WKB Method and Quantum Periods beyond Genus One

TL;DR

This paper extends topological string methods to efficiently perform WKB approximations for quantum systems with higher order potentials, involving advanced techniques for evaluating quantum periods on complex Riemann surfaces beyond genus one.

Contribution

It introduces new methods for calculating quantum periods on higher genus Riemann surfaces using Picard-Fuchs operators and holomorphic anomaly equations, expanding the applicability of topological string techniques.

Findings

Quantum periods for higher genus surfaces are computed using Picard-Fuchs operators.

Holomorphic anomaly equations are applied within the ring of quasi modular forms.

Validation is performed on a symmetric sextic potential.

Abstract

We extend topological string methods in order to perform WKB approximations for quantum mechanical problems with higher order potentials efficiently. This requires techniques for the evaluation of the relevant quantum periods for Riemann surfaces beyond genus one. The basis of these quantum periods is fixed using the leading behaviour of the classical periods. The full expansion of the quantum periods is obtained using a system of Picard-Fuchs like operators for a sequence of integrals of meromorphic forms of the second kind. Discrete automorphisms of simple higher order potentials allow to view the corresponding higher genus curves as covering of a genus one curve. In this case the quantum periods can be alternatively obtained using the holomorphic anomaly solved in the holomorphic limit within the ring of quasi modular forms of a congruent subgroup of SL$(2,\mathbb{Z})$ as we check for a symmetric sextic potential.