Higher vector bundles and multi-graded symplectic manifolds

By J. Grabowski, M. Rotkiewicz. 2007

159 citations recorded by the source corpus.

Abstract A natural explicit condition is given ensuring that an action of the multiplicative monoid of non-negative reals on a manifold F comes from homotheties of a vector bundle structure on F , or, equivalently, from an Euler vector field. This is used in showing that double (or higher) vector bundles present in the literature can be equivalently defined as manifolds with a family of commuting Euler vector fields. Higher vector bundles can be therefore defined as manifolds admitting certain N n -grading in the structure sheaf. Consequently, multi-graded (super)manifolds are canonically associated with higher vector bundles that is an equivalence of categories. Of particular interest are symplectic multi-graded manifolds which are proven to be associated with cotangent bundles. Duality for higher vector bundles is then explained by means of the cotangent bundles as they contain the collection of all possible duals. This gives, moreover, higher generalizations of the known “universal Legendre transformation” T ∗ E ≃ T ∗ E ∗ , identifying the cotangent bundles of all higher vector bundles in duality. The symplectic multi-graded manifolds, equipped with certain homological Hamiltonian vector fields, lead to an alternative to Roytenberg’s picture generalization of Lie bialgebroids, Courant brackets, Drinfeld doubles and can be viewed as geometrical base for higher BRST and Batalin–Vilkovisky formalisms. This is also a natural framework for studying n -fold Lie algebroids and related structures.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7d23c352aca8394768f47e012c6bba3b10cc673d

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Vector bundles over Lie groupoids and algebroids (2014, background): “Lie algebroids) equipped with an additional action of the monoid (R, ·), with natural compatibility conditions, in the spirit of the characterization of vector bundles in [11].”
Lie algebroid structures on double vector bundles and representation theory of Lie algebroids (2008, background): “Recently, Grabowski and Rotkiewicz [6] have studied double and n-fold vector bundles from the supergeometric point of view.”
Towards Integration on Colored Supermanifolds (2016, background): “A trivial example is the direct sum E01 ⊕ E10 ⊕ E11 of three vector bundles over M .”