BPS Skyrme submodels of the five-dimensional Skyrme model

Abstract

In this paper, we search for the BPS skyrmions in some BPS submodels of the generalized Skyrme model in five-dimensional spacetime using the BPS Lagrangian method. We focus on the static solutions of the Bogomolny’s equations and their corresponding energies with topological charge B > 0 is an integer. We consider two main cases based on the symmetry of the effective Lagrangian of the BPS submodels, i.e. the spherically symmetric and non-spherically symmetric cases. For the spherically symmetric case, we find two BPS submodels. The first BPS submodels consist of a potential term and a term proportional to the square of the topological current. The second BPS submodels consist of only the Skyrme term. The second BPS submodel has BPS skyrmions with the same topological charge B > 1, but with different energies, that we shall call “topological degenerate” BPS skyrmions. It also has the usual BPS skyrmions with equal energies, if the topological charge is a prime number. Another interesting feature of the BPS skyrmions, with B > 1, in this BPS submodel, is that these BPS skyrmions have non-zero pressures in the angular direction. For the non-spherically symmetric case, there is only one BPS submodel, which is similar to the first BPS submodel in the spherically symmetric case. We find that the BPS skyrmions depend on a constant k and for a particular value of k we obtain the BPS skyrmions of the first BPS submodel in the spherically symmetric case. The total static energy and the topological charge of these BPS skyrmions also depend on this constant. We also show that all the results found in this paper satisfy the full field equations of motions of the corresponding BPS submodels.

Some Possible Schemes in Higher Dimensional Stationary Skyrme Blackhole Formulation

“…with φ : M d+1 → S d is the Skyrme field as a map from d + 1 dimensional spacetime manifold to d dimensional sphere manifold which satisfies the O(d + 1) model constraint, namely φ a φ a = 1 [12,15,14]. In flat spacetime the map becomes φ : R 1,d → S d which contains a topological invariant.…”

“…This reparameterization of cosine coordinate works for both even and odd d cases because we can recover the even d cases constraint, i µ 2 i = 1, by choosing θ 0 = π 2 . In fact, a five dimensional multipolar ansatz like this is well known and has several unique properties such as topological degeneracy [15,22]. From this point, it is straightforward to calculate the topological degree explicitly as follows…”

BPS Skyrmions of Generalized Skyrme Model In Higher Dimensions

“…for every term we can proceed as follow Solution and features of this submodel is well-known and has been discussed in detail in [19,24].…”

“…The profile of the solution of this submodel highly depends on the form of potential and the dimension of spatial manifold. This submodel is the higher dimensional extension of the known BPS Skyrme model in four and five dimensions [16,24].…”

“…It is known that such lower bound exist in four and three dimensional Skyrme model [21,22,23]. A five dimensional BPS Skyrme submodel for a specific spherically symmetric system also have been found in [24]. Here we use the strain tensor eigenvalue combinations as the representation of higher order Skyrme terms which has been proposed in [25] to describe the geometrical properties of four dimensional Skyrmion as a harmonic map of flat spatial manifold to a three dimensional sphere manifold.…”