Superpotential algebras and manifolds

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.

31 pages, 2 figures, final version. Thanks to M. Kontsevich, V. Ginzburg, M, Van den Bergh and B. Keller for helpful comments and corrections. I've added some examples e.g. Klein bottle

References in corpus (4)

• Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
• On the Classification of Topological Field Theories
• Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants
• Double derivations and Cyclic homology

Cited by in corpus (11)

• On Generalized Cluster Categories
• Relative Calabi-Yau completions
• Deformations of algebras in noncommutative geometry
• Calabi-Yau algebras and superpotentials
• Purity and 2-Calabi-Yau categories
• The derived non-commutative Poisson bracket on Koszul Calabi-Yau algebras
• Batalin-Vilkovisky algebras and the noncommutative Poincare duality of Koszul Calabi-Yau algebras
• Exact Calabi-Yau categories and odd-dimensional Lagrangian spheres
• The Nori-Hilbert scheme is not smooth for 2-Calabi Yau algebras
• Nonexistence of exact Lagrangian tori in affine conic bundles over ${\mathbb{C}}^n$
• Persistence of unknottedness of clean Lagrangian intersections