Gauged BPS Baby Skyrme Model

• The gauged Skyrme model is a nonlinear field theory coupling an O(3) sigma model to gauge fields, resulting in topological solitons with quantized magnetic flux.
• BPS solitons arise through a superpotential framework that leads to a Bogomol'nyi energy bound and highlights key phase transitions for various potentials.
• Extensions to supersymmetric versions and higher-dimensional generalizations demonstrate the model's broad relevance in nuclear, condensed matter, and mathematical physics.

The gauged Skyrme model refers to a broad class of field theories in which the original Skyrme model—built from nonlinear $O(N+1)$ or $SU(2)$-valued maps supporting solitonic solutions with topological charge ("baryon number")—is coupled to a gauge field, typically $U(1)$ or $SO(N)$, via a covariant derivative and possibly augmented by additional gauge sector dynamics such as Maxwell or Chern-Simons terms. The gauged Skyrme model and its dimensional and structural reductions (most notably the $2+1$-dimensional "baby Skyrme" and its restricted/BPS limits) are central to the paper of topological solitons with nontrivial electromagnetic interactions and find applications in nuclear, condensed matter, and mathematical physics.

1. Fundamental Structure: Baby Skyrme Model, Gauging, and BPS Limit

The baby Skyrme model is a $(2+1)$-dimensional analogue of the standard $(3+1)$-dimensional Skyrme model. Its field $\vec\phi=(\phi_1, \phi_2, \phi_3)$ is constrained to $|\vec\phi|=1$ (i.e., mapping $\mathbb{R}^2 \rightarrow S^2$). The generic Lagrangian comprises three terms:

• A potential $V(\phi_3)$,
• A quadratic derivative, "nonlinear sigma model" term,
• A quartic Skyrme term proportional to $$[\vec\phi\cdot(\partial_\mu\vec\phi\times\partial_\nu\vec\phi)]^2$$.

In the BPS baby Skyrme model (or "restricted," "extreme" version), the sigma model term is omitted. The resulting model supports BPS (Bogomol'nyi–Prasad–Sommerfield) solitons, that is, exact solutions saturating a topological energy bound and obeying first-order differential equations. The model possesses infinitely many (non-Noether) symmetries, infinite conservation laws, and a solvable Bogomolnyi functional minimizing structure (Adam et al., 2012, Adam et al., 2012).

Gauging proceeds by promoting a global $U(1)$ subgroup (rotations in the $\phi_1$–$\phi_2$ plane) to a local symmetry. The usual derivative is replaced by the $U(1)$-covariant derivative: $$D_\mu\vec\phi = \partial_\mu\vec\phi + A_\mu(\vec n\times\vec\phi),$$ where $\vec n=(0,0,1)$ is the symmetry-breaking direction, and $A_\mu$ is the $U(1)$ gauge field. The Maxwell term for $A_\mu$ is added to the Lagrangian, enabling magnetic (and, in some couplings, electric) interactions.

2. BPS Bound, Superpotential Equation, and BPS Equations

A central feature of the gauged BPS baby Skyrme model is that, despite the gauge field coupling, a BPS bound for the static energy persists. The existence and saturation of the BPS bound hinge on a function $W(\phi_3)$ (the "superpotential") satisfying a first-order ordinary differential equation:

$$\frac{\lambda^2}{4} (W_{,h})^2 + g^2\lambda^4 W^2 = 2\mu^2 V,$$

where $V$ is the potential (as a function of $h$, typically $h=\frac{1-\phi_3}{2}$), $\lambda$, $\mu$ are model parameters, and $g$ the gauge coupling (Adam et al., 2012).

The static energy admits a Bogomolny-type completion: $E \geq \lambda^2 \int d^2x\: q\, W_{,h},$ where $q$ is the topological charge density. The BPS bound is saturated by field configurations solving the first-order equations: $$\begin{cases} Q \equiv \vec{\phi}\cdot(D_1\vec{\phi}\times D_2\vec{\phi}) = W_{,h},\ B = -g^2\lambda^2 W, \end{cases}$$ where $B=F_{12}$ is the magnetic field (Adam et al., 2012, Adam et al., 2012).

A crucial property is that for the BPS bound and solutions to exist, the superpotential $W$ must solve its equation globally (i.e., for the full physical interval $h\in[0,1]$), with $W(0)=0$ as the vacuum boundary condition (Adam et al., 2012). If the potential $V$ has more than one vacuum (e.g., two zeroes for $h$), boundary conditions tend to overconstrain $W$, and BPS solitons are generically absent.

3. Soliton Solutions and Magnetic Structure

The model supports both compacton and exponentially localized solitons, depending on the choice of potential $V(h)$. Notable examples include:

• "Old baby Skyrme" potential ($V\sim 2h$): solitons are compactons, i.e., their profiles vanish exactly at finite radius; the magnetic field is confined within the compacton, and as $g\to\infty$, magnetic flux quantization emerges with $\Phi\to -2\pi n$ for $n$ the winding number (Adam et al., 2012).
• Quartic potential ($V\sim 4h^2$): profiles decay exponentially. The energy computed for these solutions numerically matches the BPS bound with high accuracy.

A reverse-engineering approach is possible: for a given $W$, one computes the corresponding potential and can sometimes obtain exact, closed-form soliton solutions (e.g., $W=h^2/\lambda^2$ yields solutions involving the logarithmic integral function) (Adam et al., 2012).

4. Symmetries, Conservation Laws, and Integrability

The gauged BPS baby Skyrme model retains the infinite generalized integrability of the ungauged model. Its static equations admit an infinite number of area-preserving diffeomorphism (APD) symmetries on the base space and an infinite family of abelian Noether currents (not associated with the $U(1)$ gauge invariance). In the complex/projective $CP^1$ variable formulation, explicit expressions for these currents and their associated integrable structures are manifest (Adam et al., 2012).

This property puts the model in a unique position among interacting (gauge) field theories in $2+1$ dimensions, combining nontrivial topology, exact BPS structure, and generalized integrability.

5. Potentials, Multi-Vacua, and Phase Structure

The interdependence between admissible potentials and the existence of solitonic BPS solutions is an essential characteristic. For single-vacuum potentials, the superpotential equation admits the required global solution, permitting the existence and explicit construction of BPS solitons (numerical and analytic).

For double-vacuum (e.g., $V\sim (1-\phi_3^2)$) or other multi-zero potentials, the necessary boundary conditions for $W$ cannot be satisfied simultaneously, generally precluding BPS solitons in the gauged case—even if BPS solitons exist in the ungauged model (Adam et al., 2012, Adam et al., 2015). This represents a qualitative alteration in soliton stability due to electromagnetic effects.

Analytic and numerical results in related studies have demonstrated that adding external pressure or magnetic field can induce a solitonic phase in such multi-vacuum models, so the physical phase structure is richer when these control parameters are varied (Adam et al., 2015).

6. Extensions, Supersymmetry, and Higher-Dimensional Outlook

Supersymmetric completions of the gauged BPS baby Skyrme model have been constructed at both $N=1$ and $N=2$ levels (Adam et al., 2013). In these settings, the emergence of the BPS equations and saturation of the energy bound are explained via the requirement that configurations preserve a portion of the supersymmetry (one-half BPS states). The generalized superpotential equation and structure arise naturally as a restriction from the Kähler or auxiliary fields in the supersymmetric system.

The mathematical scheme underlying the BPS structure, superpotential, and Bogomolnyi reduction is strongly reminiscent of analogous constructions in supergravity and "fake supergravity" models (Adam et al., 2012). This analogy motivates investigations into higher-dimensional generalizations, specifically the $3+1$-dimensional Skyrme–Maxwell system. Whether a superpotential-based BPS bound and analytic or numerical solutions of similar character can be realized in higher dimensions is a key open direction (Adam et al., 2012).

7. Implications and Applications

The gauged BPS baby Skyrme model provides a prototype for studying:

• Topological solitons in the presence of gauge fields and non-standard kinetic terms,
• The effect of electromagnetic interactions on soliton phase structure and stability,
• Flux quantization phenomena in topological matter and potential condensed-matter realizations,
• The interplay between integrability, topology, and gauge interaction in non-linear field theories.

Related constructions, such as gauging via the Schroers or Goldstone–Wilczek currents, promise exactly solvable models with quantized flux and have potential to extend the reach of analytic soliton solutions in field theory (Adam et al., 2017).

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In summary, the gauged BPS baby Skyrme model in $2+1$ dimensions is a nonlinear field theory coupling an $O(3)$ sigma model (with the kinetic term omitted) to an abelian $U(1)$ gauge field. It admits BPS solitons saturating a topological energy bound, provided a superpotential solves a first-order ODE globally determined by the choice of potential. The model exhibits rich mathematical features, flux quantization effects, and structural dependence on the potential that reveal qualitative differences from the ungauged case. Its extensions, integrability, and implications for phase transitions, supersymmetry, and higher-dimensional generalizations make it a central—and still evolving—framework in the paper of topological field theory (Adam et al., 2012, Adam et al., 2012, Adam et al., 2013, Adam et al., 2015, Adam et al., 2017).