We consider smooth families of Lie groups (group bundles) and connections that are compatible with the group operation. We characterize the space of group connections on a group bundle as an affine space modeled over the vector space of $1$-forms with values cocycles in the Lie algebra bundle of the aforementioned group bundle. We show that group connections satisfy the Ambrose-Singer theorem and that group bundles can be seen as a particular case of associated bundles realizing group connections as associated connections. We give a construction of the Moduli space of group connections with fixed base and fiber, as an space of representations of the fundamental group of the base.
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 with the analysis of their existence and their main properties. The final part gives some examples. In particular, since this research was initially originated by some problems on geometric reduction of gauge field theories, we revisit the classical Utiyama Theorem from the perspective investigated in the article.
We characterize all LVMB manifolds X such that the holomorphic tangent bundle TX is spanned at the generic point by a family of global holomorphic vector fields, each of them having non-empty zero locus. We deduce that holomorphic connections on semi-stable holomorphic vector bundles over LVMB manifolds with this previous property are always flat.
We show how the amplitude of holonomies on a vector bundle can be controlled by the integral of the curvature of the connection on a surface enclosed by the curve.
In this article we consider the continuity of the eigenvalues of the connection Laplacian of $G$-connections on vector bundles over Riemannian manifolds. To show it, we introduce the notion of the asymptotically $G$-equivariant measured Gromov-Hausdorff topology on the space of metric measure spaces with isometric $G$-actions, and apply it to the total spaces of principal $G$-bundles equipped with $G$-connections over Riemannian 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 principal bundles and a suitably defined category of functors. From the point of view of differential geometry we detail the specialization of non-linear or Ehresmann to principal and linear connections and discuss the widely known and very useful universality of principal curvature…
In this paper, we use the direct minimizing method to find Yang-Mills connections for $SO(3)$ bundles over closed four manifolds. By constructing test connections, we prove that a minimizing sequence converges strongly to a minimizer under certain assumptions. In case the strong convergence fails, we find an anti-selfdual (or selfdual) connection.
For every codimension two spacelike submanifold of a Lorentz manifold and each choice of a normal lightlike vector field, we introduce a canonical way to construct a tractor conformal bundle. We characterize when the induced connection of a such submanifold defines a tractor connection and then, in this case, when this tractor conformal bundle with the induced connection is standard for the induced metric. Finally, the normality condition for this tractor conformal bundle endowed with the induced connection is characterized in terms of a strong relationship between the intrinsic and the extrin…
A recent paper (arxiv.org:1810.00025) studied properties of a compactification of the moduli space of irreducible Hermitian-Yang-Mills connections on a hermitian bundle over a projective algebraic manifold. In this follow-up note, we show that the Yang-Mills flow at infinity on the space of semistable integrable connections defines a continuous map to the set of ideal connections used to define this compactification. Part of the proof involves a comparison between the topologies of the Grothendieck Quot scheme and the space of smooth connections.
We construct and study general connections on Lie groupoids and differentiable stacks as well as on principal bundles over them using Atiyah sequences associated to transversal tangential distributions.
