We construct a certain algebro-geometric version $ℒ\left(X\right)$ of the free loop space for a complex algebraic variety X. This is an ind-scheme containing the scheme ${ℒ}^0\left(X\right)$ 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 $ℒ\left(X\right)$ supported in ${ℒ}^0\left(X\right)$. We also show that $ℒ\left(X\right)$ possesses a factorization structure: a certain non-linear version of a vertex algebra structure. This explains the heuristic...

A conformal block formulation for the Zhu recursion procedure in conformal field theory which allows to find $n$-point functions in terms of the lower correlations functions is introduced. Then the Zhu reduction operators acting on a tensor product of VOA modules are defined. By means of these operators we show that the Zhu reduction procedure generates explicit forms of Ward identities for conformal blocks of vertex operator algebras. Explicit examples of Ward identities for the Heisenberg and free...