Skeins, clusters and wavefunctions

Abstract

In previous work of the second- and third-named authors with Linhui Shen, cluster theory was used to construct wavefunctions for branes in threespace and conjecturally relate them to open Gromov-Witten invariants. This was done by defining a quantum Lagrangian subvariety of a quantum cluster variety, and mutating a simple solution to the defining equations in a distinguished seed. In this paper, we extend the construction to incorporate the skein-theoretic approach to open Gromov-Witten theory of Ekholm-Shende. In particular, we define a skein-theoretic version of cluster theory, including the groupoid of seeds and mutations and a skein-theoretic version of the quantum dilogarithm. We prove a pentagon relation in the skein of the closed torus in this context, and give strong evidence that its analogue holds for arbitrary surfaces. We propose face relations satisfied by the skein-theoretic wavefunction, prove their invariance under mutations, and show their solution is unique. We define a skein version of framings in the story, and use the novel cluster structure to compute wavefunctions in several examples. The skein approach incorporates moduli spaces of sheaves of higher microlocal rank and their quantizations.