Localization and wall-crossing of giant graviton expansions in AdS$_5$

TL;DR

This paper derives the $q$-expansions of 1/2-BPS indices in $ ext{AdS}_5$/CFT$_4$ by quantizing giant graviton moduli spaces, revealing wall-crossing phenomena and the effects of bulk topological branes.

Contribution

It extends previous work by explicitly quantizing the moduli space of giant gravitons and analyzing wall-crossing and topological effects in the dual string theory.

Findings

Derived $q$-expansions from bulk duals using localization.

Identified wall-crossing as an analytic continuation phenomenon.

Connected bulk topological branes to additional expansion terms.

Abstract

The $\frac12$-BPS indices of $\mathcal{N}=4$ Super Yang-Mills theory with unitary, orthogonal, and symplectic groups all admit $q$-expansions suggesting an interpretation in terms of D-branes in the dual bulk AdS$_5$ string theories. We present a derivation of these expansions in the corresponding bulk duals by quantizing the moduli space of $\frac12$-BPS giant gravitons using supersymmetric localization, extending and clarifying our study in arxiv:2312.14921. We perform a detailed analysis of the one-loop fluctuations around the maximal giants (the fixed points), and show how the Hamiltonian analysis is recovered from the functional integral for the equivariant index. We show that the analytic continuation for these giant graviton expansions observed in the literature maps precisely to a wall-crossing phenomenon for the index. In the case of orthogonal and symplectic gauge groups, the $\mathbb{Z}_2$ quotient in the bulk leads to a corresponding projection in the $q$-expansion. Additional terms in the expansion related to the Pfaffian operator arise from topologically stable branes in the bulk dual on AdS$_5 \times \mathbb{RP}^5$.