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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).

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:

Zbl 0723.55003
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References:

[1] Airault, H. and Malliavin, P. (1991). Quasi-Sure Analysis . Univ. Paris VI.
[2] Albeverio, S. (1996). Loop groups, random gauge fields, Chern–Simons models, strings: Some recent mathematical developments. In Espaces de Lacets (R. Léandre, S. Paycha and T. Wuerzbacher, eds.) 5–34. Univ. Strasbourg.
[3] Albeverio, S., Daletskii, A. and Kondratiev, Y. (2000). Stochastic analysis on product manifolds: Dirichlet operators on differential forms. J. Funct. Anal. 176 280–316. · Zbl 0970.58020
[4] Albeverio, S., Daletskii, A. and Lytvynov, Z. (2001). Laplace operator on differential forms over configuration spaces. J. Geom. Phys. 37 14–46. · Zbl 0969.60055
[5] Arai, A. and Mitoma, I. (1991). De Rham–Hodge–Kodaira decomposition in infinite dimension. Math. Ann. 291 51–73. · Zbl 0762.58004
[6] Atiyah, M. F. (1985). Circular symmetry and stationary phase approximation. Colloque en l’honneur de Laurent Schwartz (Paris 1984). Astérisque 131 311–323. · Zbl 0578.58039
[7] Bendikov, A. and Léandre, R. (1999). Regularized Euler–Poincaré number of the infinite dimensional torus. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 2 617–625. · Zbl 1043.60507
[8] Berline, N. and Vergne, M. (1983). Zéros d’un champ de vecteurs et classes caractéristiques équivariantes. Duke Math. J. 50 539–548. · Zbl 0515.58007
[9] Bismut, J. M. (1984). Large Deviations and the Malliavin Calculus . Birkhäuser, Basel. · Zbl 0537.35003
[10] Bismut, J. M. (1985). Index theorem and equivariant cohomology of the loop space. Comm. Math. Phys. 98 213–237. · Zbl 0591.58027
[11] Bismut, J. M. (1986). Localisation formulas, superconnection and the index theorem for families. Comm. Math. Phys. 103 127–166. · Zbl 0602.58042
[12] Bott, R. and Tu, L. W. (1986). Differential Forms in Algebraic Topology . Springer, New York. · Zbl 0496.55001
[13] Chen, K. T. (1973). Iterated paths integrals of differential forms and loop space homology. Ann. Math. 97 213–237. JSTOR: · Zbl 0227.58003
[14] Connes, A. (1988). Entire cyclic cohomology of Banach algebras and characters of \(\theta\)-summable Fredholm modules. \(K\)-Theory 1 519–548. · Zbl 0657.46049
[15] Driver, B. (1992). A Cameron–Martin quasi-invariance formula for Brownian motion on compact manifods. J. Funct. Anal. 110 272–376. · Zbl 0765.60064
[16] Duistermaat, J. J. and Heckman, G. J. (1982). On the variation in the cohomology of the symplectic of the reduced phase-space. Invent. Math. 69 259–269. · Zbl 0503.58015
[17] Elworthy, K. D. and Li, X. M. (2000). Special Itô maps and an \(L^2\) Hodge theory for one forms on path spaces. · Zbl 0973.58019
[18] Emery, M. and Léandre, R. (1990). Sur une formule de Bismut. Séminaire de Probabilités XXIV. Lecture Notes in Math. 1426 448–452. Springer, New York. · Zbl 0702.58081
[19] Fang, S. and Franchi, J. (1997). De Rham–Hodge–Kodaira operator on loop groups. J. Funct. Anal. 148 391–407. · Zbl 0901.58067
[20] Frölicher, A. (1982). Smooth structures. Category Theory . Lecture Notes in Math. 963 69–81. Springer, New York.
[21] Getzler, E. (1988). Cyclic cohomology and the path integral of the Dirac operator. · Zbl 0692.17007
[22] Getzler, E., Jones, J. D. S. and Petrack, S. (1991). Differential forms on a loop space and the cyclic bar complex. Topology 30 339–371. · Zbl 0729.58004
[23] Goodwillie, T. G. (1985). Cyclic homology, derivations and the free loop space. Topology 24 187–217. · Zbl 0569.16021
[24] Hida, T., Kuo, H. H., Potthoff, J. and Streit, L. (1993). White Noise: An Infinite Dimensional Calculus . Kluwer, Dordrecht. · Zbl 0771.60048
[25] Hoegh-Krohn, R. (1974). Relativistic quantum statistical mechanics in two dimensional space time. Comm. Math. Phys. 38 195–224.
[26] Iglésias, P. (1985). Thesis. Univ. Provence.
[27] Ikeda, N. and Watanabe, S. (1981). Stochastic Differential Equations and Diffusion Processes . North-Holland, Amsterdam. · Zbl 0495.60005
[28] Jones, J. D. S. (1980). Cyclic cohomology and equivariant cohomology. Invent. Math. 87 403–423. · Zbl 0644.55005
[29] Jones, J. D. S. and Léandre, R. (1991). \(L^p\) Chen forms on loop spaces. In Stochastic Analysis (M. Barlow and N. Bingham, eds.) 104–162. Cambridge Univ. Press. · Zbl 0756.58003
[30] Jones, J. D. S. and Petrack, S. (1990). The fixed point theorem in equivariant cohomology. Trans. Amer. Math. Soc. 322 35–49. JSTOR: · Zbl 0723.55003
[31] Kriegl, A. and Michor, P. W. (1997). The Convenient Setting of Global Analysis . Amer. Math. Soc., Providence, RI. · Zbl 0889.58001
[32] Kusuoka, S. (1990). De Rham cohomology of Wiener–Riemannian manifold. · Zbl 0754.60067
[33] Léandre, R. (1993). Integration by parts and rotationally invariant Sobolev calculus on free loop space. J. Geom. Phys. 11 517–528. · Zbl 0786.60074
[34] Léandre, R. (1995). Stochastic Moore loop space. Chaos: The Interplay Between Stochastic and Deterministic Behaviour . Lecture Notes in Phys. 457 479–502. Springer, Berlin. · Zbl 0849.58073
[35] Léandre, R. (1996). Cohomologie de Bismut–Nualart–Pardoux et cohomologie de Hochschild entiere. Séminaire de Probabilités XXX . Lecture Notes in Math. 1626 68–100. Springer, New York. · Zbl 0870.58011
[36] Léandre, R. (1996). The circle as a fermionic distribution. In Stochastic Analysis and Related Topics V (H. Korezlioglu, B. Oksendal and A. S. Ustunel, eds.) 233–236. Birkhäuser, Basel. · Zbl 0904.58008
[37] Léandre, R. (1997). Brownian cohomology of an homogeneous manifold. In New Trends in Stochastic Analysis (K. D. Elworthy, S. Kusuoka and I. Shigekawa, eds.) 305–347. World Scientific, Singapore.
[38] Léandre, R. (1997). Invariant Sobolev calculus on the free loop space. Acta Appl. Math. 46 267–350. · Zbl 0895.60055
[39] Léandre, R. (1998). Stochastic cohomology of the frame bundle of the loop space. J. Nonlinear Math. Phys. 5 23–41. · Zbl 0955.58032
[40] Léandre, R. (1998). Singular integral homology of the stochastic loop space. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 1 17–31. · Zbl 0945.58027
[41] Léandre, R. (1999). Stochastic cohomology and Hochschild cohomology. In Development of Infinite-Dimensional Noncommutative Analysis (A. Hora, ed.) 17–26. RIMS, Kokyuroku. · Zbl 0951.60505
[42] Léandre, R. (2000). Cover of the Brownian bridge and stochastic symplectic action. Rev. Math. Phys. 12 91–137. · Zbl 0968.58027
[43] Léandre, R. (2000). Anticipative Chen–Souriau cohomology and Hochschild cohomology. Math. Phys. Stud. 22 185–199. · Zbl 0971.55009
[44] Léandre, R. (2000). A sheaf theoretical approach to stochastic cohomology. Rep. Math. Phys. 46 157–164. · Zbl 0992.60004
[45] Léandre, R. (2001). Stochastic Adams theorem for a general compact manifold. Rev. Math. Phys. 13 1095–1133. · Zbl 1037.58025
[46] Léandre, R. (2001). Stochastic cohomology of Chen–Souriau and line bundle over the Brownian bridge. Probab. Theory Related Fields 120 168–182. · Zbl 0990.58023
[47] Léandre, R. (2001). Stochastic diffeology and homotopy. In Stochastic Analysis and Mathematical Physics (A. B. Cruzeiro and J.-C. Zambrini, eds.) 51–57. Birkhäuser, Boston. · Zbl 0990.58024
[48] Léandre, R. (2002). Analysis over loop space and topology. Math. Notes 72 212–229. · Zbl 1042.58003
[49] Léandre, R. (2003). Stochastic algebraic de Rham complexes. Acta Appl. Math. 79 217–247. · Zbl 1039.58031
[50] Léandre, R. (2004). Hypoelliptic diffusion and cyclic cohomology. In Stochastic Analysis (R. Dalang, M. Dozzi and F. Russo, eds.) 165–185. Birkhäuser, Basel. · Zbl 1087.58024
[51] Léandre, R. (2005). Speed of the Brownian loop on a manifold. Quantum Probab. (M. Bozejko, ed.). · Zbl 1109.58036
[52] Léandre, R. and Smolyanov, O. (1999). Stochastic homology of the loop space. In Analysis on Infinite Dimensional Lie Groups and Algebras (H. Heyer and J. Marion, eds.) 229–235. World Scientific, Singapore. · Zbl 0958.58030
[53] Loday, J. L. (1998). Cyclic Homology , 2nd ed. Springer, New York. · Zbl 0885.18007
[54] Nualart, D. (1995). The Malliavin Calculus and Related Topics . Springer, New York. · Zbl 0837.60050
[55] Nualart, D. and Pardoux, E. (1988). Stochastic calculus with anticipating integrands. Probab. Theory Related Fields 78 535–581. · Zbl 0629.60061
[56] Ramer, R. (1974). On the de Rham complex of finite codimensional forms on infinite dimensional manifolds. Thesis, Warwick Univ.
[57] Shigekawa, I. (1986). De Rham–Hodge–Kodaira’s decomposition on an abstract Wiener space. J. Math. Kyoto Univ. 26 191–202. · Zbl 0611.58006
[58] Smolyanov, O. G. (1986). De Rham current’s and Stoke’s formula in a Hilbert space. Soviet. Math. Dokl. 33 140–144. · Zbl 0608.58004
[59] Souriau, J. M. (1985). Un algorithme générateur de structures quantiques. Astérisque 341–399. · Zbl 0608.58028
[60] Szabo, R. J. (2000). Equivariant Cohomology and Localization of Paths Integrals in Physics . Lecture Notes in Phys. 63 . Springer, Berlin. · Zbl 0998.81520
[61] Warner, J. M. (1983). Foundations of Differentiable Manifolds and Lie Groups . Springer, New York. · Zbl 0516.58001
[62] Witten, E. (1982). Supersymmetry and Morse theory. J. Differential Geom. 17 661–692. · Zbl 0499.53056
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