## 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.

### MSC:

 58H05 Pseudogroups and differentiable groupoids 53D17 Poisson manifolds; Poisson groupoids and algebroids 22A22 Topological groupoids (including differentiable and Lie groupoids)
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