### 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 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 N =2* theory on a four-sphere by embedding the three-dimensional theory as an S-duality domain wall.

Original language | English |
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Journal | Journal of High Energy Physics |

Volume | 2014 |

Issue number | 62 |

DOIs | |

Publication status | Published - Oct 2014 |

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## Profiles

## Lotte Hollands

- School of Mathematical & Computer Sciences - Associate Professor
- School of Mathematical & Computer Sciences, Mathematics - Associate Professor

Person: Academic (Research & Teaching)

## Cite this

Hollands, L., Bullimore, M., Fluder, M., & Richmond, P. (2014). The superconformal index and an elliptic algebra of surface defects.

*Journal of High Energy Physics*,*2014*(62). https://doi.org/10.1007/JHEP10(2014)062