Superpotential algebras and manifolds

TL;DR

This paper investigates whether fundamental group algebras of acyclic manifolds can be described by superpotentials, disproving Ginzburg's conjecture for many cases and exploring implications for motivic Donaldson-Thomas theory and topological field theory.

Contribution

It proves that many fundamental group algebras of acyclic manifolds do not admit superpotential descriptions, contrary to previous conjectures, and identifies classes that do.

Findings

Fundamental group algebras of hyperbolic manifolds lack superpotential descriptions.

Certain manifolds' algebras admit superpotential presentations.

Links between these algebras, motivic Donaldson-Thomas theory, and topological field theory are discussed.

Abstract

In this paper 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 Ginzburg's conjecture 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.