Higher Chiral Algebras in a Polysimplicial Model

          @misc{ pirsa_PIRSA:25110068,
            doi = {10.48660/25110068},
            url = {https://pirsa.org/25110068},
            author = {Gui, Zhengping},
            keywords = {Mathematical physics},
            language = {en},
            title = {Higher Chiral Algebras in a Polysimplicial Model},
            publisher = {Perimeter Institute},
            year = {2025},
            month = {nov},
            note = {PIRSA:25110068 see, \url{https://pirsa.org}}
          }
          

Abstract

Vertex algebras are equivalent to translation-equivariant chiral algebras on $\mathbb{A}^1$, in the sense of Beilinson and Drinfeld. In this talk we give an algebraic construction of a chiral algebra on $\mathbb{A}^n$; this can be seen as an algebraic construction of a higher-dimensional vertex algebra. We introduce a model, in dg commutative algebras, of the derived algebra of functions on the configuration space of $k$ distinct labelled marked points in $\mathbb{A}^n$. Working in this model -- which we call the polysimplicial model-- we obtain a dg operad of chiral operations on a degree-shifted copy of the canonical sheaf. We prove that there is a quasi-isomorphism, to this dg operad, from the Lie-infinity operad. This result makes the shifted canonical sheaf into a first example of a homotopy polysimplicial chiral algebra on $\mathbb{A}^n$, in a sense which generalizes to higher dimensions Malikov and Schechtman's notion of a homotopy chiral algebra. This is joint work with Charles Young and Laura Felder.