Skip to main content Accessibility help
×
Hostname: page-component-76fb5796d-22dnz Total loading time: 0 Render date: 2024-04-27T19:36:49.987Z Has data issue: false hasContentIssue false

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

Published online by Cambridge University Press:  05 May 2013

Sylvie Paycha
Affiliation:
Universität Potsdam
Alexander Cardona
Affiliation:
Universidad de los Andes, Colombia
Iván Contreras
Affiliation:
Universität Zürich
Andrés F. Reyes-Lega
Affiliation:
Universidad de los Andes, Colombia
Get access

Summary

Image of the first page of this content. For PDF version, please use the ‘Save PDF’ preceeding this image.'
Type
Chapter
Information
Geometric and Topological Methods for Quantum Field Theory
Proceedings of the 2009 Villa de Leyva Summer School
, pp. 81 - 143
Publisher: Cambridge University Press
Print publication year: 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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.Google Scholar
[AB] M. F., Atiyah, R., Bott, The Yang–Mills equations over Riemann surfaces, Phil. Trans. R. Soc. Lond. A 308 (1982), 523–615.Google Scholar
[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.Google Scholar
[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.Google Scholar
[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.Google Scholar
[Bott] R., Bott, On the Chern–Weil homomorphism and the continuous cohomology of Lie groups, Adv. Math. 11 (1973) 289–303.Google Scholar
[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.Google Scholar
[BL] J., Brüning, M., Lesch, On the eta-invariant of certain nonlocal boundary value problems, Duke Math. J. 96 :2 (1999), 425–468.Google Scholar
[Bu] T., Burak, On spectral projections of elliptic differential operators, Ann. Scuola Norm. Sup. Pisa 3 : 22 (1968), 113–132.Google Scholar
[CDP] A., Cardona, C., Ducourtioux, S., Paycha, From tracial anomalies to anomalies in quantum field theory, Comm. Math. Phys. 242 (2003), 31–65.Google Scholar
[CFNW] M., Cederwall, G., Ferretti, B., Nilsson, A., Westerberg, Schwinger terms and cohomology of pseudodifferential operators, Comm. Math. Phys. 175 (1996), 203–220.Google Scholar
[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.Google Scholar
[CMM] A., Carey, J., Mickelsson, M., Murray, Index theory, Gerbes, and Hamiltonian quantization, Comm. Math. Phys. 183 (1997), 707–722.Google Scholar
[C1] S.-S, Chern, A simple intrinsic proof of the Gauss–Bonnet formula for closed Riemannian manifolds, Ann. Math. 45 (1944), 747–762.Google Scholar
[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.Google Scholar
[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.Google Scholar
[FGLS] B. V., Fedosov, F., Golse, E., Leichtnam, E., Schrohe, The noncommutative residue for manifolds with boundary, J. Funct. Anal. 142 (1996), 1–31.Google Scholar
[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.Google Scholar
[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.Google Scholar
[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.Google Scholar
[KK] O., Kravchenko, B., Khesin, Central extension of the Lie lagebra of (pseudo)-differential symbols, Funct. Anal. Appl. 25 (1991), 83–85.Google Scholar
[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.Google Scholar
[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.Google Scholar
[Le] J., Leslie, On a differential structure for the group of diffeomorphisms, Topology 6 (1967), 263–271.Google Scholar
[LMR] E., Langmann, J., Mickelsson, S., Rydh, Anomalies and Schwinger terms in NCG field theory models, J. Math. Phys. 42 (2001), 4779.Google Scholar
[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.Google Scholar
[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.Google Scholar
[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.Google Scholar
[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.Google Scholar
[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.Google Scholar
[N2] K.-H., Neeb, Towards a Lie theory of locally convex groups, Jap. J. Math. 1 (2006), 291–468.Google Scholar
[Ok] K., Okikiolu, The multiplicative anomaly for determinants of elliptic operators, Duke Math. J. 79 (1995), 722–749.Google Scholar
[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.Google Scholar
[OP] M.-F., Ouedraogo, S., Paycha, The multiplicative anomaly for determinants revisited; locality. Commun. Math. Anal. 12 (2012) 28–63.Google Scholar
[P1] S., Paycha, Renormalised traces as a looking glass into infinite-dimensional geometryInf. Dim. Anal. Quan. Prob. Rel. Top. 4 (2001), 221–266.Google Scholar
[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.Google Scholar
[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.Google Scholar
[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.Google Scholar
[Po2] R., Ponge, Noncommutative residue for the Heisenberg calculus and applications in CR and contact geometry, J. Funct. Anal. 252 (2007), 399–463.Google Scholar
[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.Google Scholar
[Ra] A. O., Radul, Lie algebras on differential operators, their central extensions, and W-algebras, Funct. Anal. 25 (1991), 33–49.Google Scholar
[Ro] F., Rochon, Sur la topologie de l'espace des opérateurs pseudodifférentiels inversible d'ordre 0, Ann. Inst. Fourier 58 : 1 (2008), 29–62.Google Scholar
[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.Google Scholar
[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.Google Scholar
[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.Google Scholar

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×