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Chen-Souriau calculus for rough loops. (English) Zbl 1204.58032
Mladenov, Ivaïlo M. (ed.), Proceedings of the 9th international conference on geometry, integrability and quantization, Sts. Constantine and Elena, Bulgaria, June 8–13, 2007. Sofia: Bulgarian Academy of Sciences (ISBN 978-954-8495-42-4/pbk). 252-260 (2008).
In previous papers, the author has constructed differential forms and Chen-Souriau cohomology groups on the Hölder loop space over a compact Riemannian manifold. In the present paper he considers the construction of such a diffeology over the rough loop space in the sense of the rough paths introduced in [T. J. Lyons, Rev. Mat. Iberoam. 14, No. 2, 215–310 (1998; Zbl 0923.34056)]. He shows that the cohomology groups constructed in this sense over the rough loop space are equal to the de Rham cohomology groups of the Hölder loop space. In particular, a line bundle over the rough loop space is isomorphic to a line bundle of the Hölder loop space. It is also shown that the transgression of a three form in the sense of Brylinski determines a continuous line bundle over the rough loop space.
MSC:
58J65Diffusion processes and stochastic analysis on manifolds