Superpotential algebras and manifolds

Abstract

We study a special class of Calabi–Yau algebras (in the sense of Ginzburg): those arising as the fundamental group algebras of acyclic manifolds. Motivated partly by the usefulness of ‘superpotential descriptions’ in motivic Donaldson–Thomas theory, we investigate the question of whether these algebras admit superpotential presentations. We establish that the fundamental group algebras of a wide class of acyclic manifolds, including all hyperbolic manifolds, do not admit such descriptions, disproving a conjecture of Ginzburg regarding them. We also describe a class of manifolds that do admit such descriptions, and discuss a little their motivic Donaldson–Thomas theory. Finally, some links with topological field theory are described.