Exact WKB analysis and cluster algebras

Abstract

Abstract. We develop the mutation theory in the exact WKB analysis using the framework of cluster algebras. Under a continuous deformation of the potential of the Schrödinger equation on a compact Riemann surface, the Stokes graph may change the topology. We call this phenomenon the mutation of Stokes graphs. Along the mutation of Stokes graphs, the Voros symbols, which are monodromy data of the equation, also mutate due to the Stokes phenomenon. We show that the Voros symbols mutate as variables of a cluster algebra with surface realization. As an application, we obtain the identities of Stokes automorphisms associated with periods of cluster algebras. The paper also includes an extensive introduction of the exact WKB analysis and the surface realization of cluster algebras for nonexperts.

Exact WKB Analysis and Cluster Algebras II: Simple Poles, Orbifold Points, and Generalized Cluster Algebras

“…This is a continuation of the paper [IN14], where the mutation theory in exact WKB analysis using the framework of cluster algebras was initiated. The exact WKB analysis is a method to study the WKB solutions of the Schrödinger equation using the Borel resummation.…”

“…Let us briefly summarize the main result of [IN14] highlighting some key words. We study the WKB solutions of the Schrödinger equation on a compact Riemann surface.…”

“…Along the mutation of Stokes graphs, the Voros symbols also mutate (or jump) due to the Stokes phenomenon. The main result of [IN14] is that the Voros symbols mutate as the variables of the corresponding cluster algebra. To be more precise, they mutate by signed mutations and signed pops of the extended seeds, which are certain extensions of the ordinary mutation of cluster algebras.…”

Alien derivatives of the WKB solutions of the Gauss hypergeometric differential equation with a large parameter

“…In [1] and [11], they will give a concrete form of the alien derivative of the WKB solutions of the general linear second-order differential equation with a large parameter. In this paper, we consider the alien derivatives of the WKB solutions for the Gauss hypergeometrc differential equations of the loop-type.…”

“…The WKB solutions ψ ±,1 and ψ (1) ± are Borel summable in R IV (cf. [1,11,13]). We denote the Borel transforms of ψ +,1 and ψ (1) + by ψ IV +,B and ψ is free from singularities on (3.1) (cf.…”