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Stochastic Wess-Zumino-Witten Model for the Measure of Kontsevitch

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Seminar on Stochastic Analysis, Random Fields and Applications

Part of the book series: Progress in Probability ((PRPR,volume 45))

Abstract

We define a measure over C 1 loops and give integration by parts formulas. We give different Hilbert structures over the tangent space of a loop such that the energy functional belongs to all the Sobolev spaces. We modify a little bit the measure in order that this measure is invariant under time reversal. We introduce a connection which is invariant under time reversal. This allows us to construct the Wess-Zumino-Witten supercharge and the Wess-Zumino-Witten laplacian which are associated to the energy functional.

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References

  1. S. Aida and D. Elworthy, Differential calculus on path and loop spaces, preprint.

    Google Scholar 

  2. S. Aida, S. Kusuoka and D. W. Stroock, On the support of Wiener functional, in: D. Elworthy and N. Ikeda, Eds., Asymptotic problems in probability theory: Wiener functionals and asymptotics, Longman Scientific, 284 (1993), 3–35.

    Google Scholar 

  3. H. Airault and P. Malliavin, Quasi sure analysis, publication Paris VI, (1991).

    Google Scholar 

  4. S. Albeverio and R. Hoegh-Krohn, The energy representation of Sobolev Lie groups, Compositio Math., 36 (1978), 37–52.

    MathSciNet  MATH  Google Scholar 

  5. S. Albeverio, Z. M. Ma and M. Röckner, Partition of unity in Sobolev spaces over infinite dimensional state spaces, preprint.

    Google Scholar 

  6. A. Arai, A general class of infinite dimensional operators and path representation of their index, Journal of Functional Analysis, 105 (1992), 342–408.

    Article  MathSciNet  MATH  Google Scholar 

  7. G. Ben-Arous and R. Léandre, Décroissance exponentielle du noyau de la chaleur sur la diagonale II, Probability Theory and Related Fields, 90 (1991), 377–402.

    Article  MathSciNet  MATH  Google Scholar 

  8. J. M. Bismut, Martingales, the Malliavin calculus and hypoellipticity under general Hormander condition, Z. Wahrscheinlichkeitstheorie. Verw. Geb., 56 (1981), 469–505.

    Article  MathSciNet  MATH  Google Scholar 

  9. J. M. Bismut, Large deviations and the Malliavin Calculus, Progress in Math. 45, Birkhäuser, (1984).

    MATH  Google Scholar 

  10. J. M. Bismut, The Witten complex and the degenerate Morse inequalities, J. Differ. Geometry., 23 (1986), 207–240.

    MathSciNet  MATH  Google Scholar 

  11. A. Cruzeiro and P. Malliavin, Curvatures of path spaces and stochastic analysis, preprint.

    Google Scholar 

  12. B. De Witt, Supermanifolds, Cambridge University Press, 1988.

    Google Scholar 

  13. B. Driver, A Cameron-Martin type quasi-invariance for Brownian motion on compact manifolds, Journal of Functional Analysis, 110 (1992), 272–376.

    Article  MathSciNet  MATH  Google Scholar 

  14. D. Elworthy, Stochastic Differential Equations on Manifold, L.M.S. Lectures Notes Series 20, Cambridge University Press, (1982).

    Google Scholar 

  15. O. Enchev and D. W. Stroock, Towards a Riemannian geometry on the path space over a Riemannian manifold, preprint.

    Google Scholar 

  16. E. Getzler, Dirichlet form on loop space, Bull. Sci. Maths. 2., 113 (1989), 157–174.

    MathSciNet  Google Scholar 

  17. L. Gross, Potential theory on Hilbert spaces, Journal of Functional Analysis, 1 (1967), 123–181.

    Article  MATH  Google Scholar 

  18. J. M. Guilarte, The supersymmetric sigma model, topological mechanics and knot invariants, Journal of Geometry and Physics, 7 (1990), 255–302.

    Article  MathSciNet  MATH  Google Scholar 

  19. J. M. Guilarte, Sphalerons and instantons in two dimensional field theory, preprint.

    Google Scholar 

  20. B. Hellfer and J. Sjostrand, Puits multiples en mécanique semi-classique IV, Etude du complexe de Witten, Commun. Part. Differ. Equat., 10 (1985), 245–340.

    Article  Google Scholar 

  21. M. Hino, Spectral properties of Laplacians on an abstract Wiener space with a weighted Wiener measure, J. Funct. Anal., 147 (2) (1997), 485–520.

    Article  MathSciNet  MATH  Google Scholar 

  22. L. Hormander, Hypoelliptic second order differential equations, Acta Math., 119 (1969), 147–171.

    Article  MathSciNet  Google Scholar 

  23. E. P. Hsu, Quasi-invariance of the Wiener measure on the path space over a compact Riemannian manifold, preprint.

    Google Scholar 

  24. N. Ikeda and S. Watanabe, Stochastic differential equations and diffusion processes, North-Holland, 1981.

    Google Scholar 

  25. A. Jaffe, A. Lesniewski and J. Weitsman, Index of a family of Dirac operators on loop space, Commun. Math. Phys., 112 (1987), 75–88.

    Article  MathSciNet  MATH  Google Scholar 

  26. A. Jaffe, A. Lesniewski and J. Weitsman, The loop space S 1 → R and supersymmetric quantum fields, Ann. Phys., 193 (1988), 337–351.

    MathSciNet  Google Scholar 

  27. A. Jaffe and A. Lesniewski, A priori estimates for N=2 Wess-Zumino models on a cylinder, Commun. Math. Phys., 114 (1988), 553–576.

    Article  MathSciNet  MATH  Google Scholar 

  28. J. Jones and R. Léandre, L p Chen forms over loop spaces, in: M. Barlow and N. Bingham, Eds., Stochastic Analysis, Cambridge University Press, 1991, 104–162.

    Google Scholar 

  29. J. D. S. Jones and R. Léandre, A stochastic approach to the Dirac operator over the free loop space, in: A. Sergeev, Ed., Loop Spaces, to be published.

    Google Scholar 

  30. S. Kusuoka, De Rham cohomology of Wiener-Riemannian manifolds, Proc. Int. Congr. Math., Kyoto/Japan 1990, II (1991), 1075–1082.

    MathSciNet  Google Scholar 

  31. R. Léandre, Strange behaviour of the heat kernel on the diagonal, in: S. Albeverio, Ed., Stochastic Processes, Physics and Geometry, World Scientific, 1990, 516–528.

    Google Scholar 

  32. R. Léandre, Integration by parts formulas and rotationally invariant Sobolev Calculus on the free loop space, in: R. Gielerak and A. Borowiec, Eds., XXVII Winter School of Theoretical Physics, J. of Geometry and Physics, 11 (1993), 517–528.

    Google Scholar 

  33. R. Léandre, Invariant Sobolev Calculus on the free loop space, Acta Applicandae Mathematicae, 46 (1997), 267–350.

    Article  MathSciNet  MATH  Google Scholar 

  34. R. Léandre, Brownian motion over a Kähler manifold and elliptic genera of level N, in: R. Sénéor and L. Streit, Eds., Stochastic Analysis and Applications in Physics, Nato ASI Series 449 (1994), 193–217.

    Chapter  Google Scholar 

  35. R. Léandre, Cohomologie de Bismut-Nualart-Pardoux et cohomologie de Hochschild entière, in: J. Azéma, M. Emery and M. Yor, Eds., Séminaire de Probabilités XXX in honour of P.A. Meyer et J. Neveu, Lecture Notes in Mathematics 1626 (1996), 68–100.

    Google Scholar 

  36. R. Léandre, Stochastic Wess-Zumino- Witten model over a symplectic manifold, Journal of Geometry and Physics, 21 (1997), 307–336.

    Article  MathSciNet  MATH  Google Scholar 

  37. R. Léandre, Brownian cohomology of an homogeneous manifold, in: K. D. Elworthy, S. Kusuoka and I. Shigekawa, Eds., Proceedings of the Taniguchi conference, World Scientific, 1997, 305–348.

    Google Scholar 

  38. R. Léandre, Stochastic Moore loop space, in: P. Garbaczweski, Ed., Chaos: The Interplay Between Stochastic and Deterministic Behaviour, Lecture Notes in Physics 457 (1995), 479–502.

    Chapter  Google Scholar 

  39. R. Léandre, String structure over the brownian bridge, Journal of Maths. Physics, to be published.

    Google Scholar 

  40. R. Léandre, Hilbert space of spinors fields over the free loop space, Reviews in Mathematical Physics, 9 (2) (1997), 243–277.

    Article  MathSciNet  MATH  Google Scholar 

  41. R. Léandre, Cover of the brownian bridge and stochastic symplectic action, Reviews in Mathematical Physics, to be published.

    Google Scholar 

  42. R. Léandre, Stochastic gauge transform of the string bundle, Journal of Geometry and Physics, to be published.

    Google Scholar 

  43. R. Léandre and J. Norris, Integration by parts and Cameron-Martin formulas for the free path space of a compact Riemannian manifold, in: J. Azéma, M. Emery and M. Yor, Eds., Séminaire de Probabilités XXXI, Lect. Notes in Maths 1655 (1997), 16–24.

    Chapter  Google Scholar 

  44. R. Léandre and S. S. Roan, A stochastic approach to the Euler-Poincaré number of the loop space of a developable orbifold, J. Geometry and Physics, 16 (1995), 71–98.

    Article  MATH  Google Scholar 

  45. P. Malliavin, Stochastic Calculus of variations and hypoelliptic operators, in: K. Ito, Ed., Proc. Intern. Conf. Stochastic Differential Equations, Wiley, 1978, 195–263.

    Google Scholar 

  46. J. Norris, Covariant stochastic calculus and applications to heat kernels, Probability Theory and Related Fields, 94 (1993), 525–541.

    Article  MathSciNet  MATH  Google Scholar 

  47. I. Shigekawa, Transformations of Brownian motion on a Riemannian symmetric space, Z. Wahrscheinlichkeitstheorie Verw. Geb., 65 (1984), 493–522.

    Article  MathSciNet  MATH  Google Scholar 

  48. I. Shigekawa, Differential calculus on a based loop group, in: in: K. D. Elworthy, S. Kusuoka and I. Shigekawa, Eds., Proceedings of the Taniguchi conference, World Scientific, 1997, 375–399.

    Google Scholar 

  49. D. Stroock, The Malliavin calculus and its applications, in: D. Williams, Ed., Stochastic Integrals, Lecture Notes in Mathematics, 851 (1981), 394–432.

    Chapter  Google Scholar 

  50. C. Taubes, S 1 action and elliptic genera, Commun. Math. Phys., 122 (1989), 455–526.

    Article  MathSciNet  MATH  Google Scholar 

  51. S. Watanabe, Analysis of Wiener functional and its applications to heat kernels, Ann. Probab., 15 (1) (1987), 1–37.

    Article  MathSciNet  MATH  Google Scholar 

  52. Ed. Witten, Supersymmetry and Morse theory, J. Diff. Geometry, 17 (1982), 661–692.

    MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Département de Mathématiques Institut Elie Cartan, Université Henri Poincaré Nancy 1, 54506, Vandœuvre-les-Nancy, France

    Rémi Léandre

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  1. Rémi Léandre
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Editor information

Editors and Affiliations

  1. Département de Mathématiques, Ecole Polytechnique Fédérale, CH-1015, Lausanne, Switzerland

    Robert C. Dalang

  2. Institut Elie Cartan, Université Henri Poincaré, BP 239, F-54506, Vandoeuvre-les-Nancy Cedex, France

    Marco Dozzi

  3. Département de Mathématiques Institut Galilée, Université Paris 13, F-95430, Villetaneuse, France

    Francesco Russo

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© 1999 Springer Basel AG

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Léandre, R. (1999). Stochastic Wess-Zumino-Witten Model for the Measure of Kontsevitch. In: Dalang, R.C., Dozzi, M., Russo, F. (eds) Seminar on Stochastic Analysis, Random Fields and Applications. Progress in Probability, vol 45. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8681-9_15

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  • DOI: https://doi.org/10.1007/978-3-0348-8681-9_15

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9727-3

  • Online ISBN: 978-3-0348-8681-9

  • eBook Packages: Springer Book Archive

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