Matrix models from operators and topological strings

TL;DR

This paper introduces a new class of matrix models derived from operators related to mirror curves of toric Calabi-Yau threefolds, providing a non-perturbative framework for topological string theory and verifying their predictions for specific geometries.

Contribution

The authors construct matrix models from trace class operators linked to mirror curves, establishing a non-perturbative realization of topological strings and connecting spectral properties to enumerative invariants.

Findings

Matrix models reproduce topological string free energies near conifold points.

Weak 't Hooft coupling expansion matches topological string calculations.

Non-perturbative corrections relate to Nekrasov-Shatashvili limit of refined topological strings.

Abstract

We propose a new family of matrix models whose 1/N expansion captures the all-genus topological string on toric Calabi-Yau threefolds. These matrix models are constructed from the trace class operators appearing in the quantization of the corresponding mirror curves. The fact that they provide a non-perturbative realization of the (standard) topological string follows from a recent conjecture connecting the spectral properties of these operators, to the enumerative invariants of the underlying Calabi-Yau threefolds. We study in detail the resulting matrix models for some simple geometries, like local P^2 and local F_2, and we verify that their weak 't Hooft coupling expansion reproduces the topological string free energies near the conifold singularity. These matrix models are formally similar to those appearing in the Fermi-gas formulation of Chern-Simons-matter theories, and their 1/N expansion receives non-perturbative corrections determined by the Nekrasov-Shatashvili limit of the refined topological string.