Applications of the superconformal index for protected operators and q-hypergeometric identities to [formula omitted] dual theories

Abstract

The results of Römelsberger for an 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 theory. 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.