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

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.