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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 [{\it 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. For the entire collection see [Zbl 1154.17001].
58J65Diffusion processes and stochastic analysis on manifolds