Calabi-Yau algebras and superpotentials
type: Item
instance of: scholarly article
title: Calabi-Yau algebras and superpotentials
zbMATH Open document ID: 1378.16016
author: Michel Van den Bergh
published in: Selecta Mathematica. New Series
publication date: 2015-06-01T00:00:00Z
full work available at URL: https://arxiv.org/abs/1008.0599
review text: 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.
reviewed by: Yin Tian
Mathematics Subject Classification ID: 16E45
Mathematics Subject Classification ID: 16E65
Mathematics Subject Classification ID: 16W80
zbMATH DE Number: 6441727
zbMATH Keywords: non-commutative geometry
zbMATH Keywords: superpotential
zbMATH Keywords: Calabi-Yau algebra
zbMATH Keywords: Ginzburg algebra
MaRDI profile type: Publication
OpenAlex ID: W2060863348
arXiv ID: 1008.0599
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DOI: 10.1007/S00029-014-0166-6
description: scientific article