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We present a self-gravitating, analytic and globally regular Skyrmion solution of the Einstein-Skyrme system with winding number w = ±1, in presence of a cosmological constant. The static spacetime metric is the direct product R × S 3 and the Skyrmion is the self-gravitating generalization of the static hedgehog solution of Manton and Ruback with unit topological charge. This solution can be promoted to a dynamical one in which the spacetime is a cosmology of the Bianchi type-IX with time-dependent scale and squashing coefficients. Remarkably, the Skyrme equations are still identically satisfied for all values of these parameters. Thus, the complete set of field equations for the Einstein-Skyrme-Λ system in the topological sector reduces to a pair of coupled, autonomous, nonlinear differential equations for the scale factor and a squashing coefficient. These equations admit analytic bouncing cosmological solutions in which the universe contracts to a minimum non-vanishing size, and then expands. A non-trivial byproduct of this solution is that a minor modification of the construction gives rise to a family of stationary, regular configurations in General Relativity with negative cosmological constant supported by an SU (2) nonlinear sigma model. These solutions represent traversable AdS wormholes with NUT parameter in which the only "exotic matter" required for their construction is a negative cosmological constant.

Exact configurations of the four-dimensional Skyrme model are presented. The static configurations have the profile. Which behaves as a kink and, consequently, the corresponding energy-momentum tensor describes a domain wall. Furthermore, a class of exact time-periodic Skyrmions is discovered. Within such a class, it is possible to disclose a remarkable phenomenon that is a genuine effect of the Skyrme term. For a special value of the frequency the Skyrmions admit a nonlinear superposition principle. One can combine two or more exact ''elementary'' Skyrmions (which may depend in a nontrivial way on all the spacelike coordinates) into a new exact composite Skyrmion. Because of such a superposition law, despite the explicit presence of nonlinear effects in the energy-momentum tensor, the interaction energy between the elementary Skyrmions can be computed exactly. The relations with the appearance of Skyrme crystals are discussed.

The hedgehog ansatz for spherically symmetric spacetimes in self-gravitating nonlinear sigma models and Skyrme models is revisited and its generalization for non-spherically symmetric spacetimes is proposed. The key idea behind our construction is that, even if the matter fields depend on the Killing coordinates in a nontrivial way, the corresponding energy-momentum tensor can still be compatible with spacetime symmetries. Our generalized hedgehog ansatz reduces the Skyrme equations to coupled differential equations for two scalar fields together with several constraint equations between them. Some particular field configurations satisfying those constraints are presented in several physically important spacetimes, including stationary and axisymmetric spacetimes. Incidentally, several new exact solutions are obtained under the standard hedgehog ansatz, one of which represents a global monopole inside a black hole with the Skyrme effect.

Using a remarkable mapping from the original (3+1)dimensional Skyrme model to the Sine-Gordon model, we construct the first analytic examples of Skyrmions as well as of Skyrmions-anti-Skyrmions bound states within a finite box in 3+1 dimensional flat space-time. An analytic upper bound on the number of these Skyrmions-anti-Skyrmions bound states is derived. We compute the critical isospin chemical potential beyond which these Skyrmions cease to exist. With these tools, we also construct topologically protected time-crystals: time-periodic configurations whose time-dependence is protected by their non-trivial winding number. These are striking realizations of the ideas of Shapere and Wilczek. The critical isospin chemical potential for these time-crystals is determined.

In this paper we study dynamical compactification in Einstein-Gauss-Bonnet gravity from an arbitrary dimension for generic values of the coupling constants. We show that, when the curvature of the extra-dimensional space is negative, for any value of the spatial curvature of the four-dimensional space-time one obtains a realistic behavior in which for asymptotic time both the volume of the extra dimension and expansion rate of the four-dimensional space-time tend to a constant. Remarkably, this scenario appears within the open region of parameters space for which the theory does not admit any maximally symmetric (4 þ D)-dimensional solution, which gives to the dynamical compactification an interpretation as geometric frustration. In particular there is no need to fine-tune the coupling constants of the theory so that the present scenario does not violate the ''naturalness hypothesis.'' Moreover we show that with an increase of the number of extra dimensions the stability properties of the solution are increased.

In this paper we perform a systematic classification of the regimes of cosmological dynamics in Einstein-Gauss-Bonnet gravity with generic values of the coupling constants. We consider a manifold which is a warped product of a four dimensional Friedmann-Robertson-Walker space-time with a D-dimensional Euclidean compact constant curvature space with two independent scale factors. A numerical analysis of the time evolution as function of the coupling constants and of the curvatures of the spatial section and of the extra dimension is performed. We describe the distribution of the regimes over the initial conditions space and the coupling constants. The analysis is performed for two values of the number of extra dimensions (D 6 both) which allows us to describe the effect of the number of the extra dimensions as well.

In spite of all this, until very recently no exact analytic solutions of the Skyrme model with non-trivial topological charges were known. One of the reasons is that the Skyrme-BPS bound on the energy cannot be saturated for non-trivial spherically symmetric configurations [14]. Nevertheless, many rigorous results about Skyrmions dynamics have been derived, see for instance [15][16][17].The action of the SU (2) Skyrme system in four dimensional spacetime iswhereHere t i are the SU (2) generators and we set the units = c = 1. The coupling constants K > 0 and λ > 0 are fixed by comparison with experimental data [6]. The presence of the first term of the Skyrme action (1), is mandatory to describe pions while the second is the only covariant term leading to second order field equations in time which supports the existence of Skyrmions in four dimensions.In the present paper, exact spherically symmetric solutions of the Skyrme model with both a non-trivial winding number and a finite soliton mass (topological charge) are presented. Using the formalism introduced in [18][19][20], it is shown that although the BPS bound in terms of the winding cannot be saturated, a new topological charge exists that can be saturated corresponding to a different BPS bound. The baryon number is the homotopy of the space into the group. The simplest choice would be to consider the curved background S 3 as physical space, as already considered in the pioneering papers [21,22]. The second natural choice of special sections with integer homotopy into SU (2) is S 1 × S 2 (or R × S 2 ). This can be represented by a metric of the formIn simple words, this geometry describes tridimensional cylinders whose sections are S 2 spheres of area 4πR 2 0 . The physical meaning of R 0 is that it takes into account finite volume effects. One could put the Skyrme action into, say, a cube. However, this way of proceeding often breaks symmetries. On the other hand, a spherical box of finite radius would lead to difficulties in requiring the Skyrmions approach the identity at the boundary. Therefore, it is much more convenient to choose a metric which at the same time takes into account finite volume effects and keeps the spherical symmetry. We are able construct exact Skyrmions in a finite volume but, instead of putting by hand a cut-off on the coordinates, we leave this task to the geometry. Besides, this geometry is such that the group of the isometries of (2) contains SO(3) as a subgroup and so it includes the spherical symmetry of the Skyrmion in flat space. This fact allows examining how far is the BPS bound from being saturated and to construct an energy bound which can in fact be saturated. This could be of interest both in high energy and solid state physics whose features, after the papers [21,22], have not been thoroughly investigated from the analytical viewpoint.In order to construct the exact solution of the Skyrme model, the following standard parametrization of the

In this paper the arising of Gribov copies both in Landau and Coulomb gauges in regions with non-trivial topologies but flat metric, (such as closed tubes S 1 × D 2 , or R × T 2 ) will be analyzed. Using a novel generalization of the hedgehog ansatz beyond spherical symmetry, analytic examples of Gribov copies of the vacuum will be constructed. Using such ansatz, we will also construct the elliptic Gribov pendulum. The requirement of absence of Gribov copies of the vacuum satisfying the strong boundary conditions implies geometrical constraints on the shapes and sizes of the regions with non-trivial topologies.3 Furthermore, it has been shown by Singer [5], that if Gribov ambiguities occur in Coulomb gauge, they occur in all the gauge fixing conditions involving derivatives of the gauge field (See also [6]). Other gauge fixings (such as the axial gauge, the temporal gauge, and so on) free from gauge fixing ambiguities are possible but these choices have their own problems (see, for instance, [3]).4 The condition to have a positive Faddeev-Popov operator is not enough to completely eliminate Gribov copies in the Coulomb and Landau gauges. It can be shown [11] that there exist a smaller region (called the modular region) contained in the Gribov region which is free of gauge fixing ambiguities. However, it is still not clear how to implement the restriction to the modular region in practice.5 Recently, in [42], some criticism of this approach has been proposed. However, this argument is only valid in the perturbative framework (for instance, the gluons are assumed to be asymptotic states) while the changes of the Gribov-Zwanziger approach are non-perturbative in nature.

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