We study higher-order analogues of Dirac structures, extending the multisymplectic structures that arise in field theory. We define higher Dirac structures as involutive subbundles of T M + ∧ k T M * satisfying a weak version of the usual lagrangian condition (which agrees with it only when k = 1). Higher Dirac structures transversal to T M recover the higher Poisson structures introduced in [8] as the infinitesimal counterparts of multisymplectic groupoids. We describe the leafwise geometry underlying an involutive isotropic subbundle in terms of a distinguished 1-cocycle in a natural differential complex, generalizing the presymplectic foliation of a Dirac structure. We also identify the global objects integrating higher Dirac structures.1 an isomorphism, and this recovers their well-known presymplectic foliations. We characterize various types of higher Dirac structures in T M + ∧ k T M , showing e.g. that those projecting isomorphically onto T M agree with closed (k + 1)-forms on M , while higher Poisson structures are the same as higher Dirac structures intersecting T M trivially. In Section 5, we relate higher Dirac structure to the theory of Lie groupoids by identifying their global counterparts (Theorem 5.3), extending the integration of Dirac structures by presymplectic groupoids of [10].Remark. The discussion in the paper extends with no extra cost to higher Dirac structures in T M + (∧ k T * M ⊗ R r ), which incorporates poly-symplectic [21] and poly-Poisson structures [24,28], as in [29]. (More generally, one may consider T M + (∧ k T * M ⊗ E) for a vector bundle E equipped with a (partial) flat connection, see [29, Appendix].) For simplicity, we will restrict ourselves to the case r = 1.
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The main idea of this note is to describe the integration procedure for poly-Poisson structures, that is, to find a poly-symplectic groupoid integrating a poly-Poisson structure, in terms of topological field theories, namely via the path-space construction. This will be given in terms of the poly-Poisson sigma model (P P SM ) and we prove that every poly-Poisson structure has a natural integration via relational poly-symplectic groupoids, extending the results in [8] and [26]. We provide familiar examples (trivial, linear, constant and symplectic) within this formulation and we give some applications of this construction regarding the classification of poly-symplectic integrations, as well as Morita equivalence of poly-Poisson manifolds. arXiv:1706.06014v2 [math-ph] 14 Nov 2017
Poly-Poisson manifoldsWe begin with the definition of the main structures to consider in this paper:Definition 2.1. A Poly-Poisson structure of order r, or simply an r-Poisson structure, on a manifold M is a pair (S, P ), where S → M is a vector subbundle of T * M ⊗R r and P : S → T M is a vector-bundle morphism (covering the identity) such that the following conditions hold:(i) i P (η) η = 0, for all η ∈ S,the space of section Γ(S) is closed under the bracket (2.1) η, γ := L P (η) γ − i P (γ) dη for γ, η ∈ Γ(S),
In this paper, we extend the AKSZ formulation of the Poisson sigma model to more general target spaces, and we develop the general theory of graded geometry for poly-symplectic and poly-Poisson structures. In particular, we prove a Schwarz-type theorem and transgression for graded poly-symplectic structures, recovering the action functional and the poly-symplectic structure of the reduced phase space of the poly-Poisson sigma model, from the AKSZ construction.
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