Infinite dimensional moment map geometry and closed Fedosov’s star products

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• 作者：Laurent La Fuente-Gravy
• 关键词：Symplectic connections ; Moment map ; Deformation quantization ; Closed star products ; Kähler manifolds ; 53D20 ; 53C21 ; 32Q15 ; 53D55
• 刊名：Annals of Global Analysis and Geometry
• 出版年：2016
• 出版时间：January 2016
• 年：2016
• 卷：49
• 期：1
• 页码：1-22
• 全文大小：545 KB
• 参考文献：1.Atiyah, M.F., Bott, R.: The Yang-Mills equations over Riemann surfaces. Philos. Trans. R. Soc. London 308, 523–615 (1982)MathSciNet CrossRef
  2.Aubin, T.: Nonlinear Analysis on Manifolds. Springer, New York (1982)MATH
  3.Banyaga, A.: Sur la structure du groupe des difféomorphismes qui préservent une forme symplectique. Comment. Math. Helv. 53, 174–227 (1978)MATH MathSciNet CrossRef
  4.Bayen, F., Flato, M., Fronsdal, C., Lichnérowicz, A., Sternheimer, D.: Deformation theory and quantization, Ann. Phys. 111, part I : 61–110, part II : 111–151 (1978)
  5.Bordemann, M., Waldmann, S.: A Fedosov star product of the wick type for Kähler manifolds. Lett. Math. Phys. 41(3), 243–253 (1997)MATH MathSciNet CrossRef
  6.Cahen, M., Gutt, S.: Moment map for the space of symplectic connections. In: Brackx, F., De Schepper, H. (eds.) Liber Amicorum Delanghe, pp. 27–36. Academia Press, Gent (2005)
  7.Calabi, E.: Extremal Kähler metrics II, Differerential Geometry and Complex Analysis, pp. 95–114. Springer, New York (1985)CrossRef
  8.Connes, A., Flato, M., Sternheimer, D.: Closed star products and cyclic cohomology. Lett. Math. Phys 24, 1–12 (1992)MATH MathSciNet CrossRef
  9.De Wilde, M., Lecomte, P.B.A.: Existence of star-products and of formal deformations of the Poisson Lie Algebra of arbitrary symplectic manifolds. Lett. Math. Phys. 7, 487–496 (1983)MATH MathSciNet CrossRef
  10.Donaldson, S.K.: Moment maps and diffeomorphisms. Asian J. Math. 3(1), 1–16 (1999)MATH MathSciNet
  11.Donaldson, S.K.: Remarks on Gauge Theory, Complex Geometry and 4- Manifold Topology, Fields Medalists’ Lectures. World Science Publisher, Singapore (1997)
  12.Fedosov, B.V.: A simple geometrical construction of deformation quantization. J. Differ. Geom. 40, 213–238 (1994)MATH MathSciNet
  13.Fedosov, B.V.: Deformation Quantization and Index Theory, Mathematical Topics, vol. 9. Akademie Verlag, Berlin (1996)
  14.Fedosov, B.V.: Quantization and the index. Dokl. Akad. Nauk. SSSR 291, 82–86 (1986)MathSciNet
  15.Fedosov, B.V.: On the trace density in deformation quantization, in Deformation quantization (Strasbourg, 2001), vol. 1 of IRMA Lect. Math. Theor. Phys., de Gruyter, Berlin, 67–83 (2002)
  16.Fox D.J.F.: Extremal symplectic connections on surfaces, arXiv:​1410.​1468 (2014)
  17.Fujiki, A.: The moduli spaces and Kähler metrics of polarized algebraic varieties. Sugaku Expos. 5(2), 173–191 (1992)MathSciNet
  18.Gutt, S., Rawnsley, J.: Natural star products on symplectic manifolds and quantum moment maps. Lett. Math. Phys. 66, 123–139 (2003)MATH MathSciNet CrossRef
  19.Gutt, S., Rawnsley, J.: Traces for star products on symplectic manifolds. J. Geom. Phys. 42, 12–18 (2002)MATH MathSciNet CrossRef
  20.Karabegov, A.V.: On the canonical normalization of a trace density of deformation quantization. Lett. Math. Phys. 45, 217–228 (1998)MATH MathSciNet CrossRef
  21.Kontsevitch, M.: Deformation Quantization of Poisson Manifolds. I. Preprint arXiv:​q-alg/​9709040 , September (1997)
  22.Nest, R., Tsygan, B.: Algebraic index theorem for families. Adv. Math. 113, 151–205 (1995)MATH MathSciNet CrossRef
  23.Omori, H., Maeda, Y., Yoshioka, A.: Weyl manifolds and deformation quantization. Adv. Math. 85, 224–255 (1991)MATH MathSciNet CrossRef
  
• 作者单位：Laurent La Fuente-Gravy (1)
  
  1. Institut de Recherche en Mathématique et Physique, Université catholique de Louvain, Chemin du cyclotron, 2, 1348, Louvain-la-Neuve, Belgium
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Analysis
  Statistics for Business, Economics, Mathematical Finance and Insurance
  Mathematical and Computational Physics
  Group Theory and Generalizations
  Geometry
  
• 出版者：Springer Netherlands
• ISSN：1572-9060

文摘

We study the Cahen–Gutt moment map on the space of symplectic connections of a symplectic manifold. Given a Kähler manifold \((M,\omega ,J)\), we define a Calabi-type functional \(\mathscr {F}\) on the space \(\mathcal {M}_{\Theta }\) of Kähler metrics in the class \(\Theta :=[\omega ]\). We study the space of zeroes of \(\mathscr {F}\). When \((M,\omega ,J)\) has non-negative Ricci tensor and \(\omega \) is a zero of \(\mathscr {F}\), we show the space of zeroes of \(\mathscr {F}\) near \(\omega \) has the structure of a smooth finite dimensional submanifold. We give a new motivation, coming from deformation quantization, for the study of moment maps on infinite dimensional spaces. More precisely, we establish a strong link between trace densities for star products (obtained from Fedosov’s type methods) and moment map geometry on infinite dimensional spaces. As a byproduct, we provide, on certain Kähler manifolds, a geometric characterization of a space of Fedosov’s star products that are closed up to order 3 in \(\nu \). Keywords Symplectic connections Moment map Deformation quantization Closed star products Kähler manifolds