The superconformal index and an elliptic algebra of surface defects
Mathew Bullimore, Martin Fluder, Lotte Hollands, Paul Richmond
PPrepared for submission to JHEP
The superconformal index and an elliptic algebra of surface defects
Mathew Bullimore, Martin Fluder, Lotte Hollands, Paul Richmond Perimeter Institute for Theoretical Physics,Waterloo, Ontario, N2L 2Y5, Canada. Mathematical Institute, University of Oxford,Andrew Wiles Building, Radcliffe Observatory Quarter,Woodstock Road, Oxford, OX2 6GG, UK.
E-mail: [email protected] , [email protected] , [email protected] , [email protected] Abstract:
In this paper we continue the study of the superconformal index of four-dimensional N = 2 theories of class S in the presence of surface defects. Our mainresult is the construction of an algebra of difference operators, whose elements arelabeled by irreducible representations of A N − . For the fully antisymmetric tensor rep-resentations these difference operators are the Hamiltonians of the elliptic Ruijsenaars-Schneider system. The structure constants of the algebra are elliptic generalizations ofthe Littlewood-Richardson coefficients. In the Macdonald limit, we identify the differ-ence operators with local operators in the two-dimensional TQFT interpretation of thesuperconformal index. We also study the dimensional reduction to difference operatorsacting on the three-sphere partition function, where they characterize supersymmetricdefects supported on a circle, and show that they are transformed to supersymmetricWilson loops under mirror symmetry. Finally, we compare to the difference operatorsthat create ’t Hooft loops in the four-dimensional N = 2 ∗ theory on a four-sphere byembedding the three-dimensional theory as an S-duality domain wall. a r X i v : . [ h e p - t h ] O c t ontents T ( SU (2)) 374.2.2 T ( SU ( N )) 404.3 Three-dimensional algebra 41 N = 2 ∗ theory 42 Discussion 54A Macdonald polynomials and the refined S-matrix 57
A.1 Group theory 57A.2 Schur polynomials and the modular S-matrix 58A.3 Macdonald polynomials and the refined S-matrix 59
B S-duality kernel 61
B.1 Example 62
C Factorization of Toda 3-point function 64
Surface defects are an interesting class of non-local observables in four-dimensionalgauge theories [1]. In this paper, we consider surface defects in four-dimensional N =2 superconformal field theories of class S , which are obtained by compactifying thepartially twisted six-dimensional (2 ,
0) theory on a decorated Riemann surface C [2, 3].The six-dimensional (2 ,
0) theory is characterized by a Lie algebra g of ADE type. Inthis paper we focus on the case of A N − . In this case, the six-dimensional (2 ,
0) theoryarises as the infrared limit of the worldvolume theory on a stack of N coincident M5-branes. Surface defects in four-dimensional theories of class S can be formed from bothcodimension-two and codimension-four defects in the six-dimensional parent theory.This is summarized in Table 1.Let us first discuss codimension-two defects of the (2 ,
0) theory in six-dimensions,which are labeled by embeddings ρ : su (2) → g . These defects play an important rolein the construction of theories of class S : a codimension-two defect that is inserted at apoint on the Riemann surface C and spans all four space-time dimensions correspondsto a flavor puncture in the construction of [2, 3] - see (i) of Table 1. Alternatively,wrapping the same codimension-two defect on the whole Riemann surface C leads to asurface defect in the four-dimensional theory - see (ii) of Table 1. This class of surfacedefects has been studied, for example, in [4, 5].– 1 – C Name(i) 4 0 flavor puncture(ii) 2 2 surface defect(iii) 2 0 surface defect
Table 1 . Summary of the defects in the six-dimensional (2 ,
0) theory on X × C . X is the fourdimensional space-time and C is a decorated Riemann surface. (i) and (ii) show configurationsof codimension-two defects while (iii) shows the configuration of a codimension-four defect. On the other hand, there are codimension-four defects in the (2 ,
0) theory in six-dimensions, which are expected to be labeled by an irreducible representation of g ,see for example [6] and references therein. Inserting a codimension-four defect at apoint on the Riemann surface C engineers another class of surface defects in the four-dimensional theory - see (iii) of Table 1. In this paper, we study this second class ofsurface defects in four-dimensional N = 2 theories of class S .Important evidence for the classification of codimension-four defects in terms ofirreducible representations of g comes from the correspondence between four-spherepartition functions of N = 2 theories of class S and correlation functions in Liouvilleor Toda conformal field theory on C [7, 8]. In this correspondence, flavor puncturesare represented by vertex operators labeled by non-degenerate and semi-degeneraterepresentations of the Virasoro or W N -algebra. There are also completely degeneraterepresentations labeled by two dominant integral weights of g , or equivalently, by twoirreducible representations R and R of g . Correlation functions with additional in-sertions of completely degenerate vertex operators compute the four-sphere partitionfunction in the presence of surface defects [9]. In particular, the labels R and R characterize the surface defects supported on orthogonal two-spheres.Inspired by the connection to degenerate vertex operators and the analytic struc-ture of Virasoro/ W N -algebra conformal blocks, the authors of reference [10] introduceda renormalization group flow that can be used to construct the surface defects fromvortex configurations in a larger theory. Let us consider the simplest example of thisprocedure illustrated in Figure 1.The starting point is a theory T IR with a full puncture encoding an SU ( N ) flavor– 2 – igure 1 . Schematic illustration of the renormalization group flow T UV → T IR that can beused to introduce surface defects. The white dots represent full punctures with SU ( N ) sym-metry while the black dot is a simple puncture with U (1) symmetry. The red dot representsa codimension-four defect engineering a surface defect in four dimensions. symmetry. We then form the larger theory T UV by adding a simple puncture nearbywith U (1) flavor symmetry. This corresponds to adding an additional hypermultiplet inthe bifundamental of SU ( N ) × SU ( N ) by gauging the diagonal SU ( N ). The extra U (1)symmetry corresponds to the baryonic symmetry of the bifundamental hypermultipletand the position of the simple puncture controls the gauge coupling of the gauged SU ( N ).The theories T IR and T UV are connected by a renormalization group flow that isinitiated by turning on a constant vacuum expectation value for the hypermultipletscalar. By turning on a position-dependent vacuum expectation value correspondingto a half-BPS vortex configuration in T UV , the endpoint of the renormalization groupflow is a surface defect in the original theory T IR . These surface defects are labeled by apair of positive integers ( r , r ) corresponding to the vortex numbers in orthogonal two-planes. This construction is analogous to the Toda construction of codimension-foursurface operators [9]. Hence our working conjecture is that they give a representation ofcodimensions-four surface defects labelled by a pair of symmetric tensor representationsof g .A concrete prescription was given in [10] to implement this renormalization groupflow at the level of the superconformal index. The superconformal index is a trace overstates of a superconformal field theory in radial quantization [11]. It is a much simplerobservable than the four-sphere partition function because it does not depend on themarginal couplings of the theory. For previous work on the superconformal index oftheories of class S see [12–16]. In full generality, the N = 2 superconformal indexdepends on three parameters denoted by { p, q, t } that are associated to combinations– 3 –f bosonic conserved charges commuting with a chosen supercharge. It also depends onflavor parameters { a , . . . , a N } , such that (cid:81) j a j = 1, for each global SU ( N ) symmetryand an additional parameter b for each U (1) symmetry. The superconformal index isthus denoted by I ( p, q, t, a j , b, . . . ) . (1.1)The superconformal index of the theory T IR with surface defects is obtained bycomputing a residue of the superconformal index of the theory T UV in the additionalfugacity b associated to the additional U (1) symmetry. The result is a difference oper-ator G r ,r that acts on the superconformal index of the original theory T IR by shiftingthe fugacities of the SU ( N ) flavor symmetry. Schematically, the difference operator isdefined by G r ,r · I IR ( a j , . . . ) ∼ Res b = t p r /N q r /N (cid:104) b I UV ( a j , b, . . . ) (cid:105) , (1.2)where the proportionality constant is discussed in §
2. The difference operator G r ,r corresponds to inserting a surface defect in the original theory T IR that is labeled bythe pair ( r , r ).In what follows we concentrate on the case r = 0 and simply label the differenceoperators by G r , where r ∈ Z ≥ . The label r can be thought of as denoting a symmetrictensor representation of rank r . The resulting expression for G r is G r · I ( a j ) = (cid:88) (cid:80) Nk =1 m k = r N (cid:89) j,k =1 (cid:34) m k − (cid:89) m =0 θ ( q m + m k − m j ta j /a k ; p ) θ ( q m − m k a k /a j ; p ) (cid:35) I (cid:0) a j (cid:55)→ q rN − m j a j (cid:1) , (1.3)where the theta-function θ ( z, p ) is defined in section § G R corresponding to surface defects labeled by all irreducible representations R of g . In principle, they could be constructed by starting from a theory T UV with anadditional puncture with a larger flavor symmetry. However, this would involve non-Lagrangian ingredients and, although the index can be bootstrapped as in [10], theanalytic structure needed for this approach is not manifest.Instead we follow the line of reasoning introduced in [17] and complete the algebraof difference operators. For the difference operator associated to the representation R – 4 –e make an ansatz G R · I ( a j ) = (cid:88) λ C R,λ ( p, q, t, a j ) I ( q − ( λ,h j ) a j ) , (1.4)where the sum is over the weights λ of the representation R , ( , ) is the standard innerproduct on the Cartan subalgebra of g , and h j are the weights of the fundamentalrepresentation. This ansatz is compatible with what we already know about differenceoperators G r for symmetric tensor representations R = ( r ).The coefficients C R,λ ( p, q, t, a j ) are then determined by imposing that the full setof difference operators G R is closed under composition G R ◦ G R = (cid:88) R N R ,R R ( p, q, t ) G R , (1.5)and forms a commutative algebra. Since the symmetric tensor representations form anover-complete basis, there are many compatibility conditions for the system (1.5) to besolved consistently. It is thus non-trivial that a solution exists. Nevertheless, we canfind a solution using the following method.First, we notice that all irreducible representations in the case g = su (2) aresymmetric tensor representations, so that there are no additional difference opera-tors. Even though it is not obvious and requires numerous functional identities fortheta-functions, the system (1.5) can be solved uniquely in this case. The structure co-efficients N R ,R R ( p, q, t ) turn out to be an elliptic generalization of the ( q, t )-deformedLittlewood-Richardson coefficients. In § N R ,R R ( p, q, t ) uniquely from the ( q, t )-deformed ones.If we then assume that for any rank of the gauge group the structure coefficients N R ,R R ( p, q, t ) are given by this elliptic generalization of the Littlewood-Richardsoncoefficients, the system (1.5) can be solved consistently and uniquely for all of thedifference operators G R . The coefficients C R,λ are in general sums of products of ratiosof theta-functions. Let us stress once more that the fact that we can find a consistentsolution to the system (1.5) is highly non-trivial and involves numerous identities fortheta-functions. We see this as strong evidence that a class of surface defects labeledby general irreducible representations R of g exists.In particular, we find that the difference operators G (1 r ) labeled by the rank r antisymmetric tensor representations, can be conjugated to the Hamiltonians of the– 5 – -body elliptic Ruijsenaars-Schneider integrable system. This is an extension of thefact, noted in [10], that the fundamental operator in the case of A can be conjugatedto the Hamiltonian of the two-body elliptic Ruijsenaars-Schneider integrable system.A microscopic definition of a large class of surface defects can be given by cou-pling the four-dimensional theory to two-dimensional N = (2 ,
2) degrees of freedomsupported on the surface [9, 18–20]. The superconformal index in the presence of suchsurface defects has been constructed recently in [21]. Thus it is natural to ask whetherthe surface defects introduced by the operators G R can be understood in this approach.For the rank r symmetric tensor representation, it was already noted in [21] that thetwo-dimensional degrees of freedom consist of an N = (2 ,
2) gauge theory with gaugegroup U ( r ), coupled to N fundamental and N anti-fundamental chiral fields and anadditional chiral field in the adjoint representation of U ( r ). Using the same techniques,we find that the relevant two-dimensional degrees of freedom for the rank r antisym-metric tensor representation are the same as above, but without the adjoint chiral field.For other representations, it is not clear to us whether the surface defect can be con-structed by coupling to an N = (2 ,
2) supersymmetric gauge theory. We make a fewadditional remarks about this in the discussion in § N = 2 theories of class S has a dual description interms of a two-dimensional topological quantum field theory on the surface C [14, 15].We continue in this paper by showing that the difference operators G R are naturalobjects in this two-dimensional TQFT. When we focus on the Macdonald slice { p =0 , q, t } , the TQFT is given as an analytic continuation of refined Chern-Simons theoryon S × C [22].In the Macdonald limit, the operators G (1 r ) , labeled by antisymmetric tensor repre-sentations, can be conjugated to the so-called Macdonald operators, whose eigenfunc-tions are the Macdonald polynomials P S ( a, q, t ) labeled by an irreducible representation S . We find that the eigenvalue of a general, conjugated, difference operator G cR in theMacdonald limit is given by G cR · P S ( a j , q, t ) = S R,S S ,S P S ( a j , q, t ) , (1.6)where S R,S is an analytic continuation of the modular S-matrix of refined Chern-Simonstheory, which depends on q and t . A consequence is that the surface defect introduced– 6 – igure 2 . Sequence of dualities that maps the four-dimensional T N theory (upper-left) tothe three-dimensional star-shaped quiver theory (lower-right). by the operator G cR is equivalent to a Wilson loop wrapping around the S of thethree-manifold S × C .In the Macdonald limit, the structure constants N R ,R R ( q, t ) become the ( q, t )-deformed Littlewood-Richardson coefficients and the algebra of difference operators G R is identified with the Verlinde algebra. We expect that this Verlinde algebra hasa natural interpretation in the (analytically continued) chiral boundary theory on thetwo-torus boundary near a puncture of C .We find further confirmation of the physical relevance of the difference operators G R by reducing the superconformal index to the three-sphere partition function, follow-ing [23–25]. In particular, we consider the dimensional reduction of the four-dimensional T N theory, which is obtained by compactifying the six-dimensional (2 ,
0) theory on athree-punctured sphere with three full punctures. The dimensionally reduced T N the-ory has a Lagrangian mirror description as a star-shaped quiver theory [26]. This isillustrated in Figure 2. In particular, each full puncture of the three-punctured sphereis represented by a three-dimensional linear quiver theory called T ( SU ( N )).It is expected that the surface defects introduced by the dimensional reductionof the operators G R correspond to supersymmetric Wilson loops in the representation– 7 – for the central node of the star-shaped quiver. This is in fact equivalent to thestatement that the partition function of the T ( SU ( N )) theory is an eigenfunctionof the dimensionally reduced operators G (3d) R . The partition function Z ( x, y ) of the T ( SU ( N )) theory depends on two mass parameters x and y associated to the Higgsbranch and the Coulomb branch respectively, and is symmetric under x ↔ y . For thecase of a round four-sphere, we show indeed that G (3d)(1 r ) ( y ) · Z ( x, y ) = W (1 r ) ( x ) Z ( x, y ) , (1.7)where W (1 r ) ( x ) is a supersymmetric Wilson loop in the r -th antisymmetric tensor rep-resentation.For other (non-minuscule) representations we find that this is not quite correct. Inparticular, the Wilson loops obey the algebra W R · W R = (cid:88) R N R ,R R W R , (1.8)where N R ,R R are the ordinary Littlewood-Richardson coefficients, whereas the alge-bra of the three-dimensional operators G (3d) R is not of this form. Instead, we find thatwhen the representation R is non-minuscule, the dimensionally reduced operators G (3d) R are linear combinations of operators ˜ G (3d) S , with | S | ≤ | R | , that are dual to Wilsonloop operators . This gives a simple invertible linear transformation on the algebra ofdifference operators.Finally, by embedding the three-dimensional T ( SU ( N )) theory as an S-dualitydomain wall in the four-dimensional N = 2 ∗ theory, we interpret the dimensionallyreduced difference operators G (3d) R as operators that introduce ’t Hooft defects, labeledby irreducible representations R , into the four-sphere partition function of the N = 2 ∗ theory. Again, when the representation R is an antisymmetric tensor representation, wefind perfect agreement with both localization [27] and (in the case of the fundamentalrepresentation) computations of Verlinde operators in Liouville/Toda conformal fieldtheory [9, 28–30], while for other representations we once more find an invertible lineartransformation on the algebra of operators. We defined the partial ordering of representation by | R | < | R | iff the dimension of the represen-tation R is less than the dimension of R . – 8 –he outline of this paper is as follows. In § G R by completing the algebra generated by the difference operators G r , which arelabeled by symmetric tensor representations, and we interpret the operators G R ascomputing the N = 2 superconformal index in the presence of surface defects. In § G R in the limit p = 0 as Wilson loops wrappingthe S in an analytic continuation of refined Chern-Simons theory on S × C . In § G R to three dimensions, and interpret them as operatorsthat describe line defects when added to the three-sphere partition function. In § G (3d) R to operators that introduce ’t Hooftloops into the four-sphere partition function of the four-dimensional N = 2 ∗ theory.We finish in § The superconformal index is a trace over the states of a superconformal field theory inradial quantization, or equivalently, a twisted partition function on S × S . The mostgeneral superconformal index of four-dimensional N = 2 theories is I = Tr( − F p j z − r q j w − R t r + R (cid:89) j a f j j , (2.1)where the trace is taken over states of the theory in radial quantization annihilated bya single supercharge ˜ Q , ˙ − . Here, we are parametrizing S by two complex coordinates( z, w ) obeying | z | + | w | = 1, and the generators j z and j w are rotations in theorthogonal z and w -planes respectively. The symbol r denotes the generator of thesuperconformal U (1) r and R the generator of the Cartan subalgebra of SU (2) R . The f j are generators of the Cartan subalgebra of the flavor symmetry group.The combinations of generators appearing in the powers of ( p, q, t, a j ) in equa-tion (2.1) are those combinations that commute with the supercharge ˜ Q , ˙ − . The letters p , q , t and a i are fugacities for these symmetries and obey | p | , | q | , | t | , | pq/t | < , | a j | = 1 , (2.2)– 9 –hich ensure that the index is well-defined.If there exists a weakly coupled Lagrangian, the superconformal index can be com-puted from single-letter indices by the plethystic exponential. The basic ingredientsare the single letter indices of a half-hypermultiplet and vectormultiplet, i H = √ t − pq √ t (1 − p )(1 − q ) ,i V = − p − p − q − q + pqt − t (1 − p )(1 − q ) . (2.3)For example, the superconformal index of a free hypermultiplet in the bifundamentalrepresentation of SU ( N ) × SU ( N ) is I ( a j , b j , c ) = PE (cid:34) i H N (cid:88) i,j =1 (cid:18) a i b j c + 1 a i b j c (cid:19)(cid:35) = N (cid:89) i,j =1 Γ (cid:16) √ t ( a i b j c ) ± ; p, q (cid:17) , (2.4)where PE stands for the plethystic exponential. The parameters { a i } and { b j } arefugacities for the SU ( N ) × SU ( N ) symmetry and c is the fugacity for the overall U (1)symmetry. The elliptic gamma function Γ( z ; p, q ) is defined asΓ( z ; p, q ) = ∞ (cid:89) i,j =0 (1 − z − p i +1 q j +1 )(1 − zp i q j ) . (2.5)An important operation on the superconformal index is that of gauging a globalsymmetry. Given the superconformal index I ( a ) of a theory with SU ( N ) flavor sym-metry, the superconformal index of the theory where this symmetry has been gaugedis (cid:73) ∆( a ) I V ( a ) I ( a ) , (2.6)where I V ( a ) = PE (cid:104) i V (cid:16) N (cid:88) i,j =1 a i a j − (cid:17) (cid:105) (2.7)is the superconformal index of an SU ( N ) vectormultiplet and∆( a ) = (cid:34) N − (cid:89) j =1 da j πia j (cid:35) N ! N (cid:89) i (cid:54) = j (cid:16) − a i a j (cid:17) (2.8)is the Haar measure on the maximal torus of SU ( N ).– 10 – .2 Surface defects from vortices In this section, we review the construction of the superconformal index in the presence ofa certain class of surface defects, which arise as the infinite tension limit of backgroundvortex configurations [10]. They are labeled by a nonnegative integer r , the vortexnumber, which may be interpreted as the magnetic flux through the vortex core.The starting point is any superconformal field theory T IR with a global flavorsymmetry SU ( N ). By gauging this flavor symmetry, the theory may be coupled to ahypermultiplet in the bifundamental representation of SU ( N ) × SU ( N ). The resultingsuperconformal field theory T UV has an additional baryonic U (1) symmetry acting onthe bifundamental hypermultiplet.The two theories T IR and T UV are related by a renormalization flow initiated byturning on a Higgs branch vacuum expectation value for the bifundamental scalar field Q . When this expectation value is a constant, the RG flow brings us back to the theory T IR . When the expectation value is taken to be coordinate-dependent, the theory T IR is modified along a surface and in the low energy limit we recover the theory T IR in thepresence of a surface defect.More precisely, we can introduce a vacuum expectation value for the baryon oper-ator B = det Q of the form B ( z ) = r (cid:89) i =1 ( z − z i ) , (2.9)where z is a complex coordinate in a two-plane, the degree r corresponds to the vortexnumber, and the parameters z i are the positions of the vortex strings. Taking the z i = 0,we have r coincident vortices. This construction then leads to surface defects labeledby r ∈ Z ≥ . For N = 2 superconformal field theories of class S , this construction hasan elegant interpretation in terms of the curve C - see Figure 3.This field theoretic construction of surface defects can be implemented concretelyin the superconformal index for surface defects supported on the S × S defined bythe locus { z = 0 } . Denoting the superconformal index of T IR by I IR ( a j , . . . ), then thesuperconformal index of T UV is I UV ( b j , c, . . . ) = (cid:73) ∆( a i ) I V ( a i ) I H ( a i , b j , c ) I IR ( a − i , . . . ) . (2.10)– 11 – igure 3 . The left picture illustrates the Riemann surface C corresponding to a theory T UV ,which is obtained by coupling the theory T IR to a bifundamental field. An RG flow, that isinitiated by turning on a Higgs vev for the bifundamental scalar, relates the theory T UV tothe original theory T IR with a surface defect G r . This is illustrated on the right. This has simple poles that originate from simple poles in the integrand pinching thecontour. We consider the simple poles of the integrand coming from the bifundamentalhypermultiplet index at a i = t q m i b σ ( i ) c , (2.11)where σ is a permutation of { , . . . , N } and (cid:80) i m i = r where r ∈ Z ≥ . They correspondto the chiral ring generated by derivatives of components of the bifundamental scalarfield, ( ∂ w ) m i Q σ ( i ) i . For each permutation σ , these poles pinch the contour when c = t q rN , (2.12)leading to a simple pole in the integral at this point. This pole then corresponds to thechiral ring generated by derivatives of the baryon operator ( ∂ w ) r B where B = det Q ,which is charged only under the U (1). The residue at this pole corresponds to theindex of T IR in the presence of a surface defect obtained by giving an expectation value B = z r to the baryon operator of T UV and flowing to the IR.As demonstrated in [10], the residue takes the form of a difference operator G r acting on the superconformal index of T IR . There is one term in the operator for eachdistinct set of integers { m , . . . , m N } such that (cid:80) i m i = r . The precise prescriptiondefining the difference operator is G r · I IR ( b i , . . . ) = N I V ( b i ) Res c = t q rN (cid:20) c I UV ( c, b i , . . . ) (cid:21) . (2.13)– 12 –he result of the computation is G r · I ( b i ) = (cid:88) (cid:80) Nj =1 m j = r N (cid:89) i,j =1 (cid:34) m j − (cid:89) m =0 θ ( q m + m j − m i tb i /b j ; p ) θ ( q m − m j b j /b i ; p ) (cid:35) I (cid:0) b i (cid:55)→ q rN − m i b i (cid:1) , (2.14)where the theta-function is defined as θ ( z ; p ) = ∞ (cid:89) i =0 (cid:0) − zp i (cid:1) (cid:18) − p i +1 z (cid:19) . (2.15)The difference operators G r constructed by this method are formally self-adjointwith respect to the measure ∆( a ) I V ( a ) used for gauging. They are labeled by anonnegative integer r ∈ Z ≥ . Furthermore each term in the operator can be identifiedwith a weight of the r -th symmetric tensor representation of su ( N ). In particular, thenumbers { m , m , . . . , m N } denote the number of times the integers { , . . . , N } appearin the corresponding Young tableau. Based on this observation, we associate theseoperators to surface defects labeled by the symmetric tensor representations of su ( N ).It is, however, expected that there exist surface defects labeled by arbitrary irre-ducible representations of su ( N ). The necessity of such defects becomes apparent whenthe difference operators are composed. Let us now consider the composition of two difference operators, G r ◦ G r . This can begiven a physical interpretation by coupling the theory T IR to a single hypermultiplet Q in the bifundamental representation of SU ( N ) × SU ( N ) and then to another bifun-damental hypermultiplet Q . The resulting theory T (cid:48) UV is illustrated in Figure 4. It hastwo additional flavor symmetries U (1) and U (1) that act on the two bifundamentalhypermultiplets Q and Q respectively.The original theory T IR is reached by turning on constant vacuum expectationvalues for both baryon operators B = det Q and B = det Q charged under theadditional flavor symmetries U (1) and U (1) . In the superconformal index, this corre-sponds to the residues of I UV at the simple poles c = t / and c = t / in the fugacitiesassociated to U (1) f, and U (1) f, respectively. Turning on position dependent vacuumexpectation values B = z r and B = z r corresponds to computing the residues at– 13 – igure 4 . The left picture illustrates the Riemann surface C corresponding to the theory T (cid:48) UV , which is obtained by coupling the theory T IR to two bifundamental fields. An RG flow,that is initiated by turning on Higgs vevs for both bifundamental scalars, relates the theory T (cid:48) UV to the original theory T IR with two surface defects G r and G r . This is illustrated onthe right. simple poles c = t / q r and c = t / q r . The order in which the residues are computedis irrelevant and the result G r · ( G r · I IR ) = G r · ( G r · I IR ) , (2.16)defines the (commutative) composition G r ◦ G r . This construction again has an in-terpretation in terms of the curve C for theories of class S , shown in Figure 4. The operators G r constructed above do not form a closed algebra under compositionand addition. More precisely, except for su (2), the composition G r ◦ G r cannot bedecomposed as a sum of other operators G r with coefficients that are independent ofthe flavor fugacities { a j } acted on by the operators.In order to close the algebra, we need to enlarge the set of difference operators G r .Having identified the label r with the r -fold symmetric tensor representation of su ( N ),it is natural to introduce operators G R for any irreducible representation R of su ( N )and to force them to obey the algebra G R ◦ G R = (cid:88) R N R ,R R G R , (2.17)where the coefficient N R ,R R is non-zero only when the representation R appearsin the direct sum decomposition of the tensor product R ⊗ R . Indeed, it turns out– 14 –hat this determines the operators G R and the algebra coefficients N R ,R R essentiallyuniquely, in a sense we explain in detail below. The closure of the algebra is a highlynon-trivial statement, however, depending on intricate theta-function identities.Let us explain the procedure is some more detail. For each irreducible representa-tion R of su ( N ), we make an ansatz for the operator G R . The ansatz is a sum over theweights λ of the representation R , G R · I ( a i ) = (cid:88) λ C R,λ ( p, q, t, a j ) I ( q − ( λ,h i ) a i ) (2.18)with some unknown functions C R,λ ( p, q, t, a j ). Here, the bracket ( , ) denotes the stan-dard inner product on the Cartan subalgebra normalized so that ( e i , e i ) = 2 for allsimple roots. Furthermore, h i are the weights of the fundamental representation. Theyobey ( h i , h j ) = δ i,j − /N .The weights of an irreducible representation R of su ( N ) can be represented bysemi-standard Young tableaux, that are obtained by placing a number 1 , . . . , N in eachbox of the Young diagram (as we review in Appendix A). Each weight can be writtenas a sum λ = N (cid:88) j =1 m j h j , (2.19)where m j are the filling numbers of the corresponding semi-standard Young tableau.In particular, the weights of the r -th symmetric tensor representation are given by λ = N (cid:88) j =1 m j h j , (2.20)where the numbers m i are such that (cid:80) j m j = r . Since ( λ, h i ) = m i − rN , the chosenansatz is compatible with the symmetric tensor operators G r that we already know.Now we substitute the coefficients C R,λ ( p, q, t, a j ) for the symmetric tensor opera-tors, as well as our ansatz for the remaining representations, into the algebra relations G R ◦ G R = (cid:88) R N R ,R R G R . (2.21)We first solve these relations for the su (2) coefficients N r ,r r ( p, q, t ), and propose ageneralization for the su ( N ) coefficients N R ,R R ( p, q, t ). Then we find that the remain-ing coefficients C R,λ ( p, q, t, a j ) are determined uniquely. The fact that this procedure– 15 –orks requires intricate theta-function identities, providing a strong self-consistencycheck of our ansatz.As a preliminary step, we introduce a small normalization of the operators G r labeled by r -th symmetric tensor representations. We redefine the operators by multi-plying them by the factor N r = t − r ( N − / r − (cid:89) i =0 θ ( q − − i , p ) θ ( tq i , p ) . (2.22)The purpose of the normalization is to render the leading algebra coefficient equalto one. In the Schur limit { p, q, t } → { p, q, q } this normalization factor reduces to N r → ( − r q − r ( r + N ) , in agreement with the normalization factor in [17]. A good starting point is su (2), since its irreducible representations are exhausted by r -fold symmetric products of the fundamental representation. Thus, the algebra ofdifference operators should close without introducing any new operators. In particular,we expect that the product G r ◦ G r can be decomposed according to the tensor productof the corresponding irreducible representations G r ◦ G r = r + r (cid:88) r = | r − r | N r ,r r G r , (2.23)where we can compute the OPE coefficients N r ,r r ( p, q, t ). Consistency of this struc-ture demands that the coefficients N r ,r r constructed in this way are independent ofthe fugacity parameter a .For simplicity, let us first consider the Macdonald limit p →
0. In this limit,the ratios of theta-functions in the operators are replaced by rational functions of theremaining variables q and t . The operators G r become G r · I ( a i ) = N r (cid:88) m + m = r (cid:89) i,j =1 m j − (cid:89) m =0 (cid:16) − q m + m j − m i ta i a j (cid:17)(cid:16) − q m − m j a j a i (cid:17) I (cid:0) a i (cid:55)→ q rN − m i a i (cid:1) , (2.24)where a = a and a = a − .When composing any two such rational operators G r and G r , we indeed find thatthe product G r ◦ G r decomposes according to the tensor product of the corresponding– 16 –rreducible representations, in such a way that the structure constants N r ,r r ( q, t ) arerational functions of q and t .As mentioned above, we have normalized the difference operators such that thestructure constant for the leading OPE coefficient N r ,r r + r = 1. The remainingstructure constants can be computed straightforwardly in each case. For example, G ◦ G = G + N , G , where N , ( q, t ) = (1 + t )(1 − q )(1 − qt ) . (2.25)This is a particular case of the more general decomposition G ◦ G r = G r +1 + N ,rr − G r − , (2.26)where N ,rr − ( q, t ) = (1 − t q r − )(1 − q r )(1 − tq r − )(1 − tq r ) . (2.27)Similar formulae can be derived for any other example.Remarkably, we observe that the structure constants N r ,r r ( q, t ) are equal to the( q, t )-deformed Littlewood-Richardson coefficients. In other words, the operators G r in the limit p → P r ( a, q, t ) for su (2). (We refer to Appendix A for more details regarding Macdonald polynomials and( q, t )-deformed Littlewood-Richardson coefficients.)It turns out that the structure constants of the general elliptic operator algebracan be obtained in a canonical way by “lifting” the structure constants N r ,r r ( q, t )of the Macdonald algebra. This works as follows. First we express the ( q, t )-deformedLittlewood-Richardson coefficients as rational functions consisting of factors of the form(1 − x ), where x is a monomial of the form q α t β . Then we “lift” each factor to an ellipticfunction θ ( x, p ) whose second argument is the additional parameter p . The originalcoefficients are obtained in the limit p → q, t )-deformed Littlewood-Richardson coefficients as rational functions of the form (1 − x ), such as for examplein N , ( q, t ) = (1 − t )(1 − q )(1 − t )(1 − qt ) = (1 − t )(1 − q )(1 − t )(1 − qt ) , (2.28)– 17 –he elliptic lift N , ( p, q, t ) = θ ( t , p ) θ ( q, p ) θ ( t, p ) θ ( qt, p ) . (2.29)is uniquely defined because of the theta-function identity θ ( z − ; p ) = − z θ ( z ; p ) . (2.30)Verifying the composition rules for the elliptic difference operators G r now re-quires numerous theta-function identities. For instance, checking that G ◦ G = G + N , ( p, q, t ) G requires θ ( t , p ) θ ( q, p ) θ ( t, p ) θ ( qt, p ) = + θ ( q − , p ) θ ( t − , p ) θ ( ta − , p ) θ ( ta , p ) θ ( q − , p ) θ ( q − a − , p ) θ ( q − a , p ) θ ( qt, p ) − θ ( t − , p ) θ ( ta − , p ) θ ( tq − a , p ) θ ( a − , p ) θ ( t, p ) θ ( q − a , p ) − θ ( t − , p ) θ ( ta , p ) θ ( tq − a − , p ) θ ( a , p ) θ ( t, p ) θ ( q − a − , p ) , (2.31)which can be checked for instance by expanding around p = 0.Similarly, when composing the fundamental operator G with the operator G r forany other irreducible representation of su (2), we find that another elliptic theta-functionidentity brings the non-trivial structure constant into the form N ,rr − ( p, q, t ) = θ ( t q r − , p ) θ ( q r , p ) θ ( tq r − , p ) θ ( tq r , p ) . (2.32)In fact, for any other check we did, we find that the structure constants N r ,r r are in-dependent of the fugacity parameter a and can be expressed as ratios of theta-functions.Even better, we find that they are elliptic (lifts of ( q, t )-deformed) Littlewood-Richardsoncoefficients, in the sense explained above.The elliptic operators G r thus obey an elliptic version of the Macdonald polynomialalgebra. In particular, this provides evidence for the conjecture that the surface defectslabeled by r ∈ Z ≥ are to be identified with irreducible representations of su (2). For su ( N ), with N >
2, the algebra of the difference operators G r is not closed. Weintroduce a new set of operators G R labeled by irreducible representations of su ( N ),– 18 –nd identify the difference operators G r with the operators G ( r ) labeled by the rank r symmetric tensor representation . We systematically find expressions for the noveloperators by imposing the algebra G R ◦ G R = (cid:88) R N R ,R R G R , (2.33)where we assume that the coefficients N R R ,R ( p, q, t ) are given by the elliptic (lifts of( q, t )-refined) Littlewood-Richardson coefficients, which can be found uniquely for anytriple of representations R , R and R .In the rank 2 and 3 cases, we have explicitly computed a large set of ellipticdifference operators G R , and performed ample consistency checks amongst them. Thesecomputations reveal several structures amongst the difference operators, and we are tomake some proposals for general N . Let us give a few examples here.First, consider the composition of two operators each labeled by the fundamentalrepresentation, G (1) ◦ G (1) . This representation (1) ⊗ (1) decomposes into the symmetrictensor (2) and the antisymmetric tensor (1 ,
1) representations. The coefficient of theoperator G (2) labeled by the symmetric tensor representation is one, following from ourchoice of normalization. Choose the coefficient N (1) , (1)(1 , ( p, q, t ) = θ ( q, p ) θ ( t , p ) θ ( t, p ) θ ( qt, p ) (2.34)to be the uplift of the corresponding ( q, t )-deformed Littlewood-Richardson coefficient.The difference operator G (1 , labeled by the rank-two antisymmetric tensor represen-tation of su ( N ) can then be determined from the equation G (1) ◦ G (1) = G (2) + θ ( q, p ) θ ( t , p ) θ ( t, p ) θ ( qt, p ) G (1 , . (2.35)By this method, we find that the elliptic difference operator G (1 , for the antisymmetrictensor representation is given by G (1 , · I ( a i ) = t − (cid:88) j 2) gauge theory with flavor symmetry group SU ( N ). Thistwo-dimensional theory can be coupled to the four-dimensional theory by gauging the2d flavor symmetry using the restriction of a 4d dynamical or background SU ( N )vectormultiplet to the surface S . A large class of half-BPS surface defects defined inthis way have been studied in [19, 20].The superconformal index of four-dimensional linear N = 2 quiver theories inthe presence of such surface defects can be found by combining the two-dimensionalelliptic genus with the four-dimensional superconformal index [21]. Let us consider thiscombined index in a few examples of surface defects in N = 2 superconformal QCD,i.e. a four-dimensional SU ( N ) gauge theory coupled to 2 N hypermultiplets.Before introducing surface defects, let us remind ourselves that N = 2 supercon-formal QCD has a dual description as a degeneration limit of a Riemann surface withtwo simple and two full punctures. Equivalently, its matter content can be read offfrom a linear quiver, see Figure 5. The manifest global symmetry in this presentation– 23 – igure 6 . On top (bottom): 2d-4d quiver description of a fully antisymmetric (symmetric)surface defect in N = 2 superconformal QCD. is SU ( N ) A × SU ( N ) B × U (1) A × U (1) B . If we denote the corresponding fugacities by ( a i , b i , x, y ), the superconformal index ofsuperconformal QCD is (cid:90) ∆( z j ) I V ( z j ) I H ( z − j , a i , x ) I H ( z j , b i , y ) . (2.44)Notice that we could have equivalently considered the same theory with a U ( N ) gaugegroup, since the center of mass U (1) decouples in the IR.Let us now add two-dimensional degrees of freedom to the four-dimensional super-conformal QCD theory with gauge group U ( N ). We give two examples whose 2d-4dquiver descriptions are shown in Figure 6.As a first example, we consider a two-dimensional N = (2 , 2) gauge theory withgauge group U ( r ) coupled to N fundamental and N anti-fundamental chiral fields.The two-dimensional flavor symmetry group is thus U ( N ) f × U ( N ) a . We couple the N fundamental chirals to the U ( N ) gauge symmetry, and the N anti-fundamental chirals– 24 –o the SU ( N ) B × U (1) B global symmetry, as described in [21]. The resulting quiver isillustrated on top in Figure 6. The superconformal index of the resulting 2d-4d systemis (cid:90) ∆( z j ) I V ( z j ) I H ( z − j , a i , x ) ( O r · I H ( z j , b i , y )) , (2.45)where the operator O r acts as O r · I ( z i ) = (cid:88) | I | = r (cid:89) j ∈ Ik / ∈ I θ ( tq z j /z k , p ) θ ( z k /z j , p ) I (cid:0) q − δ i,I z i (cid:1) (2.46)on the fugacities z i of the U ( N ) gauge symmetry.The terms in the above expression are in one-to-one correspondence with the (cid:0) Nr (cid:1) Higgs branch vacua of the two-dimensional theory, in which certain components of thechiral fields get a vacuum expectation value. Each term in equation (2.47) can beinterpreted as computing the index of the 2d-4d system in one of these vacua.The operator O r agrees with the elliptic difference operator G (1 r ) labeled by theantisymmetric tensor representation (1 r ) of rank r up to an overall fractional shift by q rN . Since the shifts z i (cid:55)→ q − δ i,I z i do not preserve the condition (cid:81) j z j = 1, the fugacities z j should really be interpreted as U ( N ) (instead of SU ( N )) fugacities. To find the exactoperators G (1 r ) , however, we would need to find a system that couples the same two-dimensional degrees of freedom to a 4d theory with genuine SU ( N ) symmetry groups.This is for instance required to understand surface defects in the four-dimensional T N theory, whose flavor symmetry groups cannot be enlarged to U ( N ).As a second example, we add a chiral field in the adjoint representation to the two-dimensional N = (2 , 2) theory that we considered before. The quiver description can befound on the bottom of Figure 6. The presence of the adjoint field drastically changesthe vacuum structure, which is mirrored in the expression for the superconformal index.The index of the 2d-4d system is the same as before, except that the operator O r nowacts as O r · I ( z i ) = (cid:88) (cid:80) Nj =1 m j = r N (cid:89) i,j =1 (cid:34) m j − (cid:89) m =0 θ ( q m + m j − m i tz i /z j , p ) θ ( q m − m j z j /z i , p ) (cid:35) I (cid:0) q − m i z i (cid:1) . (2.47)This expression coincides with the operator G ( r ) associated to the symmetric represen-tation ( r ) of rank r , as was already noted in [21], except that the fractional shift q rN isagain missing. – 25 –or the symmetric as well as the antisymmetric tensor representations the two-dimensional degrees of freedom on the surface defect introduced by the operators G R ,can thus be identified with certain two-dimensional N = (2 , 2) gauge theories, up tosome shifts.It would be interesting to observe whether the S partition function of these N =(2 , 2) theories can be obtained from Toda correlators with degenerate vertex operatorslabeled by highest weights of the symmetric and antisymmetric tensor representations.This has been demonstrated for the fundamental representation in [31] (see also [18,32]). In this section we identify the difference operators G R with local operators in a topo-logical quantum field theory (TQFT) of the Riemann surface C . In the case p = 0,this can be identified with an analytic continuation of refined Chern-Simons theory on S × C and the relevant local operators arise from Wilson loops in the representation R and wrapping the S . Recall that for any superconformal field theory of class S the superconformal index isindependent of marginal couplings and hence of the complex structure of the Riemannsurface C . This suggests that the superconformal index of these theories has a dualdescription as a two-dimensional TQFT on the Riemann surface C [12]. In the Schurlimit (when p → q = t ), the TQFT has been identified as q -deformed Yang-Millstheory on C in the zero area limit [14], or equivalently as an analytic continuationof Chern-Simons theory on C × S . This picture can be extended to the Macdonaldlimit ( p → 0) when the superconformal index has a dual description as an analyticcontinuation of refined Chern-Simons theory on C × S [22].In order to verify the above relation, it is necessary to extract a certain function K ( a ) from the superconformal index for each SU ( N ) flavor puncture. In what follows,we define the normalized index I ( n ) through the equation I ( a, b, . . . ) = ( K ( a ) K ( b ) · · · ) I ( n ) ( a, b, . . . ) , (3.1)– 26 –here K ( a ) = N (cid:89) i (cid:54) = j Γ( ta i /a j , p, q ) . (3.2)The normalized index I ( n ) is now gauged using the measure∆ ( n ) ( a ) = K ( a )∆( a ) = 1 N ! (cid:18) ( p, p )( q, q )Γ( t, p, q ) (cid:19) N − (cid:89) i (cid:54) = j Γ( ta i /a j , p, q )Γ( a i /a j , p, q ) . (3.3)The difference operators ¯ G R acting on the normalized index are thus obtained byconjugation ¯ G R = 1 K ( a ) ( G R · K ( a )) . (3.4)This conjugation leaves the algebra of difference operators unchanged. After a long,yet straightforward, computation we find that the conjugated operators for the fullysymmetric representations R = ( r ) are given by¯ G ( r ) · I ( n ) ( a i ) = N r (cid:88) (cid:80) Nj =1 m j = r N (cid:89) i,j =1 m j − (cid:89) m =0 θ ( tq m a i /a j , p ) θ ( q m − m i a i /a j , p ) I ( n ) (cid:0) q rN − m i a i (cid:1) (3.5)while those for the fully antisymmetric representations R = (1 r ) are¯ G (1 r ) · I ( n ) ( a i ) = t r ( r − N ) / (cid:88) | I | = r (cid:89) k ∈ Ij / ∈ I θ ( ta j /a k , p ) θ ( a j /a k , p ) I ( n ) ( q rN − δ i,I a i ) , (3.6)where the summation is over all subsets I ⊂ { , , . . . , N } of length r . Comparingwith (2.14) and (2.42) noting the reflection property θ ( z, p ) = θ ( p/z, p ) we see that theeffect of the conjugation is simply to interchange t ↔ pq/t . In summary, we have foundthat ¯ G R ( p, q, t, a ) = G R (cid:16) p, q, pqt , a (cid:17) . (3.7)Remarkably, the conjugated antisymmetric tensor operators ¯ G (1 r ) are precisely theHamiltonians of the elliptic Ruijsenaars-Schneider model, extending the observationmade in [10].We will assume that the difference operators ¯ G R admit a complete set of eigenfunc-tions { ψ S ( a i ) } , indexed by irreducible representations S of su ( N ), which are orthogonalwith respect to the measure ∆ ( n ) ( a ) and have non-degenerate eigenvalues E S ( R ) . In fact,– 27 –he eigenfunctions are determined by the fully antisymmetric operators ¯ G (1 r ) . Withthe help of these eigenfunctions { ψ S ( a i ) } the TQFT structure of the superconformalindex can be made very explicit [10].Consider for instance the sphere with three maximal punctures. The correspondingfour-dimensional conformal field theory is known as T N . It has at least SU ( N ) flavorsymmetry. Write the normalized superconformal index of the T N theory as I ( n ) ( a i , b i , c i ) , where the parameters a i , b i and c i are three sets of fugacities dual to the maximal torusof the SU ( N ) flavor symmetries. This superconformal index can be expanded in termsof the set of eigenfunctions { ψ S ( a i ) } as I ( n ) ( a i , b i , c i ) = (cid:88) S ,S ,S C S ,S ,S ψ S ( a i ) ψ S ( b i ) ψ S ( c i ) , (3.8)where C S ,S ,S are the structure constants of the two-dimensional TQFT. If we imposethat acting with any one of the operators ¯ G (1 r ) gives the same result, and assume thatthe eigenvalues are non-degenerate then the superconformal index is in fact diagonalin this basis I ( n ) ( a i , b i , c i ) = (cid:88) S C S ψ S ( a i ) ψ S ( b i ) ψ S ( c i ) . (3.9)This is illustrated in Figure 7 for the case of a sphere with three punctures. As explainedin [10], the constants C S can be found by comparing two degeneration limits of the N +1punctured sphere with N − p, q, t ) = (0 , q, t ). In this limit, the antisymmetric difference operators ¯ G (1 r ) turn intothe Macdonald operators¯ G (1 r ) · I ( n ) ( a i ) = t r ( r − N ) / (cid:88) | I | = r (cid:89) k ∈ Ij / ∈ I (1 − qa j /a k , p )(1 − a j /a k , p ) I ( n ) ( q rN − δ i,I a i ) (3.10)while the normalized vectormultiplet measure becomes∆ ( n ) ( a ) = (cid:89) i (cid:54) = j (1 − ta i /a j , q )(1 − a i /a j , q ) (3.11)– 28 – igure 7 . The superconformal index can be written as a TQFT correlator. This correlatoris diagonal in the eigenfunctions ψ S ( a i ) of the difference operators G R . and coincides with the standard Macdonald measure.The operators ¯ G (1 r ) are self-adjoint with respect to this measure on the unit circle | a | = 1 and their common eigenfunctions are the Macdonald polynomials P S ( a i ; q, t ),which are labeled by irreducible representations of su ( N ). They are by constructionorthogonal with respect to the measure ∆ ( n ) ( a ) and are normalized such that P S ( a i ; q, t ) = χ S ( a i ) + (cid:88) T 1. In this limit the structure constants C S are given by C S = 1 S ,S , (3.13)where S R,S is an analytic continuation of the modular S-matrix of refined Chern-Simonstheory. The Macdonald polynomials obey P S · P S = (cid:88) S N S ,S S P S , (3.14)– 29 –here N S ,S S are the ( q, t )-deformed Littlewood-Richardson coefficients. Remarkably,we have found that the conjugated difference operators ¯ G R obey the same algebra. Letus try to understand this fact.Consider for instance the case N = 2. The eigenvalues of the difference operators¯ G r can be computed from explicit formulae that we have found. By experimentation,we find that they are given by ¯ G r · P r ( a ) = S r ,r S ,r P r , (3.15)where S r ,r is an analytic continuation of the modular S-matrix of refined Chern-Simons theory (see Appendix A for the construction of this quantity). This formulacan be proven by a lengthy computation using the residue construction, in which theeigenvalue is given by Res a = t q r a P r ( a )Res a = t a P r ( a ) = S r ,r S ,r . (3.16)This S-matrix is known to obey the ( q, t )-deformed Verlinde formula S r ,s · S r ,s = S ,s (cid:88) r N r ,r r S r,s . (3.17)Let us now act with the composition of the operators ¯ G r and ¯ G r on the Macdonaldpolynomial P s and apply the refined Verlinde formula( ¯ G r ◦ ¯ G r ) · P s = S r ,s S ,s S r ,s S ,s P s = (cid:88) r N r ,r r S r,s S ,s P s = (cid:88) r N r ,r r ¯ G r · P s . (3.18)Since the Macdonald polynomials form a complete basis of symmetric functions, wefind that the structure constants of the difference operator algebra are indeed the( q, t )-deformed Littlewood-Richardson coefficients.Similarly, we have verified that the generalized difference operators ¯ G R , labeled byirreducible representations R of su ( N ), satisfy the eigenvalue equation¯ G R · P R = S R ,R S ,R P R (3.19)in the Macdonald slice. – 30 – .2 Wilson loops in refined Chern-Simons theory In the Macdonald slice the superconformal index is dual to an analytic continuationof the refined Chern-Simons theory on S × C . Similar to the discussion in [17] wecan identify the surface defect operators ¯ G R in this refined Chern-Simons theory as theWilson loop operator O R = Tr R P exp (cid:18) i (cid:73) S A (cid:19) (3.20)in the representation R wrapping the S . This is of course a local operator from theperspective of the two-dimensional TQFT on C , in accordance with our expectationsfrom six-dimensional engineering.Correlation functions of this operator are independent of its position on C andsimply insert a ratio S R,S /S ,S in the sum over representations S is any correlator. Forexample, inserting the operator O R in a correlator on the three-punctured sphere iscomputed as (cid:104)O R (cid:105) , = (cid:88) S P S ( a ) P S ( a ) P S ( a ) S S, S R,S S ,S , (3.21)where S S,R is an analytic continuation of the modular S-matrix of refined Chern-Simonstheory. Hence inserting the local operator O R in a TQFT correlation function is equiv-alent to acting on any of the punctures with the difference operator ¯ G R .Moreover, from the ( q, t )-deformed Verlinde formula S R ,S · S R ,S = S ,S (cid:88) R N R ,R R S R,S (3.22)we derive the operator product expansion O R · O R = (cid:88) R N R ,R R O R , (3.23)where N R ,R R are the analytically continued ( q, t )-deformed Littlewood-Richardsoncoefficients. Thus in the Macdonald limit, the algebra of the difference operators G R is equivalent to the Verlinde algebra in refined Chern-Simons theory on S × C .The general superconformal index could be taken to define a ( p, q, t )-deformedYang-Mills theory on C , whose structure constants are given in terms of the eigen-functions ψ R ( a i ; p, q, t ) of the elliptic difference operators. The difference operators– 31 –atisfy an algebra whose structure constants N R ,R R are elliptic functions. It wouldbe fascinating to understand this theory. In section § 2, we constructed the superconformal index of N = 2 theories on S × S in the presence of certain surface defects supported on S × S . These surface defectswere labeled by an irreducible representation R of su ( N ) and could be added to anysuperconformal theory with an SU ( N ) flavor symmetry. In this section, we considerthe reduction of the four-dimensional superconformal index to a partition function ona squashed three-sphere S , following [23–25]. In this limit, the surface defects becomecodimension-two defects in the three-dimensional theory wrapping an S ⊂ S .For four-dimensional theories of class S , upon dimensionally reducing on S thetheory flows to an N = 4 superconformal field theory in three-dimensions. Moreover,this has a mirror description in terms of a star-shaped quiver theory [26]. It is expectedthat the surface defects introduced by the difference operators G R become supersym-metric Wilson loops in representation R for the central node of this star-shaped quiverupon dimensional reduction. We demonstrate this explicitly for antisymmetric tensorrepresentations R = (1 r ) and the case of a round three-sphere. For non-minusculerepresentations R , however, we find that the difference operators G R introduce a linearcombination of Wilson loops in irreducible representations S with | S | ≤ | R | . The four-dimensional superconformal index on S × S can be reduced to a partitionfunction on the squashed three-sphere S , as demonstrated in [23–25]. This limit istaken by parametrizing the fugacities by p = e − βb − , q = e − βb , t = e − β(cid:15) , a j = e − iβx j , (4.1)with β > β → + . Here we have introduced the convenientnotation (cid:15) = q + im where q = b + b − .The real parameter b > b − | z | + b | w | = 1 (4.2)– 32 –nto C with complex coordinates ( z, w ). The parameters x i with (cid:80) Ni =1 x i = 0 arereal mass parameters for the global SU ( N ) symmetry that is inherited by the three-dimensional theory. It is convenient to repackage the components x j into a vector x suchthat x j = ( x, h j ). In addition, the real parameter m gives a mass to the adjoint chiralmultiplet inside the background N = 4 vectormultiplet, breaking the supersymmetryto N = 2 in three dimensions.Let us consider two important examples. Firstly, the three-dimensional limit (4.1)of the superconformal index of a free hypermultiplet is given by Z H ( x ) = S b (cid:16) (cid:15) ± ix (cid:17) . (4.3)Secondly, the superconformal index of an SU ( N ) vectormultiplet combined with theHaar measure becomes the partition function of a three-dimensional N = 4 vectormul-tiplet Z V ( x ) = N (cid:89) i 0. We use the double sine function that obeys the differenceequation S b ( x + b ± ) = 2 sin( πb ± x ) S b ( x ) and the reflection property S b ( x ) S b ( q − x ) = 1.Further properties of this special function can be found in appendix C.Let us now consider the three-dimensional limit of the difference operators G R that introduce surface defects into the four-dimensional N = 2 theory. The three-dimensional limit can be evaluated using the fact that the ratio of theta-functions witha common second argument reduces to a ratio of sine-functions, θ (cid:0) e αρ , e βρ (cid:1) θ ( e γρ , e βρ ) ρ → −−→ sin ( πα/β )sin ( πγ/β ) . (4.6)Let us first consider the difference operator G (1) labeled by the fundamental rep-resentation of su ( N ). In four dimensions this operator is given by G (1) · I ( a i , . . . ) = N (cid:88) j =1 N (cid:89) k (cid:54) = j θ ( tq a j /a k , p ) θ ( a k /a j , p ) I (cid:16) q N − δ k,i a i , . . . (cid:17) (4.7)– 33 –p to some overall t -dependent factor. Applying the above formula to each term, wefind that the three-dimensional limit of the fundamental difference operator G (1) actson the three-dimensional partition function Z ( x, . . . ) as G (3d)(1) · Z ( x, . . . ) = N (cid:88) j =1 N (cid:89) k (cid:54) = j sin πb ( (cid:15) ∗ − ix jk )sin πb ( − ix jk ) Z ( x + ibh j , . . . ) , (4.8)where we use the shorthand x jk = x j − x k . We also recall that the weights h i obey( h i , h j ) = δ ij − /N .Let us now extend this computation to the rank r antisymmetric tensor represen-tation (1 r ) of su ( N ). In section § t the correspondingdifference operator is G (1 r ) · I ( a i ) = (cid:88) | I | = r (cid:89) j ∈ Ik / ∈ I θ ( tq a j /a k , p ) θ ( a k /a j , p ) I (cid:0) q rN − δ i,I a i (cid:1) , (4.9)where the summation is over subsets I ⊂ { , . . . , N } of length | I | = r and where thesymbol δ i,I is one if i ∈ I and zero if i / ∈ I . In the three-dimensional limit, we obtainthe operator G (3d)(1 r ) · Z ( x ) = (cid:88) | I | = r (cid:89) j ∈ Ik / ∈ I sin πb ( (cid:15) ∗ − ix jk )sin πb ( − ix jk ) Z (cid:32) x + ib (cid:88) j ∈ I h j (cid:33) , (4.10)acting on the squashed three-sphere partition function Z ( x ). Similar computationscan be performed for the four-dimensional difference operators G R corresponding toany irreducible representation R of su ( N ).The dimensionally reduced operators G (3d) R have similar properties to their four-dimensional ancestors. The adjoint operator of G (3d) R with respect to the three-dimensional N = 4 vectormultiplet measure (4.4) on R N − , (cid:104) f , f (cid:105) = (cid:90) (cid:34) d N − xN ! N (cid:89) i 0. Note that the N = 2 mass deformation m in the hypermultipletcontribution appears with the opposite sign compared to equation (4.3). The reasonis that after dimensional reduction, there is a mirror symmetry required to reach thestar-shaped quiver description. Recall that Wilson loops are labeled by irreducible representations of the gauge group. Here weuse that each irreducible representation R of su ( N ) corresponds to an irreducible representation R of SU ( N ), and vice versa. – 37 –t is expected that the partition function has the following properties Z (cid:15) ( x, y ) = Z (cid:15) ( − x, y ) = Z (cid:15) ( x, − y ) Z (cid:15) ( x, y ) = Z (cid:15) ∗ ( y, x ) G (3 d )(1) ( y ) Z (cid:15) ( x, y ) = W (1) ( x ) Z (cid:15) ( x, y ) , (4.18)where G (3 d )(1) ( x ) = sin πb ( (cid:15) ∗ − ix )sin πb ( − ix ) e ib ∂ x + sin πb ( (cid:15) ∗ + 2 ix )sin πb (2 ix ) e − ib ∂ x (4.19)is the fundamental difference operator (4.8) when N = 2, and W (1) ( x ) = e πbx + e − πbx is the fundamental Wilson loop expectation value.The first line of equation (4.18) represents the enhancement of the Higgs andCoulomb branch symmetry to SU (2) × L SU (2) in the infrared. The second line illus-trates the mirror symmetry of the mass-deformed T ( SU (2)) theory. These propertieswere demonstrated in [35]. Here we would like to prove the final line of equation (4.18).Using mirror symmetry this line is equivalent to¯ G (3d)(1) ( x ) Z (cid:15) ( x, y ) = W (1) ( y ) Z (cid:15) ( x, y ) , (4.20)where ¯ G (3d)(1) ( x ) is obtained from G (3d)(1) ( x ) by the replacement m → − m . Let us provethe intertwining property in this equivalent form.As a preliminary step, we derive a few properties of the function Q ( x, z ) definedin equation (4.17). From the difference equation and the reflection property obeyed bythe double sine function S b ( x ), it is straightforward to show that e ib∂ z Q ( x, z ) = sin πb ( epsilon ∗ ± ix − iz )sin πb ( q − (cid:15) ∗ ± ix − iz ) Q ( x, z ) (4.21)and e ib ∂ x Q ( x, z ) = e − ib ∂ z (cid:34) sin πb ( (cid:15) ∗ − ix − iz )sin πb ( q − (cid:15) ∗ − ix − iz ) Q ( x, z ) (cid:35) (4.22) e − ib ∂ x Q ( x, z ) = e − ib ∂ z (cid:34) sin πb ( (cid:15) ∗ + ix − iz )sin πb ( q − (cid:15) ∗ + ix − iz ) Q ( x, z ) (cid:35) , (4.23)– 38 –here we have used the notation (cid:15) = q + im . Using these results we can now computethe action of the difference operator ¯ G (3d)(1) ( x ) on this function,¯ G (3d)(1) ( x ) · Q ( x, z ) = e − ib ∂ z (cid:34) sin πb ( (cid:15) − ix )sin πb ( − ix ) sin πb ( (cid:15) ∗ − ix − iz )sin πb ( q − (cid:15) ∗ − ix − iz )+ sin πb ( (cid:15) + 2 ix )sin πb (2 ix ) sin πb ( (cid:15) ∗ + ix − iz )sin πb ( q − (cid:15) ∗ + ix − iz ) (cid:35) Q ( x, z )= e − ib ∂ z (cid:34) πb ( (cid:15) ∗ ± ix − iz )sin πb ( q − (cid:15) ∗ ± ix − iz ) (cid:35) Q ( x, z )= ( e ib ∂ z + e − ib ∂ z ) Q ( z, x ) . (4.24)In going from the first to the second line we have applied a simple trigonometric identity.Armed with this result, we now consider the action of the difference operator¯ G (3d)(1) ( x ) on the full partition function (4.16) of the T ( SU (2)) theory. The differenceoperator can be brought inside the integral to act on Q ( x, z ) as in equation (4.24). Byshifting the contour of integration by z → z ± ib , we find¯ G (3d)(1) ( x ) · Z (cid:15) ( x, y ) = 1 S b ( q − im ) (cid:90) dz (cid:104) ( e ib ∂ z + e − ib ∂ z ) Q ( x, z ) (cid:105) e πiyz = 1 S b ( q − im ) (cid:90) Q ( x, z ) (cid:104) ( e ib ∂ z + e − ib ∂ z ) e πiyz (cid:105) = W (1) ( y ) Z (cid:15) ( x, y ) . (4.25)Using the analytic structure of the double sine function S b ( x ), it is straightforward tocheck that no poles are crossed in shifting the contours provided the mass are real.Now, applying mirror symmetry we have G (3d)(1) ( y ) Z (cid:15) ( x, y ) = W (1) ( x ) Z (cid:15) ( x, y ) , (4.26)which is the required result.An important consequence of this result, together with the similarity transforma-tion (4.12) relating the operators G (3 d )(1) and ¯ G (3 d )(1) , is that the partition function of massdeformed T ( SU (2)) theory obeys Z (cid:15) ∗ ( x, y ) = K ( y ) Z (cid:15) ( x, y ) (4.27)which is rather non-obvious from the integral representation.– 39 – .2.2 T ( SU ( N ))Let us now consider equation (4.14) for the general T ( SU ( N )) theory. In this case,we simplify the problem and prove a weaker result by taking the limit of N = 4supersymmetry ( m = 0) and a round three-sphere ( b = 0).In this limit, the operators for the fully antisymmetric representations are given by G (3d)(1 r ) · Z ( x ) = ( − r ( N − r ) (cid:88) j < ··· 12 cos ( πb ) − − (cid:19) G (3d)(0) . (4.34) Any ρ ∈ S N can be uniquely characterized by ρ ( σ ( I (cid:96) )) = π ( I ) for a unique I (cid:96) or I and σ ∈ S r , π ∈ S N − r permutations of I , Z N \ I respectively. – 41 –ince ˜ G (3d)(1) = G (3d)(1) and ˜ G (3d)(0) = G (3d)(0) = 1, we read off from equation (4.33) that˜ G (3d)(2) = G (3d)(2) + (cid:18) 12 cos ( πb ) − − (cid:19) G (3d)(0) . (4.35)The operator ˜ G (3d)(2) that is dual to a Wilson loop thus differs from the difference operator G (3d)(2) by lower order contributions.In general, the relation between the operators G (3d) R appearing in the vortex con-struction and the operators ˜ G (3d) R that are exactly dual to Wilson loops in the three-dimensional limit is given by˜ G (3d) R = G (3d) R + (cid:88) | S | < | R | c S G (3d) S . (4.36)Even though the difference operators G (3d) R are thus not exactly dual to Wilson loops,this is merely an invertible linear transformation on the algebra that these operatorsobey.The original basis of operators G R appears to be more fundamental from a four-dimensional perspective, since in the limit p → S × C . On the other hand, in thethree-dimensional limit, the basis ˜ G (3d) R seems more fundamental since it is dual to abasis of Wilson loop operators in the star-shaped quiver theories. N = 2 ∗ theory In this section, we realize the mass-deformed theory T ( SU ( N )) on a squashed three-sphere as an S-duality domain wall in four-dimensional N = 2 ∗ theory on an ellipsoid,as described in [35, 38]. We then use this observation to interpret the three-dimensionaldifference operators G (3d) R as operators that introduce supersymmetric ’t Hooft loops inthe N = 2 ∗ theory partition function on a four-sphere.The four-dimensional N = 2 ∗ theory can also be obtained by compactifying thesix-dimensional (2 , 0) theory of type A N − on a torus with a simple puncture. Aconsequence of this construction is that via the AGT correspondence [7, 8], the four-sphere partition function of the N = 2 ∗ theory can also be computed as a correlationfunction in Liouville or Toda CFT on the punctured torus. The difference operators– 42 – (3d) R can then be interpreted as Verlinde loop operators that act on suitably normalizedVirasoro or W N -algebra conformal blocks on a punctured torus. The exact partition function of N = 2 supersymmetric gauge theories on an ellipsoidhas been computed by supersymmetric localization in [39], extending the computationof Pestun for the round four-sphere S [40]. The ellipsoid geometry can be embeddedinto five-dimensional Euclidean space as x + 1 b ( x + x ) + b ( x + x ) = 1 , (5.1)where b ∈ R ≥ is a real parameter. The equator { x = 0 } is identified with the squashedthree-sphere geometry considered in the previous section by setting z = x + ix and w = x + ix .Let us concentrate on the N = 2 ∗ theory and denote the real hypermultiplet massparameter by m and the complexified gauge coupling by τ . The result of the localizationcomputation can be written as a matrix integral Z S b ( m, τ ) = (cid:90) da (cid:12)(cid:12) Z ( a, m ; τ ) (cid:12)(cid:12) (5.2)over a real slice of the Coulomb branch. In this integral Z ( a, m ; τ ) is the Nekrasovpartition function for the four-dimensional N = 2 ∗ theory in the Omega-background R (cid:15) ,(cid:15) , with equivariant parameters (cid:15) = b and (cid:15) = b − [41, 42]. It can be split into aclassical, 1-loop and instanton piece as Z ( a, m ; τ ) := Z cl ( a ; τ ) Z − loop ( a, m ) Z inst ( a, m ; τ ) . (5.3)In this paper we advertise an alternative factorization of the ellipsoid partitionfunction Z S b . We find it insightful to rewrite the matrix integral 5.2 in the form (for aderivation of this representation see appendix C) Z S b ( m ; τ ) = (cid:90) da µ ( a ) (cid:12)(cid:12) G ( a, m ; τ ) (cid:12)(cid:12) , (5.4)where µ ( a ) = (cid:89) e> πb ( e, a )) 2 sinh (cid:0) πb − ( e, a ) (cid:1) (5.5)– 43 –s the Haar measure times the partition function of a three-dimensional N = 2 vector-multiplet on the squashed three-sphere at the equator { x = 0 } .We expect that the factorization (5.4) has the following interpretation [29]. Wecan cut the ellipsoid into two half-spheres { x > } and { x < } and impose Dirichletboundary conditions on the fields in the N = 2 ∗ theory at the boundary { x = 0 } . Thisdecouples the dynamics on both half-spheres. Restricting the gauge transformationsto the identity on the boundary, leaves a flavor symmetry group SU ( N ) acting on thevalues of the fields at x = 0. We can reconstruct the partition function of an ellipsoid byinserting a three-dimensional N = 2 SU ( N ) vectormultiplet on the boundary { x = 0 } and gauging the diagonal SU ( N ) symmetry. We thus claim that G ( a, m ; τ ) in thematrix integral (5.4) is the partition function of N = 2 ∗ theory on the upper half ofthe ellipsoid { x > } with Dirichlet boundary conditions, and similarly for G ( a, m ; τ )on the lower half { x < } .Note that G ( a, m ; τ ) can be split into classical, one-loop and instanton contributionsjust like the Nekrasov partition function in (5.3). Whereas we take its classical andinstanton contributions to be the same as those of Z ( a, m ; τ ), i.e. G cl ( a ; τ ) ≡ Z cl ( a ; τ )and G inst ( a, m ; τ ) ≡ Z inst ( a, m ; τ ), the one-loop factor G − loop is not canonically deter-mined. We claim that it is fixed by imposing Dirichlet boundary conditions on thehalf-sphere, in such a way that G − loop ( a, m ) = (cid:81) w ∈ adj Γ b (cid:0) q + i ( a, w ) + im (cid:1)(cid:81) e> Γ b ( q + i ( a, e ))Γ b ( q − i ( a, e )) , (5.6)where q = b + b − and Γ b ( x ) is the Barnes’ double gamma function. The numeratorcontains the contribution from the vectormultiplet and the denominator that from theadjoint hypermultiplet with mass m in the N = 2 ∗ theory.Let us mention that via the AGT correspondence, in the case N = 2, this choice isequivalent to a commonly used normalization of Virasoro conformal blocks in Liouvilletheory, as described for example in [28]. For this choice of normalization, we will showthat the expectation values of ’t Hooft loop operators in the N = 2 ∗ theory are given byacting on G ( a, m ; τ ) with the three-dimensional difference operators G (3d) R , constructedin § 4. – 44 – .2 S-duality domain wall The three-dimensional theory T ( SU ( N )) appears as an S-duality domain wall betweentwo four-dimensional N = 4 SYM theories with gauge groups SU ( N ) and L SU ( N )respectively and equal holomorphic gauge coupling τ [33, 34]. Furthermore, the massdeformation m of the domain wall theory can be identified with the canonical massdeformation of the bulk theory to N = 2 ∗ by giving a mass to the adjoint N = 2hypermultiplet.On the ellipsoid S b , one can introduce the S-duality domain wall at the equator { x = 0 } in a way that preserves half of the supersymmetries of the bulk [29]. As above,let us assume that the normalized function G ( a, m ; τ ) corresponds to the partitionfunction of the N = 2 ∗ theory with gauge group SU ( N ) on { x < } with Dirichletboundary conditions for the vectormultiplet, and similarly that G ( a (cid:48) , m ; τ ) correspondsto the partition function of the N = 2 ∗ theory with gauge group L SU ( N ) on { x > } .Let us also denote the partition function of the T ( SU ( N )) theory on the squashed three-sphere at the equator { x = 0 } by Z ( a, a (cid:48) , m ), where a and a (cid:48) are mass parameters forthe SU ( N ) × L SU ( N ) global symmetry as in § 4. Then the combined partition functionin the presence of the S-duality domain wall is (cid:90) da da (cid:48) µ ( a ) µ ( a (cid:48) ) G ( a, m ; τ ) Z ( a, a (cid:48) , m ) G ( a (cid:48) , m ; τ ) , (5.7)where µ ( a ) µ ( a (cid:48) ) is the partition function of three-dimensional N = 2 vectormultipletson the equator { x = 0 } that gauge the symmetry SU ( N ) × L SU ( N ) (see Figure 9).Another interpretation of the same domain wall is as a Janus domain wall interpo-lating between holomorphic gauge coupling τ for { x < } and − /τ for { x > } . Thetwo pictures are related by an S-duality transformation of the theory on { x > } . An-other way of saying this is that the partition function Z ( a, a (cid:48) , m ) should be an S-dualitykernel relating the functions G ( a, m ; τ ) and G ( a, − m ; − /τ ) through the measure dµ ( a ).This statement is rather hard to check in field theory because there is in general noclosed expression for the G ( a, m ; τ ). In Appendix B, however, we demonstrate thisexplicitly in the limit m = 0 and b = 1 and explain some of the subtleties involved inmaking this statement precise.On the other hand, in the context of the AGT correspondence, it has been checkedin [35, 38] that the partition function Z ( a, a (cid:48) , m ) on a squashed three-sphere of the– 45 – igure 9 . Left: The ellipsoid partition function in the presence of an S-duality domainwall can be constructed by gluing in the domain wall partition function Z ( a, a (cid:48) , m ) in be-tween the half-sphere partition functions G ( a, m ; τ ) and G ( a (cid:48) , m ; τ ), while gauging their flavorsymmetries. Right: The same ellipsoid partition function can be constructed by gluing thehalf-sphere partition functions G ( a, m ; τ ) and G ( a, − m ; − τ ). T ( SU (2)) theory is precisely equal to the S-duality kernel of a normalized conformalblock in Liouville theory [43] under the relevant identification of parameters. We arenot aware of a similar computation for N > Since we can embed the mass-deformed T ( SU ( N )) theory as a domain wall in the four-dimensional N = 2 ∗ theory on an ellipsoid, it is natural to think that supersymmetricloop operators in the two theories on { x < } and { x > } are related. In particular,one can introduce a loop operator on one hemisphere and push it through the domainwall to find another loop operator on the other hemisphere. For an S-duality wall oneexpects that this process turns a Wilson loop operator in the four-dimensional N = 2 ∗ theory into a ’t Hooft loop operator.Let us briefly summarize a few facts that are known about supersymmetric loopoperators on the four-sphere. The four-sphere partition function can for instance beenriched with Wilson and ’t Hooft loop operators. To preserve half of the supersym-– 46 –etries such loop operators should be supported on the circle x = cos ρx = b sin ρ cos ϕx = b sin ρ sin ϕx = x = 0 , (5.8)where 0 < ϕ < π and 0 < ρ < π , or alternatively supported on the circle obtainedby interchanging b ↔ b − and { x , x } ↔ { x , x } . The support of the loop operatorlies in the squashed three-sphere at the equator { x = 0 } when ρ = π/ 2. However, theexpectation value of the loop operator is independent of ρ . Wilson loops Supersymmetric Wilson loops in the four-dimensional N = 2 ∗ theory are labeled byirreducible representations of the gauge group G . The expectation values of supersym-metric Wilson loops on the ellipsoid have been computed in [39].The expectation value for a supersymmetric Wilson loop in the irreducible repre-sentation R around a circle in the ( x , x )-plane is obtained by inserting the factor W R ( a ) = (cid:88) w ∈ R e − πb ( w,a ) (5.9)into the matrix integral. For example, for a rank r antisymmetric tensor representationof SU ( N ) we insert the factor W (1 r ) ( a ) = (cid:88) { j <... 2. Thus compatibility with S-dualitydemands that Z ( a, a (cid:48) , m ) intertwines ’t Hooft loops and Wilson loops according toequation (5.17). See Figure 10. In section § Z ( a, a (cid:48) , m ) to obey the same relation with respect to thethree-dimensional limit of the surface defect operators G (3d)(1 r ) . Thus the correspondingoperators should agree. Above we checked that this is indeed the case for a roundfour-sphere.Let us now make some comments on non-minuscule representations R . Since Wil-son loop operators labeled by R are defined by a trace over the representation R , theyobey the character algebra W R ◦ W R = (cid:88) R N R ,R R W R , (5.19)where N R ,R R are the standard Littlewood-Richardson coefficients. In particular, theycan be generated from Wilson loops labeled by fully antisymmetric tensor representa-tions by composition and addition/subtraction.– 50 –herefore, we can define a new set of operators ˆ T R by taking ˆ T (1 r ) ≡ (cid:101) T (1 r ) , orequivalently ˆ T (1 r ) ≡ G (3d)(1 r ) , for antisymmetric representations and imposing the charac-ter algebra ˆ T R ◦ ˆ T R = (cid:88) R N R ,R R ˆ T R . (5.20)The resulting operators ˆ T R automatically transform in the expected way under S-duality, and it is natural to expect that these operators encode the expectation valueof ’t Hooft loops for general representations.However, we emphasize that the ˆ T R do not seem to correspond to the expectationvalue of a ’t Hooft loop with magnetic weight given by the highest weight of therepresentation R , when the representation is non-minuscule. For example, for SU (2)the ’t Hooft loop whose magnetic weight is double that of the ’t Hooft loop of minimalcharge is given by T ◦ T rather than T ◦ T − T . This is again an invertible lineartransformation on the algebra of the operators. In this case, the origin of the basistransformation is a natural resolution of the Bogomolnyi moduli space that arises forrepresentations with non-perturbative monopole bubbling effects [27]. Once again, weemphasize that the simplest and unambiguous operators are those in antisymmetrictensor representations. All we have discussed so far in this section can also be framed in the language ofLiouville or Toda conformal field theory. This approach has the benefit that, at leastfor the ’t Hooft loop in the fundamental representation, we can compute the requiredoperator for general squashing parameter b .Let us briefly review aspects of this correspondence. For the N = 2 ∗ theory withgauge group SU ( N ), the ellipsoid partition function is related to a Liouville or type A N − Toda correlator on the punctured torus with an insertion of a semi-degenerateprimary field. The parameters on both sides of the correspondence are related asfollows:1. The geometric parameter b is a dimensionless coupling in the conformal fieldtheory and gives the central charge c = ( N − N ( N +1) q ), where q = b + b − .– 51 – igure 11 . The four-sphere partition function of the N = 2 ∗ theory is equal to a Todacorrelation function on the punctured torus with a semi-degenerate vertex operator V µ , withmomentum µ , inserted at the puncture. 2. The holomorphic gauge coupling τ is the complex structure parameter of thepunctured torus.3. The mass m of the adjoint hypermultiplet is encoded in the momentum of thesemi-degenerate primary field, µ = N (cid:16) q im (cid:17) ω N − . (5.21)Choosing a pants decomposition, the correlation function of the primary field on thepunctured torus can be written as an expansion in Liouville or W N –algebra conformalblocks (cid:90) dα C ( µ, α, Q − α ) F ( α, µ ; τ ) F ( α, µ ; τ ) , (5.22)where the integral is over non-degenerate momenta α = Q + ia , with a ∈ R N − and Q = qρ , where ρ is the Weyl vector of A N − .The conformal blocks F ( a, µ ; τ ) are normalized to contain the classical and in-stanton contributions to the Nekrasov partition function. The three-point function C ( µ, α, Q − α ) is proportional (up to an m -dependent piece that can be absorbed inthe normalization of the primary field) to the modulus squared of the 1-loop contri-bution | G − loop | times the measure µ ( a ). The meromorphic function G ( a, m ; τ ) thatwe introduced earlier corresponds to a convenient normalization of the conformal block F ( α, µ ; τ ) that absorbs the three-point functions. This is an extension to higher rankof a frequently used normalization in Liouville theory [28].Loop operators in the four-dimensional gauge theory are realized as Verlinde op-erators in the dual conformal field theory [9, 28]. The Verlinde operators act on the– 52 –pace of Virasoro or W N -algebra conformal blocks by transporting a chiral primary fieldaround a simple closed curve C on the Riemann surface. The operators constructed inthis way depend only on the homotopy class of the curve C up to a choice of ‘framing’that will not be important here.If we choose the pants decomposition of the punctured torus determined by theA-cycle in Figure 11, a supersymmetric Wilson loop in N = 2 ∗ theory in the rank r antisymmetric tensor representation corresponds to transporting a degenerate chiralprimary with momentum η = − bω j around that A-cycle. The resulting expressionchanges from the original conformal block by the factor W CFT(1 r ) = (cid:88) { j <... In this paper we generated an algebra of difference operators G R acting on the N = 2superconformal index, labeled by irreducible representations R of SU ( N ). Generalizingthe arguments of [10], we claim that these difference operators represent half-BPSsurface defects in four-dimensional N = 2 theories of class S . We discussed severalarguments in favour of this claim. Most importantly, we emphasized that it is highlynon-trivial that we indeed managed to consistently close the algebra, and that thedifference operators have a natural interpretation in various dual frames. Let us mentiona few open questions and interesting links.A microscopic gauge theory understanding of these defects is unfortunately stilllacking, either in terms of a defect description or alternatively as a description of thetwo-dimensional degrees of freedom living on the support of the defect. We did finda two-dimensional field theory description in two extreme cases: fully antisymmetricand fully symmetric representations. It is however not at all clear that there exists aLagrangian description for the two-dimensional degrees of freedom living on the supportof the surface defect for a generic representation R .The operators G R can be written as a sum of weights in the representation R , whichin the field theory description of the defect should have the interpretation as a sum overvacua. In case a weight λ appears with multiplicity one in the representation R , thecontribution to G R is a single ratio of theta-function and seems likely to have an inter-pretation as the contribution to the superconformal index in a vacuum characterizedby λ . When the weight λ appears with higher multiplicity, however, the contributionto G R is a sum of such ratios and is less likely to have such an interpretation.A similar structure can be observed from the perspective of the AGT correspon-dence. In particular, reference [31] has demonstrated that a ratio of Toda correlationfunctions involving a degenerate momentum µ = − bh captures the two-sphere parti-tion function of the N = (2 , 2) theory that we have associated to the surface defectlabelled by the fundamental representation. However, for representations with mul-tiple weight contributions, the Toda three-point function with degenerate insertions(see [45, 46]) do not appear to have the structure of one-loop contributions to thetwo-sphere partition function of an N = (2 , 2) theory.– 54 – igure 12 . Left: UV curve of the theory T UV , obtained by gluing a three-punctured spherewith two full punctures and one hook-shaped puncture to the UV curve of the theory T IR .Right: A renormalization group flow connects T UV to T IR with a possibly general surfacedefect G R . As briefly mentioned in the introduction and main text, we expect that thereis an alternative method to find the difference operators G R . Instead of coupling thetheory T IR to a bifundamental hypermultiplet, corresponding to adding a puncture with U (1) symmetry, one could add a puncture with a larger flavor symmetry group. Thisgenerically involves coupling T IR to a non-Lagrangian theory corresponding to a spherewith two full punctures and one intermediate puncture. An example is illustrated inFigure 12. The superconformal index for T UV should then contain a larger spectrum ofresidues. One might expect any difference operator G R to originate from such a residuecomputation.Again, this is analogous to the Toda perspective, where non-maximal flavor punc-tures correspond to insertions of semi-degenerate vertex operators. By analytic contin-uation correlation functions of such operators have poles, whose residues correspond toreductions to a completely degenerate vertex operator. If we insert a semi-degeneratevertex operator that corresponds to a simple U (1) puncture, we can only access com-pletely degenerate vertex operators with momentum α = − bλ − b − λ , where λ = r ω and λ = r ω are the highest weights of two symmetric tensor representations. To findcompletely degenerate vertex operators with generic weights λ and λ , one must inserta generic semi-degenerate vertex operator, corresponding to a generic flavor puncture.The difference operators G R are elliptic generalizations of the Macdonald operators.Although these operators have not been constructed mathematically for all represen-– 55 –ations R (as far as we know), the elliptic Ruijsenaars-Schneider difference operatorshave been related to exterior powers of the vector representation of the elliptic quantumgroup E τ,η ( gl N ) [47]. It would be very interesting to interpret this connection to ellip-tic quantum groups physically. This relation could appear naturally when interpretingthe difference operators G R in terms of a three-dimensional topological field theory on C × S .In the Macdonald limit p = 0 we have found that the difference operators corre-spond to Wilson loops in an analytic continuation of refined Chern-Simons theory on C × S (see § G R , suggests that when we take a Wilson loop operator closeto a puncture on C , it can be interpreted as a Verlinde loop operator on the boundarytorus. For example, taking the Schur limit t = q , and replacing q → exp (2 πi/ ( k + N ))in the modular S-matrix, we would recover the modular S-matrix elements for char-acters of integrable representations of the affine current algebra (cid:98) su ( N ) k . However, forthe superconformal index it is important that we have an analytic continuation of thisstatement to | q | < G R arerelated to operators that introduce line defects. For theories of class S , there is amirror description as a star-shaped quiver and we showed that the difference operatorsintroduce Wilson loops for the central node of the quiver, at least in the case of an-tisymmetric tensor representations. For non-minuscule representations, we found thatthere is some mixing.It would be interesting and important to understand these line defects in three-dimensions from first principles by localization. For the fully symmetric and anti-symmetric tensor representations, we expect this could be done by coupling to a su-persymmetric quantum mechanics on a circle, in a similar spirit to [21] but in onedimension lower. On the other hand, we expect that the rank- r anti-symmetric ten-sor operator has another description as a monodromy defect breaking the gauge groupto S ( U ( r ) × U ( N − r )), which might also be used to perform an exact localizationcomputation by extending the computations of [48, 49] for abelian monodromy defects.– 56 – cknowledgments We would very much like to thank Fernando Alday for collaboration on part of thisproject. The work of M.B. is supported by the Perimeter Institute for TheoreticalPhysics. Research at the Perimeter Institute is supported by the Government of Canadathrough Industry Canada and by the Province of Ontario through the Ministry ofResearch and Innovation. The work of M.F. and P.R. is supported by ERC STG grant306260. The work of L.H. is supported by a Royal Society Dorothy Hodgkin fellowship. A Macdonald polynomials and the refined S-matrix A.1 Group theory The finite dimensional irreducible representations of A N are in one-to-one correspon-dence with dominant integral weights, λ = N − (cid:88) i =1 λ i ω i whose Dynkin labels ( λ , λ , ..., λ N − ) are nonnegative integers. Equivalently, irre-ducible representations are labeled by partitions ( (cid:96) , (cid:96) , . . . , (cid:96) N ) where (cid:96) ≥ (cid:96) ≥ ... ≥ (cid:96) N = 0, such that (cid:96) i = λ i + λ i +1 + ... + λ N − . (A.1)Each partition is associated to a Young diagram whose i -th row has length (cid:96) i . Forinstance, the following diagramcorresponds to the partition (4 , , , (cid:96) , (cid:96) , . . . , (cid:96) N ) are relatedto the components of the weight in the orthogonal basis ω i = (cid:15) + · · · + (cid:15) i − iN N (cid:88) j =1 (cid:15) j (A.2)where λ = N (cid:88) i =1 κ i (cid:15) i , κ i = (cid:96) i − N N − (cid:88) j =1 j ( (cid:96) j − (cid:96) j +1 ) . (A.3)– 57 –he states in a given irreducible representation are in one-to-one correspondencewith semi-standard Young tableaux . They are obtained by filling the boxes of a Youngdiagram with the numbers { , . . . , N } , such that the numbers are non-decreasing fromleft to right and strictly increasing from top to bottom. Finally, to each semi-standardYoung tableau, we attach the labels ( n , ..., n N ), where n i denotes the number of timesthat i appears in the semi-standard tableau. As an example below we include a fewsemi-standard tableaux for the adjoint representation of SU (3) with their correspond-ing labels. 1 12 1 23 1 32 2 23(2 , , 0) (1 , , 1) (1 , , 1) (0 , , A.2 Schur polynomials and the modular S-matrix Introduce coordinates a j , for j = 1 , ..., N , obeying (cid:81) Ni =1 a i = 1. For a j = e iθ j theyare coordinates on the maximal torus of SU ( N ). The Schur polynomials form a basisof symmetric functions in the variables { a , . . . , a N } labeled by irreducible representa-tions. The Schur polynomial of the irreducible representation λ labeled by the partition( (cid:96) , . . . , (cid:96) N ) is given by the determinant formula χ λ ( a ) = det a (cid:96) i + N − ij det a N − ij . (A.4)An important property of the Schur polynomials is that they are orthonormal withrespect to the inner product on the space of symmetric functions (cid:104) f, g (cid:105) = (cid:90) ∆( a ) f ( a ) g ( a − ) (A.5)where ∆( a ) = 1 N ! N − (cid:89) i =1 da i πia i (cid:89) i (cid:54) = j (cid:18) − a i a j (cid:19) . (A.6)is the Haar measure and the integration is over the maximal torus of SU ( N ). Productsof Schur polynomials decompose according to the tensor product of the irreduciblerepresentations χ λ ( a ) χ λ ( a ) = (cid:88) µ N λ ,λ µ χ µ ( a ) (A.7)– 58 –here N λ ,λ µ are the Littlewood-Richardson numbers.In order to construct the modular S-matrix we introduce the Weyl weight ρ , whichis the highest weight of the adjoint representation of SU ( N ). Its components in theDynkin basis are ρ = (1 , , . . . , ρ = (cid:18) N − , N − , ..., − N (cid:19) , (A.8)and we will denote these components by ρ j = ( N − j + 1) / 2. Now consider twoirreducible representations λ and λ (cid:48) with components κ i and κ (cid:48) i in the orthogonal basis.Then the modular S-matrix is given by S λλ (cid:48) = S χ λ ( q ρ , ..., q ρ N ) χ ¯ λ (cid:48) ( q ρ + κ , ..., q ρ N + κ N ) , (A.9)where ¯ λ (cid:48) denotes the complex conjugate representation of λ (cid:48) . We will not need theoverall normalization S . A.3 Macdonald polynomials and the refined S-matrix The Macdonald polynomials are symmetric polynomials in the variables { a , . . . , a N } that depend on two additional complex parameters q and t . The Macdonald polyno-mials are labeled by irreducible representations λ of SU ( N ) and reduce to the corre-sponding Schur polynomials when q = t .The Macdonald polynomial labeled by the irreducible representation λ is P λ ( a, q, t ) = χ λ ( a ) + (cid:88) µ<λ c λ,µ ( q, t ) χ µ ( a ) (A.10)where c λ,µ ( q, t ) are rational functions of q and t that are uniquely determined by en-suring P λ ( a, q, t ) is a simultaneous eigenfunctions of the difference operators G r = t r (1 − N ) (cid:88) I ⊂{ ,...,N }| I | = r (cid:89) i ∈ I,j / ∈ I ta i − a j a i − a j T I , r = 1 , . . . , N − T I : a i → q − /N a i if i ∈ Iq − /N a i if i / ∈ I . (A.12)– 59 –ere we have included a background shift by q − /N compared to the standard Mac-donald difference operators in order to preserve the condition (cid:81) Ni a i = 1 relevant for SU ( N ). For example, the first few Macdonald polynomials for SU (2) are P ( a, q, t ) = 1 P ( a, q, t ) = χ ( a ) P ( a, q, t ) = χ ( a ) + q − t − qtP ( a, q, t ) = χ ( a ) + ( q − t )(1 + q )1 − tq χ ( a ) P ( a, q, t ) = χ ( a ) + ( q − t )(1 − q )(1 − q )(1 − q t ) χ ( a ) + q ( q − t )(1 + q )(1 − t )(1 − q t )(1 − q t ) . (A.13)The difference operators are self-adjoint with respect to the inner product (cid:104) f, g (cid:105) = (cid:90) ∆ q,t ( a ) f ( a ) g ( a − ) , ∆ q,t ( a ) = 1 N ! N − (cid:89) i =1 da i πia i (cid:89) i (cid:54) = j ( a i /a j ; q )( ta i /a j ; q ) (A.14)where ( a ; q ) = (cid:81) ∞ i =0 (1 − q i a ) is the q-Pochhammer symbol, and consequently, theMacdonald polynomials are non-degenerate and orthogonal with respect to the samemeasure. In order to obtain functions orthonormal with respect to the measure, anormalization factor must be included.The product of Macdonald polynomials decomposes according to the tensor productof irreducible representations P λ ( a, q, t ) P λ ( a, q, t ) = (cid:88) µ N λ ,λ µ ( q, t ) P µ ( a, q, t ) (A.15)where the N λ ,λ µ ( q, t ) are rational functions in q and t .Analogous to the modular S-matrix, the refined S-matrix is given by S λλ (cid:48) = S P λ ( t ρ , ..., t ρ N ) P ¯ λ (cid:48) ( t ρ q κ , ..., t ρ N q κ N ) . (A.16)It is then an easy exercise to check that the ratios S R,S /S ,S are indeed the eigenvaluesof the difference operators G R in the Macdonald limit, namely G R · P S ( a i , q, t ) = S R,S S ,S P S ( a i , q, t ) . (A.17)– 60 – S-duality kernel Instead of merely inserting a three-dimensional N = 2 vectormultiplet on the three-dimensional boundary { x = 0 } , we could also glue in the three-dimensional mass-deformed linear quiver theory T ( SU ( N )). In fact, its N = 4 variant was introducedas an S-dual of the Dirichlet boundary condition in the four-dimensional N = 4 theorywith gauge group SU ( N ) [34, 50].It is thus natural to expect that the mass-deformed T ( SU ( N )) theory encodes thefield theory degrees of freedom on a so-called S-duality domain wall in the N = 2 ∗ theory. Such a domain wall is defined so that the four-dimensional theories on eitherside are related by the transformation S : ( τ, m ) → ( − /τ, − m ). In this Appendix wewill verify that this is indeed the case if we assume that G b is the partition function onthe half-sphere with Dirichlet boundary conditions.Before introducing the S-duality domain wall, let us briefly consider the ellipsoidpartition function Z S b of the N = 2 ∗ theory with gauge group SU ( N ). The AGTcorrespondence relates this to a Toda correlator on the once-punctured torus. Wethus expect that the ellipsoid partition function transforms as a modular form. Moreprecisely, it should transform as [51] Z S b ( − m ; − /τ ) = | τ | m ) Z S b ( m ; τ ) , (B.1)with modular weight ∆( m ) = N ( N − (cid:18) Q m (cid:19) . (B.2)This modular property of the ellipsoid partition function is guaranteed if the half-sphere partition function G b transforms as G b ( − m, a ; − /τ ) = ( − iτ ) ∆( m ) (cid:90) da (cid:48) µ b ( a (cid:48) ) Z b ( a i , a (cid:48) , m ) G b ( m, a (cid:48) ; τ ) , (B.3)where we integrate over a real slice of the Coulomb branch (just like in all matrixintegrals in the remainder of this section). The integration kernel Z b ( a, a (cid:48) , m ) mustobey two important properties. First, it must obey the symmetry Z b ( a, a (cid:48) , m ) = Z b ( a (cid:48) , a, − m ) . – 61 –econd, it must be a unitary with respect to the measure µ b ( a ) da , in the sense that (cid:90) da µ b ( a ) Z b ( a (cid:48) , a, m ) Z b ( a, a (cid:48)(cid:48) , − m ) = µ b ( a (cid:48) ) δ ( a (cid:48) , a (cid:48)(cid:48) ) . (B.4)Now consider the ellipsoid partition function with the insertion of an S-dualitydomain wall. Assuming that G b is the half-sphere partition function of the N = 2 ∗ theory with Dirichlet boundary conditions, the S-duality partition function on thesquashed four-sphere should be given by (cid:90) da µ ( a ) G b ( m, a ; τ ) G b ( − m, a ; − /τ )= (cid:90) da µ ( a ) (cid:90) da (cid:48) µ ( a (cid:48) ) G b ( m, a ; τ ) Z b ( a, a (cid:48) , m ) G b ( m, a (cid:48) ; τ ) . (B.5)Consequently, Z b ( a, a (cid:48) , m ) should encode the gauge degrees of freedom localized on thedomain wall. Specifically, we expect that Z b ( a, a (cid:48) , m ) is the partition function of themass-deformed T ( SU ( N )) theory on a squashed three-sphere.In this context, the symmetry Z b ( a, a (cid:48) , m ) = Z b ( a (cid:48) , a, − m ) is equivalent to three-dimensional mirror symmetry. The unitary property (B.4) follows because the partitionfunction Z b ( a, a (cid:48) , m ) is an eigenfunction of the self-adjoint operator G (3d) R with respectto the measure µ b ( a ) da (see equation (4.11)).Indeed, let us denote the integral (B.4) by I ( a (cid:48) , a (cid:48)(cid:48) ) = (cid:90) da µ b ( a ) Z b ( a (cid:48) , a, m ) Z b ( a, a (cid:48)(cid:48) , − m ) . (B.6)The self-adjoint operator G (3d) R ( a (cid:48) ) can act inside this integrand in either direction,which must lead to the same answer. Consequently we find( W R ( a ) − W R ( a (cid:48)(cid:48) )) I ( a, a (cid:48)(cid:48) ) = 0 , (B.7)where W R ( a ) is the expectation value of a Wilson loop in the representation R . Thisimplies that the integral vanishes if a (cid:54) = a (cid:48)(cid:48) modulo Weyl transformations. B.1 Example Let us check the above transformation properties of the half-sphere partition function G b on the round four-sphere, when b = 1, and in the N = 4 limit, when m → G b =1 ( τ, m, a i ) for gauge group SU ( N ).Its one-loop contribution (5.6) simplifies to the formula G − loop ( m, a i ) = 1 √ π (cid:89) i 1) Gaussian integrals we expect to find the partition function Z S b =1 ( τ ) ∼ | m ( τ ) | N − Im( τ ) ( N − / , (B.12)which has the expected transformation Z S b =1 (cid:18) − τ (cid:19) = | τ | N ( N − Z S b =1 ( τ ) (B.13)under S-duality. We have indeed verified this for N = 2 , 3. In the above, we have used µ b =1 ( a ) = (cid:81) i Let us briefly review some properties of special functions we need in order to manipulateone-loop contributions. As in the main text, b ∈ R > is a real parameter and we define q ≡ b + b − .The double gamma function Γ b ( x ) is a meromorphic function of x characterized bythe functional equation Γ b ( x + b ) = √ π b bx − Γ b ( x ) / Γ( bx ) (C.1)where Γ( x ) is the Euler gamma function and its value Γ b ( q/ 2) = 1. We will also needthe double sine function, which is a meromorphic function that can be defined in termsof the double gamma function by the formula S b ( x ) ≡ Γ b ( x ) / Γ b ( q − x ). The doublesine function is characterized by the functional equation S b ( x + b ) = 2 sin( πbx ) S b ( x ) . (C.2)We will furthermore need the function Υ b ( x ) − = Γ b ( x )Γ b ( q − x ) which is entire analytic.A more complete discussion of the properties of these functions can be found, forexample, in [53].Let us begin by considering the three-point function C ( α, Q − α, ν ) in A N − Todatheory corresponding to the trivalent vertex in the pants decomposition of a torus with– 64 – simple puncture. The momentum in the internal channel α = Q + ia , with a ∈ R , isnon-degenerate and describes a delta-function normalizable state, while the momentum ν = N ( q/ im ) ω N − , with m ∈ R , is semi-degenerate. Substituting these momentainto the more general result of [45, 46] we find that C ( α, Q − α, ν ) = f ( m ) N (cid:81) i Let us express the Toda three-point function in terms of double gamma functionsand manipulate the answer into a convenient factorized form. For the hypermultiplet– 65 –ontribution, we have N (cid:89) i,j =1 Υ b (cid:16) q ia ij + im (cid:17) − = N (cid:89) i,j =1 Γ b (cid:16) q ia ij + im (cid:17) Γ b (cid:16) q − ia ij − im (cid:17) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N (cid:89) i,j =1 Γ b (cid:16) q ia ij + im (cid:17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . 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