Voros symbols as cluster coordinates

TL;DR

This paper reveals that Voros symbols in exact WKB analysis correspond to Fock-Goncharov coordinates of certain local systems, enabling their meromorphic continuation and establishing an asymptotic property of the monodromy map.

Contribution

It establishes a geometric interpretation of Voros symbols as cluster coordinates, linking WKB analysis with the theory of framed local systems.

Findings

Voros symbols are identified with Fock-Goncharov coordinates.

Borel sums of Voros symbols can be meromorphically continued.

An asymptotic property of the monodromy map is proven.

Abstract

We show that the Borel sums of the Voros symbols considered in the theory of exact WKB analysis arise naturally as Fock-Goncharov coordinates of framed $PGL_2(\mathbb{C})$-local systems on a marked bordered surface. Using this result, we show that these Borel sums can be meromorphically continued to any point of $\mathbb{C}^*$, and we prove an asymptotic property of the monodromy map introduced in collaboration with Tom Bridgeland.