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A stochastic approach to the Dirac operator over the free loop space. (English. Russian original) Zbl 0915.58097

Sergeev, A. G. (ed.) et al., Loop spaces and groups of diffeomorphisms. Collected papers. Moscow: MAIK Nauka/Interperiodica Publishing, Proc. Steklov Inst. Math. 217, 253-282 (1997); translation from Tr. Mat. Inst. Steklova 217, 258-287 (1997).
When constructing the signature operator over the free loop space of a manifold \(M\), there is formally an obstruction which is the existence of an orientation. A simpler operator is given by the Dirac operator, and the obstruction is the existence of a spin structure over \(M\) and the vanishing of the first Pontryagin class of \(M\). C. H. Taubes [Commun. Math. Phys. 122, No. 3, 455-526 (1989; Zbl 0683.58043)] gave a first approximation of such operators for infinitesimally small loops: he considered the normal bundle of the constant loop, that is, the family of loop spaces based at 0 in \(T_{x}(M)\), and the measure \(dx\otimes Q_{1,x}\), where \(Q_{1,x}\) is the Gaussian measure of covariance \(\sqrt{-\Delta}\) in the based loop space.
In the present paper, the authors repeat the construction but they take the Brownian bridge measure starting from 0 over \(T_{x}(M)\). The interest is that it can be extended over the free loop space and gives the BHK measure, by using the stochastic calculus and the theory of stochastic differential equations. In the first part, the authors treat the regularized de Rham operator over the loop space and the Lefschetz theorem. The second part involves the regularized Dirac operator over a neighbourhood of the constant loop and its equivariant index (Witten genus).
The paper contains an extensive bibliography.
For the entire collection see [Zbl 0907.00018].

MSC:

58J20 Index theory and related fixed-point theorems on manifolds
58J65 Diffusion processes and stochastic analysis on manifolds
58B20 Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds
58A10 Differential forms in global analysis

Citations:

Zbl 0683.58043
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