The Existence of Bogomolny Decompositions for Gauged $O(3)$ Nonlinear ``sigma'' Model and for Gauged Baby Skyrme Models

Abstract

The Bogomolny decompositions (Bogomolny equations) for the gauged baby Skyrme models: restricted and full one, in (2+0)-dimensions, are derived, for some general classes of the potentials. The conditions, which must be satisfied by the potentials, for each of these mentioned models, are also derived.

Bogomolny equations in certain generalized baby BPS Skyrme models

“…After applying the concept of strong necessary conditions to (2.24), we obtain the so-called dual equations (cf. [35]):…”

“…Namely, as we see, after putting (cf. [35]): Hence, we get some system of partial differential equations for V . It has turned out that V = V (ω, ω * ), and the solution of this system, for U = U (ωω * ), (cf.…”

“…It has turned out that V = V (ω, ω * ), and the solution of this system, for U = U (ωω * ), (cf. [35]), is:…”

“…In such a case, non-standard kinetic part (k-model generalization) as well as addition of a "dielectric" function to the gauge field part are typically accepted generalizations -see for example, condense matter [38] or cosmological applications [39]. Obviously, the lagrangian should be gauge invariant, however in the case of gauged models investigated in this paper, we want also to investigate (as in [35]), whether any terms analogical to Proca terms ( [40], [41]), will appear in the expressions for the potentials.…”

Baby Skyrme model and fermionic zero modes

“…The model possesses a Bogomol'nyi bound given by the topological charge, and also solutions saturating the bound when the O(3) term is absent [30]- [32]. The later situation defines the so-called BPS baby Skyrme model (BbS)…”