M-theoretic matrix models

TL;DR

This work introduces and analyzes M-theoretic matrix models—large-$N$ systems with fixed couplings—through two concrete examples: the ${N_f}$ matrix model and the polymer matrix model. It develops exact planar (and genus-one) solutions using an $O(m)$-model framework and reformulates the problems as one-dimensional Fermi gases, enabling access to non-perturbative corrections via grand potentials $J(z)$ and density matrices. The results reveal planar dominance in the M-theory regime, and expose rich non-perturbative structures such as membrane and worldsheet instantons, whose effects are captured by both perturbative expansions and worldsheet/membrane contributions in the Fermi gas language. These findings illuminate how the 't Hooft expansion misses essential physics in AdS$_4$/CFT$_3$ and related systems, and they outline concrete avenues (e.g., TBA or improved WKB analyses) to push toward a full non-perturbative understanding. The work thus provides a blueprint for exploring M-theoretic extensions beyond ABJM and for connecting localization-based matrix models to M-theory and integrable hierarchies.

Abstract

Some matrix models admit, on top of the usual 't Hooft expansion, an M-theory-like expansion, i.e. an expansion at large N but where the rest of the parameters are fixed, instead of scaling with N. These models, which we call M-theoretic matrix models, appear in the localization of Chern-Simons-matter theories, and also in two-dimensional statistical physics. Generically, their partition function receives non-perturbative corrections which are not captured by the 't Hooft expansion. In this paper, we discuss general aspects of these type of matrix integrals and we analyze in detail two different examples. The first one is the matrix model computing the partition function of N=4 supersymmetric Yang-Mills theory in three dimensions with one adjoint hypermultiplet and N_f fundamentals, which has a conjectured M-theory dual, and which we call the N_f matrix model. The second one, which we call the polymer matrix model, computes form factors of the 2d Ising model and is related to the physics of 2d polymers. In both cases we determine their exact planar limit. In the N_f matrix model, the planar free energy reproduces the expected behavior of the M-theory dual. We also study their M-theory expansion by using Fermi gas techniques, and we find non-perturbative corrections to the 't Hooft expansion.

Figures (3)

• Figure 1: The density of eigenvalue (\ref{['rho']}) for $a=1/2$ in the $x$ plane (left) and in the $z={\rm e}^{x}$ plane (right) .
• Figure 2: Comparison of the exact result for ${\rm d}^2 F_0/ {\rm d} \lambda^2$ given in (\ref{['free2']}), and plotted in a continuous blue line, with the strong and weak coupling behavior. The red dashed line represents the strong coupling behavior (\ref{['pl-strong']}), while the black dashed line represents the Gaussian weak coupling behavior (\ref{['pert-aF']}).
• Figure 3: Left: the sequence (\ref{['cseq']}) for $N_f=100$ with its $3^{\rm th}$ and $4^{\rm th}$ Richardson transform. The straight line is the analytic prediction. Right: the sequence (\ref{['aseq']}) with its $3^{\rm th}$ and $4^{\rm th}$ Richardson extrapolation, again for $N_f=100$. The straight line is the analytic prediction.