N=1 theories of class S_k

We construct classes of ${\cal N}=1$ superconformal theories elements of which are labeled by punctured Riemann surfaces. Degenerations of the surfaces correspond, in some cases, to weak coupling limits. Different classes are labeled by two integers (N,k). The k=1 case coincides with A_{N-1} ${\cal N}=2$ theories of class S and simple examples of theories with k>1 are Z_k orbifolds of some of the A_{N-1} class S theories. For the space of ${\cal N}=1$ theories to be complete in an appropriate sense we find it necessary to conjecture existence of new ${\cal N}=1$ strongly coupled SCFTs. These SCFTs when coupled to additional matter can be related by dualities to gauge theories. We discuss in detail the A_1 case with k=2 using the supersymmetric index as our analysis tool. The index of theories in classes with k>1 can be constructed using eigenfunctions of elliptic quantum mechanical models generalizing the Ruijsenaars-Schneider integrable model. When the elliptic curve of the model degenerates these eigenfunctions become polynomials with coefficients being algebraic expressions in fugacities, generalizing the Macdonald polynomials with rational coefficients appearing when k=1.


Introduction
There are several known large classes of N = 1 superconformal field theories, which often have some kind of geometric interpretation. A classical example is the class of theories associated to dimer models [1,2,3,4], labelled by bi-partite graphs drawn on a torus and a choice of rank N . These are quiver gauge theories with SU (N ) gauge groups and bi-fundamental matter, with a topology and choice of superpotential which is determined by the bipartite graph. They are associated to a string theory setup involving N D3-branes placed at the tip of a singular toric Calabi-Yau cone. The combinatorial rules for associating quivers to bipartite graphs can be extended to more general two-dimensional geometries [5,6,7], possibly including boundaries, such as disks, or higher genus Riemann surfaces. These constructions produce quiver gauge theories including both SU (N ) gauge groups and SU (N ) flavor groups, again with bi-fundamental matter.
There is another natural way to produce large classes of N = 1 superconformal field theories with a geometric origin, by a twisted compactification on a Riemann surface of six-dimensional SCFTs. Such constructions generalize the derivation of class S of N = 2 4d SCFTs from the twisted compactifications of the six-dimensional (2, 0) SCFTs on a Riemann surface decorated with punctures [8,9]. The class S construction may produce both standard gauge theories and strongly-interacting SCFTs which lack a known Lagrangian description. Geometric manipulations of the Riemann surface lead to specific manipulations of the associated SCFTs, which allow one to derive S-dualities relating in various ways the standard gauge theories and the strongly interacting SCFTs. The class S construction has an unexpected computational power, allowing for example to compute the superconformal index of all the four-dimensional class S theories, even though they might lack a Lagrangian description [10,11,12,13], and relate S 4 partition functions to 2d CFTs [14,15].
A straightforward extension of the class S construction to N = 1 gauge theories involves alternative twisted compactifications of the (2, 0) SCFTs. The basic strongly-interacting building blocks remain the same as for the N = 2 class S, but they are glued together using N = 1 gauge multiplets rather than N = 2 ones [16,17,18,19].
In this paper we are interested in a more general extension, which involves the twisted compactification of (1, 0) SCFTs. There is a rather large number of known (1, 0) SCFTs, many of which can be built through F-theory constructions [20,21]. In general, they have tensor branches of vacua where they take the appearance of six-dimensional gauge theories, coupled to matter fields which may themselves be irreducible SCFTs. A subset of the (1, 0) SCFTs become standard gauge theories on their tensor branch and often admits a D-brane engineering construction [22,23].
We focus here on the (1, 0) SCFTs T N k associated to N M5 branes sitting at the tip of an A k−1 singularity of M-theory. These theories are somewhat well understood and have several features in common with the (2, 0) theories. Furthermore, RG flows induced by vevs on the Higgs branch of these theories allow one to reach many more (1, 0) SCFTs [24]. It should be possible to extend our analysis to D-and E-type singularities, but we will not do so here.
Our main result is the conjectural definition of a new large class S k of N = 1 SCFTs associated to the compactification of the 6d SCFTs T N k on a Riemann surface with punctures. It is important to observe that the reduced amount of supersymmetry should make one very cautious in extending to N = 1 theories the intuition developed with N = 2 class S theories.
With that concern in mind, in this paper we will follow a "bottom up" approach: we focus at first on a set of conventional four-dimensional N = 1 theories which play a "data collection" role analogous to the role of linear quiver gauge theories in class S theories. By looking at the properties and S-dualities of these theories we find several pieces of evidence connecting them to their six-dimensional avatars and to a larger conjectural set of N = 1 SCFTs labelled by punctured Riemann surfaces. In the process, we learn new information about the sixdimensional SCFTs and their compactifications.
The basic set of "core" N = 1 gauge theories which are central to our analysis belongs squarely to the family of bi-partite quivers: they correspond to bi-partite hexagonal graphs drawn onto a cylinder. From that perspective, our work selects a subset of bi-partite theories which enjoy a larger than usual set of S-dualities and subjects them to a variety of manipulations to embed them into a larger family of non-bipartite N = 1 SCFTs.
The supersymmetric index plays an important role in our analysis. In particular, we are able to recast the index of class S k theories as a 2d TFT, built from wave-functions which are eigenfunctions of novel difference operators, which generalize the elliptic RS difference operators used in bootstrapping the index of class S theories. In principle, that opens the possibility to compute the index of any N = 1 SCFT in the class S k , even in the absence of a Lagrangian description. This paper is organized as follows. In section 2 we motivate our choice of "core" gauge theories by some general considerations on brane constructions. In section 3 we discuss our basic theories and dualities and set the stage for building the class S k models. In particular we introduce the the notions of theories corresponding to punctured Riemann surfaces and the basic punctures, maximal and minimal ones. This and following sections will be divided into two parts with the first part detailing general physical arguments and the second part, the index avatar, giving quantitative evidence and details using the supersymmetric index for the particular example of class S 2 A 1 theories. Next, in section 4 we analyze RG flows starting from the basic models triggered by vacuum expectation values for baryonic operators. Such RG flows lead to theories in the IR which naturally correspond to surfaces with one of the minimal punctures removed. However, unlike the class S case here the removal of a minimal puncture leads to new theory and to the notion of discrete charge labels attached to the Riemann surface. We will also start encountering new strongly coupled theories belonging to our putative classes of models. In section 5 we discuss closing maximal punctures by giving vacuum expectation values to certain mesonic operators. Such vevs lead to more general punctures and we in particular will be interested in closing maximal punctures down to minimal ones. One of the outcomes of this analysis will be a derivation of Argyres-Seiberg like duality frames for our basic duality and yet more irreducible strongly coupled SCFTs. In section 6 we will discuss how to introduce surface defects into our theories by triggering flows with space-time dependent vevs. This construction will give us certain difference operators which we expect to characterize completely the wave-functions and the 2d TFT structure of the index. We will explicitly derive the difference operators for A 1 theories of class S 2 . In section 7 we go back to the higher-dimensional perspective: we study the interplay between class S k theories and the five-dimensional gauge theories which arise from circle compactifications of the T N k theories. We will finish in section 8 with an outlook of possible further research directions. Several appendices complement the text with further technical details.

The Z k orbifold of N = 2 linear quivers
The analysis of class S theories was greatly facilitated by the existence of a core set of class S N = 2 SCFTs which admit both a Lagrangian description and a six-dimensional engineering construction: linear quiver gauge theories of unitary groups. These are theories which can be engineered by configurations of D-branes in IIA string theory, which are then lifted to M-theory to make contact with compactifications of the A N −1 (2, 0) SCFT onto a cylinder geometry.
It is thus natural to seek for some class of N = 1 SCFTs which admit a similar brane engineering construction in IIA and may be lifted to M-theory to a cylinder compactification of the T N k (1, 0) SCFTs. The latter arises on the world-volume of N M5 branes in the presence of an A k−1 singularity. We will thus take the standard NS5-D4 brane system used to engineer N = 2 linear quivers in [25] and super-impose it to an A k−1 singularity in IIA string theory.
More precisely, the D4 branes extend along directions 01234, the NS5 branes along directions 012356 and the Z k orbifold action rotates in opposite directions the 56 and 78 planes. See Figure 1 As the degrees of freedom of the original N = 2 gauge theory entirely arise from open strings, we can apply a standard orbifold construction to arrive at our candidate N = 1 theories [26]. The analysis implicitly assumes that appropriate B-fields have been turned on so that the orbifold singularity admits a perturbative description. We will come back to this point momentarily.
The orbifold procedure can be implemented in a straightforward way at the level of the gauge theory. Schematically, the orbifold group is embedded both into the global symmetries which correspond to the rotations of the internal space-time directions and into the gauge group, and all fields charged non-trivially under the orbifold action are thrown away.
We can focus at first on the D4 branes. In the absence of NS5 branes, they will support a five-dimensional N = 1 gauge theory described by a necklace quiver, the result of orbifolding N = 2 5d SYM. The Z k group acts with charges 0, 1, −1 respectively on the real scalar associated to the 9 direction and on the complex scalars associated to the 56 and 78 directions.  Figure 1: The brane configuration which engineers the core N = 1 gauge theories. We draw a setup with N = 4 D4 branes (or kN = 16 fractional D4 branes) sitting at the locus of a Z 4 singularity, intersected by 5 NS5 branes. The D4 branes in each segment between consecutive NS5 branes engineer the four necklaces of SU (N ) gauge groups. The 4-4 strings across the NS5 branes engineer the five sets of zig-zag chiral multiplets between the necklaces. The semi-infinite D4 branes can be either associated to the two necklaces of flavor groups or to five-dimensional necklace gauge theories coupled to the four-dimensional quiver theory.
The embedding of Z k in the gauge group splits the gauge fields and the neutral real scalar into k separate blocks of vector multiplets, while the complex scalar fields with charge ±1 under the embedding of Z k in the global symmetry will give bi-fundamental hypermultiplets between consecutive nodes of the 5d quiver. See Figure 2.
The six-dimensional (1, 0) SCFT T N k , compactified on a circle, is expected to give a UV completion of precisely such necklace quiver theory with gauge group SU (N ) at each node. We will review the properties of such five-dimensional theory in a later section. For now we only need to observe that it has a U (1) 2k global symmetry, which is an Abelian remnant of the SU (k) β × SU (k) γ × U (1) t global symmetry of the 6d SCFT, together with the rotation symmetry of the compact circle. The non-Abelian gauge symmetry in the UV is broken by Wilson lines for the SU (k) β × SU (k) γ global symmetry, which have to be turned on in order to have a perturbative gauge theory description in 5d, rather than an interacting 5d SCFT. In the string theory setup, this corresponds to the B-field which has to be turned on for the orbifold singularity to admit a perturbative description.
In order to arrive to our N = 1 gauge theories, we need to include the effect of the NS5 branes intersecting the D4 branes and in particular the orbifold action on the 4 − 4 strings stretched across NS5 branes. From the point of view of the four-dimensional gauge theory, we can start with an N = 2 linear quiver of four-dimensional U (kN ) gauge groups and embed Z k into the combination of SU (2) R and U (1) R-symmetries which preserves an The necklace quiver N N,k in five dimensions. It is a Z k orbifold of the maximally supersymmetric YM, with k SU (N ) nodes. Notice that each link here represents a full bi-fundamental hypermultiplet and each node a full SU (N ) vector multiplet. There is U (1) global symmetry associated with each link, rotating the bifundamental hypermultiplets. There is also a U (1) symmetry, the instanton symmetry, associated with each gauge group node. N = 1 sub-algebra, under which the 4d vector multiplet scalars (not to be confused with the 5d vector multiplets above!) transform with charge 1 and the 4d hypermultiplets with charge −1/2.
The a-th N = 2 vector multiplet will give us a necklace N a of N = 1 SU (N ) gauge groups (dropping the overall U (1) which decouple in the IR), with bi-fundamental chiral multiplets running, say, counter-clockwise along the necklace from the (i + 1)-th to the i-th gauge groups in the necklace. Because the N = 2 bi-fundalemental hypermultiplets have charge −1/2 under the R-symmetry group, we need to embed Z k in the gauge groups accordingly, with integral charges at even nodes and half-integral at odd nodes of the original linear quiver. Then each N = 2 bi-fundamental hypermultiplet is projected down to a set of bi-fundamental chiral fields which zig-zags back and forth between the nodes of consecutive necklaces, say from the i-th node of each necklace N a to the i-th node of the next necklace N a+1 and from the i-th node of N a+1 back to the (i + 1)-th of N a .
Thus if we start with a N = 2 linear quiver of n U (kN ) gauge groups, with kN flavors at each end, we end up with a N = 1 quiver of kn SU (N ) gauge groups with the topology of a tessellation of a cylinder, with triangular faces associated to cubic super-potential couplings (arising from the N = 1 superpotential coupling vectors and hypers in the original N = 2 theory) and k SU (N ) flavor groups at each end. This is the theory associated to a bi-partite honeycomb graph drawn on the cylinder, with one side of the hexagons aligned with the cylinder's axis. See Figure 3 for an example with k = 4 and four necklaces.
This N = 1 gauge theory is our candidate to describe the compactification of the T N k (1, 0) SCFT on the cylinder, decorated with (n + 1) "minimal punctures" each associated to a single transverse M5 brane wrapping two directions of the A k−1 singularity (the M-theory The "honeycomb" bi-partite graph drawn on a cylinder and the corresponding quiver gauge theory. The top and bottom lines are identified. Each cell of the bi-partite graph maps to an SU (N ) gauge group with 3N flavors corresponding to the six edges of the cell. Each node of the bi-partite graph indicates a cubic superpotential term, with sign associated to the color of the node lift of the NS5 branes in the IIA description). In analogy with the class S analysis, we hope to identify that with a compactification on a sphere, with two extra "maximal punctures" playing the role of the cylinder's ends. See Figure 4.
Here and in the next sections, we will study these "core" N = 1 gauge theories, they global symmetries, exactly marginal deformations and S-dualities, in order to find manifestations of their conjectural six-dimensional origin. In particular, we aim to find • One exactly marginal deformation parameter for each complex structure modulus of the underlying Riemann surface.
• Global symmetries matching the six-dimensional description.
• S-dualities which manifest the indistinguishability among minimal punctures and imply a 2d TFT-like associativity structure for the index • RG flows which relate different types of punctures and, in particular, relate maximal and minimal punctures, thus justifying the picture of a sphere as opposed to a cylinder.
The count of global symmetries and exactly marginal deformation parameters are closely related. As each gauge group has N f = 3N c and the super-potentials are cubic, the core theories can be thought of as deformations of a free theory. The (3n + 2)k sets of bi-fundamental hypermultiplets have each a U (1) global symmetry and there are 3nk marginal couplings. By the arguments of [27], the theory will have x exactly marginal couplings iff 2k + x of the global symmetries are unbroken by superpotential terms or mixed gauge anomalies. Figure 4: The six-dimensional lift of the brane configuration which engineers the core N = 1 gauge theories. We have N M5 branes sitting at the locus of a Z k singularity and wrapping a cylinder, intersected by 5 transverse M5 branes. The world volume theory of the N M5 branes is the T N k (1, 0) SCFT, with 5 "minimal" defects (left). We expect that the de-coupling the four-dimensional degrees of freedom can be implemented by replacing the semi-infinite ends of the cylinder by maximal punctures on a sphere (right).
The non-anomalous Abelian global symmetries of the core theories can be understood graphically as in Figure 5: each symmetry is associated to a straight sequence of chiral multiplets with alternating charges ±1. There are 2k + n + 1 generators which add up to 0. The set of (n + 1) U (1) α global symmetries associated to each of the (n + 1) blocks of chiral multiplets with the same bi-fundalental hypermultiplet ancestor can be thought of arising from the NS5 branes world volume gauge symmetry. We thus associate them to the minimal punctures in the six-dimensional description. The remaining set of U (1) 2k−1 global symmetries can be thought of as the Cartan generators of the SU (k) β ×SU (k) γ ×U (1) t global symmetry of the underlying 6d SCFT. Thus we are left with n exactly marginal couplings, precisely as in the original N = 2 theory. We associate them to the relative positions of the minimal punctures on the cylinder.

Basic building blocks
In this section we study the core N = 1 theories and their dualities. It is useful to begin by introducing some nomenclature, which helps abstracting the properties of the core N = 1 theories to the expected properties of class S k theories.
Each theory we build will be labelled tentatively by a punctured Riemann surface. Some set of "intrinsic" global symmetries will be always present independently of the choice of punctures: we can denote them as . It will be convenient to associate fugacities to these symmetries to keep track of charges of different fields and operators. We will denote the fugacities for the above mentioned global symmetries as t,  Figure 5: A simple set of generators for the Abelian global symmetries of the core theories. Each arrow represents a generator, acting on the chiral multiplets crossed by the arrow with charge ±1, depending on the sign of the crossing. The sum of all generators is 0. The vertical (red) arrows are chosen to generate the U (1) α global symmetries associated to minimal punctures. The SE pointing (green) arrows, quotiented by their diagonal, generate the U (1) k U (1) β "intrinsic" symmetries. The SW pointing (blue) arrows, quotiented by their diagonal, generate the U (1) k U (1) γ "intrinsic" symmetries.
Finally, the anti-diagonal combination of blue and green arrow give the generator of the U (1) t intrinsic symmetry. β i with i β i = 1 and γ i with i γ i = 1, respectively. We will also have a non-anomalous R-symmetry, whose fugacity we can indicate as r or, with index computations in mind, √ pq.
Each puncture will be associated to some specific set of flavor symmetries and a specific set of 't Hooft anomalies involving these flavor symmetries and the intrinsic flavor symmetries. It will also be associated to a set of canonical chiral operators with prescribed charges.
A "maximal" puncture will be associated to an SU (N ) k global symmetry. Maximal punctures will be labelled by a "color" n ∈ Z k and an "orientation" o = ±1, positive or negative, which determine the pattern of 't Hooft anomalies and the charges of the canonical chiral operators. We will have N units of U (1) t SU (N ) 2 anomaly for all SU (N ) groups, In other words, SU (N ) i has a mixed anomaly with a U (1) symmetry associated to the fugacity tβ i+n−o γ −1 i . The R-symmetry mixed anomaly should be the same as if the SU (N ) i acted on N fundamental and N anti-fundamental free chiral multiplets with R-symmetry 0 (the anomaly, of course, is computed from the fermions in the chiral multiplets, which have charge −1).
We also require a set of "mesons", chiral operators M a i+1 a i which transform as funda-mental/antifundamental of SU (N ) i+1 and SU (N ) i . 1 At a positively oriented puncture, the mesons have fugacity tβ i+n γ −1 i . A useful mnemonic rule is that they involve the β fugacity from the SU (N ) i+1 node, and the gamma fugacity from the SU (N ) i node. At a negatively oriented puncture the mesons have fugacity tβ i+n+1 γ −1 i+1 . A useful mnemonic rule is that they involve the β fugacity from the SU (N ) i node, and the gamma fugacity from the SU (N ) i+1 node. Clearly, a cyclic re-definition of the β i , with fixed γ i , will simultaneously shift the color of all the maximal punctures in a theory.

A gluing prescription
Next, we give a gluing prescription, an operation on four-dimensional theories which is interpreted in six dimensions as replacing two maximal punctures of opposite orientation and the same color with a tube.
Consider a positively oriented maximal puncture of color 0 and flavor groups SU (N ) i and a negatively oriented puncture of color 0 and flavor groups SU (N ) i . We will gauge the diagonal combinations SU (N ) g i of SU (N ) i and SU (N ) i−1 . We will also add k blocks of N 2 chiral fields Φ a i a i+1 in the fundamental/antifundamental representation of SU (N ) g i × SU (N ) g i+1 . These fields also couple to the mesonic operators associated to the maximal punctures, through cubic superpotential couplings, The superpotential couplings determine the charges of Φ a i a i+1 under the intrinsic global symmetry G k such that they have fugacities pqt −1 β −1 i γ i . The mixed anomalies at the SU (N ) g i gauge node cancel out and the intrinsic global symmetries remain non-anomalous.
It can be easily seen using Leigh-Strassler-like [28,27] arguments that this gauge theory has one exactly marginal coupling which can be continuously tuned to zero. The fact that the exactly marginal coupling can be switched off is related to the fact that for each SU (N ) gauge factor we have matter equivalent to 3N fundamental and 3N anti-fundamental chiral fields and thus the one-loop contribution to the NSVZ beta-function vanishes, and the superpotential couplings are also classically marginal.
In particular, we can count the difference between marginal parameters and broken U (1) symmetries [27]. We have 3k marginal couplings, but we break the 2(2k −2) separate intrinsic Figure 6: WRONG α POWERS The free trinion. Dots on the left an on the right represent SU (N ) groups associated to the two differnet maximal punctures. The horizontal lines are the bi-fundamentals Q and the diagonal lines Q.
symmetries of the two theories together with the k symmetries acting on the Φ i down to a diagonal combination of the intrinsic symmetries. Thus we break 3k − 1 symmetries. Each broken symmetry lifts a marginal coupling, leaving only a single exactly marginal one. We expect it to correspond roughly to the length and twist of the newly-created tube in the underlying Riemann surface.

The free trinion
Having described some generalities of the setup let us turn our attention to concrete and important examples. Our first ingredient is a free theory which we would like to label by a trinion, a sphere with three punctures. Two of the punctures are maximal and one is a "minimal" one. We will refer to this theory as the free trinion.
The free trinion is a collection of 2kN 2 free chiral fields, equipped with a specific action of the intrinsic and puncture-related global symmetries. We organize the fields in two sets of k blocks of N 2 chiral fields, denoted respectively as Q a i b i and Q It is convenient to depict the free trinion as a ring with 2k dots connected by arrows. The dots at even/odd places correspond to SU (N ) i / SU (N ) i flavor groups. See Figure 6. We also define the action of the intrinsic and of an extra U (1) α associated to the minimal puncture. Expressing the charges in terms of fugacities, we associate fugacity t It is straightforward to compute the 't Hooft anomalies of the free trinion. The U (1)SU (N ) 2 anomalies for each SU (N ) flavor group receive contributions from two blocks of chirals: we have N units of U (1) t SU (N ) 2 anomaly for all SU (N ) groups, N units of In other words, SU (N ) i has a mixed anomaly with a symmetry associated to the fugacity tβ i γ −1 i , while SU (N ) k−i+1 has a mixed anomaly with a symmetry associated to the fugacity tβ i γ −1 i−1 . There are various U (1) 3 anomalies, such as kN 2 units of U (1) 2 α U (1) t anomaly, −N 2 of U (1) α U (1) 2 β i and N 2 of U (1) α U (1) 2 γ i , and some anomalies involving intrinsic symmetries only. Finally, we have chiral operators we can associate with each puncture. For example, i . We associate them to the respective maximal punctures. On the other hand, baryon operators can be associated to the minimal puncture.
We see that the first maximal puncture, associated to the SU (N ) i , has colour 1 and positive orientation. The second maximal puncture is associated to the groups SU (N ) k−i . It has negative orientation and colour 0.

Gluing two trinions
All our core theories are produced by gluing together a sequence of n + 1 free trinions. We can focus on the simplest interacting theory built by gluing two trinions.
We can take two free trinions, shift the definition of the β i in the second trinion so that it has a positive maximal puncture of color 0 and a negative of color −1, and glue them together as described in the previous subsection. The result is our candidate for a theory with two maximal punctures, of opposite orientation and color 1 and −1, and two minimal punctures. Let us denote this theory byT k and the quiver description of the theory is in Figure 7. Throughout the paper this theory will play an important role and we will refer to it as our basic interacting theory or basic four-punctured sphere.
Our crucial claim is that the interacting theory described here enjoys an S-duality property, which corresponds to the exchange of the two minimal punctures. Notice that this is consistent with 't Hooft anomalies, as the global symmetries U (1) α and U (1) δ associated to the two minimal punctures have mixed anomalies with the intrinsic global symmetries only, and the anomalies are identical. Since we have here an exactly marginal coupling, we can switch on in a correlated manner the gauge couplings and the superpotential coupling such that the beta-functions vanish, the superconformal R-charges of the different fields are the free ones. For example the superconformal anomalies a and c are just the ones of this particular collection of free fields, Q Q Q N Figure 7: The interacting superconformal theory. The circles correspond to SU (N ) groups with the the white ones being gauged and colored ones being flavor symmetries. The fat brown line represent the mesonic operator built from the quarks which is charged under the symmetries associated to maximal puncture but is neutral under the symmetry associated to the minimal puncture. Fat blue line represents a quark from which a baryon is built which is charged under the U (1) symmetry associated with the minimal puncture but is a singlet under the symmetry associated with the maximal puncture.
We will find it convenient to assign R-charge 0 to all Q and Q and R-charge 2 for all Φ.
Notice that in our notation we are focusing on a subset of the full global symmetry of the gauge theory. The quarks and anti-quarks which are associated to a given gauge group can be re-assembled into groups of 2N flavors, so that the theory really has SU (2N ) k × U (1) k+1 global symmetry. The U (1) symmetries are associated to fugacities such as t i = tβ i γ −1 i and η 2 = αδ −1 . The S-duality transformation should invert the definition of the latter U (1) symmetry, the anti-diagonal combination of U (1) α and U (1) δ .
This structure is obviously similar to what one encounters in N = 2 SQCD, the k = 1 version of our story. As we will see momentarily, this is no coincidence: all our conjectural S-dualities can be related recursively to the N = 2 SQCD S-duality by Seiberg dualities.
Our starting point is a theoryT k defined as a necklace of SU (N ) i gauge groups, connected by bi-fundamental chiral fields Φ a i a i+1 and to fundamental fields Q c i a i and Q a i+1 c i , where the c i indices transform under SU (2N ) global symmetry groups and obvious cubic superpotential. We can take the Φ a i a i+1 to have fugacity If we apply Seiberg duality formally at the k-th node, we obtain a new theorỹ T k which has a rather simple description: it consists of a shorter necklaceT k−1 where the (k − 1)-th SU (2N ) flavor group has been gauged and coupled to 2N dual quarks and 2N dual anti-quarks. The theory also has a set of 2N × 2N mesons coupled to the latter fields by a cubic superpotential. The dual quarks and anti-quarks are rotated by the SU (2N ) k−1 and SU (2N ) k global symmetries of the original theory. Crucially, the U (1) flavor symmetry assignments work out in such a way that only fields inT k−1 transform under U (1) η .
At this point, we can immediately conclude, recursively, that say the supersymmetric index ofT k is invariant under the transformation η → η −1 : the Seiberg duality manipulation leaves the index invariant, and the index ofT k is built from the index ofT k−1 , with η only entering in the latter. AsT 1 is N = 2 SQCD with N f = 2N c , which has an S-duality which acts as η → η −1 on the flavor fugacities, the index ofT 1 is invariant under the transformation η → η −1 .
In order to believe a full S-duality statement, we need to make some assumptions about the RG flows associated to the Seiberg duality we employed, as the SU (2N ) gauge node is strongly-coupled. The RG flow of the theoryT k defines a map from the conformal manifold ofT k−1 to the conformal manifold ofT k . By induction, we can assume that the conformal manifold ofT k−1 has two weakly-coupled cusps, where the quiver gauge theory description is good. Near these cusps, the RG flow will map us to weakly-coupledT k quivers, with opposite U (1) η charge assignments. Thus the crucial assumption for the induction to hold is that the connected path between the two cusps in the conformal manifold ofT k−1 maps to a connected path in the conformal manifold ofT k . In analogy with class S, we can ask if other trinions may exist, in particular trinions which carry three maximal punctures. This is of course far from obvious. In the case of class S, a strong piece of evidence came from the existence of Argyres-Seiberg-like dualities [29].
A key step in the analysis was to find operations which "reduce" a maximal puncture to a minimal one, a Higgs branch RG flow which replaces one end of a linear quiver of SU (N ) gauge groups with a "quiver tail" which is associated to a set of minimal punctures only, all related by S-dualities. That suggests the existence of alternative S-duality frames where all minimal punctures are produced by quiver tails, attached to a conjectural SCFT with maximal punctures only.
Due to the intricacies of the quivers we study here, there is a bewildering array of possible RG flows one can trigger by a sequence of vevs for chiral operators. Correspondingly, one can modify a maximal puncture to a wide array of "smaller" punctures. Our challenge is to find a sequence which leads to a minimal puncture. We will undertake this challenge in the following sections.

The index avatar
Let us discuss the basic theories and the dualities in the language of the supersymmetric index setting the nomenclature for the discussion in the following sections. The reader can accustom him/herself with the standard definitions of the index in appendix A. The index is a very precise tool to test dualities. To be very explicit and keep our formulae readable, we will often specialize to the index of A 1 theories of class S 2 . All we will say can be rather straightforwardly extended to more general cases. Let us start from writing down the index of the free trinion. As discussed in previous subsections and depicted in Figure 9 we associate this trinion to a sphere with two maximal punctures and one minimal puncture. The two maximal punctures are of different colors, and we will return to this feature momentarily. The index can be written by collecting together the contributions of the different free field. For example, in the A 1 k = 2 case the index of the free trinion is,

Free trinion
Here the fugacities u and v correspond to SU (2) 2 and SU (2) 2 flavor symmetries associated to the maximal punctures and α is the U (1) fugacity associated to the minimal puncture. Note the index of the trinion is not symmetric under the exchange of the two maximal punctures. This is a reason why one should associate an additional parameter, color, to the maximal punctures. If we set β = 1 there is no distinction between the colors and indeed the index becomes symmetric under exchanging the two maximal punctures. In the k = 2 case we have two colors which are Z 2 valued, 0 and 1. We will also refer to the two punctures in this case as lower and upper ones.
We can glue two trinions together and obtain a theory corresponding to two maximal and two minimal punctures, and we will come to details of the gluing at the level of the index momentarily. We can also continue gluing free trinions to obtain theories corresponding to spheres with many minimal punctures and two maximal ones. The S-duality operation exchanging two punctures surely holds at the level of the index, under the usual assumption of that no new U (1) symmetry emerges accidentally in the IR.
Let us for a moment discuss implications of the S-duality for general k and N . At the level of the index, we can glue a free trinion to a theory T with a positive maximal puncture of color 0 by the formula: where I ft is the free trinion index, dµ n (u, β i , γ i , t) the measure which accounts for the gauge fields and Φ fields involved in the gluing and the u contour integral is the projection on gauge-invariant operators. This can be thought as an integral transformation on the index of T , with a kernel which depends on the fugacity α of the new minimal puncture.
The basic S-duality we derived in this section implies that if we apply the transformation twice, the result will not depend on the order of the two transformations. In other words, the integral operators associated to different values of the fugacity α commute. It is reasonable to assume that the transformations can be "diagonalized", as in the class S case [30,31], by expanding the free trinion into "wavefunctions" associated to each puncture, being the wavefunction for positive maximal punctures,ψ being the wavefunction for negative maximal punctures and φ λ (α; β i , γ i , t) being the wave function for minimal punctures, invariant under color shift, We take the maximal wavefunctions to be normalized as Then if we glue together n trinions we get an index which is explicitly invariant under S-duality. Note that gluing n free trinions we get a theory of two maximal punctures differing by n units of the Z k color.
In the rest of this subsection we will specialize to the case of k = 2 A 1 theories. As already mentioned here we have two colors for maximal punctures. The orientation of the puncture corresponds to the ordering of the two SU (2) groups. We will denote where we define (u 1 , u 2 ) † = (u 2 , u 1 ). Thus the index of the free trinion will be written as, The way to determine the functions ψ λ , ψ λ , φ λ , as well as the possible values of the index λ will be determined in section 6. These functions will turn out to be eigenfunctions of certain difference operators and thus we will refer to them as eigenfunctions. We stress that it is a rather non trivial assumption that such a representation of the index exists. We will discuss more of the physical implications of this property later on. We have the following symmetry properties satisfied by the index of the free trinion following from its explicit expression (3.3), It is natural to assume that these transformations act on single eigenfunctions and do not act on the labels λ. Under this assumption (3.11) implies that, We do not have to assume these properties for what follows but assuming them will make some of the considerations simpler and we will state explicitly when such assumption will be made. Moreover, when explicitly computing the eigenfunctions we find that these indeed are satisfied although we will not prove them mathematically.

The gauging
We gave a prescription for gluing two theories together at maximal punctures of appropriate color and orientation, by adding extra chiral fields and superpotential couplings to the mesons and gauging diagonal combinations of the flavor symmetries. The gauge group in general is SU (N ) k and in case at hand it is SU (2) 2 . Since there are two types of maximal punctures, we can glue theories along upper or lower punctures. In both cases the gauge group is the same. We identify SU (2) i of one theory with SU (2) 3−i of the other one, and we add bifundamental chiral fields of SU (2) i × SU (2) i+1 . The difference between the two gaugings is that when we glue two upper punctures the two bifundamental chirals have charges (+1, +1) and (−1, −1) under U (1) β × U (1) γ , whereas when gluing two lower punctures the charges are (+1, −1) and (−1, +1).
For the index this implies that the functions ψ λ and ψ λ are orthonormal under the following measures, , (3.14) From the free trinions we can obtain the sphere with two maximal punctures, either both upper or both lower, and two minimal punctures. The two theories are physically isomorphic and differ only by re-labeling of the flavor symmetries. The index of the theory with two lower punctures is, Here the first line comes from the gauging with the second and third lines coming from the contributions of the two trinions. This index has the duality symmetry which we should expect following our discussion in previous subsections. As discussed before this duality follows from a sequence of Seiberg and S-dualities with the relevant mathematical identities proven in [32,33,34]. Using the eigenfunctions this index is given by Gluing many trinions together we can obtain theories with arbitrary number of minimal puncture but with only two maximal punctures. Moreover if the number of minimal punctures is even the two maximal punctures are of the same color and if that number is odd they are of different color. To go beyond these constraints we will have to consider RG flows triggered by vacuum expectation values of the theories obtainable from our trinions and we are turning to that task next. Figure 11: The sphere with two maximal and two minimal punctures with the fugacities associated to the matter fields. The white nodes correspond to gauged SU (2) groups with the colored one to flavor SU (2) groups. We suppressed the pq and t fugacities.

Closing minimal punctures and discrete charges
The six-dimensional SCFTs which we conjecture lie behind our story have global symmetries which we identify with the four-dimensional intrinsic global symmetries G k . The two copies of U (1) k /U (1) actually enhance to SU (k) and in four dimensional in general only the Cartan subgroup is not broken. In the process of compactification from six to four dimensions, one has a choice of curvature for the line bundles associated to these global symmetries. In the standard class S story, there is a single global symmetry corresponding to U (1) t , and one obtains N = 2 theories for a specific choice of line bundles, and more general N = 1 theories for other choices. We will later comment on the orbifolded versions of these N = 1 theories, which should correspond to different choices of U (1) t curvature. In this section, we would like to assess the four-dimensional meaning of different choices of curvature for the U (1) β and U (1) γ bundles.
In order to do so, we look at the possibility of "closing" a minimal puncture, by giving a vev to a chiral operator charged under the U (1) α global symmetry associated to the minimal puncture. An assumption we make is that such vevs leave us inside the class of theories we are discussing, that is the theory in the IR can be associated to a certain Riemann surface. An obvious choice for the operator to receive a vev is one of the (anti)baryons built from the N 2 blocks of the corresponding free trinion. There are 2k such operators, Q N i and Q N i , and we could be giving a vev to any of them.
We can do so for any of the minimal punctures. Indeed, the S-duality properties we expect from our theories indicate that we should get the same result, up to a relabeling of minimal puncture fugacities, by turning on (anti)baryon vevs in different free trinions. For our purpose, it is particularly instructive to pick the duality frame where the free trinion we are working on is glued to a maximal puncture of some other generic theory T to give a theory T with an extra minimal puncture and a new maximal puncture. The analysis for a free trinion glued to two other theories can be done in a similar manner.
It is straightforward to see the effect of the vev. First of all, the vev Higgses the SU (N ) gauge group coupled to the (anti)quarks and identifies it with the SU (N ) flavor group of the (anti)quarks. The vev converts the cubic superpotential coupling involving that set of (antiquarks)quarks to a mass term for one of the Φ fields and for one block of (quarks)antiquarks for the nearby gauge node. We can integrate these away.
When giving a vev to the (anti)quark block, we need to re-define all the symmetry generators, including the R-symmetry one, by an appropriate multiple of the generator of U (1) α in such a way that the (anti)quark block is not charged under any of the assignments re-defined symmetries. It is easy to verify that the final theory we obtain after Higgsing has the same 't Hooft anomalies for the re-defined symmetries as the initial theory, as is obvious from anomaly matching. In particular, all the constraints we impose on 't Hooft anomalies of global symmetries associated to the punctures we did not close remain true after the Higgsing.
The resulting theory has three fewer marginal couplings: we lost two superpotential couplings involving the lifted Φ field and one gauge coupling. We also lost three blocks of fields and one global symmetry. Thus we expect to have lost an exactly marginal deformation parameter. This is consistent with the intuition that the minimal puncture is gone.
The resulting theory depends on the choice of which 2k (anti)baryons got a vev, and it is not equivalent to the theory T . Indeed, all the 2k resulting theories have the same "punctures", which differ from the punctures of T just by the colour of the maximal puncture the original free trinion was glued to. Thus we obtained 2k potential new class S k theories and discovered that class S k theories need to be labelled by extra data besides the choice of Riemann surface and punctures.
We aim to identify the extra data with a choice of curvature for the U (1) β × U (1) γ global symmetries of the underlying six-dimensional theory. Tentatively, we would say that closing a minimal puncture by a vev of a baryon charged under U (1) β i or an anti-baryon charged under U (1) γ i adds a unit of curvature for the corresponding six-dimensional global symmetry.
If this idea is correct, adding one unit of curvature for each U (1) β should be the same thing as not adding any, as the U (1) β are identified with the Cartan of a 6d SU (k) global symmetry. The same should be true when adding one unit of curvature for each U (1) γ . We will see that this is indeed the case, as long as we refine slightly our notion of "closing a minimal puncture" by adding to the theory some gauge-neutral chiral multiplets coupled linearly to the surviving baryons (if we turned on a baryon vev) or anti-baryons (if we turned on an anti-baryon vev) in the free trinion associated to the minimal puncture we closed.
As we will often have to add gauge-neutral chiral fields with linear couplings to some given chiral operator, it is useful to introduce the notion of "flipping" a chiral operator O: an operation which maps a theory with a chiral operator O to a new theory with an extra chiral multiplet φ coupled to O by a superpotential φO. The new theory lacks the operator O, but usually has a new chiral operator φ with opposite charges to O.
Let us now specialize to the k = 2 case and still hold N general while considering closing two minimal punctures. We consider a duality frame where the two minimal punctures reside in two free trinions glued to each other and also both of them glued to some general models. That is the two minimal punctures reside on a "tube" connecting otherwise generic Riemann surfaces. If we give vevs to baryons of fugacities (α −1 (4.1) The theory has a vacuum where M = Λ 2 and B =B = 0, which would precisely Higgs the left and right SU (N ) 2 gauge groups and thus bring us back to the standard theory with two fewer minimal punctures, except for the extra baryonic degrees of freedom B,B. Even at the level of the index, one finds that the N f = N node effectively produces a delta function [35] of the fugacities of the left and right SU (N ) z 2 gauge groups, multiplied by the index of the baryons B,B with fugacities (α −1 We can modify our definition of how to close a minimal puncture, by both turning on a vev for the baryon with, say, (α −1 1 √ tβ) N and a linear superpotential coupling between the other baryon B with fugacity (α −1 1 √ tβ −1 ) N and a new gauge-neutral chiral field b. The superpotential forces us at the origin of the baryonic branch of the N f = N node and insures the desired Higgsing. It also removes the undesired free baryons after the Higgsing.
We can now go back to general k. We consider a sequence of k trinions and close the corresponding punctures by giving a vev to the baryon charged under U (1) β 1 in the first trinion, U (1) β 2 in the second trinion, etc, and flipping all other baryons charged under the U (1) β . Because of S-duality, we could have picked any other permutation σ, turning on at the σ(a)-th trinion a vev for the baryon charged under U (1) βa . The order we chose simplify the analysis considerably, though.
After integrating away the chiral fields which receive mass parameters after the vevs, precisely k−1 gauge nodes end up with N f = N flavors. With the help from the superpotential couplings suppressing the corresponding baryons, the mesons for these gauge nodes get vevs, and initiate another set of Higgsing and lifting of pairs of chiral multiplets. This leads to another set of N f = N nodes, etc. At the end of the RG flow cascade, the k punctures have been completely eliminated.
Notice that there is a certain degree of correlation between the choice of punctures on a Riemann surface and the possible values for the discrete charges. Starting from a theory T with a maximal puncture of some colour and orientation, we can glue to it a chain of k free trinions and then close the resulting k minimal punctures in several different ways. This produces new theories with the same punctures as T but different discrete charges. The discrete charges, though, will necessarily add to a multiple of k.

A k = 2 example
We can give some rather explicit examples of this construction for theories in class S 2 . For example, we can start from our basic interacting theoryT 2 built from two trinions, with two minimal punctures and maximal punctures of opposite orientation and color 1 and −1 ≡ 1.
If we give a diagonal vev to, say, the block of quarks of fugacity √ tα −1 β in the second trinion, we are left with an SU (N ) gauge theory with 2N flavors: two blocks of N quarks with flavour symmetries SU (N ) 2 andSU (N ) 2 and two blocks of N anti-quarks with flavour symmetries SU (N ) 1 andSU (N ) 1 . We also have neutral chirals in bi-fundamental representations ofSU (N ) 1 ×SU (N ) 2 andSU (N ) 1 × SU (N ) 2 coupled to the (anti)quarks by cubic superpotential couplings and an additional decoupled setSU (N ) 1 × SU (N ) 1 which will enter the definition of the "mesons" at the maximal punctures of color 1 and −1. Finally, we have the single neutral chiral field we use to flip the baryon made out of the quarks of fugacity β −2 , which are charged underSU (N ) 2 . This is one of four theories which we can obtain by closing the same minimal puncture inT 2 in different ways. They all have one minimal puncture and two maximal of opposite orientation and the same color 1, but different discrete charges. They will be important in defining surface defects in section 6.
We can also consider theories obtained fromT 2 by closing both minimal punctures in such a way that the total discrete charge does not cancel out. This produces a variety of "charged tubes", i.e. theories which can be associated to a cylinder with two maximal punctures of the same type and some extra discrete charge. Such charged tubes can be glued to a maximal puncture of some other theory T to shift the discrete charges of that theory without changing the type of punctures. We will discuss some examples in appendix B.

The index avatar
Let us now translate the discussion above to the language of the index. To study RG flows generated by vacuum expectation values at the level of the index one needs to study its analytical properties. Different poles of the index correspond to operators for which a vev can be turned on with residues being indices of the theories flown to in the IR [12]. To study analytical properties of the index we first take an index of a generic theory corresponding to a Riemann surface and glue to it the free trinion. See Figure 12. The index of such a theory is given by We depict in Figure 13 the fugacities associated to different fields of the free trinion. A class of poles in the index above occurs whenever the integration contours are pinched while varying the fugacities. We look thus for pinchings of the integration contours. The poles inside and outside the integration contour coming from the free trinion are located at When some of the in poles coincide with the out poles the integration contours are pinched and the index develops poles. Different poles correspond to vevs for some protected operators.
In this section we are interested in the case of the baryonic operators obtaining a vev. We consider the pole in α, the fugacity associated to the U (1) α symmetry of a minimal puncture, at α = t 1 2 β −1 . This pole occurs when an operator with weight tβ −2 α −2 , the baryon Q 2 1 , gets a vacuum expectation value. Giving a vev to such an operator Higgses the z 1 gauge group. By turning on the vacuum expectation value we break the U (1) α symmetry, i.e. we u 2 Figure 13: The glued sphere with fugacities.
close the minimal puncture. There are similar poles at α = t 1 2 β corresponding to vev to baryon Q 2 2 , and α = t − 1 2 γ ±1 corresponding to vevs for baryons 2 and the residue of the index becomes . (4.4) Here and in what follows by residue we more precisely mean the following operation, This operation is natural as it removes the decoupled free chiral associated to the Goldstone boson index of which is I −1 V . The factor of 2 appears since the U (1) α charge of the baryonic operator which is getting a vev is 2.
In particular, if the general theory of index I({u 2 , z 2 }) is just a free trinion, the residue is the index of an N f = 4 SU (2) SQCD with additional singlet fields and superpotentials. Let us denote the field corresponding to the prefactor in the integral M , which is in bifundamental of SU (2) u 1 × SU (2) u 2 . The fields in the numerator of the integral are a quark Φ in fundamental of SU (2) u 2 and a quark Q in the fundamental of SU (2) u 1 . In I({u 2 , z 2 }) we have the contribution of additional four quarks, Q 1 and Q 1 , and four gauge singlets in a fundamental of SU (2) u 2 which we denote by M 1,2 . The superpotential then is, with M 2 being free fields. This theory enjoys an action of large duality group [36,37,38,39,40]. Under our assumptions that the RG flows generated by vacuum expectation values should leave us in our class of theories, the N f = 4 SU (2) gauge theory should be associated to a Riemann surface. In fact it can be only associated to a sphere with one minimal puncture and two maximal punctures of the same type. This is a new, interacting, trinion we discover in our bootstrap procedure. Note that here the two maximal punctures are of same color and thus the theory should be invariant under exchanging the two factors of the associated flavor symmetries. This new trinion is depicted in Figure 14. For general k such a trinion can be obtained by gluing together k free trinions and closing k − 1 minimal punctures in certain way. Let us study what this residue teaches us about the functions ψ λ , ψ λ and φ λ . As we take the residue for a minimal puncture fugacity and flip the other (anti)baryon, the wave function φ λ (α) in the sum is replaced by the insertion of certain functions of the intrinsic fugacities, which we can interpret as the contribution to the index sum of a unit of positive or negative discrete charge for U (1) β or U (1) γ . Notice that it must be true that since as we discussed closing two minimal punctures by giving vevs to the two different types of (anti)baryons leaves behind no discrete charge. Applying the residue prescription to a trinion glued to a maximal puncture, we can get a neat integral relation between wavefunctions of different color: This relation is actually known as an elliptic Fourier transform [41], and can be inverted by a second elliptic Fourier transform. The invertibility of the elliptic Fourier transform, though, is precisely the index avatar of the Seiberg duality relation for an SU (N ) gauge node with N flavors and flipped baryons, which we used to show how opposite discrete charges cancel out. Thus the elliptic inversion formula gives us the same result as directly taking a residue of a free trinion glued to a φ λ wave function: Computing residues of the basic four punctured sphere and removing the appropriate singlets we thus get new trinions with indices, This is the SU (2) N f = 4 SQCD with singlets and superpotential we discussed above. It will be very useful when we derive the difference equations satisfied by ψ λ . We can consider closing the minimal puncture in the new interacting trinion. The theories obtained in this way would correspond to spheres with two maximal punctures of same color and with non vanishing discrete charges. We will discuss briefly these constructions in appendix B.

Closing maximal punctures
Our next aim is to give evidence for existence of theories corresponding to spheres with maximal punctures only. To do so we will study RG flows triggered by the vev of chiral mesonic operators which are charged only under symmetries associated to a single maximal puncture and the intrinsic symmetries. To have a concrete example, to which we will refer in the discussion below, the generic theory glued to a free trinion in the previous section can be taken to be a sphere with two maximal and many minimal puncture, but the discussion is completely generic and example independent.
The analysis of a general case is a bit cumbersome so we choose to start our discussion with A 1 and k = 2 and gradually crank up these parameters. In the class S theories of type A 1 , maximal punctures turn out to be equivalent to minimal punctures, though they appear different in the brane construction, essentially because mesons and baryons are on the same footing in linear quivers of SU (2) gauge groups.
For k = 2 maximal and minimal punctures are clearly different. The simplest choice of chiral operators charged under the global symmetries of a maximal puncture are the meson . We refer to fugacities and fields as depicted in Figure 13. Turning on vevs for a single meson operator breaks the SU (2) u 1 × SU (2) u 2 flavor symmetry down to an U (1) δ subgroup. Thus we may hope it will result in a minimal puncture.
Without loss of generality we can focus at first on giving a vev to the operator Q 1 Q 1 , with fugacity (u 1 u 2 ) −1 t γ β . More specifically, we give a vev to the component of Q 1 with gauge charge 1 under the SU (2) 1 and Q 1 with gauge charge −1. When turning on these vevs, we need to re-define the intrinsic global symmetries and define U (1) δ by appropriate combinations of the old global and gauge symmetries, in such a way that the fields getting a vev are neutral under the new global symmetries.
At the level of fugacities, this is accomplished by setting the the SU (2) u 1 × SU (2) u 2 fugacities to (u 1 , u 2 ) = (t As we turn on the vev for these chiral fields, some other fields are lifted by the cubic superpotentials, which become mass terms. The SU (2) z 1 gauge field is Higgsed and only SU (2) z 2 is left to glue the other free trinions to the one we triggered the mesonic vev in.
Thus we find that by giving vev to a meson and adding some extra neutral chirals linearly coupled to chiral operators of the original theory we arrive to a theory which has an extra explicit symmetry, permuting the unbroken δ fugacity with (any)one of the minimal punctures fugacities. In other words, starting say from a linear quiver built by concatenating free trinions we have produced a quiver gauge theory which can be rightfully labelled by a single maximal puncture and several minimal ones. An alternative perspective is that the surviving fields in the "α" free trinion, together with the new chiral fields, define a "quiver tail" which can be appended to a negative maximal puncture with color 1 in order to convert it to two minimal punctures of fugacities α and δ.
Notice that we could have obtained a similar result starting with the second set of mesons at the original maximal puncture. That would have imposed fugacities, say, z 2 = αδ , (u 1 , u 2 ) = (t 1 2 βδ , t 1 2 1 γδ ) pole corresponding to vev for Q 2 Q 2 . This would have produced an a priori different way to reduce the maximal puncture to a minimal one.
It turns out that we can ascribe the difference between the two possible ways to map maximal to minimal to a difference in discrete charges. Indeed, we can probe the difference between these two choices further by closing the newly-created minimal puncture by giving vev to baryonic operators of fugacity δ 2 tγ ±2 or t 1 δ 2 β ±2 . At the level of fugacities, if we set, say, δ = t − 1 2 γ in the relations (u 1 , u 2 ) = (t γδ ) = (t, β γ ) due to Weyl symmetry of u 2 . Similarly, setting δ = t − 1 2 β in the first reduction is equivalent to setting δ = √ tγ −1 in the second one. A detailed analysis of the reduction procedure shows that these pairs of ways to completely close the maximal puncture are indeed equivalent, even when keeping track of the neutral chiral fields we added in the process, as long as we remove an additional singlet chiral field with fugacity β 2 γ 2 in the first way to reduce maximal to minimal and γ 2 β 2 in the second way. Thus we conclude that reducing a maximal puncture to a minimal puncture by giving a vev to a meson with fugacity proportional to γ β or to a meson with fugacity proportional to β γ give class S 2 theories with discrete charges which differ by one unit of U (1) β curvature and one unit of U (1) γ curvature.
As we have identified a "quiver tail" which can be attached to a maximal puncture to obtain two minimal punctures, it is natural to do the same step which in class S leads to the definition of non-trivial trinion theories: we can conjecture the existence of SCFTs with one puncture of color 0 and two of color 1 (and appropriate choices of discrete charges), with the property that attaching a quiver tail to one puncture of color 1 will produce our basic core theory built from two free trinions. As we have two different versions of the quiver tail, we seem to need at least two distinct SCFTs, with discrete charges differing by one unit of U (1) β curvature and one unit of U (1) γ curvature.
It is instructive to look a bit further to the combination of vevs which we expect to produce and close a minimal puncture starting from a maximal one. It corresponds to giving a vev to mesons with fugacities t γ β u −1 1 u −1 2 and t β γ u −1 1 u 2 . This implies a vev for both chiral fields with fugacity u −1 If we are working with a standard core theory built from a sequence of trinions, these vevs force us to turn on chiral fields in the next free trinion as well, because of the cubic superpotential couplings of the second and third fields to the Φ field of fugacity pq t βγz −1 1 z 2 : the extremum equations for Φ require us to turn on a vev for the meson of fugacity t 1 βγ z 1 z −1 2 in the next free trinion. Looking at the theory in detail, we find that the original free trinion has been completely eliminated, while the next free trinion is precisely subject to the vev which reduces the maximal puncture to minimal. This is just another manifestation of the S-duality relations which permute the minimal punctures of the theory.
Although we do not have an intrinsic way to compare discrete charges between theories with different types of punctures, we can use some symmetry considerations to set up some useful conventions. Let's declare that the core theories have zero discrete charges. We have several different ways to produce a theory with a single maximal puncture and several minimal ones: a minimal puncture can be produces starting from a maximal puncture of either colour, which can be reduced in two ways.
We saw that the two ways of reducing a puncture of color 0 differ by one unit of U (1) β curvature and one unit of U (1) γ curvature. In a similar fashion, the two ways of reducing a puncture of color 0 differ by one unit of U (1) β curvature and minus one unit of U (1) γ curvature. Finally, we saw that reducing a puncture of color 0 and then closing the resulting minimal puncture in a specific way is the same as reducing a puncture of color 1 in a theory with one fewer minimal punctures.
Thus it is natural to pick the following symmetric convention to compare the discrete charges of theories before and after reducing maximal punctures: giving a vev to a meson with fugacity proportional to γ ±1 β ∓1 in a color 0 puncture adds ∓1/2 unit of β charge and ±1/2 of γ charge, while giving a vev to a meson with fugacity proportional to γ ±1 β ±1 in a color 1 puncture adds ±1/2 unit of β charge and ±1/2 of γ charge.
The general k case for N = 2 is not much harder to analyze, except that now both orientation and color matter. We will proceed by analogy with our k = 2 analysis, and give a prescription to reduce a maximal puncture to minimal. Our prescription will be to give a vev to mesons of fugacity u i u −1 i+1 tβ i γ −1 i . We can give a vev to up to k − 1 of them, say for i = 1, · · · , k − 1. Thus we give a vev to chiral fields of fugacity u i z −1 i √ tβ i α −1 and z i u −1 i+1 √ tγ −1 i α and Higgs k−1 SU (2) groups, leaving only SU (2) k . The chiral field vevs enter the cubic superpotential couplings of the Φ fields between SU (2) i and SU (2) i+1 for i = 1, · · · , k − 2. They force us to also give a vev to the mesons in the nearby trinion, with fugacities z i z −1 i+1 tβ i+1 γ −1 i for i = 1, · · · , k − 2. In turn, these fugacities Higgs k − 2 SU (2) i gauge fields in the next column, for i = 1, · · · , k − 2, but also force us to turn on k − 3 mesons at the next trinion, etcetera. The result is that one end of our quiver of (k − 1)k SU (2) gauge groups is modified to take a triangular shape, with columns of k − 1, k − 2, · · · , 1 SU (2) gauge groups. See Figure 15 for illustration. Figure 15: Example of closing a maximal puncture in a linear quiver with k = 3. Giving a vev to the two mesons denoted by dashed brown lines in the left column the SU (2) 3 group denoted by filled green dots is broken down to U (1). These vevs also Higgs two of the three gauge groups in the second column. The vevs for the mesons through superpotential interactions generate vevs for a meson in the second column, again denoted by dashed brown lines. This meson Higgses another SU (2) gauge group. We end up with a "triangle" of unbroken gauge groups denoted by white dots.
We can parameterize the z i and u i in terms of a parameter δ, so that in general the z i are proportional to (αδ) −1 and the u i proportional to δ −1 . Thus in order to identify a symmetry exchanging α and δ we need first to make sure that the fields charged under SU (2) k satisfy such a symmetry. Then we can try to impose the symmetry on neutral fields by adding extra neutral chirals. The relevant fields have fugacities u ±1

Half of these receive masses by the meson vevs, the other half are
and thus δ −1 = u 1 t (k−1)/2 β −1 k . Next we can focus on the lack of symmetry exchanging the α and δ symmetries of neutral fields. The trinion fields have fugacities u ±1 √ tγ −1 i α, but the ones which do not get vevs nor masses and are not eaten by the Higgs mechanism are the ones with Half of the fields have fugacities proportional to α 2 , i.e. tβ 2 i α −2 and tγ −2 i α 2 , and are remnants of the baryons. The other half has α-independent fugacities u −2 i and u 2 i+1 proportional to δ ±2 . If we remove them through linear couplings to new neutral chirals, and replace them with chirals of fugacities tβ 2 i δ −2 and tγ −2 i δ 2 and appropriate superpotential couplings, we arrive at a theory symmetric in α and δ. For the consistency of the picture with further closing the minimal puncture we might need to decouple additional singlet fields charged only under the intrinsic symmetry; we will not analyze this here.
Thus we learned how to convert a maximal puncture to a minimal puncture, in k different ways. As for k = 2, these different procedures leave one with different amounts of discrete charges on the surface. For general k we can consider giving vevs to different combinations of mesons closing a maximal puncture down to a puncture Λ with symmetry U (1) ⊂ G Λ ⊂ SU (2) k . On the quiver such choices are classified by carving out multiple triangular wedges from the tail. We will not embark on tail classification here though it is a very interesting problem to discuss. See Figure 16 for illustration. We can call punctures accessible from maximal punctures by RG flows triggered by vevs of mesons "regular" punctures in analogy with class S. There might be other types of punctures one would want to consider but that goes beyond the analysis of this paper. Figure 16: Example of closing a maximal puncture in a linear quiver with k = 3 down to a nonminimal one. We give a vacuum expectation value to a single meson. This breaks the flavor group to SU (2) 2 × U (1) and Higgses one of the gauge groups in the next column. This, next to maximal, puncture together with the maximal and minimal ones are the only "regular" punctures of the A 1 class S 3 . This puncture comes in several varieties depending on the color of the maximal puncture we start with and the particular meson we choose to trigger the flow.
A N −1 and general k Let us now analyze the reduction of maximal punctures in A N −1 class S k theories. The basic idea is the same as in A 1 case but now we have many more mesons and thus the details are more involved. For general A N −1 even in class S there is a variety of possible punctures labelled by Young tableaux [8] and to get from a maximal one to minimal one has to turn on vevs for many mesons.
We begin our most general A N −1 class S k discussion by prescribing a vev for mesons of fugacity u (a) i+1 (u (a) ) −1 i tβ i γ −1 i . We can give a vev to up to (N − 1)(k − 1) of them, say for i = 1, · · · , k − 1 and a = 1, · · · , N − 1. Thus we give a vev to chiral fields of fugacity z i ) −1 √ tγ −1 i α and Higgs k − 1 SU (N ) groups, leaving only SU (N ) k . These vevs are not yet sufficient for our purposes, as they leave N − 1 U (1) symmetries unbroken. We will thus also turn on an extra set of N − 2 mesons, with fugacities u Thus we give a vev to chiral fields of fugacity z The vevs enter the cubic superpotentials and force us to turn on also vevs for certain mesons in the next free trinion. Namely, we need mesons with fugacities z k−1 ) −1 tβ k γ −1 k−1 with a = 1, · · · , N − 2. These vevs will Higgs the next column of SU (N ) i gauge groups to nothing, except for the last two, Higgsed again to SU (2).

These meson vacuum expectation values are implemented by vacuum expectation values for chiral fields in the next trinion of fugacities z
Here the y In terms of the SU (2) fugacityz, such thatz 2 = z k , and the flavor fugacities as (z The flavor fugacities have a ratio u k γ k β k α −2 ) −1 which we want to identify with α N /δ N . Thus we set δ −N = u With this choice of δ, we have defined our global symmetries in such a way that U (1) α and U (1) δ act in the same way on chiral fields which carry gauge charge. We will have a bunch of gauge-neutral fields charged under both symmetries and by appropriate flips the resulting theory can be made to be fully symmetric under exchanging the U (1) α and U (1) δ symmetries.
We have learned how to convert a maximal puncture into a minimal one.

The index avatar
Let us discuss the above at the level of the index. As usual we specialize to the A 1 k = 2 case.
The discussion above makes it clear that one should be able to produce φ λ , up to a normalization factor, by taking a residue of either ψ λ or ψ λ at appropriate values of the fugacities. When we defined φ λ , we have normalized in such a way that the free trinion would have a simple expansion of the form (3.10). In a 2d TFT language, it would be more natural to introduce structure constants C λ and write the index associated to a Riemann surface of genus g with n punctures p a and charges q i in the schematic form Thus if we use a convention where core theories have charge 0, we should write φ λ = C λ ψ m λ , with ψ m λ being a properly normalized minimal puncture wave-function. Then with the symmetric charge assignments discussed above, we can write We can obtain three other similar relations involving the other way to reduce ψ λ and the two other ways to reduce ψ λ , with other prefectures of the form C (β,±) λ C (γ,±) λ . Note that (5.2) together with (4.7) and (4.8) completely determine C λ and C (β/γ,±) λ . We will give explicit expressions for these in appendix B.
We can also describe at the level of the index the quiver tail which converts a maximal puncture to two minimal punctures. We start from the general expression for the index of a theory glued to a free trinion, set (u 1 , u 2 ) = (t The above manipulations imply a concrete relation between the wavefunctions corresponding to maximal punctures and to minimal ones. For the functions ψ λ , ψ λ , and ψ m λ (5.3) implies that, The relation (5.4) can be used to write the index of our core example of the four-punctured sphere in a very suggestive form. We note that This index can be interpreted as the index of a new strongly interacting trinion with two lower maximal punctures and one upper puncture and discrete charges (1/2, −1/2).
The second equality in (5.5) can be interpreted as the computation of the index in a dual description of the core theory, involving a quiver tail attached to that strongly-interacting SCFT associated to a sphere with three maximal punctures. We can invert the integral in (5.5) using the Spiridonov-Warnaar inversion formula, aka elliptic Fourier transform [41], to obtain explicitly the index of the strongly coupled theory without using the eigenfunction. This is the same inversion procedure used in [11] to obtain the index of the T 3 theory.
We will propose an additional, related, duality which connects this interacting trinion to a different Lagrangian theory in appendix D: SU (4) SQCD with four fundamental flavors and two flavors in antisymmetric representation supplemented with gauge singlets and a superpotential.
This conjectural, strongly interacting trinion theory can be used as a building block together with the free trinion (with or without closed minimal punctures) to assemble class S 2 theories labelled by a generic Riemann surfaces of genus g with arbitrary numbers of maximal and minimal punctures and arbitrary discrete charges.

Discrete charges for U (1) t
In our investigations we found that for the space of theories to be closed under gluings and RG flows we have to consider the discrete curvatures for U (1) β and U (1) γ . However we did not have to incorporate such curvatures for U (1) t . Nevertheless, we can consider turning on these curvatures too as was done for k = 1, class S, in [17]. Let us here briefly outline how to apply this generalization. To generalize our story we first should allow two types of theories. These are two copies of theories we discussed till now but with R-symmetry of the two types of theories related as (5.8) The intrinsic and puncture symmetries of the two theories are related as To trinions of type + we associate a new Z valued charge +1 and to trinions of type − we associate charge −1. We glue two trinions of the same type with the appropriate measure we discussed till now, i.e. introducing bi-fundamental fields Φ and coupling them to the mesons associated to the gauged maximal puncture through superpotential. We glue two trinions of different types along a maximal puncture without introducing any extra fields but turning on a quartic superpotential which is a product of the mesons associated to the gauged puncture from the two glued theories. At each gauge node we have here N f = 2N . See Figure 18 for an example. A way to derive the map of symmetries (5.9) is to study anomaly free gluings of two free trinions of different type as depicted in Figure 18.
If we consider only one type of theories as we did till now the new Z valued charge will have a fixed value, which we can take to be ±(g − 2 + s) with s being the total number of punctures and the sign determined by which class of trinions we use. However when we glue theories of the two types together and start triggering RG flows closing minimal punctures arbitrary values of the new Z valued charge can be achieved. It would be interesting to develop this generalization in more detail. We make some comments on the index of theories with U (1) t discrete charges in appendix E.

Surface defects
In this section we will use the strategy of [12] in order to produce difference operators which act diagonally on wavefunctions, associated to BPS surface defects produced by a "vortex construction", i.e. RG flows initiated by position-dependent vevs of chiral operators. Consider two theories T U V and T IR connected by an RG flow initiated by a constant vev of a chiral operator of charge 1 under some U (1) α global symmetry. If we couple the theory T U V to a "vortex", i.e. background connection for the U (1) α global symmetry, with n units of flux concentrated near the origin of the plane, we can give the chiral operator a vev which is constant at infinity, but has a zero of order n near the origin. An RG flow far to the IR will leave us with a surface defect in T IR .  Figure 18: An example of a theory with flux for U (1) t . The A 1 k = 2 sphere with two maximal and two minimal punctures built from two trinions of opposite type with the fugacities associated to the matter fields. The white nodes correspond to gauged SU (2) groups with the colored one to flavor SU (2) groups. Note that in our conventions the quarks in the trinion on the left have R-charge zero and on the right R-charge one. Each gauge node has four flavors. We turn on superpotentials for the two "diamond" paths in the quiver.
At the level of the index, the vortex construction of surface defect is very simple. The index of T U V has a pole at some value α 0 of the fugacity of U (1) α whose residue is the index of T IR . The pole is accompanied to an infinite family of poles located at α 0 q n , whose residues give the index of T IR in the presence of the vortex defect of order n.
Let us take a general theory in class S k and attach to it a sequence of k free trinions to produce a new theory with k extra minimal punctures and the same discrete charges. We know that we can recover the original theory by turning on a baryonic vev for each new minimal puncture, as long as we pick k baryons with distinct U (1) β i charges or anti-baryons with distinct U (1) γ i charges.
Thus we can produce 2k classes of interesting surface defects S β i ,n and S γ i ,n by making the vev of one of the k baryons position dependent, or one of the k anti-baryons.
Of course, we could also state the construction in term of the interacting trinions built from a sequence of k free trinions by closing k − 1 minimal punctures. This will save us a bit of work below.

The index avatar
The surfaces defects are very useful in the index considerations since they allow to fix the functions ψ λ (u) by specifying them as eigenfunctions of certain difference operators. As usual, we specialize here to k = 2 and N = 2.
The index of the interacting trinion is given by, We remind the reader that this theory was obtained by closing a minimal puncture of a sphere with two minimal and two maximal punctures by giving a vev to a baryon which set δ = t 1 2 β −1 . The symmetry between u and v is not manifest here, it follows from the duality property of the basic four punctured sphere. We glue this trinion to a generic theory by gauging one of the maximal punctures. The resulting index is This index has many interestng poles. If one computes the residue at α = t 1 2 β we will erase Figure 19: Gluing the intercting trinion to a general theory.
the minimal puncture and obtain precisely the index of the generic theory we glued the trinion to. As an operatorial statement we say that computing the residue at α = t 1 2 β amounts to acting with identity operator on I 0 .
Next we can consider poles in α which have additional powers of q. We claim that this index has, among many others, a pole when α = t The pole is produced by pinching the integration contours between poles of the integrand. The poles arise from the Γ e functions associated to the chiral multiplets which receive a vev. In particular, we have a collision of poles of Γ e (t 1 2 βα −1 u ±1 1 v ±1 2 ) and obtain a finite residue at 2 2 Setting α = t 1 2 βq 1 2 and u1 = q ± 1 2 v ±1 2 in (6.1) one flavor becomes massive and we can evaluate the integral in terms of the Seiberg dual mesons [42]. With the particular charges and generic fugacities for global symmetries appearing in (6.1) one of the mesons contributes zero. However tuning u2 = q ± 1 2 v ±1 1 another meson contributes a pole which cancels the zero.
Computing the residue is tedious, but straightforward. If we define , (6.4) the residue is computed by acting with the following operator on I 0 , Since the theories we consider enjoy S-duality we can act with the difference operator on any of the maximal punctures with the same outcome [12]. The functions ψ λ are eigenfunctions of S (β,−) (0,1) . This operator introduces a surface defect into the N = 1 theories of A 1 class S 2 . We can start from the trinion with opposite β discrete charge and close a minimal puncture with t 1 2 β −1 , or start with one of the trinions with γ discrete charge and close the minimal punctures appropriately. The difference operators one obtains are all related to the above, The functions ψ λ should be simultaneous eigenfunctions of all these operators and indeed it can be checked that these operators do commute. We can now in principle compute more difference operators introducing more general surface defects by computing other residues of the index above, with higher n. It is a priori straighforward but tedious exercise and we refrain from doing it here.
We leave a systematic study of the difference operators and wavefunctions with generic fugacities to future work. Here we will take some degeneration limits which simplify the analysis considerably and allow us to write down some simple, explicit formulae. Although for a general N = 1 theory the limit p → 0 of the index may not be well defined, or useful, the indices of class S k theories with the choice of fugacities used in this paper appear to have a reasonable p → 0 limit, akin to the Macdonald limit of the index of N = 2 gauge theories. In this Macdonald-like limit, the eigenfunctions of this difference operator orthonormal under the vector multiplet measure become (experimentally) polynomials up to an universal pre-factor.
The measure under which the polynomial part of the eigenfunction is orthogonal is with k = 1, N = 2, and β 1 = β, γ 1 = γ. This is a generalization of the A N −1 Macdonald measure. We will discuss a straightforward algorithm to compute the eigenfucntions by diagonalizing the difference operators in Macdonald limit in appendix B.
Let us here quote the results if we further set β, γ = 1 and take the Hall-Littlewood-like limit p, q = 0. For the first several wavefunctions we find The hat on the functions reminds us that these functions are not normalized to be orthonormal. Note that the coefficients of the polynomials here are algebraic expressions, roots of polynomial equations, as opposed to rational expressions in the N = 2 case. The N = 1 theories are in this sense irrational though algebraic. The indices are single valued sums over roots of algebraic equations. We denote the normalized functions where χ λ (z) is a polynomial. The HL index of the free trinion is then given by t) .
(6.10) -41 -As we will see in appendix B the index of the interacting trinion with three maximal punctures becomes (6.11) When β, γ = 1 there is no difference between the colors of punctures and we are blind to the different discrete charges. In appendix B we will write the general expressions for the index of any theory in class S 2 when p = 0.

Five dimensional interpretation
In this section we will re-examine our four-dimensional discussion in the language of boundary conditions and interfaces for a N = 1 five-dimensional gauge theory N N,k , the necklace quiver formed by k SU (N ) gauge groups. This model is the world-volume theory of D4 branes sitting at an A k−1 singularity and conjecturally arising from the compactification on a circle of the six-dimensional SCFTs which inspire our work. Our intuitive picture is that a Riemann surface with s semi-infinite tubes labels interfaces between s copies of N N,k defined on halfspaces.
Our first task is to review the properties and definitions of boundary conditions and interfaces for five-dimensional N = 1 gauge theories

Generalities
A five-dimensional N = 1 gauge theory is labelled by a gauge group G, a flavor group F and a quaternionic representation of G × F which specifies the hypermultiplet content. The hypermultiplets can be given real masses associated to the Cartan sub-algebra of F . The gauge couplings of G can be identified as real masses associated to "instanton" global symmetries U (1) I whose conserved currents are of the schematic form * TrF ∧ F . We can denote the full global symmetry group asF = F × U (1) I . The theory is further labelled by a choice five-dimensional Chern-Simons couplings κ, which may be integral or half-integral depending on the amount of matter fields.
Although the gauge theories may have strongly-coupled UV completions, in the IR they are free. That implies that the gauge groups can be treated as very weakly coupled when describing a boundary condition. A simple class of boundary conditions is labelled by two pieces of data: the subgroup G ∂ of the gauge symmetry preserved at the boundary and a choice of boundary condition for the hypermultiplets. Somewhat more general boundary conditions are possible, which involve a generalization of the Nahm pole which occurs in the maximally symmetric case. We will come back to these later in the section.
The boundary condition for the hypermultiplets can be strongly coupled. If the hypermultiplets pseudoreal representation is the sum of two conjugate representations, a general construction is available, which starts from some free boundary conditions and adds superpotential couplings to extra four-dimensional N = 1 degrees of freedom living at the boundary. If we denote the two halves of the hypermultiplets as X and Y , we can start from a Y = 0, ∂ ⊥ X = 0 boundary condition and add a linear superpotential coupling to an operator O X in a boundary theory B X . Alternatively, we can start from a X = 0, ∂ ⊥ Y = 0 boundary condition and add a linear superpotential coupling to an operator O Y in a boundary theory B Y . As long as we focus on F-terms only, the two constructions are essentially equivalent: B Y can be obtained from B X by "flipping" O X , i.e. by introducing a new chiral field φ with superpotential coupling φO X . Then O Y ≡ φ. A particular case is that the X = 0, ∂ ⊥ Y = 0 can be obtained from Y = 0, ∂ ⊥ X = 0 by adding a chiral with Xφ coupling, and vice versa. Boundary conditions for a five-dimensional gauge theory may have various anomalies. The boundary cubic gauge anomaly receives three contributions: • The bulk Chern-Simons coupling • The boundary theory • The boundary condition for the bulk hypermultiplet: Y = 0, ∂ ⊥ X = 0 contributes half of the anomaly of a boundary chiral field with the same charge as X. 3 This half is the reason the bulk CS coupling may sometimes be half-integral The total gauge anomaly for G ∂ must cancel out.
We also have various sources of 't Hooft anomalies. This includes cubic and mixed anomalies arising from the boundary theory, the boundary condition for the hypermultiplets and the bulk CS couplings. The R-symmetry anomaly also receives contributions from the boundary conditions for the gauge fields, which is half of what a four-dimensional G ∂ multiplet would give.

A review of N N,k
The necklace quiver theory N N,k has a U (1) 2k global symmetry: rotations of the bi-fundamental hypermultiplets and instanton symmetries. See figure 2. A natural way to parameterize these symmetries follows from the UV realization of the gauge theory as a web of fivebranes drawn on a cylinder, with k infinite NS5 branes and N circular D5 branes, see Figure 20.
The transverse position of the i-th top half-infinite NS5 brane is the mass parameter for a U (1)β i symmetry which acts with charge 1 on the X a i+1 a i fields of the i-th hypermultiplet, −1 on the Y a i a i+1 fields, 1/2 on the instanton charge at the (i + 1)-th node and −1/2 on the instanton charge at the i-th node. We define the k − 1 symmetry generators U (1) k U (1) β by quotienting by the diagonal symmetry U (1) t .
Similarly, the transverse position of the i-th bottom half-infinite NS5 brane is the mass parameter for a U (1)γ i symmetry which acts with charge −1 on the X a i+1 a i fields of the i-th hypermultiplet, 1 on the Y a i a i+1 fields, 1/2 on the instanton charge at the (i + 1)-th node and −1/2 on the instanton charge at the i-th node. We define the k − 1 symmetry generators Finally, we need to pick a U (1) p symmetry generator whose mass parameter is associated to the sum of all gauge couplings, i.e. the size of the cylinder. We pick it to act on the instanton charge at the first node.
In terms of fugacities, the hypermultiplet fields X a i+1 a i have fugacities tβ i γ −1 i , and the CS coupling at the i-th node is β i−1 γ i−1 β i γ i for i = 1, p β k γ k β 1 γ 1 otherwise. In the following we will encounter variants of this symmetry labeling, where the U (1) β i and U (1) γ i are permuted among themselves by permutations σ and τ respectively, and the U (1) p symmetry generator act on the i-th node. We can denote that as N σ,τ,i N,k . We will also often consider boundary conditions which break U (1) p . In that case, we can refer to N σ,τ, * N,k . In the low energy quiver gauge theory, the mass parameters for these global symmetries are constrained by the requirement that the gauge couplings should be positive. The UV description in terms of fivebranes suggest that symmetry enhancements should occur when some semi-infinite branes are brought together. A maximal case is when all semi-infinite branes are brought together, giving rise to a SU (k) β × SU (k) γ symmetry enhancement. Indeed, as the branes live on a cylinder there are k distinct ways to reach such a symmetry enhancement, as one bring the branes together in the order i, i + 1, · · · , i − 1. The parametrization of N σ,τ,i N,k is adapted to that order.
We should also remember the six-dimensional UV completion of the five-dimensional quiver gauge theory, in terms of a circle compactification of the (1, 0) SCFT corresponding to N M5 branes in an A k−1 singularity. Then U (1) p becomes the KK momentum and SU (k) β × SU (k) γ × U (1) t become six-dimensional global symmetries. Then the five-dimensional mass parameters are lifted to the inverse radius of the compactification circle and to Wilson lines for the six-dimensional global symmetries.
Intuitively, we may hope to find "duality walls" in the five-dimensional gauge theory, which express the invariance of the UV theory under permutation of two consecutive symmetries U (1) β i and U (1) β i+1 or U (1) γ i and U (1) γ i+1 . More precisely, such a wall would arise as the low energy limit of a Janus configuration in the UV, where the mass parameters for these symmetries are brought across each other as we move along the fifth direction.
The notion of duality walls for five-dimensional gauge theories is explored in depth in a separate publication [43]. Here we mention them because the domain walls which are associated to spheres with two punctures and non-trivial discrete charges will turn out to coincide, amazingly, with the duality walls for the N σ,τ,i N,k theories associated to permutations of the U (1) β i and U (1) γ i 5d mass parameters, i.e. of the 6d flavor Wilson lines. This provides a powerful check of our conjectural interpretation of discrete charges as curvature charges for the six-dimensional U (1) β i and U (1) γ i global symmetries.

Boundary conditions from maximal punctures
Consider any of our four-dimensional gauge theories, with a positive puncture of colour 0. We will build from it a U (1) p -breaking right boundary condition for N N,k , i.e. a boundary condition for N *

N,k
The anomalies and superpotential couplings match nicely. We can deform the X = 0 boundary conditions by coupling the Y a i a i+1 to the mesons M and on the right to N * N,k . The fact that the interface breaks U (1) p is consistent with the six-dimensional interpretation of the system as an infinite tube with a minimal puncture with a specific location on the cylinder. Upon closing the minimal puncture, though, we will find that U (1) p is restored.

Closing minimal punctures and duality walls
Next, we close the minimal puncture in the interface between N The vev Higgses the two gauge groups together, so that they form a single SU (N ) 1 gauge group stretching across the interface. The vev also couples the bifundamentals of fugacities √ tγ −1 1 α −1 = tβ 1 /γ 1 linearly to the five-dimensional hypermultiplets on the right hand side of the interface, and √ tγ −1 k α −1 = tβ 1 /γ k on the left hand side, converting the corresponding X = 0 to Y = 0 boundary conditions. The match between the global symmetries across the new interface works almost exactly as before, except for a crucial difference: the U (1) p symmetry acting on the SU (N ) 1 instanton charge both on the left and on the right of the interface is now unbroken! This is consistent with the picture that the minimal puncture has been erased. We have obtained an interface between N (2,3,··· ),(1,2,··· ),1 N,k and N 1 N,k . It turns out that such an interface can be actually decomposed into k − 1 simpler interfaces, each corresponding to a simple duality operation. The rightmost interface is between N The simple interface associated to a permutation of β 1 and β 2 lets all five-dimensional gauge groups go through the interface, except for SU (N ) L 2 and SU (N ) R 2 , which are coupled by a bi-fundamental chiral field Q of fugacity β 2 /β 1 . In turn, det Q is coupled to a gauge-neutral chiral operator b by a linear superpotential W = b det Q.
The hypermultiplets also just go through the interface, except the ones charged under SU (N ) L,R 2 . On the right, we set to zero the X R 2 field of fugacity tβ 2 /γ 2 and the Y R 1 field of fugacity t −1 β −1 1 γ 1 . On the left, we set to zero the Y L 2 field of fugacity t −1 β −1 1 γ 2 and the X L 1 field of fugacity tβ 2 γ −1 1 . We introduce superpotential couplings QY L 1 X R 1 + QX L 2 Y R 2 . This gives an interface between N (2,1,3,4,··· ), (1,2,··· ),i N,k and N i N,k for every i except when i = 2. We would like to interpret it as the duality interface for permuting β 1 and β 2 .
In a similar manner, we could consider an interface between theories N (1,2,··· ), (2,1,3,4,··· ),i N,k and N i N,k for every i except when i = 2, involving a bi-fundamental chiral fieldQ of fugacity γ 2 /γ 1 going in the opposite direction. We would need superpotential couplingsQX We would like to interpret it as the duality interface for permuting γ 1 and γ 2 . In order to test our interpretation, we should check that these interfaces fuse in a manner consistent with the expected properties of the permutation group of the β i and the γ i .
The first test is obvious: permuting β 1 and β 2 twice should give the identity, and thus concatenating two copies of the corresponding duality interface should give a trivial interface. This works indeed as expected: the SU (N ) 2 gauge theory in between the interfaces gives at low energy a four-dimensional SU (N ) gauge group with N f = N , as the boundary conditions at the interfaces eliminate completely the bulk hypermultiplets in the interval, and only the two sets of Q chiral multiplets remain, together with the baryonic couplings. We know that such a theory in the IR reduces to the QQ mesons with a non-zero vev, which Higgses the five-dimensional gauge theories on the two sides of the fused interface together.
A second test is that permuting β 1 and β 2 should commute with permuting γ 1 and γ 2 . It turns out that this follows from a neat application of Seiberg duality. The SU (N ) 2 gauge theory in between the interfaces now has N f = 2N : the Q andQ give 2N "quarks", while the bulk hypers in the interval give 2N "antiquarks". Seiberg duality at that node exchanges the setups which arise from fusing the interfaces in two different orders.
It is even easier to check that the interface for permuting of β 1 and β 2 commutes with, say, the interface for commuting γ 2 and γ 3 .
The σ 12 σ 23 σ 12 = σ 23 σ 12 σ 23 relations in the permutation group of the β i can also be checked. If we concatenate the corresponding triplets of interfaces, we get interfaces which support four-dimensional gauge theories which are related by the Seiberg duality of a SU (N ) theory with N f = 2N , up to a slight mismatch in the superpotential couplings which may perhaps explained away by operator mixing. We leave a more detailed analysis to [43].
Finally, we can comment on the relation between different ways of closing a minimal puncture and "curvature charges" in the six-dimensional puncture. By closing a minimal puncture with chirals of fugacities proportional to β i , we obtain an interface between N (2,3,··· ,1),(1,2,··· ),i N,k and N i N,k . The final order of the β j is independent of our choice, but the interfaces glue the U (1) p action on the two sides in different ways.
We can try to match the interface with the behaviour of an Abelian SU (k) connection on an infinite tube, asymptotically flat at the two ends, but with curvature (1, 0, 0, · · · ) in between. The curvature will force the Wilson line parameter in the first eigenline to vary as m 1 → m 1 + 1/R as we go from right to left, thus passing across all other Wilson line parameters and back to the initial position on the circle. This seems to roughly match what we see in the interface.

Punctures and orbifold Nahm poles
It is natural to identify a maximal puncture with a D X Dirichlet boundary condition for the five-dimensional gauge theory, setting to zero the Y a i a i+1 half of the hypermultiplets: If we map a 4d theory T with a maximal puncture to a boundary condition B by coupling it to the 5d theory, we can recover the 4d theory T from the 5d theory on a segment, with boundary condition B at one end and Dirichlet at the other end. The surviving half X a i+1 a i of the hypermultiplets provide the expected "mesons" and the boundary condition produces the desired 't Hooft anomalies at the boundary.
The five-dimensional perspective is useful in describing other types of punctures as well. For example, in class S general punctures are associated to a variant of Dirichlet boundary conditions, the Nahm pole boundary conditions. In this section we generalize that notion to the necklace quiver theories N N,k .
The boundary conditions we are after are boundary conditions for BPS equations which describe field configurations of the 5d theory which preserve four-dimensional super-Poincare invariance.
There are D-term and F-term equations. The F-term equations set the complex moment maps to be zero and require the hypermultiplet vevs to be covariantly constant under a complexified gauge connection of the form D 5 = D 5 − Φ, where Φ is the scalar super partner of the gauge bosons. The D-term equations set D 5 Φ to be equal to the real moment map for the hypermultiplets. For the necklace quiver, we can write schematically We are interested in deformations of the D X boundary conditions, which still set the Y i fields to zero but enforce a singular behavior of the X i , Φ i fields, We can call these boundary conditions orbifold Nahm poles. Notice that we can organize the A i and B i matrixes into two kN × kN matrices t 3 and t + , with t 3 block diagonal of blocks A i and t + with blocks B i under the diagonal. The t + and t 3 give an SU (2) embedding ρ into SU (kN ), with the property that t 3 commutes with a diagonal matrix Ω with N eigenvalues equal to 1, N equal to e 2πi/k , etc. while t + has charge 1 under the action of Ω.
If we start from D X and we give a vev to the "mesons", i.e. the boundary values of the X i , such that the kN × kN matrix M with blocks X i under the diagonal is nilpotent, it is natural to expect the boundary condition to flow in the IR to the orbifold Nahm pole labelled by the su(2) embedding ρ in SU (kN ) associated to M .
The vevs we used to fully close a maximal puncture are a perfect example of this setup: the "chain" of k(N −1) meson vevs engineers a nilpotent matrix M with a single, large Jordan block.

Discussion
In this paper we have given a basic description of some the properties of class S k theories. Starting from core theories which we associate to spheres with two maximal and a bunch of minimal punctures we discovered that in order to build classes of theories closed under gaugings and RG flows triggered by vevs for a very particular set of chiral operators we should consider theories which are naturally associated to spheres with more general combinations of punctures. Moreover, the theories should be labelled not just by a punctured Riemann surface, but also by a collection of discrete charges. Some of the theories which followed from our considerations are strongly coupled SCFTs. Some of these can be thought of as IR fixed points of a Lagrangian theory but some lack such a description. Using these SCFTs we in principle now can construct theories corresponding to Riemann surfaces of arbitrary genus.
Our analysis is rather incomplete in many respects and leaves room for many new insights to be uncovered. We can supplement our analysis with a very partial list of interesting questions and open problems which we would like to see addressed in the near future.

Comparison with bipartite theories
Although we have focussed on quivers drawn on a cylinder and compactifications of the sixdimensional theories on surfaces of genus 0, we could have also readily defined core theories associated to tori with kn minimal punctures, by gluing together the two maximal punctures of our standard core theories. It would be interesting to explore in full the relation between such genus 1 class S k theories and the standard bipartite quiver gauge theories associated to toric Calabi Yau singularities. See appendix C for some examples.

Classification of punctures
We have discussed at first two types of punctures, maximal and minimal, the former coming in k varieties. We have argued that more general "regular" punctures may be defined by turning vevs of collections of mesons at a maximal puncture, and proposed a five-dimensional classification in terms of su(2) embeddings in SU (kN ) commuting with a Z k subgroup. The duality walls we encountered in five dimensions also suggest that one may define maximal or other regular punctures labelled by general permutations of the β i and γ i as well. A natural open problem is to complete a systematic classification of "regular" punctures and the corresponding quiver tails.

Precision study of the spectrum
We have given a prescription to compute, at least in principle, the index for any of the theories in A N −1 class S k . It would be interesting to actually extract from the indices information about the operators of the putatively new strongly coupled SCFTs. For example, their marginal and relevant deformations [18]. It would be also interesting to find a systematic way to determine their conformal anomalies.
Extension to other (1, 0) SCFTs In this paper we discussed theories obtained, conjecturally, by reducing the T N k (1, 0) SCFTs associated to N M 5 branes on A k singularity. It would be interesting to extend the discussion to other types of (1, 0) SCFTs.
Some (1, 0) SCFTs can be obtained from RG flows initiated by Higgs branch vevs in the T N k theories. It may be possible to track the four-dimensional image of these RG flows, perhaps by giving vevs to chiral operators which are charged under intrinsic symmetries only.
It should be possible to extend our work to other 4d gauge groups and 6d SCFTs by considering "core" theories defined by brane systems enriched by orientifold planes.

Reduction to three dimensions
We can consider reducing theories of class S k to three dimensions. When we have a description of the theory in four dimensions in terms of a conformal Lagrangian, in three dimensions the description will not in general be conformal. Moreover, the reduction on a circle will produce additional superpotentials involving monopole operators [44]. Such superpotentials break explicitly symmetries in three dimensions which are anomalous in four dimensions. Keeping track of such superpotentials is crucial to have dualities work for the compactified theories.
One interesting aspect of the dimensional reduction in class S 1 was that the reduced theories possess a mirror description which was always Lagrangian, a star-shaped quiver with arms associated to the punctures of the original theory [45]. The difference operators and wave-functions [46] had a simple interpretation in the language of domain walls interpolating between two S-duality frames of N = 4 SYM with SU (N ) gauge groups [47]. The eigenvalue equation was associated to the S-duality relation between a Wilson line a 't Hooft line on the two sides of the duality wall [12,48,49].
We do not know how much of this will generalize to class S k . The difference operators do appear to be related to 't Hooft line operators in a necklace quiver. It may be that the 3d limit of the wave-functions is still related to domain walls interpolating between different duality frames of N = 2 four-dimensional necklace quivers. It would be interesting to explore this idea and perhaps build universal mirrors for the compactification of class S k theories.

Surface defects
We have discussed RG flows introducing surface defects into our theories. It would be interesting to study properties of these defects in more detail. They appear to give a broad generalization of the elliptic RS difference operators encountered in class S theories.

Holography
Class S k theories can be in principle studied in large N limit. For example one can consider theories corresponding to Riemann surfaces without punctures to avoid proliferation of flavor symmetries. Such AdS 5 backgrounds were recently considered in [50,51] generalizing some of the k = 1 results discussed in [52]. It would be interesting to study the relations between holography and our results in more detail.

Geometrization of Seiberg dualities
Starting from quiver theories with bifundamental matter and employing different types of dualities one can in principle generate gauge theory with matter in more intricate, tensor, representations. See appendix D for example. It would be interesting to understand whether class S k can serve as a natural setup to systematize the diverse variety of Seiberg dualities.

Quantum mechanical models
It will be interesting to study in more detail the quantum mechanical models for which our wavefunctions are eigenfunctions. Here we have discussed in some detail the case of A 1 and k = 2 and the generalization to higher k and N though rather straightforward might be interesting. One can also try and compute other partition functions, such as lens index [53], for class S k theories. In terms of eigenfunctions while the supersymmetric index is related to symmetric functions (and polynomials) the lens index in general is a natural generalization to non-symmetric functions [54,55]. The lens index should also provide a window into subtleties with global properties of the gauge and flavor groups [56].
Here j i are the Cartans of the SO(4) ∼ SU (2) 1 × SU (2) 2 isometry of S 3 and r is the U (1) r R-symmetry. The charges q a correspond to U (1) global flavor symmetries with the set of these symmetries denoted by F. The supersymmetric index of a free chiral field of R-charge R is given by For the sake of brevity In this paper Γ e (z) will stand for the elliptic Gamma function Γ(z; p, q) implicitly defined above. When an SU (N ) z flvor symmetry of a theory with index I(z) is gauged, the index of the gauge theory is given by, being the q-Pochhammer symbol. The index of a free vector field is given by, We will also encounter theta functions which we will define as θ(z; q) = (z; q)(qz −1 ; q) . (A.6) An important property of the supersymmetric index is that it is independent of the continuous parameters of the theory, such as marginal couplings, and also is independent of the RG scale. Thus it can be computed for example in the UV using a non-conformal description and be the same as the superconformal index of the IR CFT. We will adopt the standard convention that "ambiguous" powers in the arguments of a function denote product over all the possibilities. For example the index of the N = 2 hypermultiplet is given by, B. Eigenfunctions for A 1 class S 2 We will discuss here explicit algorithm to construct eigenfunctions for A 1 theories of class S 2 in the p = 0 limit. The simplifying factor here is that the eigenfunctions turn out to be proportional to polynomials. For p = 0, as also is the case for class S 1 , the computation is much more involved. However, in principle a perturbative computation in p around p = 0 solution can be set up. See for example [59].
For p = 0 the computation is rather straightforward. The equation we have to solve is Here S (β,−) (0,1) was explicitly given in section 6.1. A way to proceed is to make an anzats, .

(B.3)
Then we assume χ(u) to be a polynomial in u i with general coefficients of maximal degree n and symmetric in u i → u −1 i . Since our indices are invariant under u → −u we also can assume that χ(u) has well defined parity. We will have then for polynomials of order n order 1 4 n 2 free parameters, and we also have the eigenvalue as a parameter. Expanding the eigenvalue equation in u i and demanding it to hold for any monomial term will give us a system of nonlinear equations for these coefficients. The non-linearity comes because the eigenvalue can multiply other coefficients. We then should solve these system of algebraic equations which in general has a finite set of solutions.
Let us illustrate this explicitly for the lowest orders. Assuming the polynomial is of degree zero we get that and thus the order zero polynomial is an eigenfunction given that, The first eigenfunction is then given by, We have normalized it to have unit norm under the gluing measure. At next order we make an ansatz, Then we compute taking q → 0 to avoid horrendous expressions Here E = q − 1 2 E . This system of equations can be reduced to a quadratic equation in one of the two variables H or E and thus has two solutions. Ultimately we obtain, We should further normalize χ(u) to have norm one. We note that this expression is algebraic in fugacities. Taking for simplicity β, γ = 1 we obtain that here H = ±1 recovering the result advertised in section 6.1. Note also that with this value for H all the symmetry properties (3.12) are satisfied. We can continue to derive eigenfunctions in this manner. Going to higher orders we get higher order polynomial equations which do not have closed form solutions, but we can solve them perturbatively in the fugacities.

The self-adjointness of the difference operator
Let us check that the basic difference operator we computed is self adjoint under the gauging measure. That is for two sufficiently nice behaving functions f (u) and g(u) we have, We note that, .

(B.12)
Then we have that 1 )Γ e (u ±2 2 ) T (u 1 , u 2 )g(u 2 , u 1 )f (q This implies that under change of coordinates {u i } → {q 1 2 u i } the first term, out of four, on the left hand side of (B.11) in the expansion of the difference operator maps exactly to the fourth term on the right hand side. This can be repeated for the other three terms and thus implies that the difference operator is self-adjoint.
The index of generic class S 2 A 1 theory Let us here write down the index of a generic theory residing in A 1 class S 2 . We will specialize to the case p = 0 where we can write very explicitly expressions for the index.

(B.21)
Note that if we specialize fugacities for different symmetries, i.e. ignore/break corresponding symmetries the indices simplify. For example, taking γ = β, and thus identifying the two corresponding U (1) symmetries, the difference between 1 and 2 is gone and we label the theories by Riemann surface and one integer. Taking β = 1 and thus neglecting the U (1) β symmetry there is no difference between upper and lower punctures and also 1 is not a meaningful number any more. Let us mention here that setting both β = γ = 1 one can derive from (5.4) the following factorization property, Which in particular implies that switching off β and γ we cannot distinguish in the index maximal punctures from pairs of minimal ones.

Sphere with two maximal punctures
We can start from the (β, −) interacting trinion of section 4.2 and close the minimal puncture.
Note that the result is explicitly invariant under exchanging the two minimal punctures. We can attach this theory to a trinion with opposite discrete charges and three maximal punctures of the same color to obtain yet another duality frame for the basic four punctured sphere. For the eigenfunctions the above implies the following relation, 2 ) (u † , , δ) (u ±2 1 ; q)(u ±2 2 ; q)(t( γ β ) ±1 u ±1 1 u ±1 2 ; q)ψ λ (u) .
Thus we can glue our new trinion to an interacting trinion with appropriate discrete charges to obtain yet another Argyres-Seiberg like frame for our basic interacting four-punctured sphere.
One might consider closing punctures in the new trinion. We can further close the maximal puncture in two different ways to obtain a theory corresponding to sphere with three maximal punctures. We also can close a minimal puncture in four different ways and obtain a sphere with one maximal and one minimal punctures.

C. Fun with tori
Let us discuss here several simple examples of theories corresponding to torus with one minimal puncture or no puncture in the k = 2 A 1 case. Torus with one minimal puncture can be obtained for example by gluing together the two maximal punctures of the trinion with two maximal punctures of same color and a minimal puncture with one of the discrete charges, i , being ±1; the theory we obtained by closing a minimal puncture in our core example of an interacting theory. Since here we have a Lagrangian, depicted in Figure 21, we can observe that the theory has one-dimensional conformal manifold. Giving a vev to one of the chiral fields as depicted in Figure 22 we Higgs two of the gauge groups and obtain a theory which we can associate to torus with no punctures but with two units of one of the discrete charges in class S 2 . This theory has an alternative interpretation as A 1 theory of class S 1 corresponding to torus with two punctures (with additional singlet fields flipping the quadratic operators built from the two adjoints). This theory is also know as Y p=1,q=1 (modulo the extra superpotential terms) in the Y p,q nomenclature [60]. We have an additional gauge singlet field coupled to one of the bifundamental chirals connecting two nodes without an adjoint field.
Another two equivalent ways to obtain a torus with no punctures come from giving vacuum expectation value to bifundamental built from quarks connecting one of the other pairs of gauge groups. This theory has a unit of both β and γ discrete charges. The theory so obtained is the T 1,1 theory or Y p=1,q=0 theory (again with extra singlets and superpotential), SU (2) 2 gauge theory with four bifundamental chirals and a superpotential. See figure 23.
Finally we can give a vacuum expectation value to the quadratic singlet built from the adjoint field. This amounts to setting δ = t 1 2 β. This vacuum expectation value Higgses the corresponding group and gives masses to four of the quarks. The remaining theory is the SU (2) 2 with two bi-fundamental flavors and two additional singlets and is associated to torus with no punctures and no discrete charges. See figure 24. D. Class S 2 interpretation of a selfdual SU (4) SYM with tensor matter Let us take our basic interacting four punctured sphere of A 1 class S 2 and perform Seiberg duality on one of the gauge nodes. We have discussed this duality in section 3 and here let t t where this equation can be viewed as definition of Υ λ . We will derive the relation of Υ λ to the eigenfunctions soon. The fact that (D.7) can be written in terms of single sum over eigenfunction is a manifestation of S-duality or in this case the Seiberg duality of [37]. We have the following relation, . (E.3) We can derive yet another relation between eigenfunctions. Let us glue as in section 6 an interacting trinion to a general theory but now glue it to the puncture of the opposite type, i.e. glue ψ puncture toψ one without extra fields but with a quartic superpotential. Next, we close the minimal puncture residing on the interacting trinion. Assuming dualities to hold, i.e. that we can freely move around on the Riemann surface different punctures and consider various pair of pants decompositions leading to the same theory, we deduce thatψ and ψ have to be orthonormal under the measure involving only the N = 1 vector multiplets. These two facts translate into the following relation between eigenfunctions, Note that this means thatψ are eigenfunctions of the following operator, We can compute this operator to be .

(E.7)
Note also that the operators S In the k = 1 case discussed in [18] the two types of difference operators turn out to coincide but for higher k one derives two different commuting operators.