Exact Solutions to Matrix Models and String Theories: The Local Construction

TL;DR

This paper presents a unified, exact nonperturbative framework for solving hermitian matrix models and their string duals using spectral geometry and Fourier transforms, validated through diverse tests across regimes.

Contribution

It introduces a general closed-form method to derive all nonperturbative solutions from spectral geometry, incorporating anti-eigenvalues and negative-tension D-branes.

Findings

Exact solutions are valid across all parameter regimes.

Anti-eigenvalues are essential for precise numerical matches.

The approach unifies solutions for matrix models and string theories.

Abstract

Exact nonperturbative solutions to hermitian one-matrix models, their topological string duals, as well as their double-scaling limits to multicritical and minimal string theories, may be obtained via the use of resurgent transseries. These solutions are generically resonant, entailing both eigenvalues and anti-eigenvalues, or, equivalently, both D-branes and negative-tension D-branes -- but are otherwise intricate to write down, having been previously addressed on a case-by-case approach. This work shows how there is a general and rather compact way to write down all these exact and fully nonperturbative transseries solutions in closed-form, immediately starting from the spectral geometry of the matrix model or string theory at hand, in the form of a discrete Fourier or Zak transform for their partition functions. This structure is inherently associated to the existence of anti-eigenvalues or negative-tension D-branes. The validity of these solutions is testable across all values of the parameters -- from weak to strong 't Hooft coupling, from small to large N; equivalently, from semi-classical to deeply quantum regimes -- and many such nonperturbative tests are performed against diverse examples ranging from matrix models to non-critical strings, fully validating our analytical and exact expressions. In particular, anti-eigenvalues or negative-tension D-branes are absolutely required to find sharp numerical matches. In order to study these solutions globally across their phase diagrams, however, complete non-linear Stokes data is still needed -- which will be addressed in a complementary follow-up paper.