zbMATH — the first resource for mathematics

Classification of equivariant star products on symplectic manifolds. (English) Zbl 1341.53123
This paper classifies invariant star products with quantum momentum maps on symplectic manifolds. First, the authors recall basic definitions and results about invariant star products and quantum momentum maps. Then the construction of Fedosov is also briefly recalled. A bijection between the equivalence classes and the formal series in the second equivariant cohomology is established. This result allows to determine whether two given pairs of star products and corresponding quantum momentum maps are equivariantly equivalent.

53D55 Deformation quantization, star products
53D05 Symplectic manifolds, general
Full Text: DOI arXiv
[1] Arnal, D.; Cortet, J.C.; Molin, P.; Pinczon, G., Covariance and geometrical invariance in ∗-quantization, J. Math. Phys., 24, 276-283, (1983) · Zbl 0515.22015
[2] Bayen, F.; Flato, M.; Frønsdal, C.; Lichnerowicz, A.; Sternheimer, D., Deformation theory and quantization, Ann. Phys., 111, 61-151, (1978) · Zbl 0377.53024
[3] Bertelson, M.; Bieliavsky, P.; Gutt, S., Parametrizing equivalence classes of invariant star products, Lett. Math. Phys., 46, 339-345, (1998) · Zbl 0943.53051
[4] Bertelson, M.; Cahen, M.; Gutt, S., Equivalence of star products, Class. Quantum Gravity, 14, a93-a107, (1997) · Zbl 0881.58021
[5] Bordemann, M.; Brischle, M.; Emmrich, C.; Waldmann, S., Phase space reduction for star products: an explicit construction for \({{\mathbb{C} P^{n}}}\), Lett. Math. Phys., 36, 357-371, (1996) · Zbl 0849.58035
[6] Deligne, P., Déformations de l’algèbre des fonctions d’une variété symplectique: comparaison entre Fedosov et dewilde, Lecomte. Sel. Math. New Ser., 1, 667-697, (1995) · Zbl 0852.58033
[7] Dolgushev, V.A., Covariant and equivariant formality theorems, Adv. Math., 191, 147-177, (2005) · Zbl 1116.53065
[8] Fedosov, B.V., A simple geometrical construction of deformation quantization, J. Differ. Geom., 40, 213-238, (1994) · Zbl 0812.53034
[9] Fedosov, B.V.: Deformation Quantization and Index Theory. Akademie, Berlin (1996) · Zbl 0867.58061
[10] Guillemin, V.W., Sternberg, S.: Supersymmetry and Equivariant de Rham Theory. Springer, Berlin (1999) · Zbl 0934.55007
[11] 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) · Zbl 1024.53057
[12] Gutt, S.; Rawnsley, J., Natural star products on symplectic manifolds and quantum moment maps, Lett. Math. Phys., 66, 123-139, (2003) · Zbl 1064.53061
[13] Hamachi, K., Quantum moment maps and invariants for \(G\)-invariant star products, Rev. Math. Phys., 14, 601-621, (2002) · Zbl 1040.53097
[14] Jansen, S.; Neumaier, N.; Schaumann, G.; Waldmann, S., Classification of invariant star products up to equivariant Morita equivalence on symplectic manifolds, Lett. Math. Phys., 100, 203-236, (2012) · Zbl 1251.53057
[15] Kontsevich, M., Deformation quantization of Poisson manifolds, Lett. Math. Phys., 66, 157-216, (2003) · Zbl 1058.53065
[16] Müller-Bahns, M.F.; Neumaier, N., Invariant star products of Wick type: classification and quantum momentum mappings, Lett. Math. Phys., 70, 1-15, (2004) · Zbl 1065.53068
[17] 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) · Zbl 1078.53100
[18] Nest, R.: On some conjectures related to [\(Q\), \(R\)] for Hamiltonian actions on Poisson manifolds. In: Conference Talk at the Workshop on Quantization and Reduction 2013 in Erlangen (2013)
[19] Nest, R.; Tsygan, B., Algebraic index theorem, Commun. Math. Phys., 172, 223-262, (1995) · Zbl 0887.58050
[20] Nest, R.; Tsygan, B., Algebraic index theorem for families, Adv. Math., 113, 151-205, (1995) · Zbl 0837.58029
[21] Neumaier, N.: Klassifikationsergebnisse in der Deformationsquantisierung. PhD thesis, Fakultät für Physik, Albert-Ludwigs-Universität, Freiburg (2001). https://www.freidok.uni-freiburg.de/data/2100. Accessed 12 Mar 2016
[22] Neumaier, N., Local \({ν}\)-Euler derivations and deligne’s characteristic class of Fedosov star products and star products of special type, Commun. Math. Phys., 230, 271-288, (2002) · Zbl 1035.53124
[23] Tsygan, B.: Equivariant deformations, equivariant algebraic index theorems, and a Poisson version of [\(Q\), \(R\)] = 0 (2010). (Unpublished notes) · Zbl 0939.37048
[24] Waldmann S.: Poisson-Geometrie und Deformationsquantisierung. Eine Einführung. Springer, Heidelberg (2007) · Zbl 1139.53001
[25] Weinstein, A., Xu, P.: Hochschild cohomology and characteristic classes for star-products. In: Khovanskij, A., Varchenko, A., Vassiliev, V. (eds.) Geometry of Differential Equations. Dedicated to V. I. Arnold on the Occasion of his 60th Birthday, pp. 177-194. American Mathematical Society, Providence (1998) · Zbl 0837.58029
[26] Xu, P., Fedosov ∗-products and quantum momentum maps, Commun. Math. Phys., 197, 167-197, (1998) · Zbl 0939.37048
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.