nLab pre-Calabi-Yau algebra

Related notions in nnLab: Calabi-Yau algebra, Calabi-Yau category, Calabi-Yau object

  • Maxim Kontsevich, Alex Takeda, Yiannis Vlassopoulos, Pre-Calabi-Yau algebras and topological quantum field theories, arXiv:2112.14667
  • Maxim Kontsevich, Yannis Vlassopoulos, Natalia Iyudu, Pre-Calabi-Yau algebras and double Poisson brackets, arXiv:1906.07134
  • Maxim Kontsevich, Yannis Vlassopoulos, Natalia Iyudu, Pre-Calabi-Yau algebras as noncommutative Poisson structures, J. Algebra 567 (2021) 63–90 doi
  • Natalia Iyudu, Maxim Kontsevich, Pre-Calabi-Yau algebras and noncommutative calculus on higher cyclic Hochschild cohomology, arXiv:2011.11888; Pre-Calabi-Yau algebras and ξ∂-calculus on higher cyclic Hochschild cohomology, preprint IHES M-19-14 (2019) pdf
  • Maxim Kontsevich, Alex Takeda, Yiannis Vlassopoulos, Smooth Calabi-Yau structures and the noncommutative Legendre transform, arXiv:2301.01567

We elucidate the relation between smooth Calabi-Yau structures and pre-Calabi-Yau structures. We show that, from a smooth Calabi-Yau structure on an A∞-category A, one can produce a pre-Calabi-Yau structure on A; as defined in our previous work, this is a shifted noncommutative version of an integrable polyvector field. We explain how this relation is an analogue of the Legendre transform, and how it defines a one-to-one mapping, in a certain homological sense. For concreteness, we apply this formalism to chains on based loop spaces of (possibly non-simply connected) Poincaré duality spaces, and fully calculate the case of the circle.

category: algebra

Last revised on April 6, 2023 at 14:22:03. See the history of this page for a list of all contributions to it.