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

Authors: Fabrizio Del Monte, Pietro Longhi Preprint number: UUITP-14/24 Abstract: This paper studies the space of monodromy data of second order $q$-difference equations through the  framework of WKB analysis. We compute connection matrices for Stokes phenomena of WKB wavefunctions and develop a general formalism to parameterize monodromies of the $q$-difference equation. Computations of monodromies are illustrated with explicit examples, including the $q$-Mathieu equation and its degenerations. In all examples we show that the monodromy around $\mathbb{C}^*$ admits an expansion in terms of 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. We confirm these expectations against predictions from work on cluster integrable systems.