Related papers: Equivariant holonomy of U(1)-bundles

The theory of principal $G$-bundles over a Lie groupoid is an important one, unifying the various types of principal $G$-bundles, including those over manifolds, those over orbifolds, as well as equivariant principal $G$-bundles. In this…

Consider the principal $U(n)$ bundles over Grassmann manifolds $U(n)\rightarrow U(n+m)/U(m) \stackrel{\pi}\rightarrow G_{n,m}$. Given $X \in U_{m,n}(\mathbb{C})$ and a 2-dimensional subspace $\mathfrak{m}' \subset \mathfrak{m} $ $ \subset…

We give rigorous foundations for parametrized homotopy theory in this monograph. After preliminaries on point-set topology, base change functors, and proper actions of non-compact Lie groups, we develop the homotopy theory of equivariant…

The theory of frames normal for general connections on differentiable bundles is developed. Links with the existing theory of frames normal for covariant derivative operators (linear connections) in vector bundles are revealed. The…

We provide a natural generalization to submanifolds of the holographic method used to extract higher-order local invariants of both Riemannian and conformal embeddings, some of which depend on a choice of parallelization of the normal…

In this paper, we introduce a study of prolongations of homogeneous vector bundles. We give an alternative approach for the prolongation. For a given homogeneous vector bundle E, we obtain a new homogeneous vector bundle. The homogeneous…

Holonomy groups and holonomy algebras for connections on locally free sheaves over supermanifolds are introduced. A one-to-one correspondence between parallel sections and holonomy-invariant vectors, and a one-to-one correspondence between…

This is a note in which we first review symmetries of moduli spaces of stable meromorphic connections on trivial vector bundles over the Riemann sphere, and next discuss symmetries of their integrable deformations as an application. In the…

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.

We extend the notion of connection in order to be able to study singular geometric structures, namely, we consider a notion of connection on a Lie algebroid which is a natural extension of the usual concept of connection. Using connections,…

We define a type of biquandle which is a generalization of symplectic quandles. We use the extra structure of these bilinear biquandles to define new knot and link invariants and give some examples.

We explicitly determine the group of isomorphism classes of equivariant line bundles on the non-archimedean Drinfeld upper half plane for $\mathrm{GL}_2(F)$, for its subgroups of matrices whose determinant has even (respectively trivial)…

We present the fundamental properties of the K-theory groups of complex vector bundles endowed with actions of magnetic groups. In this work we show that the magnetic equivariant K-theory groups define an equivariant cohomology theory, we…

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,…

We obtain an explicit presentation of the equivariant cobordism ring of a complete flag variety. An immediate corollary is a Borel presentation of the ordinary cobordism ring. Another application is an equivariant Schubert calculus in…

Let $X$ be a complete variety over an algebraically closed field $k$ of characteristic zero, equipped with an action of an algebraic group $G$. Let $H$ be a reductive group. We study the notion of $G$-connection on a principal $H$-bundle.…

Equivariant quantization is a new theory that highlights the role of symmetries in the relationship between classical and quantum dynamical systems. These symmetries are also one of the reasons for the recent interest in quantization of…

Let $\Delta$ be a trivial knot in the three-sphere. For every finite cyclic group $G$ of odd order, we construct a $G$-equivariant Khovanov homology with coefficients in the filed $\F_{2}$. This homology is an invariant of links up to…

We use classical invariant theory to solve the biholomorphic equivalence problem for two families of plane curve singularities previously considered in the literature. Our calculations motivate an intriguing conjecture that proposes a way…

We introduce the notion of a hierarchical quandle, which is a generalisation of diquandles and multi-quandles. Using hierarchical quandle colourings, we construct a cocycle invariants for links coloured by quandles.