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...