Abstract
Results on analysis on loop spaces and topologies are surveyed. In particular, Sobolev cohomology of loop spaces, stochastic Chen--Souriau cohomology, and Witten genus are described.
Similar content being viewed by others
REFERENCES
J. M. Bismut, Large Deviations and the Malliavin Calculus, vol. 45, Progress in Math., Birkhäuser, 1984.
J. D. S. Jones and R. Léandre, “L p Chen forms on loop spaces,” in: Stochastic Analysis (M. Barlow and N. Bingham, editors), Cambridge Univ. Press, 1991, pp. 104–162.
N. Berline and M. Vergne, “Zéros d'un champ de vecteurs et classes caractéristiques équivariantes,” Duke Math. J., 50 (1983), 539–548.
J. J. Duistermaat and G. J. Heckman, “On the variation in the cohomology of the symplectic form of the reduced phase space,” Invent. Math., 69 (1982), no. 2, 259–268.
C. Taubes, “S 1 actions and elliptic genera,” Comm. Math. Phys., 122 (1989), 455–526.
M. Atiyah, “Circular symmetry and stationary phase approximation,” Conference in honor of L. Schwartz, Astérisque, 131 (1985), 43–59.
J. M. Bismut, “Index theorem and equivariant cohomology of the loop space,” Comm. Math. Phys., 98 (1985), 213–237.
R. Szabo, Equivariant Cohomology and Localization of Path Integrals in Physics, vol. 63, Lect. Notes in Physics, 2000.
E. Witten, “The index of the Dirac operator in loop space,” in: Elliptic curves and modular forms in algebraic topology (P. S. Landweber, editor), vol. 1326, Lecture Notes in Math., Springer, 1988, pp. 161–181.
G. Segal, “Elliptic cohomology,” Séminaire Bourbaki 695, Astérisque, 161–162 (1988), 187–201.
F. Hirzebruch, T. Berger, and R. Jung, Manifolds and Modular Forms, vol. 20, Asp. of Math., Vieweg, 1993.
K. Gawedzki, “Conformal field theory,” Séminaire Bourbaki 704, Astérisque, 177–178 (1989), 95–126.
R. Léandre, “Cover of the Brownian bridge and stochastic symplectic action,” Rev. Math. Phys., 12 (2000), no. 1, 91–137.
S. Aida, “Stochastic analysis on loop space,” Sugaku, 50 (1998), 265–281.
B. Driver, “Towards calculus and geometry on path spaces,” in: Stochastic Analysis (M. Cranston, R. Durrett, and M. Pinsky, editors), vol. 57, Proc. Sympos. Pure Math., 1995, pp. 405–422.
P. Malliavin, Stochastic Analysis, Springer, 1997.
D. W. Stroock, Introduction to Analysis on the Path Space on Riemannian Manifolds, vol. 74, Math. Surveys Monographs, Amer. Math. Soc., 2000.
S. Albévério, “A survey of some developments in loop space: associated stochastic processes, statistical mechanics, infinite-dimensional Lie groups, topological quantum fields,” Trudy Mat. Inst. Steklov [Proc. Steklov Inst. Math.], 217 (1997), 209–234.
R. Léandre, “Cohomologie de Bismut-Nualart-Pardoux et cohomologie de Hochschild entière,” in: Séminaire de Proba XXX. In honor of P. A. Meyer and J. Neveu (J. Azéma, M. Emery, and M. Yor, editors), vol. 1626, Lecture Notes in Math., Springer, 1996, pp. 68–100.
R. Léandre, “Brownian cohomology of an homogeneous manifold,” in: New Trends in Stochastic Analysis (K. D. Elworthy, S. Kusuoka, and I. Shigekawa, editors), World Sci., 1997, pp. 305–347.
R. Léandre, “Stochastic Adams theorem for a general compact manifold,” Rev. Math. Phys, 13 (2001), 1095–1133.
R. Léandre, “Integration by parts and rotationally invariant Sobolev calculus on free loop spaces,” in: J. Geometry Phys. (A. Borowiec and R. Gielerak, editors), vol. 11, 1993, pp. 517–528.
R. Léandre, “Invariant Sobolev calculus on free loop space,” Acta Appl. Math., 46 (1997), 267–350.
I. Ya. Aref'eva, “Non-Abelian Stokes formula,” Teoret. Mat. Fiz. [Theoret. and Math. Phys.], 43 (1980), 111–116.
H. Airault and P. Malliavin, Quasi-sure Analysis, Publication Paris VI, 1991.
S. Albévério and R. Hoegh-Krohn, “The energy representation of Sobolev Lie groups,” Compositio Math., 36 (1978), 37–52.
J. F. Adams, “On the cobar construction,” Proc. Nat. Acad. Sci. USA, 42 (1956), 346–373.
J. M. Souriau, “Un algorithme générateur de structures quantiques,” Elie Cartan et les mathématiques d'aujourd'hui, Astérisque (1985), 341–399.
R. Léandre, “Singular integral cohomology of the stochastic loop space,” Infinite-Dimensional Analysis, Quantum Probability and Related Topics, 1 (1998), no. 1, 17–31.
C. Dellacherie and P. A. Meyer, Probabilités et potentiel, vol. II, Hermann, 1980.
J. L. Brylinski, Loop Spaces, Characteristic Classes, and Geometric Quantization, vol. 107, Progress in Math., Birkhäuser, 1992.
R. Léandre, “String structure over the Brownian bridge,” J. Math. Phys., 40 (1999), no. 1, 454–479.
A. L. Carey and M. K. Murray, “String structure and the path fibration of a group,” Comm. Math. Phys., 141 (1991), 441–452.
R. Coquereaux and K. Pilch, “String structure on loop bundles,” Comm. Math. Phys., 120 (1989), 353–378.
Laughlin D. Mac, “Orientation and string structures on loop spaces,” Pacific J. Math., 155 (1992), 143–156.
R. Léandre, “Hilbert space of spinor fields over the free loop space,” Rev. Math. Phys., 9 (1997), no. 2, 243–277.
R. Léandre, “Stochastic cohomology of the frame bundle of the loop space,” J. Nonlinear Math. Phys., 5 (1998), no. 1, 23–41.
R. Léandre, “Stochastic gauge transform of the string bundle,” J. Geometry Phys., 26 (1998), 1–25.
R. Léandre, “A unitary representation of the basical central extension of a loop group,” Nagoya Math. J., 159 (2000), 113–124.
R. Léandre, “Stochastic cohomology of Chen-Souriau and line bundle over the Brownian bridge,” Probability Theory Related Fields, 120 (2001), 168–182.
R. Léandre, “A sheaf theoretical approach to stochastic cohomology,” XXXI Symposium of Mathematical Physics (Torun) (R. Mrugala, editor), Rep. Math. Phys., 46 (2000), no. 1/2, 157–164.
R. Bonic, J. Frampton, and A. Tromba, “Λ-Manifolds,” J. Funct. Anal., 3 (1969), 310–320.
R. Bonic and J. Frampton, “Smooth functions on Banach manifolds,” J. Math. Mech., 15 (1966), no. 5, 877–898.
F. W. Warner, Foundations of Differentiable Manifolds and Lie Groups, Springer, 1983.
R. Bott and L. W. Tu, Differential Forms in Algebraic Topology, Springer, 1986.
K. D. Elworthy and X. M. Li, “Special Itô maps and an L 2 Hodge theory for one forms on path spaces,” in: Stochastic Processes, Physics and Geometry: New Interplay (I. M. Roeckner, editor), vol. 28, C.M.S. Conference Proceedings, 2001, pp. 145–162.
K. D. Elworthy and M. Yor, “Conditional expectations for derivatives of certain stochastics flows,” in: Séminaire de Proba XXVII (J. Azéma, P. A. Meyer, and M. Yor, editors), vol. 1557, Lecture Notes in Math., Springer, 1993, pp. 159–172.
A. Arai and I. Mitoma, “De Rham-Hodge-Kodaira decomposition in infinite dimensions,” Math. Ann., 291 (1991), 51–73.
S. Fang and J. Franchi, “A differentiable isomorphism between Wiener space and path groups,” in: Séminaire de Proba XXXI (J. Azéma, P. A. Meyer, and M. Yor, editors), vol. 1655, Lecture Notes in Math., 1997, pp. 54–61.
H. Airault and P. Malliavin, Integration on Loop Groups, Publication Paris VI, 1991.
S. Fang and J. Franchi, “De Rham-Hodge-Kodaira-Kodaira operator on loop groups,” J. Funct. Anal., 148 (1997), 391–407.
R. Léandre and I. Volovich, “The stochastic Lévy Laplacian and Yang-Mills equation on manifolds,” Infinite-Dimensional Analysis, Quantum Probability and Related Topics, 4 (2001), no. 2, 161–172.
L. Accardi, P. Gibilisco, and I. Volovich, “Yang-Mills gauge fields as harmonic functions for the Lévy Laplacian,” Russian J. Math. Phys., 2 (1994), 235–250.
O. Alvarez, T. P. Killingback, M. Mangano, and P. Windey, “String theory and loop space index theorem,” Comm. Math. Phys., 111 (1997), 1–10.
A. Jaffe, A. Lesniewski, and J. Weitsman, “Index of a family of Dirac operators on loop space,” Comm. Math. Phys., 112 (1987), 75–88.
L. Dixon, J. Harvey, C. Vafa, and E. Witten, “Strings on orbifold,” Nucl. Phys. B., 261 (1985), 678–686.
R. Dijkgraaf, G. Moore, E. Verlinde, and H. Verlinde, “Elliptic genera of symmetric products and second quantized strings,” Comm. Math. Phys., 185 (1997), 197–209.
A. Arai, “A general class of infinite-dimensional operators and path representation of their index,” J. Funct. Anal., 105 (1992), 304–408.
J. D. S. Jones and R. Léandre, “A stochastic approach to the Dirac operator over free loop space,” Trudy Mat. Inst. Steklov [Proc. Steklov Inst. Math.], 217 (1997), 258–287.
R. Léandre and S. S. Roan, “A stochastic approach to the Euler-Poincaré number of the loop space of an orbifold,” J. Geometry Phys., 16 (1995), 71–98.
R. Léandre, “Quotient of a loop group and Witten genus,” J. Math. Phys., 42 (2001), no. 3, 1364–1383.
P. Gilkey, Invariance Theory, the Heat Equation and the Atiyah-Singer Index Theorem, vol. 11, Publish or Perish, 1984.
R. Léandre, A Stochastic Approach of Witten's Explanation of Rigidity Relation, Preprint, Univ. de Nancy, 2000.
B. Driver and M. Roeckner, “Construction of diffusion on path and loop spaces of compact Riemannian manifolds,” C. R. Acad. Sci. Paris Sér. I, 315 (1992), 603–608.
S. Albévério, R. Léandre, and M. Roeckner, “Construction of a rotational invariant diffusion on the free loop space,” C. R. Acad. Sci. Paris Sér. I, 316 (1993), 287–292.
S. Fang and T. Zhang, “Large deviations for the Brownian motion on loop groups,” J. Theoretical Probability, 14 (2001), 463–483.
G. Felder, K. Gawedzki, and A. Kupiainen, “Spectra of Wess-Zumino-Witten models with arbitrary simple groups,” Comm. Math. Phys., 117 (1988), no. 1, 127–159.
Z. Brzezniak and R. Léandre, “Horizontal lift of an infinite-dimensional diffusion,” Potential Analysis, 12 (2000), 249–280.
Z. Brzezniak and D. Elworthy, “Stochastic differential equations on Banach manifolds,” Methods Functional Analysis Topology, 6 (2000), no. 1, 43–84.
Z. Brzezniak and R. Léandre, Stochastic Pants over a Riemannian Manifold, Preprint, Université de Nancy I, 2001.
K. T. Chen, “Iterated paths integrals of differential forms and loop space homology,” Ann. Math., 97 (1973), 213–237.
R. Léandre, “Anticipative Chen-Souriau cohomology and Hochschild cohomology,” in: Conference in Memory of M. Flato II (G. Dito and D. Sternheimer, editors), vol. 22, Math. Phys. Studies, 2000, pp. 185–199.
I. Shigekawa, “De Rham-Hodge-Kodaira's decomposition on an abstract Wiener space,” J. Math. Kyoto Univ., 26 (1986), 191–202.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Léandre, R. Analysis on Loop Spaces and Topology. Mathematical Notes 72, 212–229 (2002). https://doi.org/10.1023/A:1019850028562
Issue Date:
DOI: https://doi.org/10.1023/A:1019850028562