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Categorical bundles provide a natural framework for gauge theories involving multiple gauge groups. Unlike the case of traditional bundles there are distinct notions of triviality, and hence also of local triviality, for categorical…
We prove that the category of abelian gerbes with connection over a smooth manifold is equivalent to a certain category of principal bundles over the free loop space. These bundles are equipped with a connection and with a "fusion" product…
Connections and curvings on gerbes are beginning to play a vital role in differential geometry and mathematical physics -- first abelian gerbes, and more recently nonabelian gerbes. These concepts can be elegantly understood using the…
Given a double vector bundle $D\to M$, we define a bigraded `Weil algebra' $\mathcal{W}(D)$, which `realizes' the algebra of smooth functions on the supermanifold $D[1,1]$. We describe in detail the relations between the Weil algebras of…
A new global approach in the study of duality transformations is introduced. The geometrical structure of complex line bundles is generalized to higher order U(1) bundles which are classified by quantized charges and duality maps are…
First, we extend the notion of second order differential equations (SODE) on a smooth manifold to anchored Banach vector bundles. Then we define the Banach Lie algebroids as Lie algebroids structures modeled on anchored Banach vector…
We investigate relative holomorphic connections on a principal bundle over a family of compact complex manifolds. A sufficient condition is given for the existence of a relative holomorphic connection on a holomorphic principal bundle over…
We define a general notion of abstract double Lie algebroid. We show (1) that the double Lie algebroid of a double Lie groupoid is a double Lie algebroid in this sense; (2) that the double cotangent constructed from Lie algebroid structures…
We develop a ready-to-use comprehensive theory for (super) 2-vector bundles over smooth manifolds. It is based on the bicategory of (super) algebras, bimodules, and intertwiners as a model for 2-vector spaces. We discuss symmetric monoidal…
We generalize the Donagi and Witten construction of a first obstruction class for splitting of a supermanifold via differential operators using the theory of $n$-fold vector bundles and graded manifolds. Applying the generalized…
Odd exact Courant algebroids constitute a simple class of transitive Courant algebroids. Their underlying vector bundle is of odd rank and differs from a generalized tangent bundle by the addition of a line bundle. In this article we study…
In this paper we propose an algebraic formalization of connectors in the quantitative setting, in order to address their non-functional features in architectures of component-based systems. We firstly present a weighted Algebra of…
We study finite-rank left-translation invariant algebraic $D$-modules on complex affine algebraic groups. Using the standard description of these objects as left-invariant flat algebraic connections on the trivial vector bundle, modulo…
In this note we prove that the moduli stack of vector bundles on a curve, with a fixed determinant is $\mathbb{A}^1$-connected. We obtain this result by classifying vector bundles on a curve upto $\mathbb{A}^1$-concordance. Consequently we…
In this paper, first we introduce the notion of a $\VB$-Lie $2$-algebroid, which can be viewed as the categorification of a $\VB$-Lie algebroid. The tangent prolongation of a Lie $2$-algebroid is a $\VB$-Lie $2$-algebroid naturally. We show…
We define and explore the notion of linear weightings for vector bundles, extending the recent work by Loizides and Meinrenken. We construct weighted normal bundles and deformation spaces in the category of vector bundles. We explain how a…
A Q-manifold is a graded manifold endowed with a vector field of degree one squaring to zero. We consider the notion of a Q-bundle, that is, a fiber bundle in the category of Q-manifolds. To each homotopy class of ``gauge fields'' (sections…
We study generic graded contractions of Lie algebras from the perspectives of group cohomology, affine algebraic geometry and monoidal categories. We show that generic graded contractions with a fixed support are classified by a certain…
We introduce and investigate a functorial construction which associates coherent sheaves to finite dimensional (restricted) representations of a restricted Lie algebra $\mathfrak g$. These are sheaves on locally closed subvarieties of the…
In general terms, Gelfand duality refers to a correspondence between a geometric, topological, or analytical category, and an algebraic category. For example, in smooth differential geometry, Gelfand duality refers to the topological…