Applications of the Superconformal Index for Protected Operators and q-Hypergeometric Identities to N=1 Dual Theories

TL;DR

This paper analyzes the superconformal index for N=1 theories, demonstrating its invariance under dualities like Seiberg and Kutasov-Schwimmer, and connects it to q-hypergeometric identities, providing a detailed operator content expansion.

Contribution

It extends the superconformal index analysis to various gauge groups and dualities, proving identities using elliptic hypergeometric integrals and linking to mathematical q-series results.

Findings

Index invariance under Seiberg duality verified

Elliptic hypergeometric integrals relate electric and magnetic theories

Explicit protected operator content expansion obtained

Abstract

The results of Romelsberger for a N=1 superconformal index counting protected operators, satisfying a BPS condition and which cannot be combined to form long multiplets, are analysed further. The index is expressible in terms of single particle superconformal characters for N=1 scalar and vector multiplets. For SQCD, involving SU(N_c) gauge groups and appropriate numbers of flavours N_f, the formula used to construct the index may be proved to give identical results for theories linked by Seiberg duality using recently proved theorems for q-series elliptic hypergeometric integrals. The discussion is also extended to Kutasov-Schwimmer dual theories in the large N_c, N_f limit and to dual theories with Sp(N) and SO(N) gauge groups. For the former, a transformation identity for elliptic hypergeometric integrals directly verifies that the index is the same for the electric and magnetic theories. For SO(N) theories the corresponding result may also be obtained from the same basic identity. An expansion of the index to several orders is also obtained in a form where the detailed protected operator content may be read off. Relevant mathematical results are reviewed.