Analytic solutions in the gauged skyrme model and conservation laws in gauge theories with applications to black hole mechanics.

Abstract

The present thesis consists of two parts. Part I is devoted to the construction of the first analytic examples of topologically non-trivial solutions of the U(1) gauged Skyrme model within a finite box in (3 + 1)-dimensional flat space-time. There are two types of gauged solitons. The first type corresponds to gauged Skyrmions living within a finite volume. The second corresponds to gauged time-crystals (smooth solutions of the U(1) gauged Skyrme model whose periodic time-dependence is protected by a winding number). The notion of electromagnetic duality can be extended for these two types of configurations in the sense that some of the electric and magnetic field components can be interchanged. These analytic solutions show very explicitly the Callan-Witten mechanism (according to which magnetic monopoles may “swallow” part of the topological charge of the Skyrmion) since the electromagnetic field contribute directly to the topological charge of the gauged Skyrmions. As it happens in superconductors, the magnetic field is suppressed in the core of the gauged Skyrmions. On the other hand, the electric field is strongly suppressed in the centre of gauged time crystals. Part II is concerned with studying a new derivation of surface charges in gauge theories. Part of the focus is on reviewing the method to compute quasi-local surface charges for gauge theories to clarify conceptual issues and their range of applicabil ity. The surface charges found are quasi-local, explicitly coordinate independent, and gauge invariant. Many surface charge formulas for gravity theories are expressed in metric, tetrads-connection, and even Chern-Simons connection. For most of them, the language of differential forms is exploited and contrasted with the more popular metric components language. The study focuses on General Relativity theory coupled with matter fields as Maxwell, Skyrme, and spinors. To derive the sur face charges, we specify the phase space by identifying the symplectic structure. We use the formulation of the covariant phase space method. Here the symplectic structure has two parts: the standard Lee-Wald term plus a contribution from the boundary term read from the action. The latter is fixed by requiring the on-shell and linearized equations of motion condition, and exact symmetry condition. These conditions guarantees the conservation of the symplectic structure in phase space, and leads to the new concept of “symplectic symmetry”. Given the “conservation law” satisfying the symplectic structure, we construct the corresponding charges, the “symplectic symmetry generators”. The explicit expression of the charges corresponds to a function over the phase space. We find the remarkable property that, in contrast with usual Noether procedures to compute charges, the boundary terms and even topological terms do not affect the surface charges. On the other hand, by studying two concrete examples, we also examine how torsion affects the surface charges. Both of them conclude that the torsion field does not affect the general formula for the surface charges. Furthermore, three examples with ready-to-download Mathematica notebook codes show the method in full action. The charges and their associated first law of thermodynamics are derived for: the BTZ black hole, the charged rotating (3 + 1)-black hole, and the Lorentzian rotating Taub-NUT space-time.