Skip to main content Accessibility help
×
Home
  • Get access
    Check if you have access via personal or institutional login
  • Cited by 1
  • Print publication year: 2013
  • Online publication date: May 2013

3 - Paths towards an extension of Chern–Weil calculus to a class of infinite dimensional vector bundles

Related content

Powered by UNSILO
References
[Ad] M., Adler, On a trace functional for formal pseudodifferential operators and the symplectic structure of the Korteweg–de Vries type equation, Invent. Math. 50 (1987), 219–248.
[AB] M. F., Atiyah, R., Bott, The Yang–Mills equations over Riemann surfaces, Phil. Trans. R. Soc. Lond. A 308 (1982), 523–615.
[ARS] M. R., Adams, T., Ratiu, R., Schmidt, The Lie group structure of diffeomorphism groups and invertible Fourier integral operators, with applications. In Infinite-dimensional groups with applications, (ed. V., Kac), Springer, 1985, pp. 1–69.
[AS] M. F., Atiyah, I. M., Singer, The index of elliptic operators: IV, Ann. Math. 93 (1971), 119–149.
[BGV] N., Berline, E., Getzler, M., Vergne, Heat kernels and Dirac operators, Grundlehren Math. Wiss. 298, Springer Verlag, 1996.
[B] J.-M, Bismut, The Atiyah–Singer theorem for families of Dirac operators: two heat equation proofs, Invent. Math. 83 (1986), 91–151.
[BW] B., Booss-Bavnbek, K., Wojciechowski, Elliptic boundary problems for Dirac operators, Mathematics: Theory and Applications, Birkhäuser, 1993.
[Bo] J.-B, Bost, Principe, d'Oka, K-théorie et systèmes dynamiques non commutatifs, Invent. Math. 101 (1990), 261–333.
[Bott] R., Bott, On the Chern–Weil homomorphism and the continuous cohomology of Lie groups, Adv. Math. 11 (1973) 289–303.
[BT] R., Bott, L. W., Tu, Differential forms in algebraic topology, Springer Verlag, 1982.
[BGJ] R., Bott, Lectures on characteristic classes and foliations. In Lectures on algebraic and differential topology (ed. R., Bott, S., Gitler, I. M., James), Lecture Notes in Math. 279, Springer, 1972, pp. 1–94.
[BG] J. L., Brylinski, E., Getzler, The homology of algebras of pseudodifferential symbols and non commutative residues, K-theory 1 (1987), 385–403.
[BL] J., Brüning, M., Lesch, On the eta-invariant of certain nonlocal boundary value problems, Duke Math. J. 96 :2 (1999), 425–468.
[Bu] T., Burak, On spectral projections of elliptic differential operators, Ann. Scuola Norm. Sup. Pisa 3 : 22 (1968), 113–132.
[CDP] A., Cardona, C., Ducourtioux, S., Paycha, From tracial anomalies to anomalies in quantum field theory, Comm. Math. Phys. 242 (2003), 31–65.
[CFNW] M., Cederwall, G., Ferretti, B., Nilsson, A., Westerberg, Schwinger terms and cohomology of pseudodifferential operators, Comm. Math. Phys. 175 (1996), 203–220.
[CDMP] A., Cardona, C., Ducourtioux, J.-P., Magnot, S., Paycha, Weighted traces on algebras of pseudodifferential operators and geometry on loop groups, Inf. Dim. Anal. Quan. Prob. Rel. Top. 5 (2002), 1–38.
[CMM] A., Carey, J., Mickelsson, M., Murray, Index theory, Gerbes, and Hamiltonian quantization, Comm. Math. Phys. 183 (1997), 707–722.
[C1] S.-S, Chern, A simple intrinsic proof of the Gauss–Bonnet formula for closed Riemannian manifolds, Ann. Math. 45 (1944), 747–762.
[C2] S.-S, Chern, Topics in differential geometry, Institute for Advanced Study, mimeographed lecture notes (1951).
[CM] R., Cirelli, A., Manià, The group of gauge transformations as a Schwartz–Lie group, J. Math. Phys. 26 (1985), 3036–3041.
[D] C., Ducourtioux, Weighted traces on pseudodifferential operators and associated determinants, PhD Thesis, Université Blaise Pascal, Clermont-Ferrand, 2001 (unpublished).
[F] D., Freed, The geometry of loop groups, J. Diff. Geom. 28 (1988), 223–276.
[FGLS] B. V., Fedosov, F., Golse, E., Leichtnam, E., Schrohe, The noncommutative residue for manifolds with boundary, J. Funct. Anal. 142 (1996), 1–31.
[Gi] P., Gilkey, Invariance theory, the heat equation and the Atiyah–Singer index theorem, Studies in Advanced Mathematics, CRC Press, 1995.
[Gl] H., Glöckner, Algebras whose groups of units are Lie groups, Studia Math. 153 (2002), 147–177.
[GN] H., Glöckner, K.-H., Neeb, Introduction to infinite-dimensional Lie groups, Vol. 1, in preparation.
[GR] L., Guieu, C., Roger, L'algèbre et le groupe de Virasoro: Aspects géometriques et algébriques (French) [Algebra and the Virasoro group: Geometric and algebraic aspects, generalizations], Les Publications CRM, 2007.
[Gu] V., Guillemin, Residue traces for certain algebras of Fourier integral operators, J. Funct. Anal. 115 : 2 (1993), 391–417.
[H] L., Hörmander, The analysis of linear partial differential operators III. Pseudodifferential operators, Grundlehren Math. Wiss. 274, Springer Verlag, 1994.
[Ka] M., Karoubi, K-theory (An introduction), Grundlehren Math. Wiss. 226, Springer Verlag, 1978.
[Kas] Ch., Kassel, Le résidu non commutatif (d'après M. Wodzicki), Séminaire Bourbaki, Astérisque 177–178 (1989), 199–229.
[KK] O., Kravchenko, B., Khesin, Central extension of the Lie lagebra of (pseudo)-differential symbols, Funct. Anal. Appl. 25 (1991), 83–85.
[KV1] M., Kontsevich, S., Vishik, Determinants of elliptic pseudodifferential operators, Max Planck Preprint (1994) (unpublished) arXiv-hep-th/9404046.
[KV2] M., Kontsevich, S., Vishik, Geometry of determinants of elliptic operators. In Functional analysis on the Eve of the 21st century, Vol. 1 (ed. S., Gindikin, J., Lepowsky, R., Wilson). Progress in Mathematics 131. Birkhäuser Boston, 1994, pp. 173–197.
[KM] A., Kriegel, P., Michor, The convenient setting of global analysis, Mathematical Surveys and Monographs 53, American Mathematical Society, 1997.
[La] A., Larrain-Hubach, Explicit computations of the symbols of order 0 and −1 of the curvature operator of ΩG, Lett. Math. Phys. 89 (2009) 265–275.
[LRST] A., Larrain-Hubach, S., Rosenberg, S., Scott, F., Torres-Ardila, Characteristic classes and zeroth order pseudodifferential operators. In Spectral theory and geometric analysis (ed. M., Braverman, L., Friedlander, Th., Kappeler, P., Kuchment, P., Topalov and J., Weitsman), Cont. Math. 532, American Mathematical Society, 2011, pp. 141–158.
[L] M., Lesch, On the non commutative residue for pseudodifferential operators with log-polyhomogeneous symbols, Ann. Global Anal. Geom. 17 (1998), 151–187.
[Le] J., Leslie, On a differential structure for the group of diffeomorphisms, Topology 6 (1967), 263–271.
[LMR] E., Langmann, J., Mickelsson, S., Rydh, Anomalies and Schwinger terms in NCG field theory models, J. Math. Phys. 42 (2001), 4779.
[LN] M., Lesch, C., Neira-Jimenez, Classification of traces and hypertraces on spaces of classical pseudodifferential operators, J. Noncomm. Geom. in press.
[LP] J.-M., Lescure, S., Paycha, Traces on pseudo-differential operators and associated determinants, Proc. Lond. Math. Soc. 94 : 2 (2007), 772–812.
[Man] Yu. I., Manin, Aspects algébriques des équations différentielles non linéaires, Itogi Nauk. i Tekhn. Sovrem. Probl. Matematik. 11 (1978) 5–152 (in Russian); Engl. transl. J. Soviet Math. 11 (1979) 1–122.
[MRT] Y., Maeda, S., Rosenberg, F., Torres-Ardila, Riemannian geometry on loop spaces, arXiv:0705.1008 (2007).
[MSS] L., Maniccia, E., Schrohe, J., Seiler, Uniqueness of the Kontsevich–Vishik trace, Proc. Amer. Math. Soc. 136 (2008), 747–752.
[MN] R., Melrose, N., Nistor, Homology of pseudo-differential operators I. Manifolds with boundary, funct-an/9606005 (1999) (unpublished).
[Mich] P., Michor, Gauge theory for fiber bundles, Monographs and Textbooks in Physical Science 19. Bibliopolis, 1991.
[Mick1] J., Mickelsson, Second quantization, anomalies and group extensions, Lecture notes given at the “Colloque sur les Méthodes Géométriques en physique”, C.I.R.M, Luminy, June 1997.
[Mick2] J., Mickelsson, Noncommutative residue and anomalies on current algebras. In Integrable models and strings (ed. A., Alekseevet al.), Lecture Notes in Physics 436, Springer Verlag, 1994.
[Mil] J., Milnor, Remarks on infinite dimensional Lie groups. In Relativity, groups and topology II In (ed. B., De Witt and R., Stora), North Holland, 1984.
[MP] J., Mickelsson, S., Paycha, Renormalised Chern–Weil forms associated with families of Dirac operators, J. Geom. Phys. 57 (2007), 1789–1814.
[MS] J., Milnor, J., Stasheff, Characteristic classes, Annals of Mathematics Studies 76, Princeton University Press, University of Tokyo Press, 1974.
[N1] K.-H., Neeb, Central extensions of infinite-dimensional Lie groups, Ann. Inst. Fourier 52 (2002), 1365–1442.
[N2] K.-H., Neeb, Towards a Lie theory of locally convex groups, Jap. J. Math. 1 (2006), 291–468.
[Ok] K., Okikiolu, The multiplicative anomaly for determinants of elliptic operators, Duke Math. J. 79 (1995), 722–749.
[Om] H., Omori, On the group of diffeomorphisms of a compact manifold. In Global analysis, Proc. Sympos. Pure Math. 15. American Mathematical Society, 1970, pp. 167–183. See also Infinite dimensional Lie groups, AMS Translations of Mathematical Monographs 158, 1997.
[OMYK] H., Omori, Y., Maeda, A., Yoshida, O., Kobayashi, On regular Fréchet–Lie groups: Several basic properties, Tokyo Math. J. 6 (1986), 39–64.
[OP] M.-F., Ouedraogo, S., Paycha, The multiplicative anomaly for determinants revisited; locality. Commun. Math. Anal. 12 (2012) 28–63.
[P1] S., Paycha, Renormalised traces as a looking glass into infinite-dimensional geometryInf. Dim. Anal. Quan. Prob. Rel. Top. 4 (2001), 221–266.
[P2] S., Paycha, The uniqueness of the Wodzicki residue and the canonical trace in the light of Stokes' and continuity properties, arXiv:0708.0531 (2007).
[P3] S., Paycha, Regularised traces, integrals and sums; an analytic point of view, American Mathematical Society University Lecture Notes 59, American Mathematical Society, 2012.
[PR1] S., Paycha, S., Rosenberg, Curvature on determinant bundles and first Chern forms, J. Geom. Phys. 45 (2003), 393–429.
[PR2] S., Paycha, S., Rosenberg, Traces and characteristic classes in loop groups. In Infinite dimensional groups and manifolds (ed. T., Wurzbacher), I.R.M.A. Lectures in Mathematical and Theoretical Physics 5. De Gruyter, 2004, pp. 185–212.
[PS1] S., Paycha, S., Scott, A Laurent expansion for regularised integrals of holomorphic symbols, Geom. Funct. Anal., 17 :2 (2005), 491–536.
[PS2] S., Paycha, S., Scott, Chern–Weil forms associated with superconnections. In Analysis, geometry and topology of ellipitc operators (ed. B., Booss-Bavnbeck, S., Klimek, M., Lesch, W., Zhang), World Scientific, 2006, pp. 79–104.
[Po1] R., Ponge, Spectral asymmetry, zeta functions and the noncommutative residue, Int. J. Math. 17 (2006), 1065–1090.
[Po2] R., Ponge, Noncommutative residue for the Heisenberg calculus and applications in CR and contact geometry, J. Funct. Anal. 252 (2007), 399–463.
[Po3] R., Ponge, Traces on pseudodifferential operators and sums of commutators, arXiv:0707.4265v2 [math.AP] (2008).
[PS] A., Pressley, G., Segal, Loop groups, Oxford Mathematical Monographs, Oxford University Press, 1986.
[Q] D., Quillen, Superconnections and the Chern character, Topology 24 (1985), 89–95.
[Ra] A. O., Radul, Lie algebras on differential operators, their central extensions, and W-algebras, Funct. Anal. 25 (1991), 33–49.
[Ro] F., Rochon, Sur la topologie de l'espace des opérateurs pseudodifférentiels inversible d'ordre 0, Ann. Inst. Fourier 58 : 1 (2008), 29–62.
[Rog] C., Roger, Sur les origines du cocycle de Virasoro (2001). Published as a historical appendix in [GR].
[Schm] S., Schmid, Infinite dimensional Lie groups with applications to mathematical physics, J. Geom. Symm. Phys. 1 (2004), 1–67.
[Schr] E., Schrohe, Wodzicki's noncommutative residue and traces for operator algebras on manifolds with conical singularities. In Microlocal analysis and spectral theory (ed. L., Rodino), Proceedings of the NATO Advanced Study Institute, Il Ciocco, Castelvecchio Pascoli (Lucca), Italy, 1996, NATO ASI Ser. C, Math. Phys. Sci. 490. Kluwer Academic Publishers, 1997, pp. 227–250.
[Sc] S., Scott, Zeta-Chern forms and the local family index theorem, Trans. Amer. Math. Soc. 359 : 5 (2007), 1925–1957.
[Se] R. T., Seeley, Complex powers of an elliptic operator. In Singular integrals. (Proc. Symp. Pure Math., Chicago) American Mathematical Society, 1966, pp. 288–307.
[Sh] A., Shubin, Pseudodifferential operators and spectral theory, Springer Verlag, 1980.
[T] M. E., Taylor, Pseudodifferential operators, Princeton University Press, 1981.
[Tr] F., Trèves, Introduction to Pseudodifferential and Fourier integral operators, vol. 1, Plenum Press, 1980.
[W1] M., Wodzicki, Spectral asymmetry and noncommutative residue (in Russian). Habilitation thesis, Steklov Institute (former) Soviet Academy of Sciences, Moscow, 1984.
[W2] M., Wodzicki, Non commutative residue, Chapter 1. Fundamentals, K-theory, arithmetic and geometry, Springer Lecture Notes 1289. Springer, 1987, pp. 320–399.
[W3] M., Wodzicki, Report on the cyclic homology of symbols. Preprint, IAS Princeton, Jan. 87, Available online at http://math.berkeley.edu/wodzicki.
[Woc] Ch., Wockel, Lie group structures on symmetry groups of principal bundles, J. Funct. Anal. 251 (2007), 254–288.