Calabi-Yau algebras and superpotentials
In this paper, the author introduces the notion of exact Calabi-Yau algebras and proves that Calabi-Yau algebras of this type are derived from superpotentials. The notion of d-Calabi-Yau algebras is introduced by Ginzburg which can be viewed as a DG (differential graded) enhancement of Jacobi algebras. 3-Calabi-Yau algebras derived from superpotentials are particularly important and have many applications in the theory of cluster algebras and stability conditions. The goal of the paper is to give a description of certain d-Calabi-Yau algebras which are derived from superpotentials. The tool used in the paper is the notion of exact Calabi-Yau algebras, which is a strengthening of Calabi-Yau algebras by including more higher homotopy information, involving Hochschild and cyclic homologies. Then the author restricts to ``complete'' cases, i.e., sufficiently local cases. It is proved that a ``complete'' d-Calabi-Yau algebra is exact. The main result of the paper is to show that a complete exact d-Calabi-Yau algebras under certain conditions is derived from a superpotential.
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