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The superconformal index and an elliptic algebra of surface defects

Bullimore, Mathew; Fluder, Martin; Hollands, Lotte; Richmond, Paul

The superconformal index and an elliptic algebra of surface defects Thumbnail


Martin Fluder

Lotte Hollands

Paul Richmond


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.


Bullimore, M., Fluder, M., Hollands, L., & Richmond, P. (2014). The superconformal index and an elliptic algebra of surface defects. Journal of High Energy Physics, 2014(10), Article 062.

Journal Article Type Article
Acceptance Date Sep 25, 2014
Online Publication Date Oct 9, 2014
Publication Date Oct 9, 2014
Deposit Date Jun 7, 2018
Publicly Available Date Jun 8, 2018
Journal Journal of High Energy Physics
Print ISSN 1126-6708
Publisher Scuola Internazionale Superiore di Studi Avanzati (SISSA)
Peer Reviewed Peer Reviewed
Volume 2014
Issue 10
Article Number 062


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Copyright Statement
© The Author(s) 2014 Open Access This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

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