Integration of twisted Dirac brackets
Henrique Bursztyn, Marius Crainic, Alan Weinstein, and Chenchang Zhu
Source: Duke Math. J. Volume 123, Number 3
(2004), 549-607.
Abstract
Given a Lie groupoid G over a manifold M, we show that multiplicative 2-forms on G relatively closed with respect to a closed 3-form ϕ; on M correspond to maps from the Lie algebroid of G into T*M satisfying an algebraic condition and a differential condition with respect to the ϕ-twisted Courant bracket. This correspondence describes, as a special case, the global objects associated to ϕ-twisted Dirac structures. As applications, we relate our results to equivariant cohomology and foliation theory, and we give a new description of quasi-Hamiltonian spaces and group-valued momentum maps.
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Permanent link to this document: http://projecteuclid.org/euclid.dmj/1086957716
Digital Object Identifier: doi:10.1215/S0012-7094-04-12335-8
Mathematical Reviews number (MathSciNet): MR2068969
Zentralblatt MATH identifier: 02103766
References
A. Alekseev, A. Malkin, and E. Meinrenken, Lie group valued moment maps, J. Differential Geom. 48 (1998), 445--495.
Mathematical Reviews (MathSciNet): MR1638045
Project Euclid: euclid.jdg/1214460860
M. F. Atiyah and R. Bott, The moment map and equivariant cohomology, Topology 23 (1984), 1--28.
Mathematical Reviews (MathSciNet): MR0721448
Digital Object Identifier: doi:10.1016/0040-9383(84)90021-1
Zentralblatt MATH: 0521.58025
K. Behrend, P. Xu, and B. Zhang, Equivariant gerbes over compact simple Lie groups, C. R. Math. Acad. Sci. Paris 336 (2003), 251--256.
Mathematical Reviews (MathSciNet): MR1968268
Digital Object Identifier: doi:10.1016/S1631-073X(02)00024-9
Zentralblatt MATH: 1068.58010
R. Bott, H. Shulman, and J. Stasheff, On the de Rham theory of certain classifying spaces, Advances in Math. 20 (1976), 43--56.
Mathematical Reviews (MathSciNet): MR0402769
Digital Object Identifier: doi:10.1016/0001-8708(76)90169-9
Zentralblatt MATH: 0342.57016
H. Bursztyn and O. Radko, Gauge equivalence of Dirac structures and symplectic groupoids, Ann. Inst. Fourier (Grenoble) 53 (2003), 309--337.
Mathematical Reviews (MathSciNet): MR1973074
A. Cannas da Silva and A. Weinstein, Geometric Models for Noncommutative Algebras, Berkeley Math. Lect. Notes 10, Amer. Math. Soc., Providence, 1999.
Mathematical Reviews (MathSciNet): MR1747916
A. S. Cattaneo and G. Felder, ``Poisson sigma models and symplectic groupoids'' in Quantization of Singular Symplectic Quotients, ed. N. P. Landsman, M. Pflaum, and M. Schlichenmaier, Progr. Math. 198, Birkhäuser, Basel, 2001, 61--93.
Mathematical Reviews (MathSciNet): MR1938552
A. S. Cattaneo and P. Xu, Integration of twisted Poisson structures, preprint.}
arXiv: {math.SG/0302268
Mathematical Reviews (MathSciNet): MR2077298
Digital Object Identifier: doi:10.1016/S0393-0440(03)00086-X
Zentralblatt MATH: 1159.53347
A. Coste, P. Dazord, and A. Weinstein, Groupoïdes symplectiques, Publ. Dép. Math. Nouvelle Sér. A 2, Univ. Claude-Bernard, Lyon, 1987, 1--62.
Mathematical Reviews (MathSciNet): MR0996653
T. J. Courant, Dirac manifolds, Trans. Amer. Math. Soc. 319 (1990), 631--661.
Mathematical Reviews (MathSciNet): MR0998124
Digital Object Identifier: doi:10.2307/2001258
Zentralblatt MATH: 0850.70212
T. Courant and A. Weinstein, ``Beyond Poisson structures'' in Actions hamiltoniennes de groupes: Troisème théorème de Lie (Lyon, 1986), Travaux en Cours 27, Hermann, Paris, 1988, 39--49.
Mathematical Reviews (MathSciNet): MR0951168
M. Crainic, Differentiable and algebroid cohomology, van Est isomorphisms, and characteristic classes, Comment. Math. Helv. 78 (2003), 681--721.
Mathematical Reviews (MathSciNet): MR2016690
Digital Object Identifier: doi:10.1007/s00014-001-0766-9
Zentralblatt MATH: 1041.58007
M. Crainic and R. L. Fernandes, Integrability of Lie brackets, Ann. of Math. (2) 157 (2003), 575--620.
Mathematical Reviews (MathSciNet): MR1973056
Digital Object Identifier: doi:10.4007/annals.2003.157.575
JSTOR: links.jstor.org
Zentralblatt MATH: 1037.22003
--------, Integrability of Poisson brackets, preprint.}
arXiv: {math.DG/0210152
M. Crainic and C. Zhu, Integration of Jacobi manifolds, preprint.}
arXiv: {math.DG/0403268
M. Duflo and M. Vergne, ``Cohomologie équivariante et descente'' in Sur la cohomologie équivariante des variétés differentiables, Astérisque 215, Soc. Math. France, Montrouge, 1993, 5--108.
Mathematical Reviews (MathSciNet): MR1247060
P. J. Higgins and K. Mackenzie, Algebraic constructions in the category of Lie algebroids, J. Algebra 129 (1990), 194--230.
Mathematical Reviews (MathSciNet): MR1037400
Digital Object Identifier: doi:10.1016/0021-8693(90)90246-K
Zentralblatt MATH: 0696.22007
J. Huebschmann and L. C. Jeffrey, Group cohomology construction of symplectic forms on certain moduli spaces, Internat. Math. Res. Notices 1994, no. 6, 245--249.
Mathematical Reviews (MathSciNet): MR1277051
Digital Object Identifier: doi:10.1155/S1073792894000279
Zentralblatt MATH: 0816.58017
F. W. Kamber and P. Tondeur, ``Foliations and metrics'' in Differential Geometry (College Park, Md., 1981/1982), Progr. Math. 32, Birkhäuser, Boston, 1983, 103--152.
Mathematical Reviews (MathSciNet): MR0702530
C. Klimčik and T. Strobl, WZW-Poisson manifolds, J. Geom. Phys. 43 (2002), 341--344.
Mathematical Reviews (MathSciNet): MR1929911
Digital Object Identifier: doi:10.1016/S0393-0440(02)00027-X
Zentralblatt MATH: 1027.70023
Z.-J. Liu, A. Weinstein, and P. Xu, Manin triples for Lie bialgebroids, J. Differential Geom. 45 (1997), 547--574.
Mathematical Reviews (MathSciNet): MR1472888
Project Euclid: euclid.jdg/1214459842
J.-H. Lu, ``Momentum mappings and reduction of Poisson actions'' in Symplectic Geometry, Groupoids, and Integrable Systems (Berkeley, 1989), ed. P. Dazord and A. Weinstein, Math. Sci. Res. Inst. Publ. 20, Springer, New York, 1991, 209--226.
Mathematical Reviews (MathSciNet): MR1104930
K. Mackenzie, Lie Groupoids and Lie Algebroids in Differential Geometry, London Math. Soc. Lecture Note Ser. 124, Cambridge Univ. Press, Cambridge, 1987.
Mathematical Reviews (MathSciNet): MR0896907
K. Mikami and A. Weinstein, Moments and reduction for symplectic groupoids, Publ. Res. Inst. Math. Sci. 24 (1988), 121--140.
Mathematical Reviews (MathSciNet): MR0944869
Digital Object Identifier: doi:10.2977/prims/1195175328
Zentralblatt MATH: 0659.58016
I. Moerdijk and J. Mrčun, On integrability of infinitesimal actions, Amer. J. Math. 124 (2002), 567--593.
Mathematical Reviews (MathSciNet): MR1902889
Digital Object Identifier: doi:10.1353/ajm.2002.0019
Zentralblatt MATH: 1013.58010
J.-S. Park, ``Topological open $p$-branes'' in Symplectic Geometry and Mirror Symmetry (Seoul, 2000), World Sci., River Edge, N.J., 2001, 311--384.
Mathematical Reviews (MathSciNet): MR1882334
P. Ševera and A. Weinstein, ``Poisson geometry with a $3$-form background'' in Noncommutative Geometry and String Theory (Yokohama, 2001), ed. Y. Maeda and S. Watamura, Progr. Theoret. Phys. Suppl. 144, Kyoto Univ., Kyoto, 2001, 145--154.
Mathematical Reviews (MathSciNet): MR2023853
A. Weinstein, Symplectic groupoids and Poisson manifolds, Bull. Amer. Math. Soc. (N.S.) 16 (1987), 101--104.
Mathematical Reviews (MathSciNet): MR0866024
Digital Object Identifier: doi:10.1090/S0273-0979-1987-15473-5
Project Euclid: euclid.bams/1183553676
Zentralblatt MATH: 0618.58020
--. --. --. --., Coisotropic calculus and Poisson groupoids, J. Math. Soc. Japan 40 (1988), 705--727.
Mathematical Reviews (MathSciNet): MR0959095
Digital Object Identifier: doi:10.2969/jmsj/04040705
Project Euclid: euclid.jmsj/1230129807
Zentralblatt MATH: 0642.58025
--. --. --. --., ``The symplectic structure on moduli space'' in The Floer Memorial Volume, ed. H. Hofer, C. H. Taubes, A. Weinstein, and E. Zehnder, Progr. Math. 133, Birkhäuser, Basel, 1995, 627--635.
Mathematical Reviews (MathSciNet): MR1362845
--------, The geometry of momentum, to appear in Geometry in the 20th Century: 1930--2000 (Paris, 2001), preprint.
arXiv: math.SG/0208108
A. Weinstein and P. Xu, Extensions of symplectic groupoids and quantization, J. Reine Angew. Math. 417 (1991), 159--189.
Mathematical Reviews (MathSciNet): MR1103911
P. Xu, Dirac submanifolds and Poisson involutions, Ann. Sci. École Norm. Sup. (4) 36 (2003), 403--430.
Mathematical Reviews (MathSciNet): MR1977824
Digital Object Identifier: doi:10.1016/S0012-9593(03)00013-2
Zentralblatt MATH: 1047.53052
--------, Momentum maps and Morita equivalence, preprint.}
arXiv: {math.SG/0307319
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