nLab Dirac operator on smooth loop space

Idea

The traditional definition of a Dirac operator is formulated for operators acting on sections of spinor bundles over Riemannian manifolds, not however directly for bundles over infinite dimensional manifolds.

When the conceptual importance of the Dirac-Ramond operator in the superstring worldsheet 2d SCFT was realized (an operator in a (sheaf of) super vertex operator algebra) via the relation in the large volume limit of its index (suitably regarded) to the Witten genus, then it seemed suggestive that it should be possible to regard the Dirac-Ramond operator as an actual Dirac operator on the infinite-dimensional smooth loop space of the underlying manifold, and regard its $S^1$-equivariant index of a Dirac operator in some precise sense.

A Dirac operator on a smooth loop space would also serve to make precise the sense in which superstring quantum dynamics is supersymmetric quantum mechanics on the smooth loop space, an observation that initially motivated the work by Witten on supersymmetric quantum mechanics.

The definition of Dirac operators on smooth loop spaces is technically tricky, but constructions do exist. It remains however unclear how these constructions relate to the Dirac-Ramond operator and a rigorous derivation of the Witten genus as an (equivariant) index of a Dirac operator along these lines seems to remain open.

Related concepts

• stringor bundle

References

The motivation for constructing Dirac operators on smooth loop spaces is mainly due to the observations about the universal elliptic genus (the Witten genus) in

• Edward Witten, The Index Of The Dirac Operator In Loop Space Proc. of Conf. on Elliptic Curves and Modular Forms in Algebraic Topology Princeton (1986) (spire)

On formal loop space

Since either due to arguing that $S^1$-equivariance localizes the index of a loop space Dirac operator on constant loops, or else arguing that it sees only the “large volume limit”, it is only the formal loop space that actually enters the computation of the Witten genus. Therefore large parts of this literature focus on some version of this formal loop space while often still speaking of just “loop space”. This includes the early articles

• C.H. Taubes, $S^1$-actions and elliptic genera, Communications in mathematical physics 122 (1989), no. 3, pages 455-526, doi:10.1007/BF01238437.

• Jean-Luc Brylinski, Representations of loop groups, Dirac operators on loop space, and modular forms, Topology, 29(4):461–480, (1990) doi:10.1016/0040-9383(90)90016-D

• J. D. S. Jones and R. Léandre. A stochastic approach to the Dirac operator over the free loop space, Tr.Mat. Inst. Steklova, 217(Prostran. Petel i Gruppy Diffeomorf.):258–287, 1997

• Gregory Landweber, Dirac operators on loop space PhD thesis (Harvard 1999) (pdf)

Via the relation of formal loop space to chiral differential operators similar comments apply modern “geometric” constructions of the Witten genus in terms of sheaves of vertex operator algebras of chiral differential operators and similar.

On genuine smooth loop space

One construction (and hence possibly the only available) that produces an operator genuinely on the full smooth loop space over a manifold with string structure and with symbol as it should be for a Dirac operator is given in

• Andrew Stacey, The co-Riemannian Structure of Smooth Loop Spaces (arXiv:0809.3108) and How to Construct a Dirac Operator in Infinite Dimensions (arXiv:0809.3104)