On the wall-crossing formula for quadratic differentials

TL;DR

This paper proves an analytic version of the wall-crossing formula for quadratic differentials, characterizes related automorphisms, and applies these results to compute Stokes automorphisms in WKB analysis.

Contribution

It introduces an analytic formulation of the wall-crossing formula and connects it with birational automorphisms via Fock-Goncharov coordinates, with applications to WKB analysis.

Findings

Derived an explicit analytic wall-crossing formula

Characterized birational automorphisms using Fock-Goncharov coordinates

Computed Stokes automorphisms for Voros symbols

Abstract

We prove an analytic version of the Kontsevich-Soibelman wall-crossing formula describing how the number of finite-length trajectories of a quadratic differential jumps as the differential is varied. We characterize certain birational automorphisms of an algebraic torus appearing in this wall-crossing formula using Fock-Goncharov coordinates. As an application, we compute the Stokes automorphisms for the Voros symbols appearing in the exact WKB analysis of Schr\"odinger's equation.