Abstract
We subject the baby Skyrme model to a Moyal deformation, for unitary or Grassmannian target spaces and without a potential term. In the Abelian case, the radial BPS configurations of the ordinary noncommutative sigma model also solve the baby Skyrme equation of motion. This gives a class of exact analytic noncommutative baby Skyrmions, which have a singular commutative limit but are stable against scaling due to the noncommutativity. We compute their energies, investigate their stability and determine the asymptotic two-Skyrmion interaction.
Highlights
It is known that the two-dimensional CP 1 σ -model [1] possesses metastable states which when perturbed may shrink or spread out due to the conformal invariance of the model [2,3,4]
In the CP 1 baby Skyrme model, the target manifold S2 is parametrized by a three-dimensional isovector scalar φ subject to the constraint |φ|2 = 1
In this Letter, we present a noncommutative baby Skyrme model,1 without potential term, for group- or Grassmannianvalued targets, and explicitly obtain a class of exact analytic solitonic solutions, which have no analogues in the commutative theory
Summary
It is known that the two-dimensional CP 1 σ -model [1] possesses metastable states which when perturbed may shrink or spread out due to the conformal (scale) invariance of the model [2,3,4]. All terms are needed to prevent the collapse of topological configurations which yield to Skyrmion solutions In this situation, a noncommutative deformation (for reviews see [11]) may serve as a substitute for the potential term (or Zeeman interaction), because it introduces a new length scale into the theory, which stabilizes solitons against collapse or spreading. In this Letter, we present a noncommutative baby Skyrme model, without potential term, for group- or Grassmannianvalued targets, and explicitly obtain a class of exact analytic solitonic solutions, which have no analogues in the commutative theory This surprising feat succeeds because certain BPS configurations of the Moyal-deformed ordinary sigma model extremize the Skyrme part of the energy as well. We compute their static energy, discuss their stability and evaluate the two-Skyrmion interaction potential at large distances
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