The Superconformal Index of Class S Theories of Type D

We consider the superconformal index of class S theories of type D, which arise by compactification of the (2,0) D_n theories on a punctured Riemann surface C. We also allow for the presence of twist lines on C associated to the Z_2 outer automorphism of D_n. For a two-parameter slice (p=0, q, t) in the space of superconformal fugacities, we determine the 2d TQFT that computes the index.


Introduction
Class S theories are a family of four-dimensional gauge theories with N = 2 supersymmetry, which arise by (partially twisted) compactification of the six-dimensional (2, 0) theories on a punctured Riemann surface [1,2]. There is a beautiful dictionary relating supersymmetric observables of the 4d theory with quantities defined on the surface C. Notably, the complex structure moduli of C correspond to the exactly marginal 4d gauge couplings, while the punctures are associated to flavor symmetries. The partition function of the 4d theory on the possibility of Z 2 twist lines on C. In type D theories, twist lines are very natural, indeed they are necessary for the description of all the non-trivial examples with a Lagrangian, such as the superconformal linear quivers with alternating SO/USp gauge groups [7,8]. Ideally, one would generalize the approach of [15], and derive the general answer with no guesswork. However this seems technically challenging, because conformal tails of type D have no analog of the U (1) flavor punctures that were used in [15] to derive the difference operators. Thinking about the physics of surface operators may provide the right clues, but we leave this for the future. Here we generalize instead the approach of [14], and look for a diagonal representation of the index for p = 0 in the Macdonald basis. The presence of twist lines makes the story richer, leading to an interesting extension of 2d TQFT structure. As in [15], we get some mileage by considering the action of Macdonald difference operators. In particular we use these difference operators to argue that the index of free hypermultiplets has a diagonal expansion in the Macdonald basis. In this paper the difference operators serve an auxiliary technical role, but it is natural to expect that they also have a physical interpretation in terms of BPS surface defects.
The rest of the paper is organized as follow. In section 2 we review some facts about the class S theories of type D, with or without Z 2 twist lines. In section 3, after briefly recalling the definition of the superconformal index and the TQFT approach, we consider the special cases of D 2 and D 3 theories, and extrapolate from them our general proposal for the D n case (with maximal and empty punctures). In section 4 we consider the extension to partially closed punctures. In appendix A we collect some technical background material on Macdonald theory. Finally appendix B contains the main intertwining identity that shows diagonality of the free hyper index in the Macdonald basis.
Note added: Last night, the interesting article [23] appeared on the ArXiv. The authors of [23] focus on the evaluation of the index twisted 3 by the outer automorphism group along the temporal S 1 , while we focus instead on the ordinary (untwisted) index of type D theories but with twist lines on C, so our results are largely complementary to theirs. There is some partial overlap in the discussion of the purely SO theories.

Class S Theories of Type D
The superconformal theories that we consider were constructed in [7,8] following [1]. Let us briefly summarize the relevant points.
First recall that an N = 2 hypermultiplet in representation R of the gauge group G can be decomposed in two N = 1 chiral multiplets sitting in complex conjugate representations R and R * . The flavor symmetry of N f such hypermultiplets depends on the reality properties of the representation R. For R complex one has flavor symmetry U(N f ); for R real, the flavor symmetry is enhanced to USp(2N f ); for R pseudoreal the flavor symmetry is enlarged to SO(2N f ). For pseudoreal R, there is no need to double the multiplet to satisfy CPT, and one can consider a single chiral multiplet in representation R as an N = 2 multiplet. This is the so-called half-hypermultiplet. However, to avoid Witten's global anomaly [ 26] one cannot have an odd number of half-hypermultiplets. N f half-hypermultiplets have SO(N f ) flavor symmetry. In order to construct superconformal theories, all couplings need to be marginal. For gauge group SO(m) with hypermultiplets in the vector representation this implies that N f = m − 2, and then the flavor symmetry is USp(2m − 4). For gauge group USp(2n) with hypermultiplets in the fundamental representation the requirement is that N f = 2n+2, and the flavor symmetry is given by SO(4n + 4). It is possible to consider half-hypermultiplets in this case.
Given these constraints, one can start constructing superconformal quiver gauge theories. In order to be superconformal, these will in general have alternating gauge groups. Let us briefly introduce our quiver conventions which follow [7]. A gauge group is depicted by a circle, grey when SO and black for USp. Numbers inside the circles indicate which SO or USp gauge group is considered. Lines connecting two gauge groups represent bifundamental 4 half-hypermultiplets. Additional hypermultiplets are denoted by squares. Grey squares indicate that they carry SO flavor symmetry, and black ones USp flavor symmetry. Numbers inside the boxes indicate which flavor symmetry they have. The simplest example is to consider an SO(2n) gauge theory coupled to N f = 2n − 2 hypermultiplets. The flavor symmetry is USp(4n − 4). This theory is depicted in figure 1a. In figure 1b one focuses on a USp(2n − 2) × USp(2n − 2) subgroup of this flavor symmetry group. This quiver theory can also be depicted in terms of the curve C in figure 1c. Here the SO gauge group is depicted by a cylinder and each three-punctured sphere represents a free half-hypermultiplet transforming under the fundamental of USp(2n − 2), corresponding to the puncture denoted by the symbol •, and the vector of SO(2n), depicted by the symbol •. In this case the diagonal SO group is gauged. Also notice the presence of a Z 2 twist line connecting the USp puncture and the empty USp flavor symmetry puncture (drawn as ×), in the curve. Similarly one can consider a USp(2n − 2) gauge theory coupled to N f = 2n Figure 3: Construction of the T SO(2n) theory. The · · · in the middle of the top quiver stand for a sequence of alternating SO(2n) and USp(2n − 2) gauge groups. The other · · · on the left/right stand for a sequence of alternating odd SO and USp gauge groups of increasing/decreasing rank. The symbol ⊂ means that we gauge a SO(2n − 1) subgroup of SO(2n).
hypermultiplets. Figure 2 depicts this theory. Here the USp gauge group is denoted by a cylinder with a twist line.
Starting with linear quivers, which have a Lagrangian description, one can make use of dualities to obtain new interacting theories. Figure 3 summarizes the construction of the so-called T SO(2n) theory [7]. It is described by a three-punctured sphere with three SO(2n) punctures. The effective number of hyper-and vectormultiplets for this theory is [7] Notice that for n = 2 one finds that the effective number of vectormultiplets is zero. Figure 4 summarizes the construction of another interacting theory [27], denoted asT SO(2n), which is described by a sphere with two USp(2n − 2) punctures and one SO(2n) puncture. The effective number of hyper-and vectormultiplets for this theory is [27] (2.2) Figure 4: Construction of theT SO(2n) theory. The · · · in the middle of the top quiver stand for a sequence of alternating SO(2n) and USp(2n − 2) gauge groups. The other · · · on the left/right stand for a sequence of alternating odd SO and USp gauge groups of increasing/decreasing rank. The symbol ⊂ means that we gauge a SO(2n − 1) subgroup of SO(2n).
Finally, there is one more basic three-punctured sphere we can introduce and that will only appear as part of a bigger theory, namely a sphere with an SO(2n) puncture and two empty USp punctures ×.
Gluing two punctures by a cylinder corresponds as always to gauging the diagonal flavor group. By gluing the basic three-punctured spheres introduced so far, see figure 5, we can generate punctured surfaces of arbitrary topology, subject to the condition that the USp punctures must come in an even number, and be connected pairwise by twist lines. There can also be twist lines wrapping cycles of the surface. Different duality frames correspond to different degeneration limits of the same surface [1]. Basic examples are given in figure 6.
The class of surfaces generated by the three-punctured spheres figure 5 correspond to the smallest subclass of theories that contains the alternating linear quivers with SO(2n) and USp(2n−2) gauge groups and is closed under S-dualities. More general superconformal tails will be incorporated in section 4 by considering more general punctures. , . Figure 5: The three-punctured spheres with maximal SO punctures and maximal or empty USp punctures. We indicate the flavor fugacity assignments and our notation for the corresponding index.

A TQFT with Twist Lines
In this section we determine the 2d TQFT that computes the p = 0 limit of the index for the theories described in the previous section. Our starting point is the plausible guess that the Macdonald basis should diagonalize the structure constants of the TQFT. The presence of twist lines and of punctures of both SO and USp type leads however to a more involved structure. Indeed we will have to deal with these complications already in the basic example of free hypers. Nevertheless, we are led to a natural proposal that passes several checks.

Preliminaries
Let us recap some basic fact about the N = 2 superconformal index. We follow the notations and conventions of [14]. The index [5] is defined by a trace over the states on S 3 (in the standard radial quantization of the CFT), Here j 1,2 are the Cartans of the Lorentz group SU (2) 1 × SU (2) 2 , r is the U (1) r generator, R is the Cartan of SU (2) R and f i are flavor symmetry charges. The combinations of superconformal generators in (3.1) commute with the superchargeQ 1− which carries quantum numbers j 1 = 0, j 2 = − 1 2 , r = − 1 2 and R = 1 2 and with its conjugate. The index counts states in the cohomology ofQ 1− . In particular only states satisfying contribute. For theories with a Lagrangian description the index can be computed as a matrix integral, which can be written schematically as Here U collectively denotes all gauge group elements and V all flavor group elements, while [dU ] denotes the invariant Haar measure for all gauge groups 5 . The sum over j runs over all the N = 2 supermultiplets in the Lagrangian. The symbol R j denotes the representation of the j-th supermultiplet under the flavor and gauge groups, and χ R j is its character. Finally f (j) denotes the single-letter indices, which are given , (3.5) for an N = 2 vectormultiplet and an N = 2 half-hypermultiplet respectively. Several limits of the index (3.1) with enhanced supersymmetry can be considered. For p → 0, one obtains what was called the Macdonald index in [14]. This index counts states annihilated by bothQ 1− and Q 1+ , and thus is a 1 4 BPS object. If in addition one takes q → 0, one obtains the Hall-Littlewood index, which counts states annihilated byQ 1− , Q 1+ and Q 1− . Finally, taking q = t and p arbitrary results in the Schur index, which in fact turns out to be independent of p. In this paper we will focus on these limits of enhanced supersymmetry.

TQFT structure
Following the logic of [4], we wish to identify the TQFT associated to the index of type D theories of class S, in the enhanced supersymmetry limit p = 0 (the "Macdonald index"). In this section we restrict to theories with maximal and empty punctures. The basic building blocks are the three-punctured spheres in figure 5 and the cylinders (propagators).
There are two types of propagators: with or without a twist line running along the cylinder, corresponding respectively to USp and SO gauge groups. They are given by where ∆ SO (a) and ∆ USp (a) denote the SO(2n) and USp(2n−2) Haar measures respectively. The index for a general vectormultiplet is known explicitly: Here the plethystic exponential PE is defined as , and the q-Pochhammer symbol as (a; q) ∞ = ∞ j=0 (1 − aq j ). Usually, we omit the subscript in the plethystic exponential and take it with respect to all parameters. Moreover, f V is the single letter partition function for the vectormultiplet (3.4) in the Macdonald limit (p → 0), χ adj = r + α∈R e α the adjoint character, and r denotes the rank of the group. More concretely, one has for the SO(2n) and USp(2n − 2) vectormultiplet indices where a i , i = 1, . . . , n, and b α , α = 1, . . . , n − 1, denote fugacities of SO and USp respectively. Here and throughout the paper we use the condensed notation that ± means we take the product over all sign choices and (x 1 , . . . , x l ; q) ∞ = l k=1 (x k ; q) ∞ . The indices of the different three-punctured spheres are parametrized as in figure 5. All of these are a priori unknown functions of the flavor symmetry fugacities and q and t, except for the free half-hypermultiplet index I (a, b), which reads Here χ v SO(2n) and χ f USp(2n−2) are the characters of the vector representation of SO(2n) and the fundamental representation of USp(2n − 2), respectively.
Following [14], we expand the free hypermultiplet index in two complete bases of functions {f λ SO (a)} and {f λ USp (b)}, labeled by SO(2n) and USp(2n − 2) representations respectively, These functions are chosen to be orthonormal under the measure which appears in each of the propagators We will now show that we can find bases such that the structure constants, C λ,λ , are "diagonal" in the sense that only some of the SO(2n) representations appear in the sum, which are in one-to-one correspondence with all representations of USp(2n − 2). As in [14] we consider the following Ansätze with the factors K chosen such that the polynomials P λ are orthonormal under the new measures∆ We shall see that if we take the two complete sets of functions P λ 6 to be the Macdonald polynomials for the SO and USp groups, the structure constants are "diagonal". This means that∆ will be the Macdonald measure defined in equation (A.4), and this fixes the K-factors to be where we made use of (A.5) and (3.7) and the obvious invariance under negating roots. To show the "diagonality", we act on the index of the free hyper with both the SO(2n) and USp(2n − 2) Macdonald operators (equations (A.16) and (A.18)), conjugated by the respective K-factor, and find that (see appendix B) Since the Macdonald polynomials are eigenfunctions of D, the functions f λ will be eigenfunctions of KDK −1 with non-degenerate eigenvalues c λ . Using the expansion of the index this means that , where λ i denotes the Dynkin labels of the representation λ . This, together with the non-degeneracy of the eigenvalues c SO λ , implies that the expansion of the index in equation (3.11) is "diagonal" in the sense that for any USp(2n − 2) representation λ = λ 1 , . . . , λ n−1 hence the sum runs over all USp(2n − 2) representations. The index of T SO(2n) can also be expanded in the set of functions {f λ SO (a)} Motivated by the structure of the T N theory index of [14,15] we assume that the basis of functions which diagonalizes the C λ,λ also diagonalizes the B structure constants, 7 Note that the sum in (3.20) runs over all SO(2n) representations. The S-dualities shown in figure 6 can be written in simple form if we also expand the indices of the three-punctured spheres in figures 5c and 5d in the {f λ SO (a)} and {f λ USp (b)} bases, and can be used to constrain the structure constants of these spheres. We write the indices of these spheres as respectively. The S-duality shown in figure 6b written in terms of the structure constants reads Using the "diagonality" of C λλ and B λ 1 ,λ 2 ,λ 3 this implies that Here we used the shorthand notation that λ = λ means λ = λ 1 , . . . , λ n−1 , λ n−1 and δ λ,λ imposes the same condition. Now we find that A λ 2 ,λ 1 ,λ 3 vanishes unless λ 1 = λ 3 and λ 2 = λ 1 , This relation also fixes the value of the A structure constants as This would follow at once from the approach of [15], namely if one could derive the action of D on the index of theory T by extracting residues (in flavor fugacities) in the index of a bigger theory T . Taking residues in different duality frames of T one would get the action of D on different flavor punctures of T . Invariance of the index under S-duality would then imply diagonality in the basis of the eigenfunctions of D. As in [15], we also expect that acting with D on the index amounts to decorating the 4d theory with a BPS surface defect.
The "diagonality" of the A structure constants means that the sum in equation (3.22) runs only over USp(2n − 2) representations.
A similar reasoning will allow us to obtain the E structure constants from the S-duality in figure 6a, which reads The "diagonality" of C and B implies that E λ vanishes unless it is of the form λ = (λ 1 , . . . , λ n−1 , λ n−1 ), and so the sum in equation (3.23) does not run over all SO(2n) representations. The value of E is also fixed by this duality to be Having introduced the structure constants for the different basic spheres, we now turn to evaluating them in some special cases. In our calculations we have made extensive use of the LieART Mathematica package of [28], and use the determinantal formula of [29] to compute the Macdonald polynomials.

D 2 theories
Although not an honest member of the D-series, we will start our discussion of the structure constants with D 2 ≡ A 1 ×A 1 . Due to the low rank, it is possible to obtain exact results in the Hall-Littlewood and Schur limits. These theories also allow for an interesting interpretation in terms of A 1 theories.
Without much ado, we write the index of the free hypermultiplet for the D 2 theory as The equality of the above expression with equation (3.10) for q = 0 was checked by performing the sum over representations, which is geometric. We can read off the structure constant C λ=λ ,λ to be where we used the Hall-Littlewood limit of the SO(4) (q, t)-dimension formula. In general, the (q, t)-dimension (or Macdonald dimension) for SO(2n) is given by The usual limits apply: q → 0 gives the Hall-Littlewood or t-dimension dim SO t and q = t gives the Schur dimension dim SO q , also known as the q-dimension. In the Hall-Littlewood limit we write t = τ 2 .
The expression (3.31) for the structure constant, as well as the ones below for the other spheres, will hold in the other limits and for higher rank if naturally modified and generalized.
Since USp(2) = SU (2), we can apply the procedure of [15] to close the USp(2) puncture and obtain the index of the sphere with one SO(4) puncture and two closed USp punctures. This index is computed as 2 I V Res b=τ . (3.33) Notice that the index for this sphere vanishes when summed over representations. Since it has n V = −3 and n H = 0 it only makes sense as part of a larger theory, e.g. in figure 6a.
Considering the relations imposed by S-duality in (3.28) and (3.26) one can now also write down the index for the T SO(4) theory and theT SO(4) theory, respectively, as These results were also checked against the construction in figure 3 and 4. The index for the linear quivers was expanded in powers of τ and compared to the expansion of the T SO(4) (respectivelyT SO(4) ) theory glued to its three tails. On the other hand, the independent construction of the T SO(4) andT SO(4) provides evidence of the S-duality in figure 6b. The structure constants for these theories are . (3.37) For the T SO(4) theory, we recall that, from equation (2.1), the effective number of hypermultiplets in this theory is 8, and the effective number of vectormultiplets is 0. In fact, we claim that it corresponds to the index of one half-hypermultiplet in the 2 × 2 × 2 and another in the2 ×2 ×2 of the flavor symmetry group SO(4) 3 . This was checked in a τ expansion, and numerically after performing the sum over representations. We will discuss this further in Sec. 3.3.4.

Schur limit
The index for the free half-hypermultiplet in the Schur limit is given by The equality of the above expression with equation (3.10) for q = t was proved exactly by comparing the analytic structure of both sides. In fact, as we will see later in Sec. 3.3.4, a redefinition of the fugacities maps this theory onto the T 2 theory discussed in [14], and therefore the proof is immediate from the similar proof there. The index of the T SO(4) theory is Again we checked this expression using the construction of figure 3 by expanding the index for both sides in q. S-duality then fixes the index for the other three-punctured spheres. The index forT SO(4) was independently checked against the construction of figure 4 in a q expansion.

Macdonald limit
Let us finally state our claim in the Macdonald limit. For the free half-hypermultiplet we have Some of the structure constants were computed in an expansion in q and t, and the equality was also checked numerically, truncating the sum at high enough λ . For the T SO(4) theory we have where the sum runs over all SO(4) representations.

D 2 theories in terms of A 1 theories
Since locally SO(4) = SU(2) × SU(2), USp(2) = SU (2) and in the superconformal tails SO(3) = SU(2), one expects that the D 2 theories can also be described in terms of the A 1 theory [7]. As mentioned before, one naturally considers alternating quivers (see also section 4.1), which for D 2 theories have gauge groups Calling the number of SO(4) gauge groups s, the corresponding Riemann surface is a sphere with 2s + 6 empty USp (2) punctures × connected in pairs by s + 3 twist lines. In A 1 language, it corresponds to a genus s + 2 Riemann surface, which however, since the couplings of the two SU(2)s within SO(4) are not independent, is hyperelliptic [7]. In writing SO(4) as the product of two SU(2)s, the Macdonald polynomials decompose as As a consequence, an SO(4) puncture splits into two SU(2) punctures, and the SO(4) propagator turns into two SU(2) propagators connecting the two SU(2) punctures separately.
Let us now rewrite the index for the basic D 2 spheres in terms of A 1 quantities. First, we have that the index of the free D 2 half-hypermultiplet I D 2 trivially equals the index of the T 2 theory I T 2 if the fugacities are appropriately identified: Second, the index for the T SO(4) theory is, as was mentioned already, equal to the index of one half-hypermultiplet in the 2 × 2 × 2 and another in the2 ×2 ×2 of the flavor symmetry group SO(4) 3 . This can be rewritten in terms of a product of indices of the T 2 theory.
I T SO(4) ((a 1 , a 2 ), (A 1 , A 2 ), (α 1 , α 2 )) = This factorized expression can also be easily checked by applying (3.43) to the expression for the index in (3.42) and noticing that the K-factors and A decompose nicely in the product of the corresponding SU(2) quantities. The result (3.45) can be understood from the construction of the T SO(4) theory in figure 3. Here we have the three-punctured sphere with three maximal SO(4) punctures, of which from each the diagonal SO(3) subgroup is gauged. The curve corresponding to the linear quiver describing this theory is a sphere with six empty USp punctures ×. The corresponding A 1 hyperelliptic curve is a genus two surface, which can be thought of as gluing together two three-punctured spheres. One can now observe that each of these three-punctured spheres contains one of the two SU (2) punctures in which each of the three SO(4) punctures split and gauging the diagonal SO(3) subgroup is indeed obtained by gluing these together. Third, the index for theT SO(4) theory is equal to the A 1 index of the four-punctured sphere if again the SO(4) fugacities are appropriately identified, Notice that the number of effective hyper and vectormultiplets is the same for both theories, as can be inferred from (2.2), namely eight and three respectively. We can also understand the result (3.46) from the construction in figure 4. There we have the threepunctured sphere with one maximal SO(4) puncture, of which the diagonal SO(3) subgroup is gauged, and two maximal USp(2) punctures, which each are connected to an SO(3) gauge group. The linear quiver describing this theory corresponds to a sphere with eight empty USp punctures ×, and the corresponding A 1 hyperelliptic curve is a genus three surface. This surface can be thought of as originating from a four-punctured sphere of which two punctures are glued to each other and the two other punctures are glued to a torus. As before, gluing the two punctures to each other corresponds to gauging the diagonal SO (3) subgroup, and gluing a torus corresponds to connecting to an SO(3) gauge group (since the adjoint of SO(3) equals the adjoint of SU (2)). Finally, one can also interpret the partially closed colored box-shaped SO(4) puncture in A 1 language. We will do so in section 4.2.2.

D 3 theories
We claim that in the D 3 case the index of the free hypermultiplet is given by In the Hall-Littlewood limit this equality was checked numerically after performing the sum exactly. In the Schur limit these coefficients were checked in a q expansion for several representations, and the equality of the index in this limit was also checked numerically, truncating the sum over representations at a high enough λ . Since SO(6) = SU(4), the T SO(6) theory is the same as the T 4 theory (the sphere with 3 maximal SU(4) punctures), the index of which was computed in [14]. The index for this theory can then just be found from equation (7.11) of [14] by re-writing the SU(4) Macdonald polynomials as SO (6)  whereλ denotes the SU(4) representation corresponding to the λ SO(6) representation. The index then becomes, with the sum running over all SO (6) representations, S-duality then fixes the index of theT SO (6) , and as in the D 2 case this index was independently checked using the construction of figure 4 both in the Hall-Littlewood and Schur limits (in a τ and q expansion respectively).

D n theories
Let us finally state our conjecture for the D n theory in the Macdonald limit 8 . The index associated to the free hypermultiplet in the bifundamental of SO(2n) × USp(2n − 2) is This expression was checked numerically for D 4 in the Hall-Littlewood and Schur limits by truncating the sum over representations. For the T SO(2n) theory we conjecture where the sum runs over all SO(2n) representations. The genus g partition function for the pure SO(2n) theory is As in [14], for the slice t = q β this result appears to be related to the partition function Z(C × S 1 ) of a refinement of level k Chern-Simons theory discussed in [30]. For a genus g Riemann surface, C, Z is given by where S 0λ is given by (3.51) and S 00 by 9 The parameters q, t are identified as q = exp 2πi k+βy and t = exp 2πiβ k+βy . Also note that we consider the Macdonald polynomials to be normalized, and so dim SO q,t (λ) here is dim SO q,t (λ)/ √ g λ in [30], where g λ denotes the norm of the polynomials there. It is now easy to verify that our result coincides with this partition function, up to the overall factor ((q; q) ∞ (t; q) ∞ ) n(g−1) .

Partially Closed Punctures
In this section we discuss more general theories with partially closed punctures.

Classification
The classification of superconformal tails for type D theories was discussed in [7]. Let us briefly summarize the main points. One naturally considers an alternating quiver of SO and USp gauge groups as in figure 7. In such a quiver, the requirement that every coupling is marginal implies that Following [14] we only kept the interesting q and t dependent part of S00 (see also equation (7.5) of [30]). The full three-sphere partition function is where |∆+| is the number of positive roots, y is the dual coxeter number and |P/Q| denotes the number of elements in the fundamental cell of the quotient lattice P/Q. Note that with (3.52) there is an extra overall factor when comparing the partition function on C × S 1 with the superconformal index for the Riemann surface C. This is precisely the same factor that appears when comparing the A-type index of [14] with the complete refined Chern-Simons partition function on C × S 1 for SU(N ).
Since k a is a non-negative integer one finds that a general quiver necessarily has The part to the left of gauge group l and to the right of gauge group r are the two tails of this quiver. We denote d l = . . . = d r = 2n and introduce d 0 = d m+1 = 0. Focusing on the right tail 2n = d r > d r+1 > . . . > d m and observing that from (4.1) d a − d a+1 is monotonically non-decreasing, we can associate a Young tableau to this quiver as follows.
If the quiver ends in a USp gauge group, one considers a grey -i.e. SO -tableau with length of rows given by d r −d r+1 , d r+1 −d r+2 , . . . , which has a total number of boxes of 2n. If on the other hand the quiver ends in a SO gauge group, then one can construct a black tableau with length of rows given by which has a total number of boxes equal to 2n − 2.
The Young tableau thus constructed encodes as usual the embedding of SU(2) in SO(2n) (USp(2n − 2)), the commutant of which captures the flavor symmetry information. The embedding is given by the decomposition of the vector representation 2n of SO(2n) (fundamental representation 2n − 2 of USp(2n − 2)) under SU (2). Let us denote the number of columns of height h by l h . Then the relation between the Young tableau and the decomposition is given for a grey tableau by 2n → 1 + 1 + . . . + 1 Note that the reality of the vector representation requires that l h is even for even h. The flavor symmetry associated to such a tail is then where the USp groups have an even argument indeed. Following [10] we call decompositions (4.3) in which only even dimensional representations appear "very even". Since even dimensional representations must occur with even multiplicities, this case only occurs for even n. Actually, such tableaux correspond to two different punctures which are exchanged by the Z 2 outer automorphism and were colored red and blue in [10]. Note that if the decomposition of the vector representation into SU(2) representations is of the form 2n → (2n−k)+k for arbitrary odd k, or k = n if n even, the above algorithm will give rise to a "USp(0)" gauge group. The occurrence of such a gauge group can be understood from the brane picture, see [7].
For a black tableau one has 2n − 2 → 1 + 1 + . . . + 1 Note that the pseudo-reality of the fundamental representation requires that l h is odd for even h. The flavor symmetry associated to such a tail is then The curve corresponding to these superconformal tails has m − r + 4 punctures (or simply said, the number of gauge groups plus three) of which one is a maximal SO puncture, one puncture is the partially closed one under consideration and the other m − r + 2 punctures are empty USp punctures ×. Note that a possible "USp(0)" gauge group is also included in this counting of gauge groups.

The index with partially closed punctures
Following [14], we generalize the structure of the superconformal index in section 3.5 to include partially closed punctures. For a three punctured sphere with punctures labeled by a Young tableau Y I we propose schematically (1, t, . . . , t n−1 | q, t) . (4.7) Here the color of the Young tableau determines whether one considers SO or USp polynomials and corresponding K-factors. Due to the twist line there are either no or two USp punctures. If USp punctures are present the sum is over USp representations and the occurring SO representations are restricted in the sense discussed above. The denominator is always the (q, t)-dimension of type SO. Given this structure, we need to determine for each allowed Young tableau which fugacities one needs to plug in the polynomials and what the K-factor associated to that puncture is.

Fugacities corresponding to Young diagram
To find which fugacities appropriately describe a partially closed puncture we use that a partially closed SO(2n) (USp(2n − 2)) puncture is classified by an embedding of SU(2) determined by the decomposition of the vector (fundamental) representation of this group and that the multiplicities with which the different SU(2) representations occur encode the corresponding flavor symmetry. The fugacities are then determined by looking at the following equality. On one side of the equality, one simply writes the character of the vector (fundamental) representation in terms of its n (n − 1) fugacities. On the other side of the equality we write its decomposition determined by the puncture in SU(2) characters with as SU(2) fugacity τ = t 1/2 , where we replace the multiplicities by the character of the vector/fundamental of the flavor symmetry determined by that multiplicity. One can now simply read off the relevant fugacities. How one identifies the fugacities does not matter in USp, i.e. neither the order nor if one chooses X or X −1 is relevant, because these operations precisely correspond to Weyl symmetries, under which the Macdonald polynomials are invariant. For the SO punctures, the order of the fugacities again corresponds to a Weyl symmetry. However, in this case Weyl invariance only means we can invert fugacities in pairs. For odd rank, the tableau always contains at least two columns of odd height. This implies that the two apparently different choices one obtains after exploiting the Weyl symmetries correspond to a renaming of the fugacities. For even rank, there also are odd height columns if the decomposition is not "very even". Then again, there is a unique Figure 8: An example of a "very even" SO(8) puncture with USp(2) flavor symmetry. (2) fugacity. The two essentially inequivalent choices are a 1 = bτ 3 , a 2 = bτ , a 3 = b/τ , a 4 = b/τ 3 , and a 1 = bτ 3 , a 2 = bτ , a 3 = b/τ , a 4 = τ 3 /b. They correspond to the red and blue punctures respectively. 10 choice of fugacities up to a renaming. However, for the "very even" case, there are two essentially inequivalent choices. These are interchanged by the Z 2 outer-automorphism, and they correspond to the red and blue punctures. We illustrate this in figure 8. This distinction disappears once one considers the puncture as part of a Riemann surface containing also USp punctures, since then the sum over SO(2n) representations is restricted due to "diagonality", i.e. only representations for which the last orthogonal weight is zero ( n = 0) appear. In this case inverting any fugacity becomes a symmetry of the polynomial.

K-factors
We now turn attention to the K-factors, which are independent of the red and blue coloring. For a general maximal puncture, the K-factor is given in (3.16) in terms of the roots. For the empty USp puncture ×, the K-factor was given in general in (3.48). For the L-shaped empty SO puncture, one can easily find the K-factor by demanding that one obtains a delta function with respect to the propagator measure by closing one puncture in T SO . The resulting K-factor is then Note that in general, there are more punctures in both SO and USp which do not carry flavor symmetry. All of these factors can be determined by the following procedure. As was discussed above, each puncture is constructed as a certain superconformal tail and as such is naturally associated to a certain Riemann surface. By comparing the index of the tail, which is straightforward to write down since it is Lagrangian, with the index of the Riemann surface associated to it, one can fix, in principle, all K(Y I ).
The index of a Riemann surface of genus g and with s punctures, labeled by Young tableaux Y I is given by (4.10) As before, the color of the Y I encodes whether the Macdonald polynomials correspond to SO or USp groups. Here the sum over λ is restricted as soon as USp punctures appear, or there is a twist line wrapping a cycle. Riemann surfaces associated to partially closed punctures have genus zero, one maximal SO puncture, several empty USp punctures and the partially closed puncture itself. To implement the above construction, it is convenient to consider a general Ansatz for the K-factors, as follows. The K-factor for the maximal puncture diverges if one plugs in the fugacities associated to a non-maximal Young tableau. However, from [15,31] we expect to obtain the K-factor associated with a Young tableau from K for the maximal puncture by removing some factors, including the divergent ones, and multiplying by an appropriate power of I V = (t; q) ∞ (q; q) ∞ . Taking out the divergent factors, and allowing for an extra arbitrary power of (t; q) ∞ (q; q) ∞ thus provides an Ansatz for each K(Y I ), and one can determine which terms must be removed by comparing the index of the superconformal tail described above to the corresponding Riemann surface.
Equipped with a general recipe, it would be interesting to check the dualities of [32] by comparing the index in different frames, especially for the cases where enhanced flavor symmetries appear.
We now discuss some simple examples. to the reduced flavor symmetries. Using the above procedure, these results were also checked in a q (t) expansion for the Schur (Hall-Littlewood) limit, and thus serve as a proof of principle. Here a, b, c denote respectively SO(2), USp(2) and SO(3) fugacities. (4.11) The three-punctured sphere consisting of the second puncture (which has SO(3) flavor symmetry) and two maximal punctures gives the E 7 SCFT of Minahan-Nemeschansky [33], and its index can be independently (and in agreement with the above expression) computed by using the S-duality described in [7].
Note that the tail corresponding to the last puncture (SO(2) flavor symmetry) ends on a "USp(0)" gauge group, as shown in figure 10a, which also shows the corresponding Riemann surface 10b. This happens for the class of punctures described in section 4.1, but the procedure outlined above is still applicable. We find K for the USp punctures with USp(2) and SO(2) flavor symmetries to be respectively given by Here b denotes an USp(2) and a an SO(2) fugacity. These factors were checked in the Schur and Hall-Littlewood limits by comparing the index of the tail with the one for the curve, in an expansion in q and τ respectively. The superconformal tail for the SO(2) flavor symmetry is shown in figure 10c, along with the corresponding curve 10d. The index for this linear quiver is the same as the index for the one shown in figure 10a, since the "USp(0)" gauge group does not contribute to the index. From the curve this implies that (after integrating against the SO(6) polynomials) 13) where λ 1 and λ 2 are Dynkin labels. This identity is true for any Dynkin labels λ 1 and λ 2 , and so taking λ 1 = λ 2 = 0 the representation dependent part cancels on its own and we get a relation between the K-factors, (4.14) We also get a relation between the SO and USp polynomials. In general, any SO puncture labeled by a Young diagram with only two columns can similarly be related to a USp puncture labeled by the Young diagram one obtains by removing the bottom two boxes of the SO Young diagram and coloring it black, thus giving a relation between the K-factors and an identity involving Macdonald polynomials. D 2 theory. Another simple example of this relation, in D 2 , is given by the colored boxshaped SO puncture with USp(2) flavor symmetry. The K-factor for this puncture is We also have 1 dim SO q,t ((λ, λ)) P (λ,λ) M SO at 1/2 , a t 1/2 | q, t P  At this point we can return to the A 1 interpretation of D 2 theories and include this partially closed puncture. Namely, this partial closing of the SO(4) puncture amounts to fully closing one of the two SU(2) punctures in which it decomposes in A 1 language. The coloring is simply encoded in the choice of which SU(2) becomes fully closed. More in detail, one has for the red puncture M SU (a, 1/a | q, t) , (4.18) and for the blue puncture Note that when one imposes "diagonality", the coloring becomes as expected irrelevant. For the (color-independent) K-factor one has Also note that when using (4.18) (or (4.19)), rewriting the Macdonald dimension in terms of SU Macdonald dimensions using (3.43), and naturally thinking of the USp(2) polynomials in terms of SU(2) polynomials, equation (4.17) is trivial.
An alternative characterization of the Weyl characters is as the unique polynomials satisfying two conditions, namely that they can be written in terms of the orbit-sums for some coefficients K λµ as which constrains the leading behavior, and that they are orthogonal under the Haar measure 11 . It is these two criteria which are generalized to define the Hall-Littlewood and Macdonald polynomials for arbitrary root systems. Let us start by introducing a 2-parameter generalization of the Haar measure where we used the q-Pochhammer symbols (a; q) ∞ = ∞ j=0 (1 − aq j ). We will call this measure the Macdonald measure. For t = q it reduces to the Haar measure, The definition of the scalar product of two functions f = λ∈P f λ e λ and g = λ∈P g λ e λ is most easily written down if we first write them in terms of the x-variables introduced above. To do so we write λ = i i i in the orthogonal basis, and thus e λ = i x i i . We also introduceḡ = λ∈P g λ e −λ = λ∈P g λ i x − i i , and the shorthand notation where f (x) = f (x 1 , x 2 , . . .) and so forth, |W | is the order of the Weyl group and∆ denotes one of the measures introduced above. The generalization we were alluding to is now as follows. There exists a unique basis for A W of functions {P λ | λ ∈ P + } such that P λ = m λ + µ∈P + , µ<λ u λµ m µ , (A. 8) where the coefficients are rational functions of q and t, such that they are orthogonal under what we called the Macdonald measure.
Note that instead of normalization, one favors a constrained leading behavior. We reserve the notationP for polynomials satisfying this requirement. In the main text we will use polynomials orthonormal under (A.7) with the Macdonald measure (A.4), which we denote by P.
As mentioned above already, for q = t the polynomials are the Weyl characters (A.3). For q → 0, one has the Hall-Littlewood polynomials, which are given explicitly bỹ where W λ (t) = w∈W wλ=λ t l(w) . (A.10) For general t, q, a simple expression as above for the Hall-Littlewood polynomials or Weyl-characters is absent. However, the polynomials can be generated quite easily through a determinantal formula [29]. This formula makes use of the proposition that there exists a linear operator D : A W → A W such that 1. D is selfadjoint, i.e. Df, g = f, Dg for all f, g ∈ A W ; 2. D is triangular relative to the basis m λ , i.e. for each λ ∈ P + , Dm λ is of the form Dm λ = µ≤λ c λµ m µ ; 3. the eigenvalues of D are distinct, i.e. if λ = µ ∈ P + then c λλ = c µµ .
It is easy to understand how the existence theorem follows from this proposition. Namely, given an operator D satisfying these three properties, one can consider for each λ ∈ P + the eigenfunctionP λ with eigenvalue c λλ . One can normalize this eigenfunction such that the coefficient of m λ equals 1. Moreover, using the selfadjointness of D and the nondegeneracy of its eigenvalues, one can argue that c λλ P λ ,P µ = DP λ ,P µ = P λ , DP µ = c µµ P λ ,P µ , which implies that for µ = λ one has P λ ,P µ = 0.
The proof of the proposition is given in [34] and is simply based on the construction of an operator satisfying the above three properties. To that purpose one starts by constructing for each minuscule weight π for the dual root system R ∨ 12 the operator D π as D π = 1 |W π | w∈W   α∈R + 1 − t π,α e w(α) 1 − e w(α)   T w(π),q , (A.11) where T x,q is defined by its action on the exponentials e λ as T x,q e λ = q λ,x e λ . (A.12) One can prove that these operators satisfy the requirements 1. and 2. The property 3. is also satisfied except for the D-series. In order to lift the degeneracy of the eigenvalues in the D-case, one constructs an appropriate linear combination of the operators D π 1 and D π 2 , where π 1 and π 2 are the two minuscule weights of R ∨ D = R D , namely the fundamental weights corresponding to the two spinor representations, i.e. ω n and ω n−1 . For any integer N > 1 2 n(n − 1), where n is the rank, one considers which now also satisfies property 3. The eigenvalues of the operators D π can be written in general as c λλ (π) = t π,ρ τ ∈W (π) t τ,ρ q τ,λ . (A.14) One thus has for the eigenvalues of D SO in (A.13) t τ,ρ q τ,λ + (t N − 1)t π 2 ,ρ τ ∈W (π 2 ) t τ,ρ q τ,λ   .

(A.15)
A.2 More explicit expressions for the C-and D-series Let us write the Weyl group, the Macdonald operator and its eigenvalues a little more explicit.
A.2.1 The case C n = USp(2n) The Weyl group of C n is given by all possible permutations and sign changes of the orthogonal weights. The dual root lattice R ∨ of C n equals the root lattice of B n . The unique minuscule weight of R ∨ is then the fundamental weight ω n of B n . The Macdonald operator then reads explicitly [35]