Mikołaj Rotkiewicz

Higher vector bundles and multi-graded symplectic manifolds

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.

Graded bundles and homogeneity structures

Abstract We introduce the concept of a graded bundle which is a natural generalization of the concept of a vector bundle and whose standard examples are higher tangent bundles T n Q playing a fundamental role in higher order Lagrangian formalisms. Graded bundles are graded manifolds in the sense that we can choose an atlas whose local coordinates are homogeneous functions of degrees 0 , 1 , … , n . We prove that graded bundles have a convenient equivalent description as homogeneity structures , i.e. manifolds with a smooth action of the multiplicative monoid ( R ≥ 0 , ⋅ ) of non-negative reals. The main result states that each homogeneity structure admits an atlas whose local coordinates are homogeneous. Considering a natural compatibility condition of homogeneity structures we formulate, in turn, the concept of a double ( r - tuple , in general) graded bundle –a broad generalization of the concept of a double ( r -tuple) vector bundle. Double graded bundles are proven to be locally trivial in the sense that we can find local coordinates which are simultaneously homogeneous with respect to both homogeneity structures.

Bundle-theoretic methods for higher-order variational calculus

We present a geometric interpretation of the integration-by-parts formula on an arbitrary vector bundle. As an application we give a new geometric formulation of higher-order variational calculus.

A note on actions of some monoids

Smooth actions of the multiplicative monoid (R,⋅) ( R , ⋅ ) of real numbers on manifolds lead to an alternative, and for some reasons simpler, definitions of a vector bundle, a double vector bundle and related structures like a graded bundle (Grabowski and Rotkiewicz (2011) [10] ). For these reasons it is natural to study smooth actions of certain monoids closely related with the monoid (R,⋅) ( R , ⋅ ) . Namely, we discuss geometric structures naturally related with: smooth and holomorphic actions of the monoid of multiplicative complex numbers, smooth actions of the monoid of second jets of punctured maps (R,0)→(R,0) ( R , 0 ) → ( R , 0 ) , smooth actions of the monoid of real 2 by 2 matrices and smooth actions of the multiplicative reals on a supermanifold. In particular cases we recover the notions of a holomorphic vector bundle, a complex vector bundle and a non-negatively graded manifold.

Polarisation of Graded Bundles

We construct the full linearisation functor which takes a graded bundle of degreek (a particular kind of graded manifold) and produces ak-fold vector bundle. We fully characterise the image of the full linearisation functor and show that we obtain a subcategory ofk-fold vector bundles consisting of symmetric k-fold vector bundles equipped with a family of morphisms indexed by the symmetric group Sk. Interestingly, for the degree 2 case this additional structure gives rise to the notion of a symplectical double vector bundle, which is the skew-symmetric analogue of a metric double vector bundle. We also discuss the related case of fully linearising N-manifolds, and how one can use the full linearisation functor to superise a graded bundle.

Duality for Graded Manifolds

We study the notion of duality in the context of graded manifolds. For graded bundles, somehow like in the case of Gelfand representation and the duality: points vs. functions, we obtain natural dual objects which belongs to a different category than the initial ones, namely graded polynomial (co)algebra bundles and free graded Weil (co)algebra bundles. Our results are then applied to obtain elegant characterizations of double vector bundles and graded bundles of degree 2. All these results have their supergeometric counterparts. For instance, we give a simple proof of a nice characterisation of

Prototypes of higher algebroids with applications to variational calculus

N

Double Affine Bundles

-manifolds of degree 2, announced in the literature.