Monodromies of Second Order $q$-difference Equations from the WKB Approximation

TL;DR

This paper develops a framework using WKB analysis to compute and parameterize monodromies of second order $q$-difference equations, with applications to quantum field theory and integrable systems.

Contribution

It introduces a general method to analyze monodromies of $q$-difference equations via WKB, including explicit computations and connections to physical theories.

Findings

Monodromies can be expanded in Voros symbols with integer coefficients.

Explicit examples include the $q$-Mathieu equation and its degenerations.

Monodromy traces relate to $q$-Painlevé Hamiltonians.

Abstract

This paper studies the space of monodromy data of second order $q$-difference equations through the framework of WKB analysis. We compute the connection matrices associated to the Stokes phenomenon of WKB wavefunctions and develop a general framework to parameterize monodromies of $q$-difference equations. Computations of monodromies are illustrated with explicit examples, including a $q$-Mathieu equation and its degenerations. In all examples we show that the monodromy around the origin of $\mathbb{C}^*$ admits an expansion in terms of Voros symbols, or exponentiated quantum periods, with integer coefficients. Physically these monodromies correspond to expectation values of Wilson line operators in five dimensional quantum field theories with minimal supersymmetry. In the case of the $q$-Mathieu equation, we show that the trace of the monodromy can be identified with the Hamiltonian of a corresponding $q$-Painlev\'e equation.