Exact multisoliton solutions in the four-dimensional Skyrme model

Abstract

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

Cosmological Einstein-Skyrme solutions with nonvanishing topological charge

“…Our result can also be seen as a generalization of the hedgehog ansatz discussed in [27], that allows for the construction of exact multi-Skyrmion configurations composed by elementary spherically symmetric Skyrmions with non-trivial winding number in four-dimensions [29,30].…”

“…Following [27][28][29][30], [32], [34], we adopt the standard parametrization of the SU (2)-valued scalar U (x µ ) as…”

“…Until a few years ago, no analytic solutions of the Skyrme model in flat space had been found. Quite recently, however, the generalized hedgehog ansatz (introduced in [27] and its generalizations in [28][29][30][31][32][33][34][35]) allowed for the construction of exact multi-Skyrmion configurations as well as the first analytic gravitating Skyrmions [34]. Moreover, these approaches also work in the Yang-Mills case [35].…”

Multi-Skyrmions with orientational moduli

“…The conclusions of [14] have been strengthened in [15]. Using these results, in [16] the first exact multi-Skyrmionic solutions of arbitrary Baryon number, living within a finite volume and composed of interacting elementary SU (2) Skyrmions, have been constructed. The useful mathematical trick to construct these configurations has been to place them within a suitable tube-shaped finite-volume space-time (which, in the context of the Skyrme model, has been also considered in [17] although with different motivations).…”

“…This analysis reveals many interesting novel features. First of all, the orientational moduli of [25], when finite volume effects are taken into account as in [16], tend to decrease the repulsive interactions among Skyrmions which characterize the multi-Skyrmionic configurations living in the tube-shaped regions mentioned above. Moreover the appearance of moduli is energetically favorable if finite volume effects are present.…”