Finite $N$ indices and the giant graviton expansion

TL;DR

This paper extends the giant graviton expansion of the superconformal index in $ ext{N}=4$ super-Yang Mills theory to multiple giant gravitons, providing explicit formulas and analyzing differences from supergravity predictions.

Contribution

It derives explicit formulas for the superconformal index involving an arbitrary number of giant gravitons and examines the non-uniqueness of the expansion beyond a single graviton.

Findings

Explicit formulas for multiple giant graviton contributions

Demonstration of differences between matrix integral and supergravity expansions

Evidence of non-uniqueness in the giant graviton expansion

Abstract

The superconformal index of $\mathcal N=4$ super-Yang Mills theory with $U(N)$ gauge group can be written as a matrix integral over the gauge group. Recently, Murthy demonstrated that this integral can be reexpressed as a sum of terms corresponding to a giant graviton expansion of the index, and provided an explicit formula for the case of a single giant graviton. Here we give similar explicit formulae for an arbitrary number, $m\ge1$, of giant gravitons. We provide 1/2 and 1/16 BPS index examples up to the order where three giant gravitons contribute and demonstrate that the expansion of the matrix integral differs from the giant graviton expansion computed in the supergravity dual. This shows that the giant graviton expansion is not necessarily unique once two or more giant gravitons start appearing.