nLab pre-Calabi-Yau algebra

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

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 infA_\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.