Integration of twisted Dirac brackets.

*(English)*Zbl 1067.58016Dirac 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.

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.

Reviewer: Bernd Ammann (Vandœuvre-les-Nancy)

##### MSC:

58H05 | Pseudogroups and differentiable groupoids |

53D17 | Poisson manifolds; Poisson groupoids and algebroids |

22A22 | Topological groupoids (including differentiable and Lie groupoids) |

##### Keywords:

Dirac structures; Poisson structures; twisted Dirac structures; integration of Lie algebroids
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\textit{H. Bursztyn} et al., Duke Math. J. 123, No. 3, 549--607 (2004; Zbl 1067.58016)

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