arXiv research

Defines hybrid systems on principal bundles and studies impact effects.

A (smooth) dynamical system with transformation group $\mathbb{T}^n$ is a triple $(A,\mathbb{T}^n,α)$, consisting of a unital locally convex algebra $A$, the $n$-torus $\mathbb{T}^n$ and a group homomorphism $α:\mathbb{T}^n\rightarrow\Aut(A)$, which induces a (smooth) continuous action of $\mathbb{T}^n$ on $A$. In this…

The paper presents new formulations of gauge and gravity theories using dynamical principal bundles.

Constructs a triple on an Atiyah algebroid with connection.

This paper develops a Hamiltonian reduction method for field theories over affine principal bundles.

Develops a unified theory of Yang-Mills and GR using generalized principal bundles.

From a geometrical point of view it is, so far, not sufficiently well understood what should be a "noncommutative principal bundle". Still, there is a well-developed abstract algebraic approach using the theory of Hopf algebras. An important handicap of this approach is the ignorance of topological and geometrical aspe…

Develops a new calculus for studying operators on principal bundles.

Paper untwists gauge fields, linking two bundles.

Let $Λ$ be a finite abelian group. A dynamical system with transformation group $Λ$ is a triple $(A,Λ,α)$, consisting of a unital locally convex algebra $A$, the finite abelian group $Λ$ and a group homomorphism $α:Λ\rightarrow\Aut(A)$, which induces an action of $Λ$ on $A$. In this paper we present a new, geometricall…

Study of semi-principal bundles using group actions and wreath products.

In this paper we investigate some connections between Topological Dynamics, the theory of G-Principal Bundles, and the theory of Locally Trivial Groupoids.

Survey on advanced gauge theory concepts.

Introduces generalized principal bundles and connections, linking them to standard gauge theories.

The paper explores noncommutative geometry of frame bundles using C*-algebras.

Unified geometric framework for adiabatic quantum mechanics.

The equivalence of principal bundles with transitive Lie groupoids due to Ehresmann is a well known result. A remarkable generalisation of this equivalence, due to Mackenzie, is the equivalence of principal bundle extensions with those transitive Lie groupoids over the total space of a principal bundle, which also admi…

Study Lie algebroid connections on principal bundles over complex projective varieties.

Holonomies match for higher local systems and principal 2-bundles.

Let $X$ be a compact connected Kähler manifold equipped with an anti-holomorphic involution which is compatible with the Kähler structure. Let $G$ be a connected complex reductive affine algebraic group equipped with a real form $σ_G$. We define pseudo-real principal $G$--bundles on $X$; these are generalizations of re…

In this paper, we initiate the study of a parametrised version of Rieffel's strict deformation quantization. We apply it to give a classification of noncommutative principal torus bundles, in terms of parametrised strict deformation quantization of ordinary principal torus bundles. The paper also contains a putative de…

We define the pull-back of a smooth principal fibre bundle, and show that it has a natural principal fibre bundle structure. Next, we analyse the relationship between pull-backs by homotopy equivalent maps. The main result of this article is to show that for a principal fibre bundle over a paracompact manifold, there i…

A principal Higgs bundle $(P,φ)$ over a singular curve $X$ is a pair consisting of a principal bundle $P$ and a morphism $φ:X\to\text{Ad}P \otimes Ω^1_X$. We construct the moduli space of principal Higgs G-bundles over an irreducible singular curve $X$ using the theory of decorated vector bundles. More precisely, given…

We develop the theory of smooth principal bundles for a smooth group $G$, using the framework of diffeological spaces. After giving new examples showing why arbitrary principal bundles cannot be classified, we define $D$-numerable bundles, the smooth analogs of numerable bundles from topology, and prove that pulling ba…

In this paper we introduce a notion of parallel transport for principal bundles with connections over differentiable stacks. We show that principal bundles with connections over stacks can be recovered from their parallel transport thereby extending the results of Barrett, Caetano and Picken, and Schreiber and Waldof f…

Paper introduces Atiyah sequence for Lie groupoids and studies gauge transformations.

Study characterizes martingales on fiber bundles for harmonic map analysis.

Logarithmic connections on principal bundles over normal varieties are studied.

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

Defines sheaves and Čech cohomology for diffeological spaces and classifies principal bundles.

Study connections on complex Riemann surfaces for Lie algebroid structures.

Let M be a simply connected Riemannian symmetric space, with at most one flat direction. We show that every Riemannian (or unitary) vector bundle with parallel curvature over M is an associated vector bundle of a canonical principal bundle, with the connection inherited from the principal bundle. The problem of finding…

Let $P$ be a parabolic subgroup of a connected simply connected complex semisimple Lie group $G$. Given a compact Kähler manifold $X$, the dimensional reduction of $G$-equivariant holomorphic vector bundles over $X\times G/P$ was carried out by the first and third authors. This raises the question of dimensional reduct…

The paper bridges diffeological bundle theory with higher topos theory.

Reduces field theories on principal bundles by a subgroup, deriving reduced equations.

The paper proves a stronger Petersen--Wilhelm conjecture for principal bundles.

In this paper, we introduce the classification of equivariant principal bundles over the 2-sphere. Isotropy representations provide tools for understanding the classification of equivariant principal bundles. We consider a $Γ$-equivariant principal $G$-bundle over $S^2$ with structural group $G$ a compact connected Lie…

Gradient flow solves optimal mass transport for covariance matrices.

Geometric Arbitrage Theory reformulates a generic asset model possibly allowing for arbitrage by packaging all assets and their forwards dynamics into a stochastic principal fibre bundle, with a connection whose parallel transport encodes discounting and portfolio rebalancing, and whose curvature measures, in this geom…

We develop the concept of a double (more generally n-tuple) principal bundle departing from a compatibility condition for a principal action of a Lie group on a groupoid.

The paper generalizes bundle gerbes over groupoids and their correspondence with PB groupoids.

Introduces Lie semiheaps and their relation to Lie groups and bundles.

We construct manifestly superconformal field theories in six dimensions which contain a non-Abelian tensor multiplet. In particular, we show how principal 3-bundles over a suitable twistor space encode solutions to these self-dual tensor field theories via a Penrose-Ward transform. The resulting higher or categorified …

In this paper, we introduce the concept of principal bundles on statistical manifolds. After necessary preliminaries on information geometry and principal bundles on manifolds, we study the $α$-structure of frame bundles over statistical manifolds with respect to $α$-connections, by giving geometric structures. The man…

In this paper we propose a geometrization of the non-relativistic quantum mechanics for mixed states. Our geometric approach makes use of the Uhlmann's principal fibre bundle to describe the space of mixed states and as a novelty tool, to define a dynamic-dependent metric tensor on the principal manifold, such that the…

The paper revisits a claim about a principal bundle over a contractible base and finds it non-trivial.

The study examines connections and their curvatures on different types of bundles.

We give a complete characterization of invariant integrable complex structures on principal bundles defined over hermitian symmetric spaces, using the Jordan algebraic approach for the curvature computations. In view of possible generalizations, the general setup of invariant holomorphic principal fibre bundles is desc…