Spectral Theory and Mirror Symmetry

TL;DR

This paper explores the deep connection between spectral theory and mirror symmetry in string theory, revealing how spectral properties encode geometric invariants and provide a non-perturbative framework for topological string theory.

Contribution

It introduces a new class of solvable spectral problems linked to mirror curves of toric Calabi-Yau threefolds and connects their spectral data to enumerative geometry and topological strings.

Findings

Fredholm determinants expressed via Gromov-Witten invariants

Spectral spectrum determined by quantum theta function

Matrix integral representations of topological string partition functions

Abstract

Recent developments in string theory have revealed a surprising connection between spectral theory and local mirror symmetry: it has been found that the quantization of mirror curves to toric Calabi-Yau threefolds leads to trace class operators, whose spectral properties are conjecturally encoded in the enumerative geometry of the Calabi-Yau. This leads to a new, infinite family of solvable spectral problems: the Fredholm determinants of these operators can be found explicitly in terms of Gromov-Witten invariants and their refinements; their spectrum is encoded in exact quantization conditions, and turns out to be determined by the vanishing of a quantum theta function. Conversely, the spectral theory of these operators provides a non-perturbative definition of topological string theory on toric Calabi-Yau threefolds. In particular, their integral kernels lead to matrix integral representations of the topological string partition function, which explain some number-theoretic properties of the periods. In this paper we give a pedagogical overview of these developments with a focus on their mathematical implications