The Giant Graviton Expansion

TL;DR

This work proposes and tests a universal giant-graviton expansion for the superconformal index of 4d ${ m U}(N)$ ${ m N}=4$ SYM and a broad class of index-like quantities, expressing $Z_N$ as $Z_00$ plus a convergent series of corrections controlled by $x^{kN}$. The corrections are given by auxiliary indices ${ ilde Z}_k$ and ${ hat Z}_k$, which are interpreted holographically as fluctuations of $k$ maximal giant gravitons, encoded by worldvolume theories of the giants and connected to a dual auxiliary counting problem. The authors develop a fermionization framework to count finite determinant modifications, extend it to determinants of adjoint, bifundamental, and fundamental letters, and demonstrate the expansion across several 4d index variants (including the single-matrix, ${1 extstyle/4}$-BPS, Schur) as well as a 3d M2-brane index with monopole sectors. While the expansion works broadly and exposes a deep combinatorial-holographic structure, it has limits, as shown by counterexamples, and the full formulation remains subtle in index-like settings lacking a single bosonic letter. Overall, the giant-graviton expansion provides a new non-perturbative, combinatorial handle on SCFT indices and their holographic interpretations, with potential localization-based explanations and extensions to defects and higher dimensions.

Abstract

We propose and test a novel conjectural relation satisfied by the superconformal index of maximally supersymmetric $U(N)$ gauge theory in four dimensions. Analogous relations appear to be also valid for the superconformal indices of a large collection of other gauge theories, as well as for a broad class of index-like generating functions. The relation expresses the finite $N$ index as a systematic series of corrections to a large $N$ answer. Individual corrections have an holographic interpretation as the analytic continuation of contributions from "giant graviton" branes fixed by a specific symmetry generator.