Jacobi algebra presentations for fundamental group algebras

TL;DR

This paper proves a special case of Davison's conjecture, showing that the fundamental group algebra of certain mapping tori can be described as a Jacobi algebra of a quiver with potential.

Contribution

It establishes a superpotential description of fundamental group algebras for mapping tori of Riemann surfaces with finite order automorphisms, confirming a specific case of Davison's conjecture.

Findings

Fundamental group algebra of mapping tori is described by a Jacobi algebra.

Superpotential structure is given by a quiver with potential.

Supports a special case of Davison's conjecture.

Abstract

We prove a special case of a conjecture of Davison which pertains to superpotential descriptions of fundamental group algebras $k[\pi_1(X)]$. We consider the case in which the manifold $X$ is the mapping torus $M_{g, \varphi}$ of a genus $g$ Riemann surface $\Sigma_g$ and a finite order automorphism $\varphi$, and the superpotential structure is given by the Jacobi algebra of a quiver with potential.