (6.44) For the N " 1 case we can identify M mes " C 3 {pZ ˆZ k q, so the Hilbert series for the N " 1 case is given by (6.5). Therefore, the Hilbert series for general N is given by Let us now consider the deconstruction limit. Taking the limit on the N " 1 result, recalling that |τ 1,2,3 | ă 1, gives The coordinate-ring of M for this theory in the k, Ñ 8 limit is therefore simply CrM p1q , M p2q , . . . , M pN q s.

Computation of the 1 2 -BPS Partition
Function for p1, 1q LST Let us now move to the computation of the 6d quantity that we would like to match to the 4d quantity (6.47). We can use the fact that, at low energies, the N " p1, 1q LST admits an effective description as 6d maximally supersymmetric SYM theory with gauge group U pN q. The on-shell degrees of freedom of the p1, 1q SYM theory contains a 2-form gauge field strength F P r0, 1, 1s p0,0q 2 , scalars X P r0, 0, 0s p 1 2 , 1 2 q 2 and fermions λ P r0, 1, 0s , r λ P r0, 0, 1s all in the adjoint of g " upN q. The supersymmetry transformations of interest to us are We can therefore construct 1 2 -BPS multiplets whose highest weight state is annihilated by both Q 1 and r Q 9 1 . The supersymmetric primary of these multiplets is given by which are independent for A " 1, . . . , N . By acting with all possible supersymmetries Q, r Q and sop6q ' sup2q R 1 ' sup2q R 2 generators we can generate the entire 1 2 -BPS multiplet by acting on the highest weight state O h.w.
A . We can define a 1 2 -BPS partition function (Hilbert series) by passing to the scalar sector of the Q 1 X r Q A . (6.52) These operators can be expected to be found as a subset of those counted by the S 1ˆS5 partition function. By the Nahm classification [71], there is no superconformal algebra associated to N " p1, 1q SUSY in 6d so there is no superconformal index associated to this theory, nevertheless we can define the S 1ˆS5 partition function of the theory where H " tQ, Q : u " E´h 1´h2´h3´4 R 1 with Q :" Q 1´´. The partition function (6.53) can be computed using the elliptic genus method [72][73][74] or using the refined topological string [75,76]. Z p1,1q S 1ˆS5 is expected to receive both perturbative contributions as well as non-perturbative contributions from 6d SYM instanton string states.
If we consider taking xy " p 1 " p 2 " 1 then Z p1,1q receives extra shortening and is annihilated by Q and r Q 9 1´´, 1´`. Therefore the unrefined limit (6.54) receives non-zero contributions only from states with h 1 " h 2 " h 3 " E´4R 2 and E`2R 1 " 6R 2 . H is clearly contained as a subset of those states when E " 4R 2 .
We also note that (6.50) is equal to the 1 2 -BPS limit of the index of the N " p2, 0q theory of type g " upN q, which falls into the general result that, in all known examples, the 1 2 -BPS partition function seems to be a universal quantity in all maximally supersymmetric theories in 3, 4 and 6 dimensions [77]. Additionally, the p1, 1q and p2, 0q LSTs are related by T-duality. T-duality exchanges winding and momentum modes along the temporal S 1 . Since, by definition, Z p1,1q

Interacting Trinion Theories in Class S k"2
This Section is devoted to the study of the strongly interacting Trinion theories in Class S k"2 that correspond to three punctured Riemann spheres and have no Lagrangian description.
As pictured in Figure 10 the basic A 1 four punctured sphere in class S 2 admits three 'S-dual' decomposition. The first is the most familiar being the standard Lagrangian frame and is given in Figure 11. The other descriptions involve strongly interacting SCFTs, associated to spheres with three maximal punctures, with an SU p2q gauging to a quiver tail. These SCFTs, denoted T A & T B in [79] carry global symmetries of, at least sup2q 2 z ' Finally, there is a simpler Trinion for Class S 2 and N " 2 than the T A and T B which was studied in [80], however, we will not study it in this paper 12 .
The field content of the B-and A-type quiver tails is listed in Table 7 and Table 8, respectively. The superpotential of the tail, in these cases, will include The respective T B & T A SCFTs have mesonic operators M ȗ , M v , M z which are in the bifundamental representation of the corresponding sup2q cˆs up2q c symmetry with c " tz, v, uu and in the`1 representation of up1q t . We list such operators, with the corresponding charges 12 Our main tool in this Section is the Spiridonov-Warnaar inversion formula [81]. The simpler Trinion of [80] is not obtained via an inversion and it involves a new type of a puncture.
γβ pq t Figure 11: Quiver diagram of the theory associated to a sphere with two minimal punctures and two maximal punctures of colour c l " c r " 0.
under the flavor symmetry group, in Table 9 and in Table 10 for the T B and T A theory respectively. In the case of the T B theory, as depicted in Table 9, M ȗ , M v have charge˘1 under up1q γ and¯1 up1q β while M z has˘1,˘1. In the case of the T A theory M ȗ , M v , M z all have charge˘1 under up1q γ and¯1 up1q β as in Table 10. The tails couple to their respective trinions through superpotentials Ultimately we have the following identity between the indices in the three different frames Table 7: Field content associated to the type B 'quiver tail' in Class S 2 corresponding to a sphere with two minimal punctures α, δ and a 'pinched' maximal puncture pz 1 " αδ, z 2 q which can be glued to a maximal puncture to convert it to two minimal punctures. Each field is an N " 1 chiral multiplet. In the final columns we firstly list the value of δ 1˘" r`2j 2´4 3 q t˘2 j 1 and r δ 2 9 " 4j 2`2 r`4 3 q t of the corresponding field. Table 8: Field content associated to the type A 'quiver tail' in class S 2 corresponding to a sphere with two minimal punctures α, δ and a 'pinched' maximal puncture pz 1 " αδ, z 2 q which can be glued to a maximal puncture to convert it to two minimal punctures. Each field is an N " 1 chiral multiplet. In the final columns we firstly list the value of δ 1˘" r`2j 2´4 3 q t˘2 j 1 and r δ 2 9 " 4j 2`2 r`4 3 q t of the corresponding field. and we set z " pαδ, z 2 q. The expression for I v uδα is given in (4.10). The final two expressions read [6,79] I pBqz δα¨I where we defined s " δ α and z 1 " αδ and the function δpx, y; T q is defined in (B.19). We now need to apply the Spiridonov-Warnaar inversion formula [81] to invert the above integrals. The inverse formula is reviewed and explained in Appendix B and in the form that we will use it is presented in equation (B.21).

(7.15)
Finally we conclude this Section by computing the Coulomb index for the T B theory, i.e. we consider the limit (p, q, t Ñ 0, pq t Ñ T ), which was originally given in [10]. We have that 16) The Coulomb index for the 4-punctured sphere was given in (3.13) and is given by notice that it is independent of s and therefore we can easily compute the above integral by collecting the residues of the poles inside C z 2 at s " T β 2 z 1 , T γ 2 z 1 , γ β z˘1 2 . The final result is This result is in agreement with the row number six of Table 1 in [10]. We conclude that the Coulomb branch for the T B theory is freely generated and it is generated by three bosonic operators. In particular the first operator has charge`2 under both up1q β and up1q γ , the second has charge´2 under both up1q β and up1q γ , while the third has charge`2 under up1q β and charge´2 under up1q γ .
Let us explain why we get this minus sign from multiplets recombination. The expansion of the Hall-Littlewood limit of the index of the four punctured sphere theory has all positive coefficients, in particular it does not contain a C 2 3 p0,0q multiplet with q t " q β " q γ " 2. This could mean that the C 2 3 p0,0q multiplet may be "recombined" in a long multiplet A, whose Hall-Litllewood index I HL pAq is equal to zero. As we approach the complete decoupling limit where the SU p2q z 2 is ungauged the four punctured sphere theory decomposes into the T A SCFT plus the A-type quiver tail theory. As we are taking this limit a long multiplet A 3 2 3 ,p0,0q of the four-punctured sphere theory decomposes via the recombination rule (C.4) Using the relations (C.14)-(C. 16 as it must be since A 3 2 3 ,p0,0q is a long multiplet. Therefore as long as the C 2 3 ,p0,0q and the B 8 3 ,p0,0q multiplet recombine their contribution to the HL index mutually cancels. However, as we have seen, the C 2 3 p0,0q multiplet appear in the HL index of the T A theory. On the other hand, we are then led to expect that, the B 8 3 ,p0,0q multiplet with q t " q β " q γ " 2 lives in the A-type quiver tail theory. As a matter of fact, going through the list of Hall-Littlewood (δ 1˘" 0) operators with p´1q F "`1, q t " q β " q γ " 2 in the tail theory Table 8, we obtain an operator in the 2 b 2 -3 ' 1 of the enhanced SU p2q s"δ{α B IJ :"˜B 1,``B1,`´B1,``B1,´B 1,´`B1,`´B1,´´B1,`´¸I J " B pIJq`BrIJs (7.24) with I, J " 1, 2 SU p2q s indices. Since the T A theory contains only SU p2q s singlets we are instructed to take the 1 in the decomposition and we conclude that the operator is the top component of the B 8 3 ,p0,0q multiplet in the above recombination rule. Therefore we conclude that the origin of the minus sign in the expansion (7.20) is due to fact that the long multiplet A 3 2 3 ,p0,0q splits into two short multiplets that do not recombine anymore. Due to this, the negative contribution coming from the C 2 3 p0,0q multiplet, i.e. the term´γ 2 β 2 τ 4 , is not cancelled by the positive contribution arising from the index of the B 8 3 p0,0q multiplet of the A-type theory. It is then natural to expect that, also at higher Table 10: Higgs branch operators of the T A SCFT that appear at the order τ 2 of the expansion.
orders of the expansions, new negative coefficients will appear. A similar feature has been observed in a 4d N " 2 context in [50]. The conformal R-symmetry of the T A theory is [79] r c " r`0.0689pq γ`qβ q´0.044777q t . (7.26) Therefore, operators contributing to the Hall-Littlewood limit of the index have conformal energy By the same arguments as the ones made for the T B theory the operators with q t " 1 appearing in the expansion (7.20) have j 2 " 0. Therefore the operators of Table 10 comprise, at least a subset of, the possible operators on the Higgs branch of the T A SCFT. In the notation of Table 9 Table 10 is in fact the complete list of Higgs branch generators for the T A theory and that new generators do not appear at higher orders. We were able to obtain a closed expression in the unrefined z " v " u " 1, γβ´1 " 1 limit: r P ps, γβ, τ qds ps´βγq 2 pβγs´1q 2 pβγs´τ 2 q 5 psτ 2´β γq 5 ps´βγτ 2 q 3 pβγsτ 2´1 q 3 "β 5 γ 5 τ 15 Q A pτ, βγq´Q A pτ´1, β´1γ´1q p1´τ 2 q 19 p1´γ 2 β 2 τ 2 q 4 (7.28) where r P ps, γβ, τ q is a polynomial in s and in the second line we took the residues at s " βγ, pβγq˘1τ 2 and Q A pτ, βγq is a polynomial of degree 15 in τ . We list the full expression for it in (E.4) and quote here only the result for γ " β " 1 which reads HL pT A q1 11ˇγ "β"1 " 1 p1´τ 2 q 18`τ 20`3 8τ 18`4 74τ 16`2 582τ 14`6 895τ 12 9516τ 10`6 895τ 8`2 582τ 6`4 74τ 4`3 8τ 2`1˘, (7.29) again, note the palindromic structure of the numerator. Also note that in the fully unrefined limit HL . This is expected since the T A & T B SCFTs differ only by choices for fluxes for U p1q γ and U p1q β . This equality no longer holds with refinement turned on, since T A and T B have different global symmetry algebras. The Coulomb index for the T A theory can also be computed, the computation is identical to the T B case and it reads The above result is in agreement with row number five of table 1 in [10]. We observe that the Coulomb branch is generated by three bosonic operators without relations among them.
In particular the first operator has charge`2 under both up1q β and up1q γ , while the second and the third have charge´2 under both up1q β and up1q γ .

Outlook and Open Problems
In this paper we studied several aspects of Class S k SCFTs. We review the notion of moduli space of vacua for this class of theories and we provide a consistent definition of its Coulomb branch and its Higgs branch. We then introduce the mathematical tools that we use for the characterization of these theories, namely the Higgs branch and Coulomb branch Hilbert series and the Superconformal index along with some of its limits. First, we analyse Lagrangian Class S k theories at genus g " 0. For this subclass of theories we prove the equality between the Hall-Littlewood limit of the index and the corresponding Higgs branch Hilbert series. Then, we consider Lagrangian Class S k theories at genus g " 1. We compute the Higgs branch Hilbert series and the Hall-Littlewood limit of the index for some of these theories. As in the case of N " 2 class S theories at genus g " 1 the Hall-Littlewood limit of the index and the Higgs branch Hilbert series are different. The reason is that the Higgs branch Hilbert series counts only B r,p0,0q and D p0,0q multiplets, while the Hall-Littlewood limit of the index, as discussed in Appendix C, counts also C r,p0,j 2 q ,Ĉ p0,j 2 q and D pj 2`1 q pj 2 ,0q multipltes. We show that the Hilbert series of the mesonic branch of the theory arising on a stack of N -D3 branes probing a C 3 {pZ kˆZl q singularity in the limit l, k Ñ 8 reproduces the 1/2-BPS partition function of the p1, 1q LST. This equality provides an important check of the deconstruction proposal for the (1,1) LST.
Finally, we consider the A 1 four punctured sphere in class S 2 . This theory admits three pants decompositions corresponding to the three distinct S-duality frames. In two of these, the T A and T B theories, which are strongly coupled SCFTs appear. Using the Spiridonov-Warnaar inversion formula we compute the Hall-Littlewood and the Coulomb limit of the index for these two theories. We observe that all the coefficients arising in the expansion of the HL index of the T B theory are positive. On the other hand in the expansion of the HL index of the T A theory are present negative coefficients. From an index perspective these negative coefficients arise because a long multiplet hits a unitary bound and splits into short multiplets. Moreover this also implies that in general, for non-Langrangian class S k theories, the HL limit of the index is not equal to the Higgs branch Hilbert series.
In the future it would be interesting to study whether it is possible to define a Coulomb branch and a Higgs branch for theories in the Class S Γ [11]. For S Γ"D,E , a direct generalisation of our work is not immediate since there is no analogue of U p1q t . Naively, the reason is that A N type singularity has a U p1q isometry whereas D N , E N do not. For example C 2 {A N has defining equations xy " w N`1 , we can give U p1q charges`1,´1, 0 to x, y, w, but C 2 {D N`2 has x 2`y2 w " w N`1 and has empty isometry group.
Finally, an important question that we didn't address in this paper, is whether a two dimensional chiral algebra [29] for the 4d N " 1 theories discussed in this paper exist. In this correspondence a central role is played by the Schur limit I S of the N " 1 index (4.21). In Section 4 we observed that the current j µ 11 , whose top component is identified with the stress-energy tensor of the corresponding chiral algebra, is not present in I S . This fact seems to lead to the conclusion that such chiral algebra does not have a stress-energy tensor. It has been observed in [82][83][84] that in the context of 4d N " 2 SCFT with defects it is possible to construct a chiral algebra without a stress-energy tensor. Therefore, it would be interesting to clarify if a similar phenomenon could take place also for the 4d N " 1 theories considered in this paper. As a possibly smaller step in this direction it would be interesting to clarify which of the properties, valid for N " 2 4d Higgs branches, discussed [30] have a counterpart for the N " 1 theories discussed in this paper.

Acknowledgments
It is a pleasure to thank Antoine Bourget for useful discussions, and Shlomo Razamat for not only important discussions but also for reading and commenting on our draft in its final stages. The work of the authors was partially supported by the DFG via the Emmy Noether program "Exact results in Gauge theories" and the GIF Research Grant I-1515-303./2019.

A The Algebra-Geometry Correspondence
In this Appendix we review the correspondence between complex algebraic geometry and affine varieties (we refer the interested reader to [85] for a more broad introduction to this topic). Given an affine complex variety V Ă C n , we can associate to it an ideal IpVq Ă Crx 1 , ..., x n s using the following map on the other hand given an ideal I Ă Crx 1 , ...x n s we can associate to it a complex variety VpIq in the following way Therefore these two maps provide a way to relate ideals and complex affine varieties. However in general this correspondence (more precisely the map (A.1)) is not one to one. 13 Table 11: Summary of the algebra-geometry correspondence .
order understand under which conditions we can get a bijection let us introduce the notion of the radical of a given ideal I Ă Crx 1 , ..., x n s, that will be denoted by ? I, and it is defined as ?
From its definition it follows I Ă ?
I (since f P I implies that f 1 P I). Moreover the radical of an ideal I is always a radical ideal 14 [85]. The Hilbert's Nullstellensatz theorem states that if I is an ideal in Crx 1 , ...x n s then Therefore this theorem and the maps (A.2)-(A.1) provide us a one to one correspondence between complex varietes and algebraic quantities given by radical ideals. This dictionary can be extended reformulating in algebraic terms geometrical problems, see Table 11. A particular useful class of ideals is provided by the so called prime ideals 15 . It's easy to prove that every prime ideal is also a radical ideal, and that moreover there is a one-to-one correspondence between irreducible varieties 16 and prime ideals [85].

A.1 The F-flat Moduli Space
Therefore, using the dictionary summarized in the previous subsection, the problem concerning the characterization of the different branches of the master space F can be recast into an algebraic problem. As a matter of fact the full information contained in F considered as a variety can be equivalently encoded in the quotient ring between the ring R :" CrQ i s of all the polynomials with complex coefficients that can be written starting from the scalar fields Q i that are taking a VEV and the ideal I, that is enconding the F-terms constraints. Schematically we can write Moreover since R is a Noetherian ring 17 , using the Lasker-Noether theorem, we can always decompose and ideal of R as an irredundant intersection of a finite set tJ i u of primary ideals [85], this procedure is called primary decomposition. In particular we can take into account the above ideal I and we get We can then consider the different radical ideals ? J i associated to each of the primary ideals in (A.6) that turn out to be a prime ideals. Therefore using the algebra-geometry dictionary, we have a one to one correspondence between affine irreducible complex variates and the radical ideals ? J i . This implies that the master space F, considered as a complex variety, can be written as in this way the algebraic approach turns out to be very powerful since it provides a systematic way to decompose the F-terms moduli space of a theory into different irreducible branches.
Remarkably we can also establish when the space F is a complete intersection. It holds the following [86] Theorem 1 Given the ring of polynomials R " Crx 1 , ...x n s and the ideal I Ă R then the algebraic variety associated to the quotient ring R{I is a complete intersection if and only if I is generated by a regular sequence of homogeneous polynomials.
Therefore if the ideal is generated by a regular sequence 18 of polynomials then we can use letter counting for the computation of the corresponding Hilbert series. For application of the above theorem in a different context see [65], we also refer the interested reader to Appendix A of that paper for a more detailed discussion related to this issue.

B Identities and Special Functions
The Plethystic exponential of a function f pxq, with f p0q " 0 is defined to be PE rf pxqs :" exp˜8 there exists an n such that In " In`1 " . . . and µpnq is the Möbius µ function. Some basic identities are PErts " 1 1´t , PEr´ts " 1´t , The Elliptic Gamma function is defined as Γ e pzq :" Γpz; p, qq " An obvious, yet important, identity is Γ e pzqΓ e ppq{zq " 1 .
We will often use the shorthand notation f pz˘nq " f pz n qf pz´nq . SU pN q characters Highest weight, irreducible representations R pd 1 ,d 2 ,...,d N´1 q of SU pN q may be labelled by a Young diagram λ of length pλq " N where the relations between the Dynkin labels of the representation and the Young diagram is The conjugate representation R pd 1 ,d 2 ,...,d N´1 q " R pd N´1 ,d N´2 ,...,d 1 q is therefore associated to the Young diagram λ with λ A " ř N´1 r"A d N´r " ř N´1 r"A pλ N´r´λN´r`1 q " λ 1´λN´A`1 . The characters for the representation R pd 1 ,d 2 ,...,d N´1 q are given by Schur polynomials with λ N " 0 and ś N A"1 x A " 1. We also often abuse notation and denote these characters simply by their Dynkin labels χ pd 1 ,d 2 ,...,d N´1 q pxq " rd 1 , d 2 , . . . , d N´1 s. The representation labelled by λ has dimension Schur polynomials are orthogonal with respect to the Haar measure of SU pN q xs λ , s µ y :" ¿ dµpxqs λ pxqs µ pxq " δ λ,µ , One fact that we will often use is, that for any class function f : SU pN q Ñ C that is also invariant under the Weyl group of SU pN q we can write where c µ νη are Littlewood-Richardson coefficients, we can write For Schur polynomials of type A 1 the Littlewood-Richardson coefficients are, of course, simply SOp2N q characters We will also sporadically make use of SOp2N q characters. These are given byŝ where the λ 1 ě λ 2 ě, . . . , ě |λ N | ě 0 is related to the Dynkin labels rd 1 , d 2 , . . . , d N s by   where C w is a deformation of the unit circle that includes the poles at z " T´1w˘1 but excludes those at T w˘1. Note that, if lim p,qÑ0 pf,f , T q :" pf ,f, T q is smooth, (B.21) implies pzq .
By construction the right-handed index of all multiplets of the type XLrj 1 , j 2 s prq E is zero. In computing the above one must carefully deal with equations of motion. If any given sop4, 2q representation appearing in a multiplet saturates the unitarity bound then there will be a corresponding equation of motion which must enter the index with opposite statistics. We list the the possible null states of sop4, 2q in Table 16.

C.1 N " 1 Index Equivalence Classes
By either examining the recombination rules (C.4) (C.5), or, by directly observing the contribution to the index from each multiplet (C.6)-(C.8) we can immediately read off the (left)right-handed index equivalence classes [50][51][52]. That is, the set of multiplets with equal contribution to the (left)right-handed index. Here we focus only on the right-handed index. The equivalence classes (leaving implicit the quantum number inequalities of Table  14) are rr r, j 1 s`:" tC r,pj 1 , r r´r 2 q |r r´r P 2Z ě0 u Y tĈ pj 1 , 3r r´2j 1 4 q |3r r´2j 1 P 4Z ě0 u (C.9) rr r, j 1 s´:" tC r,pj 1 , r r´r 2 q |r r´r P´1`2Z ě0 u Y tĈ pj 1 , 3r r´2j 1 4 q |3r r´2j 1 P 2`4Z ě0 u (C. 10) and their contributions to the index are I rr r,j 1 s`"´Irr r,j 1 s´" p´1q 2j 1`1 ppqq r r 2`1 p1´pqp1´qq χ 2j 1 .
(C.11) Multiplet q t bound B r,pj 1 ,0q r 2´2 3 q t ě j 1 C r,pj 1 ,j 2 q r 2`j 2`1´2 3 q t ě j 1 C pj 1 ,j 2 q 1´2 3 q t ě 2 3 pj 1´j2 q D p0,j 2 q j 2`1´qt ě 0 D pj 1 ,0q´2 j 1`1´qt ě 0 Table 17: Restrictions imposed on the up1q t representations implied by the existence of the Hall-Littlewood limit of the index. In order that a multiplet contributes to the Hall-Littlewood index it must have j 1 " 0 and saturate the bound.
The cases in which we can extract the most information regarding the spectrum from the index are those in which the equivalence class contains a small number of representatives. For example, for example if r r P p2j 1 {3, 4{3`2j 1 {3s then rr r, j 1 s`is empty and rr r, j 1 s´can contain only B r r´1,pj 1 ,0q . The multiplets D p0,j 2 q and D pj 1 ,0q are free fields and sit in their own equivalence classes. Finally the multipletsĈ p 1 2 ,0q andĈ p0, 1 2 q are the only representatives within the classes r 1 3 , 1 2 s`and r 2 3 , 0s´, respectively. r 1 3 , 1 2 s´and r 2 3 , 0s`also contain only a single representatives, being B 7 3 p 1 2 ,0q and C 2 3 p0,0q respectively. We also defined the net degeneracy NDrr r, j 2 s :" #rr r, j 2 s`´#rr r, j 2 s´. (C.12)

C.2 Hall-Littlewood Limit
The indices (C.6)-(C.8) can of course enter into the character expansion of (4.1) with factors of pτ {ρσq 2qt{3 " pt{ppqq 2{3 q qt . By construction the Hall-Littlewood limit of the index (4.14) counts only those operators with 2j 2 "´r`4 3 q t and j 1 " 0. Assuming that this limit always exists for theories in class S k we can extract bounds for the value of up1q t charges for given multiplets appearing the character expansion of the index. In particular, using the fact that with j 1 ě 0 so that the limit exists only if a ě j 1 . Therefore, from (C.6)-(C.8) one can see that the the up1q t charges of the multiplets contributing to the right handed index must obey the constraints of Table 17. By appying conjugation it is possible to find similar bounds for multiplets appearing in the character expansion of the left-handed index. One may also repeat such an exercise with the Macdonald limit (σ Ñ 0) of the index. Defining HL M pq t q " lim σ,ρÑ0´τ σρ¯2 qt{3 I M and assuming that the bounds of Table 17 are satisfied here, r X :" r IR rXs. Taking the PE yields [90] I X pp, qq :" PEri X pp, qqs " Γ e´p pqq r X 2¯,
(D.6) The vector multiplet also comes with an integral over the gauge group (B.12).