arXiv research

This paper shows vector bundles and differential bundles are equivalent in smooth manifolds.

We present a construction of a 2-Hilbert space of sections of a bundle gerbe, a suitable candidate for a prequantum 2-Hilbert space in higher geometric quantisation. We introduce a direct sum on the morphism categories in the 2-category of bundle gerbes and show that these categories are cartesian monoidal and abelian.…

Tangent categories are categories equipped with a tangent functor: an endofunctor with certain natural transformations which make it behave like the tangent bundle functor on the category of smooth manifolds. They provide an abstract setting for differential geometry by axiomatizing key aspects of the subject which all…

Proposes a new category of bundles for Lagrangian reduction in field theory.

We construct a prequantum 2-Hilbert space for any line bundle gerbe whose Dixmier-Douady class is torsion. Analogously to usual prequantisation, this 2-Hilbert space has the category of sections of the line bundle gerbe as its underlying 2-vector space. These sections are obtained as certain morphism categories in Wald…

Given a compact Kähler manifold $X$, there is an equivalence of categories between the completely reducible flat vector bundles on $X$ and the polystable Higgs bundles $(E,\, θ)$ on $X$ with $c_1(E)= 0= c_2(E)$ \cite{SimC}, \cite{Cor}, \cite{UY}, \cite{DonI}. We extend this equivalence of categories to the context of c…

We show that the category of affine bundles over a smooth manifold M is equivalent to the category of affine spaces modelled on projective finitely generated C^\infty(M)-modules. Using this equivalence of categories, we are able to give an alternate proof of the main result of [13], showing that the characterization of…

New equivalence found for flat vector bundles without extra conditions.

Parallel transport of a connection in a smooth fibre bundle yields a functor from the path groupoid of the base manifold into a category that describes the fibres of the bundle. We characterize functors obtained like this by two notions we introduce: local trivializations and smooth descent data. This provides a way to…

In this paper, we construct a category of short exact sequences of vector bundles and prove that it is equivalent to the category of double vector bundles. Moreover, operations on double vector bundles can be transferred to operations on the corresponding short exact sequences. In particular, we study the duality theor…

This work defines a categorical notion of principal bundles.

Study multiplicity-free covering of graded manifolds, proving equivalence of categories.

Vector bundles and double vector bundles, or $2$-fold vector bundles, arise naturally for instance as base spaces for algebraic structures such as Lie algebroids, Courant algebroids and double Lie algebroids. It is known that all these structures possess a unified description using the language of super\-geometry and g…

Study on topological rigidity of ALE vector bundles with specific conditions.

Turaev-Viro invariants match for certain surface bundles.

The present paper is a contribution to categorial index theory. Its main result is the calculation of the Pfaffian line bundle of a certain family of real Dirac operators as an object in the category of line bundles. Furthermore, it is shown how string structures give rise to trivialisations of that Pfaffian.

Universal connection constructed using diffeology theory.

This paper introduces $\infty$- and $n$-fold vector bundles as special functors from the $\infty$- and $n$-cube categories to the category of smooth manifolds. We study the cores and "n-pullbacks" of $n$-fold vector bundles and we prove that any $n$-fold vector bundle admits a non-canonical isomorphism to a decomposed …

We make the category BGrb_M of bundle gerbes on a manifold M into a 2-category by providing 2-cells in the form of transformations of bundle gerbe morphisms. This description of BGrb_M as a 2-category is used to define the notion of a bundle 2-gerbe. To every bundle 2-gerbe on M is associated a class in H^4(M;Z). We de…

Projectively flat bundles matched with semi-stable Higgs bundles on Kähler manifolds.

Extends Gelfand duality to various geometric and analytical categories.

In this note, we consider a Lie group G acting on a manifold M. We prove that the category of bundles with connection on the differential quotient stack is equivalent to the category of G-equivariant bundles on M with G-invariant connection.

We discuss nonabelian bundle gerbes and their differential geometry using simplicial methods. Associated to any crossed module there is a simplicial group NC, the nerve of the 1-category defined by the crossed module and its geometric realization |NC|. Equivalence classes of principal bundles with structure group |NC| …

The Serre-Swan theorem in differential geometry establishes an equivalence between the category of smooth vector bundles over a smooth compact manifold and the category of finitely generated projective modules over the unital ring of smooth functions. This theorem is here generalized to manifolds of bounded geometry. I…

Quantizes Kähler manifolds using sheaves and differential operators.

Let $X$ be a compact real analytic manifold, and let $T^*X$ be its cotangent bundle. Let $Sh(X)$ be the triangulated dg category of bounded, constructible complexes of sheaves on $X$. In this paper, we develop a Fukaya $A_\infty$-category $Fuk(T^*X)$ whose objects are exact, not necessarily compact Lagrangian branes in…

Constructs diffeological moduli stacks for Higgs and flat bundles on Kähler manifolds

Mathematical study supports connection between 3D manifolds and modular tensor categories.

Constructs 2-representations and 2-vector bundles for Lie 2-groups.

This paper introduces tangent display maps to simplify tangent category theory.

The abstract generalizes a construction for splitting supermanifolds and studies Lie supergroup cases.

The paper geometrizes N-manifolds using symmetric vector bundles.

The paper classifies Lie algebroids and their connections, modulating principal objects.

A 2-category categorifies complex Lagrangians in hyperkähler manifolds.

Perhaps the most important contribution of gauge theory to general mathematics is to point out the importance of association functors. Emphasizing category theory we characterize association functors by two of their natural properties and use this characterization to establish an equivalence between the category of pri…

This paper describes an equivalence of the canonical category of $\mathbb N$-manifolds of degree $2$ with a category of involutive double vector bundles. More precisely, we show how involutive double vector bundles are in duality with double vector bundles endowed with a linear metric. We describe then how special sect…

Study of gauge theory and parallel transport in Lie 2-group bundles over Lie groupoids.

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 with respect to triples of paths. The equivalence is established by explicit fun…

Three definitions of graded vector bundles are shown to be equivalent.

A stratified bundle is a fibered space in which strata are classical bundles and in which attachment of strata is controlled by a structure category of fibers. Well known results on fibre bundles are shown to be true for stratified bundles; namely the pull back theorem, the bundle theorem and the principal bundle theor…

Completes reduction scheme in Lagrange-Poincaré category.

Extends Higgs fields theory to complex fiber bundles.

Study homotopy sheaves on categories and their presheaves, proving descent properties.

Introduces principal bundles in a new geometric category.

We show that conically smooth stratified spaces embed fully faithfully into $\infty$-categories. This articulates a stratified generalization of the homotopy hypothesis proposed by Grothendieck. As such, each $\infty$-category defines a stack on conically smooth stratified spaces, and we identify the descent conditions…

We study a natural map from representations of a free (resp. free abelian) group of rank g in GL_r(C), to holomorphic vector bundles of degree zero over a compact Riemann surface X of genus g (resp. complex torus X of dimension g). This map defines what is called a Schottky functor. Our main result is that this functor…

We prove that the algebra of singular cochains on a smooth manifold, equipped with the cup product, is equivalent to the A-infinity structure on the Lagrangian Floer cochain group associated to the zero section in the cotangent bundle. More generally, given a pair of smooth manifolds of the same dimension with embeddin…

The paper bridges diffeological bundle theory with higher topos theory.