# Calabi-Yau algebras and superpotentials

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## Abstract

We prove that complete \(d\)-Calabi-Yau algebras in the sense of Ginzburg are derived from superpotentials.

## Keywords

Non-commutative geometry Superpotential Calabi-Yau algebra Ginzburg algebra## Mathematics Subject Classification

16E55 16E45## Notes

### Acknowledgments

The author wishes to thank Bernhard Keller for generously sharing his insights on Calabi-Yau algebras and in particular for explaining his strengthening of the Calabi-Yau property during a 2006 Paris visit. In addition, he thanks Bernhard Keller for technical help with the bar cobar formalism. This paper was furthermore strongly influenced by ideas of Ginzburg [13], Kontsevich and Soibelman [21] and Lazaroiu [23]. The author thanks Maxim Kontsevich for pointing out to him that “exact” Calabi-Yau is a better terminology than “strongly” Calabi-Yau which was used in the first version of this article. Finally, the author thanks the referee for his very thorough reading of the manuscript.

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