nLab
formal loop space

Contents

Idea

The space of infinitesimal loops in a given space XX – hence the formal neighbourhood of the constant loops XLXX \hookrightarrow L X – is often called the formal loop space (as in formal geometry).

Formal loop spaces have application for instance in the theory of elliptic genera, since it was shown that the Witten genus is the S 1S^1-equivariant index of a Dirac-Ramond operator which is like Dirac operator on smooth loop space, only that due to S 1S^1-equivariance (orelse due to a large volume limit) the index is fully determined by the restriction of that operator to formal loops.

Properties

Relation to chiral differential operators

With XX a suitable scheme, its formal loop space L infXL_inf X in the sense of (Kapranov-Vasserot I) has a Tate structure? and hence an associated determinantal gerbe Det L infXDet_{L_{inf} X} with band? 𝒪 L infX *\mathcal{O}^\ast_{L_{inf} X}. According to (Kapranov-Vasserot IV) this gerbe is essentially identified with the gerbe CDO XCDO_X of chiral differential operators on XX.

References

In the context of algebraic geometry formal loop spaces have been introduced and studied in

Tentative aspects of a generalization to differential geometry are discussed in

Revised on March 25, 2014 02:07:59 by Urs Schreiber (82.113.98.43)