Abstract
Chern–Weil and Chern–Simons theory extend to certain infinite-rank bundles that appear in mathematical physics. We discuss what is known of the invariant theory of the corresponding infinite-dimensional Lie groups. We use these techniques to detect cohomology classes for spaces of maps between manifolds and for diffeomorphism groups of manifolds.
To Joe Wolf, with great appreciation
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References
Adler, M., van Moerbeke, P., and Vanhaecke, P., Algebraic Integrability, Painlevé Geometry and Lie Algebras, Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics, Vol. 47, Springer-Verlag, Berlin, 2004.
Atiyah, M., Circular symmetry and stationary phase approximation, Astérisque 131 (1984), 43–59.
Atiyah, M. and Singer, I. M., The index of elliptic operators. IV., Annals of Math. 93 (1971), 119–138.
Berline, N., Getzler, E., and Vergne, M., Heat Kernels and Dirac Operators, Grundlehren der Mathematischen Wissenschaften 298, Springer-Verlag, Berlin, 1992.
Bismut, J. M., The Atiyah-Singer index theorem for families of Dirac operators: two heat equation proofs, Inventiones Math. 83 (1986), 91–151.
Chern, S.-S., Complex Manifolds without Potential Theory, Springer-Verlag, New York, 1979.
Chern, S.-S. and Simons, J., Characteristic forms and Geometric Invariants, Annals of Math. 99 (1974), no. 1, 48–69.
Chevalley, C. and Eilenberg, S., Cohomology groups of Lie groups and Lie algebras, Trans. AMS 63 (1948), 85–124.
Dieudonné, J., A History of Algebraic and Differential Topology 1900–1960, Birkhäuser, Boston, 1989.
Dupont, J. L., Curvature and Characteristic Classes, Lect. Notes Math. 640, Springer-Verlag, Berlin, 1978.
Eells, J., A setting for global analysis, Bull. Amer. Math. Soc. 72 (1966), 751–807.
Fedosov, B., Golse, F., Leichtnam, E., and Schrohe, E., The noncommutative residue for manifolds with boundary, J. Funct. Analysis 142 (1996), 1–31.
Freed, D., Geometry of Loop Groups, J. Diff. Geom. 28 (1988), 223–276.
Garland, H., Murray, M. K., Kac-Moody algebras and periodic instantons, Commun. Math. Phys. 120 (1988), 335–351.
Gauntlett, J.P., Martelli, D., Sparks, J., Waldram, D., Sasaki-Einstein metrics on S 2 ×S 3, Adv. Theor. Math. Phys. 8 (2004), 711, hep–th/0403002.
Guest, M., Harmonic Maps, Loop Groups, and Integrable Systems, LMS Student Texts, Vol. 38, Cambridge U. Press, Cambridge, 1997.
Hamilton, R., Nash-Moser implicit function theorems, Bull. Amer. Math. Soc. 7 (1986), 65–222.
Hatcher, A., Algebraic Topology, Cambridge U. Press, Cambridge, UK, 2002, www.math.cornell.edu/~hatcher/AT/ATpage.html.
Husemoller, D., Fibre Bundles, 1st ed., Springer-Verlag, New York, 1966.
Larrain-Hubach, A., Explicit computations of the symbols of order 0 and -1 of the curvature operator of \(\Omega G\), Letters in Math. Phys. 89 (2009), 265–275.
Larrain-Hubach, A., Paycha, S., Rosenberg, S., and Scott, S., in preparation.
Larrain-Hubach, A., Rosenberg, S., Scott, S., and Torres-Ardila, F., in preparation.
__________ , Characteristic classes and zeroth order pseudodifferential operators, Spectral Theory and Geometric Analysis, Contemporary Mathematics, Vol. 532, AMS, 2011.
Lawson, H. Blaine and Mickelsohn, M., Spin Geometry, Princeton U. Press, Princeton, NJ, 1989.
Lesch, M. and Neira Jimenez, C., Classification of traces and hypertraces on spaces of classical pseudodifferential operators, arXiv:1011.3238.
Lescure, J.-M. and Paycha, S., Uniqueness of multiplicative determinants on elliptic pseudodifferential operators, Proc. London Math. Soc. 94 (2007), 772–812.
Maeda, Y., Rosenberg, S., and Torres-Ardila, F., Riemannian geometry on loop spaces, arXiv:0705.1008.
Milnor, J., Characteristic Classes, Princeton U. Press, Princeton, 1974.
Misiolek, G., Rosenberg, S., and Torres-Ardila, F., in preparation.
Murray, M. K. and Vozzo, R., The caloron correspondence and higher string classes for loop groups, J. Geom. Phys. 60 (2010), 1235–1250.
Omori, H., Infinite-Dimensional Lie Groups, A.M.S., Providence, RI, 1997.
Paycha, S., Chern-Weil calculus extended to a class of infinite dimensional manifolds, arXiv:0706.2554.
Ponge, R., Traces on pseudodifferential operators and sums of commutators, arXiv:0607.4265.
Pressley, A. and Segal, G., Loop Groups, Oxford University Press, New York, NY, 1988.
Reyman, A.G. and Semenov-Tian-Shansky, M.A., Integrable Systems II: Group-Theoretical Methods in the Theory of Finite-Dimensional Integrable Systems, Dynamical systems. VII, Encyclopaedia of Mathematical Sciences, Vol. 16, Springer-Verlag, Berlin, 1994.
Rochon, F., Sur la topologie de l’espace des opérateurs pseudodifférentiels inversibles d’ordre 0, Ann. Inst. Fourier 58 (2008), 29–62.
Rosenberg, S., The Laplacian on a Riemannian Manifold, Cambridge U. Press, Cambridge, UK, 1997.
Scott, S., Traces and Determinants of Pseudodifferential Operators, Oxford U. Press, Oxford, 2010.
Witten, E., Quantum field theory and the Jones polynomial, Commun. Math. Phys. 121 (1989), 351–399.
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Rosenberg, S. (2013). Chern–Weil Theory for Certain Infinite-Dimensional Lie Groups. In: Huckleberry, A., Penkov, I., Zuckerman, G. (eds) Lie Groups: Structure, Actions, and Representations. Progress in Mathematics, vol 306. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4614-7193-6_15
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