Multi-matrix correlators and localization

TL;DR

This paper extends the Harish-Chandra-Itzykson-Zuber integral to multiple matrices to compute correlators of BPS states in $ ext{N}=4$ super Yang-Mills, enabling analysis of more complex operators at finite N.

Contribution

It introduces a generalization of the HCIZ integral for multiple matrices, applicable to BPS operators, and connects it with the restricted Schur polynomial basis.

Findings

Derived the four-matrix HCIZ integral for U(2)

Proposed a method for general N>2

Facilitated computation of multi-matrix correlators

Abstract

We study generating functions of $\frac{1}{4}$-BPS states in $\mathcal{N}=4$ super Yang-Mills at finite $N$ by attempting to generalize the Harish-Chandra-Itzykson-Zuber integral to multiple commuting matrices. This allows us to compute the overlaps of two or more generating functions; such calculations arise in the computation of two-point correlators in the free-field limit. We discuss the four-matrix HCIZ integral in the $U(2)$ context and lay out a prescription for finding a more general formula for $N>2$. We then discuss its connections with the restricted Schur polynomial operator basis. Our results generalize readily to arbitrary numbers of matrices, opening up the opportunity to study more generic BPS operators.