Near BPS skyrmions and restricted harmonic maps

作者：Speight J M^*

来源：Journal of Geometry and Physics, 2015, 92: 30-45.

DOI：10.1016/j.geomphys.2015.02.001

• Skyrmions
• Topological solitons
• Harmonic maps

Motivated by a class of near BPS Skyrme models introduced by Adam, Sanchez-Guillen and Wereszczynski, the following variant of the harmonic map problem is introduced: a map phi : (M, g) -> (N, h) between Riemannian manifolds is restricted harmonic if it locally extremizes E-2 on its SDiff(M) orbit, where SDiff(M) denotes the group of volume preserving diffeomorphisms of (M, g), and E-2 denotes the Dirichlet energy. It is conjectured that near BPS skyrmions tend to restricted harmonic maps in the BPS limit. It is shown that phi is restricted harmonic if and only if phi*h has exact divergence, and a linear stability theory of restricted harmonic maps is developed, from which it follows that all weakly conformal maps are stable restricted harmonic. Examples of restricted harmonic maps in every degree class R-3 -> SU (2) and R-2 -> S-2 are constructed. It is shown that the axially symmetric BPS skyrmions on which all previous analytic studies of near BPS Skyrme models have been based, are not restricted harmonic, casting doubt on the phenomenological predictions of such studies. The problem of minimizing E-2 for phi : R-k -> N over all linear volume preserving diffeomorphisms is solved explicitly, and a deformed axially symmetric family of Skyrme fields constructed which are candidates for approximate near BPS skyrmions at low baryon number. The notion of restricted harmonicity is generalized to restricted Fcriticality where F is any functional on maps (M, g) -> (N, h) which is, in a precise sense, geometrically natural. The case where F is a linear combination of E-2 and E-4, the usual Skyrme term, is studied in detail, and it is shown that inverse stereographic projection R-3 -> S-3 equivalent to SU(2) is stable restricted F-critical for every such F.

• 出版日期2015-6