Abstract
In this paper we continue the study of the superconformal index of four-dimensional \( \mathcal{N} \) =2 theories of class \( \mathcal{S} \) in the presence of surface defects. Our main result is the construction of an algebra of difference operators, whose elements are labeled by irreducible representations of A N −1. For the fully antisymmetric tensor representations these difference operators are the Hamiltonians of the elliptic Ruijsenaars-Schneider system. The structure constants of the algebra are elliptic generalizations of the Littlewood-Richardson coefficients. In the Macdonald limit, we identify the difference operators with local operators in the two-dimensional TQFT interpretation of the superconformal index. We also study the dimensional reduction to difference operators acting on the three-sphere partition function, where they characterize supersymmetric defects supported on a circle, and show that they are transformed to supersymmetric Wilson loops under mirror symmetry. Finally, we compare to the difference operators that create ’t Hooft loops in the four-dimensional \( \mathcal{N} \) =2* theory on a four-sphere by embedding the three-dimensional theory as an S-duality domain wall.
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Bullimore, M., Fluder, M., Hollands, L. et al. The superconformal index and an elliptic algebra of surface defects. J. High Energ. Phys. 2014, 62 (2014). https://doi.org/10.1007/JHEP10(2014)062
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DOI: https://doi.org/10.1007/JHEP10(2014)062