2d Index and Surface operators

TL;DR

This paper computes the superconformal index of 2d (2,2) supersymmetric gauge theories, explores its invariance under dualities and flop transitions, and studies surface operators and their dualities in N=2 superconformal theories.

Contribution

It introduces explicit calculations of the 2d superconformal index, demonstrating invariance under dualities and flop transitions, and extends the analysis to surface operators and their dualities in 4d theories.

Findings

The 2d index is invariant under flop transitions and CY-LG correspondence.

The index confirms Seiberg-type dualities for non-abelian theories.

Surface operator indices are invariant under generalized S-duality.

Abstract

In this paper we compute the superconformal index of 2d (2,2) supersymmetric gauge theories. The 2d superconformal index, a.k.a. flavored elliptic genus, is computed by a unitary matrix integral much like the matrix integral that computes 4d superconformal index. We compute the 2d index explicitly for a number of examples. In the case of abelian gauge theories we see that the index is invariant under flop transition and CY-LG correspondence. The index also provides a powerful check of the Seiberg-type duality for non-abelian gauge theories discovered by Hori and Tong. In the later half of the paper, we study half-BPS surface operators in N=2 superconformal gauge theories. They are engineered by coupling the 2d (2,2) supersymmetric gauge theory living on the support of the surface operator to the 4d N=2 theory, so that different realizations of the same surface operator with a given Levi type are related by a 2d analogue of the Seiberg duality. The index of this coupled system is computed by using the tools developed in the first half of the paper. The superconformal index in the presence of surface defect is expected to be invariant under generalized S-duality. We demonstrate that it is indeed the case. In doing so the Seiberg-type duality of the 2d theory plays an important role.