Supersymmetry and the formal loop space

TL;DR

This paper develops a framework for super-ind-schemes of formal loops on algebraic super-manifolds, establishing a quasi-isomorphism of de Rham complexes that advances the understanding of chiral differential operators.

Contribution

It introduces the super-ind-scheme of formal loops and proves a key quasi-isomorphism relating de Rham complexes, aiding classification of chiral differential operators.

Findings

The transgression map is a quasi-isomorphism between truncated de Rham complexes.

The super-manifold SSX and sl(1|2) action are crucial in the proof.

Provides a geometric approach to classifying sheaves of chiral differential operators.

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.