Vertex algebras and the formal loop space

Abstract

We construct a certain algebro-geometric version $\mathcal{L}(X)$ of the free loop space for a complex algebraic variety X. This is an ind-scheme containing the scheme $\mathcal{L}^{0}(X)$ of formal arcs in X as studied by Kontsevich and Denef-Loeser. We describe the chiral de Rham complex of Malikov, Schechtman and Vaintrob in terms of the space of formal distributions on $\mathcal{L}(X)$ supported in $\mathcal{L}^{0}(X)$ . We also show that $\mathcal{L}(X)$ possesses a factorization structure: a certain non-linear version of a vertex algebra structure. This explains the heuristic principle that “all” linear constructions applied to the free loop space produce vertex algebras.