Supersymmetry and the formal loop space

Abstract

For any algebraic super-manifold M we define the super-ind-scheme LM of formal loops and study the transgression map (Radon transform) on differential forms in this context. Applying this to the super-manifold M=SX, the spectrum of the de Rham complex of a manifold X, we obtain, in particular, that the transgression map for X is a quasi-isomorphism between the [2,3)-truncated de Rham complex of X and the additive part of the [1,2)-truncated de Rham complex of LX. The proof uses the super-manifold SSX and the action of the Lie superalgebra sl(1|2) on this manifold. This quasi-isomorphism result provides a crucial step in the classification of sheaves of chiral differential operators in terms of geometry of the formal loop space.