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Exact WKB analysis and cluster algebras

Kohei Iwaki, Tomoki Nakanishi

TL;DR

The work builds a precise bridge between exact WKB analysis and cluster algebras by showing that mutations of Stokes graphs, triggered by deformations of the Schrödinger potential, correspond to seed mutations in a surface-realized cluster algebra. Voros symbols, capturing monodromy data from WKB analysis, mutate as cluster variables (both $x$- and $\hat{y}$-coordinates) under these mutations, with explicit jump formulas (DDP-type) accounting for Stokes phenomena. The authors develop a comprehensive framework including signed flips/pops, the surface realization via ideal and tagged triangulations, and the mutation of simple paths/cycles, culminating in Stokes-automorphism identities tied to cluster algebra periodicities. This synthesis provides a rigorous, geometry-powered approach to wall-crossing phenomena and DT-type identities, and offers extensive introductory material to both exact WKB analysis and surface cluster algebras. The framework yields concrete, computable identities for Stokes automorphisms associated with cluster-periodic data, enriching both the analytic and algebraic sides of the theory.

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

TL;DR

The work builds a precise bridge between exact WKB analysis and cluster algebras by showing that mutations of Stokes graphs, triggered by deformations of the Schrödinger potential, correspond to seed mutations in a surface-realized cluster algebra. Voros symbols, capturing monodromy data from WKB analysis, mutate as cluster variables (both - and -coordinates) under these mutations, with explicit jump formulas (DDP-type) accounting for Stokes phenomena. The authors develop a comprehensive framework including signed flips/pops, the surface realization via ideal and tagged triangulations, and the mutation of simple paths/cycles, culminating in Stokes-automorphism identities tied to cluster algebra periodicities. This synthesis provides a rigorous, geometry-powered approach to wall-crossing phenomena and DT-type identities, and offers extensive introductory material to both exact WKB analysis and surface cluster algebras. The framework yields concrete, computable identities for Stokes automorphisms associated with cluster-periodic data, enriching both the analytic and algebraic sides of the theory.

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.

Paper Structure

This paper contains 50 sections, 60 theorems, 216 equations, 60 figures, 1 table.

Table of Contents

  1. Introduction
  2. Exact WKB analysis
  3. Schrödinger equations and associated quadratic differentials
  4. Turning points and singular points
  5. Riccati equation
  6. WKB solutions
  7. Borel resummation method and Stokes phenomenon
  8. Trajectories, Stokes curves, and Stokes graphs
  9. Orientation of trajectories
  10. Borel summability of WKB solutions
  11. Connection formula for WKB solutions
  12. Voros symbols and Stokes automorphisms
  13. Homology groups and Voros symbols
  14. Saddle class associated with saddle trajectory
  15. Saddle reduction
  16. ...and 35 more sections

Key Result

Proposition 2.7

(a).The equality holds, and the even part is given by the logarithmic derivative of the odd part: (b). The (formal series valued) 1-form $S_{\rm odd}(z,\eta)dz$ is invariant under coordinate transformations. That is, the odd part $\tilde{S}_{\rm odd}(\tilde{z},\eta)$ of a formal solution of the Riccati equation associated with eq:Sch-w is given by if we choose the square root in eq:S-plus-minus

Figures (60)

  • Figure 1: Outlines of previous and this works
  • Figure 2: Normalization at a simple turning point.
  • Figure 3: Examples of Stokes graphs. The rational functions represent the function $Q_0(z)$.
  • Figure 4: Foliation around a simple zero.
  • Figure 5: Patterns of foliation around a double pole.
  • ...and 55 more figures

Theorems & Definitions (171)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.4
  • Example 2.6
  • Proposition 2.7
  • proof
  • Proposition 2.8
  • proof
  • Definition 2.9
  • Definition 2.10
  • ...and 161 more