The problem of the gauge hierarchy is brought up in a hypercomplex scheme for a U (1) field theory; in such a scheme a compact gauge group is deformed through a γ-parameter that varies along a non-compact internal direction, transverse to the U (1) compact one, and thus an additional SO(1, 1) gauge symmetry is incorporated. This transverse direction can be understood as an extra internal dimension, which will control the spontaneous symmetry breakdown, and will allow us to establish a mass hierarchy. In this mechanism there is no brane separation to be estabilized as in the braneworld paradigm, however, a different kind of fine-tuning is needed in order to generate the wished electroweak/Planck hierarchy. By analyzing the effective self-interactions and mass terms of the theory, an interesting duality is revealed between the real and hybrid parts of the effective potential. This duality relates the weak and strong self-interaction regimes of the theory, due to the fact that both mass terms and self-coupling constants appear as one-parameter flows in γ. Additionally the γ-deformation will establish a flow for the electromagnetic coupling that mimics the renormalization group flow for the charge in QED.

We prove that some basic aspects of gravity commonly attributed to the modern connection-based approaches, can be reached naturally within the usual Riemannian geometry-based approach, by assuming the independence between the metric and the connection of the background manifold. These aspects are: 1) the BFlike field theory structure of the Einstein-Hilbert action, of the cosmological term, and of the corresponding equations of motion; 2) the formulation of Maxwellian field theories using only the Riemannian connection and its corresponding curvature tensor, and the subsequent unification of gravity and gauge interactions in a four dimensional field theory; 3) the construction of four and three dimensional geometrical invariants in terms of the Riemann tensor and its traces, particularly the formulation of an anomalous Chern-Simons topological model where the action of diffeomorphisms is identified with the action of a gauge symmetry, close to Witten's formulation of threedimensional gravity as a Chern-Simon gauge theory. 4) Tordions as propagating and non-propagating fields are also formulated in this approach. This new formulation collapses to the usual one when the metric connection is invoked, and certain geometrical structures very known in the traditional literature can be identified as remanent structures in this collapse.

By using the Atiyah-Singer theorem through some similarities with the instanton and the antiinstanton moduli spaces, the dimension of the moduli space for two and four-dimensional BF theories valued in different background manifolds and gauge groups scenarios is determined. Additionally, we develop Dirac's canonical analysis for a four-dimensional modified BF theory, which reproduces the topological YM theory. This framework will allow us to understand the local symmetries, the constraints, the extended Hamiltonian and the extended action of the theory.

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Using a hyperbolic complex plane, we study the realization of the underlying hyperbolic symmetry as an internal symmetry that enables the unification of scalar fields of cosmological and particle physics interest. Such an unification is achieved along the universal prescriptions used in physics, avoiding the use of concepts as Euclideanization, non-canonical Lagrangians and hidden structures, that have appeared in other approaches. The scalar potentials constructed within the present scheme are bounded from below, and the realization of the spontaneous symmetry breaking of the aforementioned noncompact symmetry is studied. The profiles of these potentials with exact/broken hyperbolic symmetry replicate qualitative aspects of those ones used in inflationary models, and then a detailed comparison is made. Moreover, the homotopy constraints of the topology induced on the corresponding vacuum manifolds, restricts the existence of topological defects associated with continuous symmetries, allowing only those defects associated with discrete symmetries; the consistency of these results is contrasted with current observational tests from the LIGO/Virgo collaboration, and terrestrial experiments based on a synchronized network of atomic magnetometers. At the end, the nonrelativistic limit of the model is identified with a hyperbolic version of the nonlinear Schrödinger equation.