Heterotic Surface Defects and Dualities from 2d/4d Indices

TL;DR

This paper investigates surface defects in 4d supersymmetric gauge theories using superconformal indices, confirming their IR limits as vortices via elliptic genera, and explores how 4d dualities influence the world sheet theories of these defects.

Contribution

It extends the understanding of surface defects in 4d N=1 theories by connecting superconformal indices, vortex interpretations, and dualities to 2d world sheet theories and their elliptic genera.

Findings

Residue contributions in indices correspond to surface defects.

Elliptic genera of 2d theories match superconformal algebra predictions.

4d N=1 dualities induce dualities in surface defect theories.

Abstract

Starting with the superconformal indices for 4d N=2 and N=1 supersymmetric gauge theories, which are related by superpotential deformation, we perform the contour integrations and isolate the residue contributions which can be attributed to the surface defects. These defects can be interpreted as the IR limit of dynamical vortices. Given the 2d N=(2,2) and N=(0,2) world sheet theories for these surface defects, we then verify this statement by explicitly computing their elliptic genera and identifying their fugacity parameters through superconformal algebras. We show them precisely match, and the results extend previous prescriptions for inserting surface defects into 4d supersymmetric partition functions to N=1 setting. We also study how 4d N=1 IR dualities descend into the N=(0,2) world sheet theories of their surface defects, and extend the N=(2,2) triality observed earlier to other N=(0,2) surface defects unrelated to dynamical vortices.