Spectral determinants and quantum theta functions

Understand

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.