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
Marion Boucrot, Morphisms of pre-Calabi-Yau categories and morphisms of cyclic -categories, arXiv:2304.13661
A. Sharapov, E. Skvortsov, R. Van Dongen, Strong homotopy algebras for chiral higher spin gravity via Stokes theorem, J. High Energ. Phys. 2024, 186 (2024) doi
Chiral higher spin gravity is defined in terms of a strong homotopy algebra of pre-Calabi-Yau type (noncommutative Poisson structure). All structure maps are given by the integrals over the configuration space of concave polygons and the first two maps are related to the (Shoikhet-Tsygan-)Kontsevich Formality. As with the known formality theorems, we prove the A∞-relations via Stokes’ theorem by constructing a closed form and a configuration space whose boundary components lead to the A∞-relations. This gives a new way to formulate higher spin gravities and hints at a construct encompassing the known formality theorems.