Integration of twisted Dirac brackets

Abstract

Given a Lie groupoid G over a manifold M , we show that multiplicative 2forms 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.Motivated by the relationship between symplectic realizations of Poisson manifolds and hamiltonian actions [9], we study presymplectic realizations of twisted Dirac structures. Just as in usual Poisson geometry, these presymplectic realizations carry natural actions of presymplectic groupoids. In fact, it is this property that determines our definition of presymplectic realizations. An important example of twisted Dirac structure is described in [27, Example 4.2]: any nondegenerate invariant inner product on the Lie algebra h of a Lie group H induces a natural Dirac structure on H, twisted by the invariant Cartan 3-form; we call such structures Cartan-Dirac structures. We show that presymplectic realizations of Cartan-Dirac structures are equivalent to the quasi-hamiltonian h-spaces of Alekseev, Malkin and Meinrenken [1] in such a way that the realization maps are the associated groupvalued momentum maps. It also follows from our results that the transformation groupoid H ⋉ H corresponding to the conjugation action carries a canonical twisted presymplectic structure, which we obtain explicitly by "integrating" the Cartan-Dirac structure. As a result, we recover the 2-form on the "double" D(H) of [1] and the AMM groupoid of [4]. (Closely related forms were introduced earlier in [18,30].) A unifying approach to momentum map theories based on Morita equivalence of presymplectic groupoids has been developed by Xu in [34]; much of our motivation for considering quasi-hamiltonian spaces comes from his work. Our results indicate that Dirac structures provide a natural framework for the common description of various notions of momentum maps (as e.g. in [1,22], see also [31]).We illustrate our results on multiplicative 2-forms and Dirac structures in many examples. In the case of action groupoids, we obtain an explicit formula for the natural map from the cohomology of the Cartan model of an H-manifold [2, 16] to (Borel) equivariant cohomology [2,3] in degree three; for the monodromy groupoid of a foliation F, we show that multiplicative 2-forms are closely related to the usual cohomology and spectral sequence of F [19].The entire discussion of relatively closed multiplicative 2-forms on groupoids may be embedded in the more general context of a van Est theorem for the "bar-de Rham" double complex of forms on the simplicial space of composable sequences in a groupoid G, whose total complex computes the cohomology of the classifying space...

On dual pairs in Dirac geometry

“…give a dual pair [10]. However, even if L is not integrable, the Banach manifold P (L) of L-paths carries a canonical two-form ω P (L) which is basic for the homotopy foliation, and, in the integrable case, it is the pullback of ω G(L) .…”

“…Specifically, let φ ∈ Ω 3 (M ) be closed, and consider the φ-twisted Dorfman bracket Twisted Dirac structures go back, in one form or another, to [27,32,34,35]. A crucial example of such structures is given by Cartan-Dirac structures associated to nondegenerate, invariant inner products on the Lie algebra g of a Lie group G [35, Example 4.2], and whose presymplectic realizations correspond to the quasiHamiltonian g-spaces of [1] (see [10]). …”

“…Twisted Dirac structures are in particular Lie algebroids, whose global objects correspond (in the integrable case) to the twisted presymplectic groupoids of [10], or quasi-symplectic groupoids in the language of [41].…”

“…[9,28,39,40]. This line of thought was greatly extended in [41], which introduces Morita equivalence for quasi-symplectic groupoids (or twisted presymplectic groupoids in the sense of [10]), which plays an analogous role for twisted Dirac manifolds as Morita equivalence of symplectic groupoids in Poisson geometry. As in the Poisson case, this notion only makes sense in the integrable case, but some recent 'stacky' versions of Morita equivalence forgo the integrability hypothesis [8,9,12] (cf.…”

Integrability of Jacobi and Poisson structures

“…Examples of "integrations" are given by Lie's third theorem (which integrates finite dimensional Lie algebras), by Palais' work on integrability of infinitesimal Lie algebra actions [18], by Weinstein's symplectic groupoids which integrate Poisson structures [19] (and variations, e.g. Dirac structures [2]) or by the integrability of general "Lie brackets of geometric type", i.e. Lie algebroids [6].…”