Charging the Superconformal Index

TL;DR

The paper introduces a charged refinement of the four-dimensional superconformal index by inserting the charge conjugation operator $C$, defining a charged character $K_C$ whose trace on real $SU(N)$ representations maps to ordinary characters of $SO(N+1)$ (for even $N$) or $SP(N-1)$ (for odd $N$). This leads to a matrix integral for the charged index $\,\mathcal{I}_C$ over $G(N)$, replacing $SU(N)$ characters with traces involving $K_C$ and determinants evaluated in the adjoint representation; a simple conjecture equates $\,\mathrm{Tr}_{\mathbf{d}} K_C$ with the corresponding $G(N)$ characters via a one-to-one map of Dynkin labels. The authors perform nontrivial checks: exact tests for small $N$ up to $N=7$ confirm the charged-character relation and determinant structure, while a large-$N$ saddle-point analysis yields a planar expression for $\,\mathcal{I}_C$ that matches the Polya-counted planar charged index and reduces to the known single-letter planar result in the simple adjoint-scalar limit. The work demonstrates that the charged index shares the large-$N$ independence property of the ordinary index and opens avenues for generalizations to other gauge groups and connections to $1/16$ BPS partitions in AdS/CFT. Overall, the paper provides a coherent framework to refine protected-state counts in holographic theories by incorporating discrete symmetries through charged characters and group-theoretic matrix integrals. $

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 -> infinity. Like the ordinary superconformal index, for N=4 SYM the charged index is independent of N in the large-N limit.