Voros symbols as cluster coordinates

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 $PG{L}_2(\mathbb{C})$-local systems on a marked bordered surface. Using this result, we show that these Borel sums can be meromorphically continued to any point of ${\mathbb{C}}^{\ast }$, and we prove an asymptotic property of the monodromy map introduced in collaboration with Tom Bridgeland.

46 pages. Version 2: Accepted for publication in Journal of Topology. Version 3: Clarifications added to Theorem 1.4 and its proof, superseding the published version