Exact WKB analysis and cluster algebras II: Simple poles, orbifold points, and generalized cluster algebras

TL;DR

This paper extends the mutation theory in exact WKB analysis to include simple poles and orbifold points, demonstrating how Stokes graph mutations relate to generalized cluster algebra variables on Riemann surfaces.

Contribution

It introduces a framework connecting orbifold triangulations with mutation of Stokes graphs and Voros symbols in the context of simple poles in WKB analysis.

Findings

Orbifold triangulations describe mutation of Stokes graphs with simple poles.

Voros symbols mutate as variables of generalized cluster algebras.

Framework applies to Schrödinger equations on compact Riemann surfaces.

Abstract

This is a continuation of developing mutation theory in exact WKB analysis using the framework of cluster algebras. Here we study the Schrodinger equation on a compact Riemann surface with turning points of simple-pole type. We show that the orbifold triangulations by Felikson, Shapiro, and Tumarkin provide a natural framework of describing the mutation of Stokes graphs, where simple poles correspond to orbifold points. We then show that under the mutation of Stokes graphs around simple poles the Voros symbols mutate as the variables of generalized cluster algebras introduced by Chekhov and Shapiro.