Wilson lines on the moduli space of $G$-local systems on a marked surface

Abstract: For a marked surface $\Sigma$, there are two kinds of extensions of moduli spaces of local systems on $\Sigma$, written as $\mathcal{A}_{\widetilde{G}, \Sigma}$ and $\mathcal{P}_{G, \Sigma}$, where $\widetilde{G}$ is a connected simply-connected complex simple algebraic group and $G=\widetilde{G}/Z(\widetilde{G})$ its adjoint group. These are introduced by Fock--Goncharov and Goncharov--Shen respectively, and it is known that the pair $(\mathcal{A}_{\widetilde{G}, \Sigma}, \mathcal{P}_{G, \Sigma})$ forms a cluster ensemble. In this talk, we formulate a class of $\widetilde{G}$ or $G$-valued morphisms defined on these moduli spaces, which we call Wilson lines. I explain their basic properties and application. In particular, we give an affirmative answer to the $\mathrm{A}=\mathrm{U}$ problem for the cluster algebras arising from the cluster $K_2$-structures on $\mathcal{A}_{\widetilde{G}, \Sigma}$ under some assumptions on $G$ and $\Sigma$. This talk is based on a joint work with Tsukasa Ishibashi and Linhui Shen.

mathematical physicsalgebraic geometrygeometric topologyquantum algebra

Audience: researchers in the discipline

Geometry, Algebra and Physics at KIAS