Related papers: Introduction to graded bundles
Usually bundle gerbes are considered as objects of a 2-groupoid, whose 1-morphisms, called stable isomorphisms, are all invertible. I introduce new 1-morphisms which include stable isomorphisms, trivializations and bundle gerbe modules.…
Several representations of geometric shapes involve quotients of mapping spaces. The projection onto the quotient space defines two sub-bundles of the tangent bundle, called the horizontal and vertical bundle. We investigate in these notes…
Lagrangian formalism on graded manifolds is phrased in terms of the Grassmann-graded variational bicomplex, generalizing the familiar variational bicomplex for even Lagrangian systems on fiber bundles.
In this paper we study cobordism categories consisting of manifolds which are endowed with geometric structure. Examples of such geometric structures include symplectic structures, flat connections on principal bundles, and complex…
We study fiber bundles where the fibers are not a group $G$, but a free $G$-space with disjoint orbits. These bundles closely resemble principal bundles, hence we call them semi-principal bundles. The study of such bundles is facilitated by…
In this work, generalized principal bundles modelled by Lie group bundle actions are investigated. In particular, the definition of equivariant connections in these bundles, associated to Lie group bundle connections, is provided, together…
This paper presents a transformative framework for artificial neural networks over graded vector spaces, tailored to model hierarchical and structured data in fields like algebraic geometry and physics. By exploiting the algebraic…
We discuss multi-graded nilpotent tuples of multi-graded vector spaces which are a generalization of graded nilpotent pairs. The multi-grading yields a natural notion of a shape of such tuple and our main interest is to answer the question…
Just as $\Cstar$ principal bundles provide a geometric realisation of two-dimensional integral cohomology; gerbes or sheaves of groupoids, provide a geometric realisation of three dimensional integral cohomology through their Dixmier-Douady…
In this paper we establish the principal bundle counterpart of the well-known and widely applied notion of vector bundle groupoid (VB-groupoid). In particular, we provide a general notion of principal bundle groupoid (PB-groupoid) as a…
We introduce and examine the notion of principal $\mathbb{Z}_2^n$-bundles, i.e., principal bundles in the category of $\mathbb{Z}_2^n$-manifolds. The latter are higher graded extensions of supermanifolds in which a $\mathbb{Z}_2^n$-grading…
When the action of a reductive group on a projective variety has a suitable linearisation, Mumford's geometric invariant theory (GIT) can be used to construct and study an associated quotient variety. In this article we describe how…
A first step towards a systematic theory of relative line bundles over SUSY-curves is presented. In this paper we deal with the case of relative line bundles over families of ordinary Riemann surfaces. Generalizations of the Gauss-Bonnet…
We describe a number of geometric contexts where categorification appears naturally: coherent sheaves, constructible sheaves and sheaves of modules over quantizations. In each case, we discuss how "index formulas" allow us to easily perform…
In this review the foundations of Geometric Quantization are explained and discussed. In particular, we want to clarify the mathematical aspects related to the geometrical structures involved in this theory: complex line bundles, hermitian…
The intention of this article is to make an attempt of classification of transitive Lie algebroids and on this basis to construct a classifying space. The realization of the intention allows to describe characteristic classes of transitive…
To understand the structure of an algebraic variety we often embed it in various projective spaces. This develops the notion of projective geometry which has been an invaluable tool in algebraic geometry. We develop a perfectoid analog of…
A VB-algebroid is essentially defined as a Lie algebroid object in the category of vector bundles. There is a one-to-one correspondence between VB-algebroids and certain flat Lie algebroid superconnections, up to a natural notion of…
This talk introduces a Cartan-geometric framework for generalised geometries governed by a differential graded Lie algebra. In contrast to ordinary Cartan geometry, the tangent bundle is extended and qu both a global duality group and a…
We introduce a notion of elliptic differential graded Lie algebra. The class of elliptic algebras contains such examples as the algebra of differential forms with values in endomorphisms of a flat vector bundle over a compact manifold, etc.…