Higher Chiral Algebras in a Polysimplicial Model

TL;DR

This paper constructs a higher-dimensional analogue of vertex algebras called homotopy polysimplicial chiral algebras on a1^n, using dg commutative algebras and operad theory, extending the concept to higher dimensions.

Contribution

It introduces a novel algebraic model for higher-dimensional chiral algebras via dg operads and establishes a quasi-isomorphism with the Lie-infinity operad.

Findings

Established a dg operad of chiral operations on a1^n

Proved a quasi-isomorphism with the Lie-infinity operad

Constructed a homotopy polysimplicial chiral algebra in higher dimensions

Abstract

Vertex algebras are equivalent to translation-equivariant chiral algebras on $\mathbb{A}^1$, in the sense of Beilinson and Drinfeld. In this paper 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.