M-Theory of Matrix Models

TL;DR

This work advances an M-theory perspective for eigenvalue matrix models, proposing a global partition function that unifies Hermitian, Kontsevich, and complex matrix models through dualities and multi-level expansions in $g$ and coupling constants. It introduces intertwiners that realize instanton/meron–type decompositions, yielding explicit but partially constructed factorization formulas such as $Z_G(t)=\hat{U}_{G|K}(t|\tau_\pm)\{Z_K(\tau_+)\otimes Z_K(\tau_-)\}$ and their generalizations to $Z_{W,f}$, while showing how bar-coordinate reformulations render genus contributions polynomial. The group-theoretic framing via Virasoro constraints and loop equations provides a unifying algebraic structure for these dualities, with further support from explicit non-linear realizations (e.g., $SO(2)$ and $T$-duality) and a spectral-surface approach that recasts dualities in terms of a global loop operator on a spectral curve, a KN algebra, and Seiberg–Witten–type kernels. Collectively, the paper lays out a coherent framework for relating multiple matrix-model branches within a single M-theory, with practical implications for Kontsevich/DV-type correspondences and the broader landscape of non-perturbative matrix-model partition functions.

Abstract

Small M-theories unify various models of a given family in the same way as the M-theory unifies a variety of superstring models. We consider this idea in application to the family of eigenvalue matrix models: their M-theory unifies various branches of Hermitean matrix model (including Dijkgraaf-Vafa partition functions) with Kontsevich tau-function. Moreover, the corresponding duality relations look like direct analogues of instanton and meron decompositions, familiar from Yang-Mills theory.