On a gauge-invariant functional (original) (raw)
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Differential Geometry and its Applications, 2003
Given a principal bundle P → M we classify all first order Lagrangian densities on the bundle of connections associated to P that are invariant under the Lie algebra of infinitesimal automorphisms. These are shown to be variationally trivial and to give constant actions that equal the characteristic numbers of P if dim M is even and zero if dim M is odd. In addition, we show that variationally trivial Lagrangians are characterized by the de Rham cohomology of the base manifold M and the characteristic classes of P of arbitrary degree.
Gauge invariant functions of connections
Proceedings of the American Mathematical Society, 1994
It is shown that, for certain gauge groups, central functions of the holonomy variables do not determine connections up to gauge equivalence. It is also shown that, for a large class of compact groups, such functions do determine connections up to gauge equivalence. It is then shown that, for the latter type of gauge groups, the Euclidean quantum gauge field measure is determined by the expectation values of the Wilson loop variables (products of characters evaluated on holonomies).
On the Natural Gauge Fields of Manifolds
Modern Physics Letters A, 2000
The gauge symmetry inherent in the concept of manifold has been discussed. Within the scope of this symmetry the linear connection or displacement field can be considered as a natural gauge field on the manifold. The gauge-invariant equations for the displacement field have been derived. It has been shown that the energy–momentum tensor of this field conserves and hence the displacement field can be treated as one that transports energy and gravitates. To show the existence of the solutions of the field equations, we have derived the general form of the displacement field in Minkowski space–time which is invariant under rotation and space and time inversion. With this ansatz we found spherically-symmetric solutions of the equations in question.
On the regularity implied by the assumptions of geometry II -- Connections on vector bundles
arXiv (Cornell University), 2021
We establish optimal regularity and Uhlenbeck compactness for connections on vector bundles over arbitrary base manifolds, including both Riemannian and Lorentzian manifolds, and allowing for both compact and noncompact Lie groups. The proof is based on the discovery of a non-linear system of elliptic equations, (the RT-equations), which provides the coordinate and gauge transformations that lift the regularity of a connection to one derivative above its L p curvature. This is a new mathematical principle. It extends theorems of Kazdan-DeTurck and Uhlenbeck from Riemannian to the Lorentzian geometry of General Relativity, including Yang-Mills connections of relativistic Physics. CONTENTS 1. Introduction 1 2. Statement of Results 4 3. Preliminaries 10 4. The RT-equations for vector bundles 12 5. Existence theory for the RT-equations-Proof of Theorem 2.2 18 6. Weak formalism 27 7. Proof of optimal regularity and Uhlenbeck compactness 30 Appendix A. Basic results from elliptic PDE theory 33 References 34
A smooth Gauge model on tangent bundle
2007
The tangent manifold T M of a smooth, paracompact manifold M , fibered over M by the natural projection π , carries an integrable distribution kerπ∗, called vertical distribution. If one takes a supplementary distribution of it, called horizontal, an almost complex structure FS appears. One endowes the vertical distribution with a Riemmanian metric γ . Then γ can be prolonged to a Riemannian metric G S on T M such that the pair (FS ,GS) becomes an almost Hermitian structure. In this paper some deformations of FS and GS are proposed and new almost Hermitian structures are determined. With respect to some gauge objects, some properties of these structures are pointed out. For a smooth gauge geometrical model determined by one of these almost Hermitian structures, the general form of the corresponding Einstein-Yang Mills equations is obtained.
Yang–Mills theory and conjugate connections
Differential Geometry and Its Applications, 2003
We develop a new Yang-Mills theory for connections D in a vector bundle E with bundle metric h, over a Riemannian manifold by dropping the customary assumption Dh = 0. We apply this theory to Einstein-Weyl geometry (cf. M.F. Atiyah, et al., Self-duality in four-dimensional Riemannian geometry, Proc. Roy. Soc. London 362 (1978) 425-461, and H. Pedersen, et al., Einstein-Weyl deformations and submanifolds, Internat. J. Math. 7 (1996) 705-719) and to affine differential geometry (cf. F. Dillen, et al., Conjugate connections and Radon's theorem in affine differential geometry, Monatshefts für Mathematik 109 (1990) 221-235). We show that a Weyl structure (D, g) on a 4-dimensional manifold is a minimizer of the functional (D, g) → 1 2 M R D 2 v g if and only if * R D = ±R D * , where D * is conjugate to D. Moreover, we show that the induced connection on an affine hypersphere M is a Yang-Mills connection if and only if M is a quadratic affine hypersurface.
Journal of Geometry and Physics, 1984
Some aspects of the geometry of gauge theories are sketched in this review. We deal essentially with Yang-Mills theory, discussing the structure of the space of gauge orbits and the geometrical interpretation of ghosts and anomalies. Occasionally we deal also with classical egauge theories, of gravitation and in particular we study the action of the group of diffeomorphisms on the space of linear connections. Finally we comment on the mathematical interpretation of anomalies in field theories.
1 on the Gauge Structure of the Calculus of Variations with Constraints
2016
A gauge-invariant formulation of constrained variational calculus, based on the introduction of the bundle of affine scalars over the configuration manifold, is presented. In the resulting setup, the "Lagrangian" L is replaced by a section of a suitable principal fibre bundle over the velocity space. A geometric rephrasement of Pontryagin's maximum principle, showing the equivalence between a constrained variational problem in the state space and a canonically associated free one in a higher affine bundle, is proved.
Generalised connections and curvature
Mathematical Proceedings of the Cambridge Philosophical Society, 2005
The concept of generalised (in the sense of Colombeau) connection on a principal fibre bundle is introduced. This definition is then used to extend results concerning the geometry of principal fibre bundles to those that only have a generalised connection. Some applications to singular solutions of Yang-Mills theory are given.