International Journal of Geometric Methods in Modern Physics

In gravitational Higgs mechanism graviton components acquire mass via spontaneous diffeomorphism breaking by scalar vacuum expectation values. We point out that in the massless limit the resulting theory is not Einstein–Hilbert gravity (EHG) but constrained gravity (CG). Consequently, massive solutions in the massless limit must be compared to those in CG (as opposed to EHG). We discuss spherically symmetric solutions in this context. The Schwarzschild solution in EHG can be coordinate-transformed such that it is also a solution in CG. The resulting solutions are non-perturbative in the asymptotic regime, and are reproduced in the massless limit of asymptotic massive solutions, hence no van Dam–Veltman–Zakharov discontinuity. We point out that higher curvature terms must be included to obtain non-singular spherically symmetric massive solutions and discuss a suitable framework.

Due to a suitable Higgs mechanism, a standard Anti-de Sitter gauge theory becomes spontaneously broken. The resulting Lorentz invariant gravitational action includes the Hilbert–Einstein term of ordinary Einstein–Cartan gravity with cosmological constant, plus contributions quadratic in curvature and torsion, and a scalar Higgs sector.

A definition of space-time metric deformations on an n-dimensional manifold is given. We show that such deformations can be regarded as extended conformal transformations. In particular, their features can be related to the perturbation theory giving a natural picture by which gravitational waves are described by small deformations of the metric. As further result, deformations can be related to approximate Killing vectors (approximate symmetries) by which it is possible to parameterize the deformed region of a given manifold. The perspectives and some possible physical applications of such an approach are discussed.

The compensating Utiyama's method including space-time symmetries is revisited as well as the gauge gravitational theories associated with translation, Poincaré and Weyl groups. Then we propose an extension of the gauge symmetry, allowing for the incorporation of the gauge group parameters into the theory as dynamical fields by considering the jet-gauge group as fundamental symmetry. As a consequence, a natural mass-generating mechanism for the gauge potentials arises without damaging gauge invariance. We also present, as a simple example, some sort of generalized Stueckelberg model for the Weyl group, thus accounting for massive dilatonic gauge field. Finally, the standard diffeomorphism symmetry of gravitation is extended by resorting to the jet-diffeomorphism group, formalism which helps to fix the Hilbert-Einstein Lagrangian in the teleparallelism version.

Any quantum-mechanical system possesses a U(1) gerbe naturally defined on configuration space. Acting on Feynman's kernel exp(iS/ħ), this U(1) symmetry allows one to arbitrarily pick the origin for the classical action S, on a point-by-point basis on configuration space. This is equivalent to the statement that quantum mechanics is a U(1) gauge theory. Unlike Yang–Mills theories, however, the geometry of this gauge symmetry is not given by a fibre bundle, but rather by a gerbe. Since this gauge symmetry is spontaneously broken, an analogue of the Higgs mechanism must be present. We prove that a Heisenberg-like noncommutativity for the space coordinates is responsible for the breaking. This allows to interpret the noncommutativity of space coordinates as a Higgs mechanism on the quantum-mechanical U(1) gerbe.

The internal structure of the tetrads in a Poincaré nonlinear gauge theory of gravity is considered. Minkowskian coordinates becomes dynamical degrees of freedom playing the role of Goldstone bosons of the translations. A critical length allowing a covariant expansion similar to the weak field approach is deduced, the zeroth order metric being maximally symmetric (Minkowskian in some cases).

As a historical remark the similarities are pointed out between a form of chiral action introduced by Schwinger and the formalism used in the noncommutative extensions of electromagnetism.

By virtue of the well-known theorem, a structure Lie group K of a principal bundle P → X is reducible to its closed subgroup H iff there exists a global section of the quotient bundle P/K → X. In gauge theory, such sections are treated as Higgs fields, exemplified by pseudo-Riemannian metrics on a base manifold X. Under some conditions, this theorem is extended to principal superbundles in the category of G-supermanifolds. Given a G-supermanifold M and a graded frame superbundle over M with a structure general linear supergroup, a reduction of this structure supergroup to an orthogonal-symplectic supersubgroup is associated to a supermetric on a G-supermanifold M.