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Integration of twisted Dirac brackets. (English) Zbl 1067.58016
Dirac structures were introduced by T. J. Courant [Trans. Am. Math. Soc. 319, No. 2, 631–661 (1990; Zbl 0850.70212), and with A. Weinstein [Trav. Cours 27, 39–49 (1988; Zbl 0698.58020)] as a generalization of Poisson structures, presymplectic forms and regular foliations. Courant’s main motivation was to study constrained mechanical systems. Connection of Poisson structures to topological sigma-models led to the definition of Poisson structures twisted by a closed \(3\)-form and more generally twisted Dirac structures [P. Severa and A. Weinstein, Prog. Theor. Phys., Suppl. 144, 145–154 (2001; Zbl 1029.53090)].
In [M. Crainic and R. L. Fernandes, Ann. Math. (2) 157, No. 2, 575–620 (2003; Zbl 1037.22003)] it was shown that many Lie algebroids are integrable, i.e. there is a Lie groupoid such that the given Lie algebroid is the Lie algebroid to this Lie groupoid. This result by Crainic and Fernandes should be viewed as a generalization of Lie’s third theorem stating that an abstract Lie algebra is always the Lie algebra to a Lie group. The integration of such a Lie algebroid is not only an enourmous conceptual progress, but it rapidly turned out that this integration is very useful for solving analytical problems on non-compact, complete manifolds and on foliated manifolds.
In a similar way, it was shown in [A. Coste, P. Dazord and A. Weinstein, Publ. Dép. Math., Nouv. Sér., Univ. Claude Bernard, Lyon 2/A, 1–62 (1987; Zbl 0668.58017), M. Crainic and R. L. Fernandes, J. Differ. Geom. 66, No. 1, 71–137 (2004; Zbl 1066.53131)] that the infinitesimal object “integrable (twisted) Poisson structure” could be integrated to a (twisted) symplectic groupoid.
In view of this, it is natural to ask for an integration of the more general twisted Dirac structures. This is the subject of the article under review. A twisted Dirac structure gives rise to a Lie algebroid structure. This Lie algebroid structure can be integrated to a Lie groupoid. The main task is now to determine the additional structure on the Lie groupoid that arises by integrating the twisted Dirac structure. A Lie groupoid with such an additional structure is called \(\phi\)-twisted presymplectic groupoid.
To give some more details: Let \(G\) be a \(2n\)-dimensional Lie groupoid over an \(n\)-dimensional manifold \(M\) with target map \(t\) and source map \(s\). We assume that \(\phi\) is a closed \(3\)-form on \(M\). Then \((G,\omega,\phi)\) is called a \(\phi\)-twisted presymplectic groupoid if \(\omega\) is a multiplicative \(2\)-form on \(G\) such that \[ d\omega = s^*\phi -t^*\phi \] and \[ \text{ ker\;}(\omega_x)\cap \text{ ker}(ds)_x \cap \text{ ker}(dt)_x=\{0\} \] for all \(x\in M\).
The main result of the article is that modulo integrability issues, there is a \(\phi\)-twisted presymplectic groupoid for any \(\phi\)-twisted Dirac structure, and that \(\phi\)-twisted Dirac structure arise infinitesimally from \(\phi\)-twisted presymplectic groupoids. This correspondence is bijective if we only consider \(s\)-simply connected Lie groupoids.
The article starts with a well written introduction to the subjects, which provides motivation, the historical background and suitable references. Section 2 gives definitions and basic properties of the main objects of the article such as “twisted Dirac structures”, “groupoids” and “multiplicative \(2\)-forms”. At the end of this section, the main results are properly stated. The following sections are devoted to the proof. After having proved several technical preliminaries about multiplicative \(2\)-forms in section 3, the authors describe the passage from twisted presymplectic groupoids to twisted Dirac structures (section 4), and the inverse procedure in section 5.
Section 6 is mainly devoted to examples. The example of Cartan-Dirac structures on Lie groups leads to section 7, in which the associated Alekseev-Malkin-Meinrenken groupoid is studied. Finally, section 8 studies multiplicative \(2\)-forms on foliation groupoids, and the connections to the spectral sequence of this foliation.

58H05 Pseudogroups and differentiable groupoids
53D17 Poisson manifolds; Poisson groupoids and algebroids
22A22 Topological groupoids (including differentiable and Lie groupoids)
Full Text: DOI
[1] A. Alekseev, A. Malkin, and E. Meinrenken, Lie group valued moment maps , J. Differential Geom. 48 (1998), 445–495. · Zbl 0948.53045
[2] M. F. Atiyah and R. Bott, The moment map and equivariant cohomology , Topology 23 (1984), 1–28. · Zbl 0521.58025
[3] K. Behrend, P. Xu, and B. Zhang, Equivariant gerbes over compact simple Lie groups , C. R. Math. Acad. Sci. Paris 336 (2003), 251–256. · Zbl 1068.58010
[4] R. Bott, H. Shulman, and J. Stasheff, On the de Rham theory of certain classifying spaces , Advances in Math. 20 (1976), 43–56. · Zbl 0342.57016
[5] H. Bursztyn and O. Radko, Gauge equivalence of Dirac structures and symplectic groupoids , Ann. Inst. Fourier (Grenoble) 53 (2003), 309–337. · Zbl 1026.58019
[6] A. Cannas da Silva and A. Weinstein, Geometric Models for Noncommutative Algebras , Berkeley Math. Lect. Notes 10 , Amer. Math. Soc., Providence, 1999. · Zbl 1135.58300
[7] 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. · Zbl 1038.53074
[8] A. S. Cattaneo and P. Xu, Integration of twisted Poisson structures , preprint. · Zbl 1159.53347
[9] A. Coste, P. Dazord, and A. Weinstein, Groupoïdes symplectiques , Publ. Dép. Math. Nouvelle Sér. A 2 , Univ. Claude-Bernard, Lyon, 1987, 1–62. · Zbl 0668.58017
[10] T. J. Courant, Dirac manifolds , Trans. Amer. Math. Soc. 319 (1990), 631–661. · Zbl 0850.70212
[11] 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. · Zbl 0698.58020
[12] M. Crainic, Differentiable and algebroid cohomology, van Est isomorphisms, and characteristic classes , Comment. Math. Helv. 78 (2003), 681–721. · Zbl 1041.58007
[13] M. Crainic and R. L. Fernandes, Integrability of Lie brackets , Ann. of Math. (2) 157 (2003), 575–620. JSTOR: · Zbl 1037.22003
[14] ——–, Integrability of Poisson brackets , preprint.
[15] M. Crainic and C. Zhu, Integration of Jacobi manifolds , preprint.
[16] 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. · Zbl 0795.57014
[17] P. J. Higgins and K. Mackenzie, Algebraic constructions in the category of Lie algebroids , J. Algebra 129 (1990), 194–230. · Zbl 0696.22007
[18] 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. · Zbl 0816.58017
[19] 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.
[20] C. Klimčik and T. Strobl, WZW-Poisson manifolds , J. Geom. Phys. 43 (2002), 341–344. · Zbl 1027.70023
[21] Z.-J. Liu, A. Weinstein, and P. Xu, Manin triples for Lie bialgebroids , J. Differential Geom. 45 (1997), 547–574. · Zbl 0885.58030
[22] 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. · Zbl 0735.58004
[23] K. Mackenzie, Lie Groupoids and Lie Algebroids in Differential Geometry , London Math. Soc. Lecture Note Ser. 124 , Cambridge Univ. Press, Cambridge, 1987. · Zbl 0683.53029
[24] K. Mikami and A. Weinstein, Moments and reduction for symplectic groupoids , Publ. Res. Inst. Math. Sci. 24 (1988), 121–140. · Zbl 0659.58016
[25] I. Moerdijk and J. Mrčun, On integrability of infinitesimal actions , Amer. J. Math. 124 (2002), 567–593. · Zbl 1013.58010
[26] J.-S. Park, “Topological open \(p\)-branes” in Symplectic Geometry and Mirror Symmetry (Seoul, 2000) , World Sci., River Edge, N.J., 2001, 311–384. · Zbl 1024.81043
[27] 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. · Zbl 1029.53090
[28] A. Weinstein, Symplectic groupoids and Poisson manifolds , Bull. Amer. Math. Soc. (N.S.) 16 (1987), 101–104. · Zbl 0618.58020
[29] –. –. –. –., Coisotropic calculus and Poisson groupoids , J. Math. Soc. Japan 40 (1988), 705–727. · Zbl 0642.58025
[30] –. –. –. –., “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. · Zbl 0834.58011
[31] ——–, The geometry of momentum , to appear in Geometry in the 20th Century: 1930–2000 (Paris, 2001) ,
[32] A. Weinstein and P. Xu, Extensions of symplectic groupoids and quantization , J. Reine Angew. Math. 417 (1991), 159–189. · Zbl 0722.58021
[33] P. Xu, Dirac submanifolds and Poisson involutions , Ann. Sci. École Norm. Sup. (4) 36 (2003), 403–430. · Zbl 1047.53052
[34] ——–, Momentum maps and Morita equivalence , preprint. · Zbl 1106.53057
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