Abstract
In this note we consider a quantum reduction scheme in deformation quantization on symplectic manifolds proposed by Bordemann, Herbig and Waldmann based on BRST cohomology. We explicitly construct the induced map on equivalence classes of star products which will turn out to be an analogue to the Kirwan map in the Cartan model of equivariant cohomology. As a byproduct, we shall see that every star product on a (suitable) reduced manifold is equivalent to a reduced star product.
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References
Atiyah, M.F.: Complex analytic connections in fibre bundles. Trans. Am. Math. Soc. 85, 181–207 (1957)
Bayen, F., Flato, M., Frønsdal, C., Lichnerowicz, A., Sternheimer, D.: Deformation theory and quantization. Ann. Phys. 111, 61–151 (1978)
Bertelson, M., Cahen, M., Gutt, S.: Equivalence of star products. Class. Quantum Gravity 14, A93–A107 (1997)
Bordemann, M.: (Bi)Modules, morphisms, and reduction of star-products: the symplectic case, foliations, and obstructions. Trav. Math. 16, 9–40 (2005)
Bordemann, M., Herbig, H.-C., Waldmann, S.: BRST cohomology and phase space reduction in deformation quantization. Commun. Math. Phys. 210, 107–144 (2000)
Bott, R., Tu, L.W.: Differential Forms in Algebraic Topology. Graduate Texts in Mathematics, vol. 82. Springer, New York (1982)
Cartan, H.: La transgression dans un groupe de Lie et dans un espace fibré principal. In: Colloque de topologie (espaces fibrés), Bruxelles, 1950, pp. 57–71. Georges Thone, Liège; Masson et Cie., Paris (1951)
Cattaneo, A.S., Felder, G.: Coisotropic submanifolds in Poisson geometry and branes in the Poisson sigma model. Lett. Math. Phys. 69, 157–175 (2004)
Cattaneo, A.S., Felder, G.: Relative formality theorem and quantisation of coisotropic submanifolds. Adv. Math. 208, 521–548 (2007)
Deligne, P.: Déformations de l’Algèbre des Fonctions d’une Variété Symplectique: Comparaison entre Fedosov et DeWilde, Lecomte. Sel. Math. New Ser. 1(4), 667–697 (1995)
Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950)
Duistermaat, J.J., Kolk, J.A.C.: Lie Groups. Springer, Berlin (2000)
Fedosov, B.V.: A simple geometrical construction of deformation quantization. J. Differ. Geom. 40, 213–238 (1994)
Fedosov, B.V.: Deformation Quantization and Index Theory. Akademie Verlag, Berlin (1996)
Forger, M., Kellendonk, J.: Classical BRST cohomology and invariant functions on constraint manifolds I. Commun. Math. Phys. 143, 235–251 (1992)
Glößner, P.: Star product reduction for coisotropic submanifolds of codimension 1. Preprint Freiburg FR-THEP-98/10. arXiv:math.QA/9805049 (1998)
Gotay, M.J., Tuynman, G.M.: \({{R}}^{2n}\) is a universal symplectic manifold for reduction. Lett. Math. Phys. 18(1), 55–59 (1989)
Guillemin, V.W., Sternberg, S.: Supersymmetry and Equivariant de Rham Theory. Springer, Berlin (1999)
Gutt, S., Rawnsley, J.: Equivalence of star products on a symplectic manifold; an introduction to Deligne’s Čech cohomology classes. J. Geom. Phys. 29, 347–392 (1999)
Gutt, S., Waldmann, S.: Involutions and representations for reduced quantum algebras. Adv. Math. 224, 2583–2644 (2010)
Hamachi, K.: Quantum moment maps and invariants for \(G\)-invariant star products. Rev. Math. Phys. 14(6), 601–621 (2002)
Kalkman, J.: BRST model applied to symplectic geometry. Ph.D. thesis, Utrecht U. (1993)
Kirwan, F.: Cohomology of Quotients in Symplectic and Algebraic Geometry. Mathematical Notes. Princeton University Press, Princeton (1984)
Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry I. Interscience Tracts in Pure and Applied Mathematics, vol. 15. Wiley, New York (1963)
Kontsevich, M.: Deformation quantization of Poisson manifolds. Lett. Math. Phys. 66, 157–216 (2003)
Lu, J.-H.: Moment maps at the quantum level. Commun. Math. Phys. 157, 389–404 (1993)
Marsden, J., Weinstein, A.: Reduction of symplectic manifolds with symmetry. Rep. Math. Phys. 5, 121–130 (1974)
Müller-Bahns, M.F., Neumaier, N.: Invariant star products of wick type: classification and quantum momentum mappings. Lett. Math. Phys. 70, 1–15 (2004)
Müller-Bahns, M.F., Neumaier, N.: Some remarks on \(\mathfrak{g}\)-invariant Fedosov star products and quantum momentum mappings. J. Geom. Phys. 50, 257–272 (2004)
Nest, R., Tsygan, B.: Algebraic index theorem. Commun. Math. Phys. 172, 223–262 (1995)
Nicolaescu, L.I.: On a theorem of Henri Cartan concerning the equivariant cohomology. An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.) 45.1, 17–38 (1999)
Reichert, T., Waldmann, S.: Classification of equivariant star products on symplectic manifolds. Lett. Math. Phys. 106, 675–692 (2016)
Waldmann, S.: Poisson-Geometrie und Deformationsquantisierung. Eine Einführung. Springer, Heidelberg (2007)
Weinstein, A., Xu, P.: Hochschild cohomology and characteristic classes for star-products. In: Khovanskij, A., Varchenko, A., Vassiliev, V. (eds.) Geometry of Differential Equations, pp. 177–194. American Mathematical Society, Providence (1998). Dedicated to V. I. Arnold on the occasion of his 60th birthday
Xu, P.: Fedosov \(*\)-products and quantum momentum maps. Commun. Math. Phys. 197, 167–197 (1998)
Acknowledgements
The author would like to thank Stefan Waldmann for numerous helpful discussions, James Stasheff for useful advice on the preprint, the anonymous referees for their constructive input, Jonas Schnitzer for assistance with proofreading the text as well as Marco Benini and Alexander Schenkel for valuable remarks.
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Reichert, T. Characteristic classes of star products on Marsden–Weinstein reduced symplectic manifolds. Lett Math Phys 107, 643–658 (2017). https://doi.org/10.1007/s11005-016-0921-z
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DOI: https://doi.org/10.1007/s11005-016-0921-z
Keywords
- Deformation quantization
- Marsden–Weinstein reduction
- Quantum momentum map
- Equivariant cohomology
- Cartan model