On contravariant functors from the category of preschemes over a field into the category of abelian groups

Jaap P. Murre

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