Jacobi algebra presentations for fundamental group algebras

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 $M_{g,\phi }$ of a genus $g$ Riemann surface ${\mathrm{\Sigma }}_g$ and a finite order automorphism $\phi$, and the superpotential structure is given by the Jacobi algebra of a quiver with potential.