Pure and Applied Mathematics Quarterly

Volume 19 (2023)

Number 1

Special Issue in honor of Don Zagier

Guest editors: Benedict H. Gross, Ken Ono, and Fernando Rodriguez Villegas

Unitary matrix models, free fermions, and the giant graviton expansion

Pages: 299 – 340

DOI: https://dx.doi.org/10.4310/PAMQ.2023.v19.n1.a12


Sameer Murthy (Department of Mathematics, King’s College London, United Kingdom)


We consider a class of matrix integrals over the unitary group $U(N)$ with an infinite set of couplings characterized by a series $f(q) = \sum_{n \geq 1} a_n \, q^n$, with $a_n \in \mathbb{Z}$. Such integrals arise in physics as the partition functions of free four-dimensional gauge theories on $S^3$ and, in particular, as the superconformal index of super Yang–Mills theory. We show that any such model can be expressed in terms of a system of free fermions in an ensemble parameterized by the infinite set of couplings. Integrating out the fermions in a given quantum state leads to a convergent expansion as a series of determinants, as shown by Borodin–Okounkov many years ago. By further averaging over the ensemble, we obtain a formula for the matrix integral as a $q$-series with successive terms suppressed by $q^{\alpha N + \beta}$ where $\alpha,\beta$ do not depend on $N$. This provides a matrix-model explanation of the giant graviton expansion that has been observed recently in the literature.

The author is supported by the ERC Consolidator Grant N. 681908, “Quantum black holes: A microscopic window into the microstructure of gravity”, and by the STFC grant ST/P000258/1.

Received 6 February 2022

Received revised 13 September 2022

Accepted 19 September 2022

Published 3 April 2023