Related notions in $n$Lab: 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.

- Alex Takeda,
*The noncommutative Legendre transform and Calabi-Yau structures*, Purdue Topology Seminar youtube - Johan Leray, Bruno Vallette,
*Pre-Calabi–Yau algebras and homotopy double Poisson gebras*, arXiv:2203.05062 - Wai-Kit Yeung,
*Pre-Calabi-Yau structures and moduli of representations*, arXiv:1802.05398;*Ribbon dioperads and modular ribbon properads*, arXiv:2202.13269 - David Fernández, Estanislao Herscovich,
*Double quasi-Poisson algebras are pre-Calabi-Yau*, arXiv:2002.10495 - Marion Boucrot,
*Morphisms of pre-Calabi-Yau categories and morphisms of cyclic $A_\inf$-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.

- Alexandre Quesney,
*Balanced infinitesimal bialgebras, double Poisson gebras and pre-Calabi-Yau algebras*, arXiv:2312.14893

category: algebra

Last revised on July 18, 2024 at 19:42:37. See the history of this page for a list of all contributions to it.