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We discover a new class of topological solitons. These solitons can exist in a space of infinite volume like, e.g., $\mathbb{R}^n$, but they cannot be placed in any finite volume, because the resulting formal solutions have infinite energy. These objects are, therefore, interpreted as totally incompressible solitons. As a first, particular example we consider (1+1) dimensional kinks in theories with a nonstandard kinetic term or, equivalently, in models with the so-called runaway (or vacummless) potentials. But incompressible solitons exist also in higher dimensions. As specific examples in (3+1) dimensions we study Skyrmions in the dielectric extensions both of the minimal and the BPS Skyrme models. In the the latter case, the skyrmionic matter describes a completely incompressible topological perfect fluid., Comment: reference added

We continue the investigation of supersymmetric extensions of baby Skyrme models in d=2+1 dimensions. In a first step, we show that the CP(1) form of the baby Skyrme model allows for the same N=1 SUSY extension as its O(3) formulation. Then we construct the N=1 SUSY extension of the gauged baby Skyrme model, i.e., the baby Skyrme model coupled to Maxwell electrodynamics. In a next step, we investigate the issue of N=2 SUSY extensions of baby Skyrme models. We find that all gauged and ungauged submodels of the baby Skyrme model which support BPS soliton solutions allow for an N=2 extension such that the BPS solutions are one-half BPS states (i.e., annihilated by one-half of the SUSY charges). In the course of our investigation, we also derive the general BPS equations for completely general N=2 supersymmetric field theories of (both gauged and ungauged) chiral superfields, and apply them to the gauged nonlinear sigma model as a further, concrete example., Comment: 32 pages, Latex file

We consider Lifshitz field theories with a dynamical critical exponent z equal to the dimension of space d and with a large group of base space symmetries, concretely space coordinate transformations with unit determinant ("Special Diffeomorphisms"). The field configurations of the theories considered may have the topology of skyrmions, vortices or monopoles, although we focus our detailed investigations on skyrmions. The resulting Lifshitz field theories have a BPS bound and exact soliton solutions saturating the bound, as well as time-dependent topological Q-ball solutions. Finally, we investigate the U(1) gauged versions of the Lifshitz field theories coupled to a Chern-Simons gauge field, where the BPS bound and soliton solutions saturating the bound continue to exist., Comment: 30 pages, Latex, 3 figures; v2: typos corrected, references added

The BPS Skyrme model is a specific subclass of Skyrme-type field theories which possesses both a BPS bound and infinitely many soliton solutions (skyrmions) saturating that bound, a property that makes the model a very convenient first approximation to the study of some properties of nuclei and hadrons. A related property, the existence of a large group of symmetry transformations, allows for solutions of rather general shapes, among which some of them will be relevant to the description of physical nuclei. We study here the classical symmetries of the BPS Skyrme model, applying them to construct soliton solutions with some prescribed shapes, what constitutes a further important step for the reliable application of the model to strong interaction physics., Comment: Latex, 17 pages

We demonstrate that in the supersymmetric extensions of a class of generalized (or K) field theories introduced recently, the static energy satisfies a BPS bound in each topological sector. Further, the corresponding soliton solutions saturate the bound. We also find strong indications that the BPS bound shows up in the SUSY algebra as a central extension, as is the case in the well-known supersymmetric field theories with standard kinetic terms., Comment: Latex, 16 pages

The BPS baby Skyrme models are submodels of baby Skyrme models, where the nonlinear sigma model term is suppressed. They have skyrmion solutions saturating a BPS bound, and the corresponding static energy functional is invariant under area-preserving diffeomorphisms (APDs). Here we show that the solitons in the BPS baby Skyrme model, which carry a nontrivial topological charge $Q_{b} \in \pi_2(S^2)$ (a winding number), are dual to vortices in a BPS vortex model with a topological charge $Q_{v}\in \pi_1(S^1)$ (a vortex number), in the sense that there is a map between the BPS solutions of the two models. The corresponding energy densities of the BPS solutions of the two models are identical. A further consequence of the duality is that the dual BPS vortex models inherit the BPS property and the infinitely many symmetries (APDs) of the BPS baby Skyrme models. Finally, we demonstrate that the same topological duality continues to hold for the U(1) gauged versions of the models., Comment: Latex, 18 pages, references added, typos corrected

The baby Skyrme model is a well-known nonlinear field theory supporting topological solitons in two space dimensions. Its action functional consists of a potential term, a kinetic term quadratic in derivatives (the "nonlinear sigma model term") and the Skyrme term quartic in first derivatives. The limiting case of vanishing sigma model term (the so-called BPS baby Skyrme model) is known to support exact soliton solutions saturating a BPS bound which exists for this model. Further, the BPS model has infinitely many symmetries and conservation laws. Recently it was found that the gauged version of the BPS baby Skyrme model with gauge group U(1) and the usual Maxwell term, too, has a BPS bound and BPS solutions saturating this bound. This BPS bound is determined by a superpotential which has to obey a superpotential equation, in close analogy to the situation in supergravity. Further, the BPS bound and the corresponding BPS solitons only may exist for potentials such that the superpotential equation has a global solution. We also briefly describe some properties of soliton solutions., Comment: Latex, 4 pages, Contribution to the "International Conference on Mathematical Modeling in Physical Sciences (ICMSQUARE)", Budapest, September 2012

We analyze the vector meson formulation of the BPS Skyrme model in (3+1) dimensions, where the term of sixth power in first derivatives characteristic for the original, integrable BPS Skyrme model (the topological or baryon current squared) is replaced by a coupling between the vector meson $\omega_\mu$ and the baryon current. We find that the model remains integrable in the sense of generalized integrability and almost solvable (reducible to a set of two first order ODEs) for any value of the baryon charge. Further, we analyze the appearance of topological solitons for two one-parameter families of one vacuum potentials: the old Skyrme potentials and the so-called BPS potentials. Depending on the value of the parameters we find several qualitatively different possibilities. In the massless case we have a parameter region with no skyrmions, a unique compact skyrmion with a discontinuous first derivative at the boundary (equivalently, with a source term located at the boundary, which screens the topological charge), and Coulomb-like localized solitons. For the massive vector meson, besides the no-skyrmion region and a unique C-compact soliton, we find exponentially as well as power-like localized skyrmions. Further, we find (for a specific potential) BPS solutions, i.e., skyrmions saturating a Bogomolny bound (both for the massless and massive vector mesons) which are unstable for higher values of the baryon charge. The properties of the model are finally compared with its baby version in (2+1) dimensions, and with the original BPS Skyrme model, contributing to a better understanding of the latter., Comment: Latex, 26 pages, 5 figures

We investigate the relation between the BPS baby Skyrme model and its vector meson formulation, where the baby Skyrme term is replaced by a coupling between the topological current $B_\mu$ and the vector meson field $\omega_\mu$. The vector model still possesses infinitely many symmetries leading to infinitely many conserved currents which stand behind its solvability. It turns out that the similarities and differences of the two models depend strongly on the specific form of the potential. We find, for instance, that compactons (which exist in the BPS baby Skyrme model) disappear from the spectrum of solutions of the vector counterpart. Specifically, for the vector model with the old baby Skyrme potential we find that it has compacton solutions only provided that a delta function source term effectively screening the topological charge is inserted at the compacton boundary. For the old baby Skyrme potential squared we find that the vector model supports exponentially localized solitons, like the BPS baby Skyrme model. These solitons, however, saturate a BPS bound which is a nonlinear function of the topological charge and, as a consequence, higher solitons are unstable w.r.t. decay into smaller ones, which is at variance with the more conventional situation (a linear BPS bound and stable solitons) in the BPS baby Skyrme model., Comment: 20 pages, 4 figures

The baby Skyrme model is a well-known nonlinear field theory supporting topological solitons in two space dimensions. In the limit where the term quadratic in derivatives (the "sigma model term") vanishes some additional structure emerges. The resulting ("extreme" or "restricted" or "BPS") baby Skyrme model has exact soliton solutions saturating a BPS bound which exists for this restricted model. Further, the restricted model has infinitely many symmetries and infinitely many conservation laws. Here we consider the gauged version of the restricted baby Skyrme model with gauge group U(1) and the usual Maxwell term for the gauge field. We find that, again, there exists a BPS bound and BPS solutions saturating this bound. We further find that the whole problem is essentially determined by a new kind of superpotential equation. The BPS bound and the corresponding BPS solitons only may exist for potentials such that the superpotential equation has a solution which exists globally, i.e., on the whole target space. We also calculate soliton solutions both exactly and numerically, completely confirming our qualitative analytical results., Comment: Latex, 53 pages, 15 figures; an incorrect comment on symmetries (base space diffeomorphisms) in Section III corrected; precision of numerical calculations improved; presentation of figures improved; some typos corrected

We review the recently developed supersymmetric extensions of field theories with non-standard kinetic terms (so-called K field theories) in two an three dimensions. Further, we study the issue of topological defect formation in these supersymmetric theories. Specifically, we find supersymmetric K field theories which support topological kinks in 1+1 dimensions as well as supersymmetric extensions of the baby Skyrme model for arbitrary nonnegative potentials in 2+1 dimensions., Comment: Contribution to the Proceedings of QTS7, Prague, August 2011

We construct supersymmetric K field theories (i.e., theories with a non-standard kinetic term) in 1+1 and 2+1 dimensions such that the bosonic sector just consists of a nonstandard kinetic term plus a potential. Further, we study the possibility of topological defect formation in these supersymmetric models. Finally, we consider more general supersymmetric K field theories where, again, topological defects exist in some cases., Comment: Latex, 6 figures, 27 pages

We construct a method to supersymmetrize higher kinetic terms and apply it to the baby Skyrme model. We find that there exist N=1 supersymmetric extensions for baby Skyrme models with arbitrary potential., Comment: 9 pages, Latex

A joint characterisation of the controllability and observability of a particular kind of discrete system has been developed. The key idea of the procedure can be reduced to a correct choice of the sampling sequence. This freedom, owing to the arbitrary choice of the sampling instants, is used to improve the sensitivity of system controllability and observability, by exploiting an adequate geometric structure. Some qualitative examples are presented for illustrative purposes., Comment: There are several equations that must be corrected

A restriction of the baby Skyrme model consisting of the quartic and potential terms only is investigated in detail for a wide range of potentials. Further, its properties are compared with those of the corresponding full baby Skyrme models. We find that topological (charge) as well as geometrical (nucleus/shell shape) features of baby skyrmions are captured already by the soliton solutions of the restricted model. Further, we find a coincidence between the compact or non-compact nature of solitons in the restricted model, on the one hand, and the existence or non-existence of multi-skyrmions in the full baby Skyrme model, on the other hand., Comment: latex, 18 pages, 2 figures; some typos corrected

The Skyrme model is a low-energy effective field theory for QCD, where the baryons emerge as soliton solutions. It is, however, not so easy within the standard Skyrme model to reproduce the almost exact linear growth of the nuclear masses with the baryon number (topological charge), due to the lack of Bogomolny solutions in this model, which has also hindered analytical progress. Here we identify a submodel within the Skyrme-type low energy effective action which does have a Bogomolny bound and exact Bogomolny solutions, and therefore, at least at the classical level, reproduces the nuclear masses by construction. Due to its high symmetry, this model qualitatively reproduces the main features of the liquid droplet model of nuclei. Finally, we discuss under which circumstances the proposed sextic term, which is of an essentially geometric and topological nature, can be expected to give a reasonable description of properties of nuclei., Comment: 11 pages, 2 figures, latex. v3: Extended and revised version, some clarifications added. Some references and 2 figures added. v4: matches published version

The strongly coupled limit of the Skyrme-Faddeev-Niemi model (i.e., without quadratic kinetic term) with a potential is considered on the spacetime S^3 x R. For one-vacuum potentials two types of exact Hopf solitons are obtained. Depending on the value of the Hopf index, we find compact or non-compact hopfions. The compact hopfions saturate a Bogomolny bound and lead to a fractional energy-charge formula E \sim |Q|^{1/2}, whereas the non-compact solitons do not saturate the bound and give E \sim |Q|. In the case of potentials with two vacua compact shell-like hopfions are derived. Some remarks on the influence of the potential on topological solutions in the full Skyrme-Faddeev-Niemi model or in (3+1) Minkowski space are also made., Comment: Latex, 25 pages, 9 figures, extensive revision; a section on the comparison with the full model added

For the baby Skyrme model with a specific potential, compacton solutions, i.e., configurations with a compact support and parabolic approach to the vacuum, are derived. Specifically, in the non-topological sector, we find spinning Q-balls and Q-shells, as well as peakons. Moreover, we obtain compact baby skyrmions with non-trivial topological charge. All these solutions may form stable multi-soliton configurations provided they are sufficiently separated., Comment: 23 pages, 17 figures, some comments and some references added

Using a nonlocal field transformation for the gauge field known as Cho--Faddeev--Niemi--Shabanov decomposition as well as ideas taken from generalized integrability, we derive a new family of infinitely many conserved currents in the self-dual sector of SU(2) Yang-Mills theory. These currents may be related to the area preserving diffeomorphisms on the reduced target space. The calculations are performed in a completely covariant manner and, therefore, can be applied to the self-dual equations in any space-time dimension with arbitrary signature., Comment: 12 pages, extensively revised version. One section on previous results and some references added

We review the developments of a recently proposed approach to study integrable theories in any dimension. The basic idea consists in generalizing the zero curvature representation for two-dimensional integrable models to space-times of dimension $d+1$ by the introduction of a $d$-form connection. The method has been used to study several theories of physical interest, like self-dual Yang-Mills theories, Bogomolny equations, non-linear sigma models and Skyrme-type models. The local version of the generalized zero curvature involves a Lie algebra and a representation of it, leading to a number of conservation laws equal to the dimension of that representation. We discuss the conditions a given theory has to satisfy in order for its associated zero curvature to admit an infinite dimensional (reducible) representation. We also present the theory in the more abstract setting of the space of loops, which gives a deeper understanding and a more simple formulation of integrability in any dimension., Comment: Invited talk at the Meeting ''Trends in Theoretical Physics II'', Buenos Aires, Argentina, December 1998