Spectral determinants and quantum theta functions

TL;DR

This paper explores the modular properties of spectral determinants related to mirror curves, demonstrating their invariance and expanding around a specific point to connect quantum theta functions with classical theta functions.

Contribution

It provides a detailed analysis of the symplectic and modular properties of spectral determinants and introduces explicit expansions around hbar=2pi, linking topological string theory with spectral theory.

Findings

Spectral determinants are modular invariant around hbar=2pi.

Explicit first terms of the expansion of spectral determinants are derived.

New test of duality between topological strings and spectral theory is proposed.

Abstract

It has been recently conjectured that the spectral determinants of operators associated to mirror curves can be expressed in terms of a generalization of theta functions, called quantum theta functions. In this paper we study the symplectic properties of these spectral determinants by expanding them around the point hbar=2pi, where the quantum theta functions become conventional theta functions. We find that they are modular invariant, order by order, and we give explicit expressions for the very first terms of the expansion. Our derivation requires a detailed understanding of the modular properties of topological string free energies in the Nekrasov-Shatashvili limit. We derive these properties in a diagrammatic form. Finally, we use our results to provide a new test of the duality between topological strings and spectral theory.