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Unitary matrix models, free fermion ensembles, and the giant graviton expansion

Sameer Murthy

TL;DR

The paper develops a unifying framework linking unitary matrix integrals with an infinite set of couplings to free-fermion ensembles, enabling a determinant-based expansion via a Hubbard–Stratonovich transformation. By integrating out fermions and averaging over the auxiliary bosonic variables, the authors derive a convergent $q$-series for the index, with a precise giant-graviton expansion $I^f_N(q)/I^f_\infty(q)=\sum_m G^{(m)}_{f,N}(q)$, where each term is suppressed by $q^{\alpha mN+\alpha m(m+1)/2}$ for a power-series order $\alpha$ of the single-letter index $f(q)$. The main technical tool is the Borodin–Okounkov determinant expansion, recast in a free-fermion language and organized into HS-transformed multi-point kernels; the first nontrivial term ($m=1$) is encoded by a generating function $\widehat K_f(\zeta,q)$ that captures the giant-graviton contributions. The results provide a concrete matrix-model explanation for giant-graviton expansions observed in AdS/CFT contexts and yield explicit formulas for several BPS indices, with numerical checks verifying the predicted order relations and structure.

Abstract

We consider a class of matrix integrals over the unitary group $U(N)$ with an infinite set of couplings characterized by a series $f(q)=\sum_{n \ge 1} a_n q^n$, 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 $q^{αN + β}$ where $α$, $β$ do not depend on $N$. This provides a matrix-model explanation of the giant graviton expansion that has been observed recently in the literature.

Unitary matrix models, free fermion ensembles, and the giant graviton expansion

TL;DR

The paper develops a unifying framework linking unitary matrix integrals with an infinite set of couplings to free-fermion ensembles, enabling a determinant-based expansion via a Hubbard–Stratonovich transformation. By integrating out fermions and averaging over the auxiliary bosonic variables, the authors derive a convergent -series for the index, with a precise giant-graviton expansion , where each term is suppressed by for a power-series order of the single-letter index . The main technical tool is the Borodin–Okounkov determinant expansion, recast in a free-fermion language and organized into HS-transformed multi-point kernels; the first nontrivial term () is encoded by a generating function that captures the giant-graviton contributions. The results provide a concrete matrix-model explanation for giant-graviton expansions observed in AdS/CFT contexts and yield explicit formulas for several BPS indices, with numerical checks verifying the predicted order relations and structure.

Abstract

We consider a class of matrix integrals over the unitary group with an infinite set of couplings characterized by a series , with . Such integrals arise in physics as the partition functions of free four-dimensional gauge theories on 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 -series with successive terms suppressed by where , do not depend on . This provides a matrix-model explanation of the giant graviton expansion that has been observed recently in the literature.
Paper Structure (8 sections, 6 theorems, 126 equations, 2 figures)

This paper contains 8 sections, 6 theorems, 126 equations, 2 figures.

Table of Contents

  1. Introduction, philosophy, and the main statement
  2. The partition sum as a sum over partitions
  3. Hubbard-Stratonovich transformation
  4. The determinant expansion
  5. The giant graviton expansion
  6. Examples
  7. The matrix integral as a sum over partitions
  8. Review of bosonization and fermionic correlation functions

Key Result

Theorem 2.1

(Graviton stability) For any single-particle index $f(q)$ as in defis of order $\alpha$, the $q$-series coefficients in indtrace of the integral defined by Uact, defis obeys where the infinite-$N$ index is given by Iinfty, Iinftycoeffs.

Figures (2)

  • Figure 1: Each partition is associated with a state in the free-fermion Hilbert space. Here the partition $(5,3,1,1,1)$ (rotated by $45^\circ$) is associated with $\psi_{-\frac{3}{2}} \, \psi_{-\frac{9}{2}} \, \overline \psi_{-\frac{1}{2}} \, \overline \psi_{-\frac{9}{2}} \,\mid \! 0 \, \rangle$
  • Figure 2: There is no hole in the bottom of the sea. For finite $N$, we can excite any number of holes at depths less than $N$ in the Fermi sea and can excite the same number of electrons at arbitrary heights above the sea. Some allowed configurations are shown for $N=3$.

Theorems & Definitions (6)

  • Theorem 2.1
  • Theorem 4.1
  • Proposition 5.1
  • Proposition 5.2
  • Theorem 5.1
  • Theorem 5.2