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.
Alex Takeda, The noncommutative Legendre transform and Calabi-Yau structures, Purdue Topology Seminar youtube