Multisymplectic Geometry and Lie Groupoids

Bursztyn, Henrique; Cabrera, Alejandro; Iglesias, David

doi:10.1007/978-1-4939-2441-7_3

Cited by 9 publications

(19 citation statements)

References 35 publications

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“…a closed (k + 1)-form ω which is non-degenerate in the sense that the vector bundle morphism T N → ∧ k T * N, X → i X ω is an embedding. As expected, degree one multisymplectic NQ-manifolds are equivalent to Lie algebroids equipped with an IM multisymplectic structure, also called higher Poisson structure in [4]. The latter are infinitesimal counterparts of multisymplectic groupoids.…”

Section: 2supporting

confidence: 55%

“…The latter have been recently introduced in [11] (see also [25]) as infinitesimal counterparts of multiplicative vector valued forms on Lie groupoids. In particular, degree one multisymplectic NQ-manifolds are equivalent to Lie algebroids equipped with an IM multisymplectic structure [4] (Theorem 39). We stress that we do only consider differential forms with values in vector bundles generated in one single degree (which, up to a shift, are actually generated in degree zero).…”

Section: Introductionmentioning

confidence: 99%

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Vector bundle valued differential forms on ℕQ-manifolds

Vitagliano

2016

Pacific J. Math.

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Abstract. Geometric structures on NQ-manifolds, i.e. non-negatively graded manifolds with an homological vector field, encode non-graded geometric data on Lie algebroids and their higher analogues. A particularly relevant class of structures consists of vector bundle valued differential forms. Symplectic forms, contact structures and, more generally, distributions are in this class. We describe vector bundle valued differential forms on non-negatively graded manifolds in terms of non-graded geometric data. Moreover, we use this description to present, in a unified way, novel proofs of known results, and new results about degree one NQ-manifolds equipped with certain geometric structures, namely symplectic structures, contact structures, involutive distributions (already present in literature) and locally conformal symplectic structures, and generic vector bundle valued higher order forms, in particular presymplectic and multisymplectic structures (not yet present in literature).

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Section: 2supporting

confidence: 55%

Section: Introductionmentioning

confidence: 99%

Vector bundle valued differential forms on ℕQ-manifolds

Vitagliano

2016

Pacific J. Math.

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“…Higher analogues of Dirac structures have been considered in field theory [33], in Nambu geometry [22,5], as well as in the study of p-branes in string theory [6]; a more systematic treatment is developed in [34], which was one of the motivations for our work. Here we present another viewpoint to the subject, inspired by the theory of Lie groupoids: as discussed in [8], just as Poisson structures are infinitesimal versions of symplectic groupoids [15] (analogously to how Lie algebras linearize Lie groups), one is led to a natural notion of higher Poisson structure by considering the infinitesimal counterparts of multisymplectic groupoids (i.e., Lie groupoids equipped with compatible higher-order symplectic structures). In this paper, we take such higher Poisson structures as the starting point to develop a notion of higher Dirac structure.…”

Section: Introductionmentioning

confidence: 99%

“…For example, such subbundles satisfying L ∩ ∧ k T * M = {0} correspond to closed (k + 1)-forms on M . The condition L ∩ ∧ k T * M = {0}, however, falls short of describing higher Poisson structures: as observed in [8], these are not given by lagrangian subbundles. This led us to develop a new viewpoint to higher Dirac structures that weakens the lagrangian condition, in such a way that the resulting notion encompasses both closed higher-degree forms and higher Poisson structures, hence displaying a richer collection of examples.…”

Section: Introductionmentioning

confidence: 99%

“…This paper is organized as follows. We first consider higher Poisson structures [8] in Section 2; we illustrate how they arise from multisymplectic structures in the presence of symmetries and provide a natural example in field theory, analogous to the Poisson brackets of classical mechanics. We then introduce higher Dirac structures at the linear level, in Section 3.…”

Section: Introductionmentioning

confidence: 99%

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On Higher Dirac Structures

Bursztyn

Alba

Rubio

2017

International Mathematics Research Notices

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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. Contents

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Drinfel’d double of bialgebroids for string and M theories: dual calculus framework

Çatal-Özer,

Doğan,

Yetişmişoğlu

2024

J. High Energ. Phys.

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We extend the notion of Lie bialgebroids for more general bracket structures used in string and M theories. We formalize the notions of calculus and dual calculi on algebroids. We achieve this by reinterpreting the main results of the matched pairs of Leibniz algebroids. By examining a rather general set of fundamental algebroid axioms, we present the compatibility conditions between two calculi on vector bundles which are not dual in the usual sense. Given two algebroids equipped with calculi satisfying the compatibility conditions, we construct its double on their direct sum. This generalizes the Drinfel’d double of Lie bialgebroids. We discuss several examples from the literature including exceptional Courant brackets. Using Nambu-Poisson structures, we construct an explicit example, which is important both from physical and mathematical point of views. This example can be considered as the extension of triangular Lie bialgebroids in the realm of higher Courant algebroids, that automatically satisfy the compatibility conditions. We extend the Poisson generalized geometry by defining Nambu-Poisson exceptional generalized geometry and prove some preliminary results in this framework. We also comment on the global picture in the framework of formal rackoids and we slightly extend the notion for vector bundle valued metrics.

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Multisymplectic Geometry and Lie Groupoids

Cited by 9 publications

References 35 publications

Vector bundle valued differential forms on ℕQ-manifolds

Vector bundle valued differential forms on ℕQ-manifolds

On Higher Dirac Structures

Drinfel’d double of bialgebroids for string and M theories: dual calculus framework

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