Stochastic equivariant cohomologies and cyclic cohomology. (English) Zbl 1093.58010
Consider the free loop space \(L(M)\) over a Riemann manifold \(M\), and its canonical Killing vector field \(X\), which generates the circle action. The equivariant exterior derivative \((d+i_X)\) defines a complex on the set of forms which are invariant under rotation, defining thus the so-called “equivariant cohomology” of \(L(M)\), which was used by Bismut for its proof of Atiyah’s Index Theorem. J. D. S. Jones and S. B. Petrack [Trans. Am. Math. Soc. 322, No. 1, 35–49 (1990; Zbl 0723.55003)] proved that, for the smooth loop space, this equivariant cohomology equals the cohomology of \(M\).
The author establishes a generalization to the non-smooth case, of the result by Jones and Petrack (loc. cit.). For that, he uses the natural rotation-invariant measure on \(L(M)\) deduced from the pinned Wiener measure on \(M\), together with a so-called “diffeology” (in the Chen-Souriau sense).
The author establishes a generalization to the non-smooth case, of the result by Jones and Petrack (loc. cit.). For that, he uses the natural rotation-invariant measure on \(L(M)\) deduced from the pinned Wiener measure on \(M\), together with a so-called “diffeology” (in the Chen-Souriau sense).
Reviewer: Jacques Franchi (Strasbourg)
MSC:
| 58J65 | Diffusion processes and stochastic analysis on manifolds |
| 46L80 | \(K\)-theory and operator algebras (including cyclic theory) |
| 60H07 | Stochastic calculus of variations and the Malliavin calculus |
| 55N91 | Equivariant homology and cohomology in algebraic topology |
Citations:
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