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 $PGL_2(\mathbb{C})$-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.