G-principal Bundle Research Articles

Some Aspects of Positive Kernel Method of Quantization

We discuss various aspects of the positive kernel method of quantization of the one-parameter groups tau _t in text{ Aut }(P,vartheta ) of automorphisms of a G-principal bundle P(G,pi ,M) with a fixed connection form vartheta on its total space P. We show that the generator {hat{F}} of the unitary flow U_t = e^{it {hat{F}}} being the quantization of tau _t is realized by a generalized Kirillov–Kostant–Souriau operator whose domain consists of sections of some vector bundle over M, which are defined by a suitable positive kernel. This method of quantization applied to the case when G=hbox {GL}(N,{mathbb {C}}) and M is a non-compact Riemann surface leads to quantization of the arbitrary holomorphic flow tau _t^{mathrm{hol}} in text{ Aut }(P,vartheta ). For the above case, we present the integral decompositions of the positive kernels on Ptimes P invariant with respect to the flows tau _t^{mathrm{hol}} in terms of the spectral measure of {hat{F}}. These decompositions generalize the ones given by Bochner’s Theorem for the positive kernels on {mathbb {C}} times {mathbb {C}} invariant with respect to the one-parameter groups of translations of complex plane.

Notes on equivariant bundles

We compare two notions of G-fiber bundles and G-principal bundles in the literature, with an aim to clarify early results in equivariant bundle theory that are needed in current work of equivariant algebraic topology. We also give proofs of some equivariant generalizations of well-known non-equivariant results involving the classifying space.

Geometric foundations of Cartan gauge gravity

We use the theory of Cartan connections to analyze the geometrical structures underpinning the gauge-theoretical descriptions of the gravitational interaction. According to the theory of Cartan connections, the spin connection ω and the soldering form θ that define the fundamental variables of the Palatini formulation of general relativity can be understood as different components of a single field, namely a Cartan connection A = ω + θ. In order to stress both the similarities and the differences between the notions of Ehresmann connection and Cartan connection, we explain in detail how a Cartan geometry (PH→ M, A) can be obtained from a G-principal bundle PG→ M endowed with an Ehresmann connection (being the Lorentz group H a subgroup of G) by means of a bundle reduction mechanism. We claim that this reduction must be understood as a partial gauge fixing of the local gauge symmetries of PG, i.e. as a gauge fixing that leaves "unbroken" the local Lorentz invariance. We then argue that the "broken" part of the symmetry — that is the internal local translational invariance — is implicitly preserved by the invariance under the external diffeomorphisms of M.

Conley indexes and stable sets for flows on flag bundles

Consider a continuous flow of automorphisms of a G-principal bundle which is chain transitive on its compact Hausdorff base. Here G is a connected non-compact semi-simple Lie group with finite centre. The finest Morse decomposition of the induced flows on the associated flag bundles were obtained in previous articles. Here we describe the stable sets of these Morse components and, under an additional assumption, their Conley indexes.

Generalized orbifold Euler characteristics of symmetric orbifolds and covering spaces

Let G be a finite group and let M be a G-manifold. We introduce the concept of generalized orbifold invariants of M/G associated to an arbitrary group Gamma, an arbitrary Gamma-set, and an arbitrary covering space of a connected manifold Sigma whose fundamental group is Gamma. Our orbifold invariants have a natural and simple geometric origin in the context of locally constant G-equivariant maps from G-principal bundles over covering spaces of Sigma to the G-manifold M. We calculate generating functions of orbifold Euler characteristic of symmetric products of orbifolds associated to arbitrary surface groups (orientable or non-orientable, compact or non-compact), in both an exponential form and in an infinite product form. Geometrically, each factor of this infinite product corresponds to an isomorphism class of a connected covering space of a manifold Sigma. The essential ingredient for the calculation is a structure theorem of the centralizer of homomorphisms into wreath products described in terms of automorphism groups of Gamma-equivariant G-principal bundles over finite Gamma-sets. As corollaries, we obtain many identities in combinatorial group theory. As a byproduct, we prove a simple formula which calculates the number of conjugacy classes of subgroups of given index in any group. Our investigation is motivated by orbifold conformal field theory.

Some remarks on Lagrangian and Poisson reduction for field theories

Given a Hamiltonian system on a fiber bundle, the Poisson covariant formulation of the Hamilton equations is described. When the fiber bundle is a G-principal bundle and the Hamiltonian density is G-invariant, the reduction of this formulation is studied thus obtaining the analog of the Lie-Poisson reduction for field theories. The relation of this reduction with the Lagrangian reduction and the Lagrangian and Poisson reduction for electromagnetism are also analyzed.

Théorie de jauge et symétries des fibrés

Let ξ be a smooth G-principal bundle over a manifold M (G being a compact Lie group). Given an action of a compact Lie group Γ on M, one asks the question whether it comes from an action on the bundle ξ. In this paper, this question is shown to be essentially equivalent to the existence of fixed points for the naturally induced actions of Γ on various moduli spaces of G-connections on M.

Surgery obstructions of fibre bundles

In a previous paper we obtained an algebraic description of the transfer maps p∗:L n( Z[π 1(B)]→L n+d( Z[π 1(e)]) induced in the Wall surgery obstruction groups by a fibration ▪ with the fibre F a d-dimensional Poincaré complex. In this paper we define a Π 1( B)-equivariant symmetric signature σ∗(F, ω)ϵL d(π 1(b), Z) depending only on the fibre transport ω: Π 1( B)→[ F, F], and prove that the composite p ∗p∗:L n( Z[π 1(B)])→L n+d( Z[π 1(b)] is the evaluation σ∗(F, ω)⊗? of the product ⊗:L d(π 1(B), Z⊗L n( Z[π 1(B)])→L n+d( Z[π 1(B)]) . This is applied to prove vanishing results for the surgery transfer, such as p∗=0 if F= G is a compact connected d-dimensional Lie group which is not a torus, and ▪ is a G-principal bundle. An appendix relates this expression for p∗p ∗ to the twisted signature formula of Atiyah, Lusztig and Meyer.

Reduction of some classical non-holonomic systems with symmetry

Two types of nonholonomic systems with symmetry are treated: (i) the configuration space is a total space of a G-principal bundle and the constraints are given by a connection; (ii) the configuration space is G itself and the constraints are given by left-invariant forms. The proofs are based on the method of quasicoordinates. In passing, a derivation of the Maurer-Cartan equations for Lie groups is obtained. Simple examples are given to illustrate the algorithmical character of the main results.

1-forms over the moduli space of irreducible connections defined by the spectrum of Dirac operators

Let M ( P) be the moduli space of irreducible connections of a G-principal bundle P over a closed Riemannian spin manifold M. Let D A be the Dirac operator of M coupled to a connection A of P and f a smooth function on M. We consider a smooth variation A( u) of A with tangent vector ω and denote T ω:= d du ( D A ( u) − f) ( u=0. The coefficients of the asymptotic expansion of trace ( T ω · e - t( D A − f) 2 ) near t=0 define 1-forms a ( k) f , k=0, 1, 2, … on M ( P). In this paper we calculate aa (0) f , a (1) f , a (2) f and study some of their properties. For instance using the 1-form a (2) f for suitable functions f we obtain a foliation of codimension 5 of the space of G-instantons of S 4.