Matrix models from operators and topological strings, 2

TL;DR

This paper derives an explicit matrix model from operators associated with mirror curves of local P1xP1, showing it captures all-genus topological string free energy and generalizes known matrix models.

Contribution

It provides an explicit integral kernel for the trace class operator and establishes a new O(2) matrix model that encodes topological string theory on local P1xP1.

Findings

Explicit integral kernel in terms of Faddeev's quantum dilogarithm.

Exact planar limit of the matrix model.

Evidence that the 1/N expansion reproduces all-genus topological string free energy.

Abstract

The quantization of mirror curves to toric Calabi--Yau threefolds leads to trace class operators, and it has been conjectured that the spectral properties of these operators provide a non-perturbative realization of topological string theory on these backgrounds. In this paper, we find an explicit form for the integral kernel of the trace class operator in the case of local P1xP1, in terms of Faddeev's quantum dilogarithm. The matrix model associated to this integral kernel is an O(2) model, which generalizes the ABJ(M) matrix model. We find its exact planar limit, and we provide detailed evidence that its 1/N expansion captures the all genus topological string free energy on local P1xP1.