Charging the superconformal index

Abstract

The superconformal index is an important invariant of superconformal field theories. In this note we refine the superconformal index by inserting the charge conjugation operator C. We construct a matrix integral for this charged index for N = 4 SYM with SU(N) gauge group. The key ingredient for the construction is a “charged character”, which reduces to Tr(C) for singlet representations of the gauge group. For each irreducible real SU(N ) representation, we conjecture that this charged character is equal to the standard character for a corresponding representation of SO(N + 1) or SP(N − 1), for N even or odd respectively. The matrix integral for the charged index passes tests for small N and for N → ∞. Like the ordinary superconformal index, for N = 4 SYM the charged index is independent of N in the large-N limit.