Near BPS skyrmions and restricted harmonic maps

Abstract

Motivated by a class of near BPS Skyrme models introduced by Adam, Sánchez-Guillén and Wereszczyński, the following variant of the harmonic map problem is introduced: a map ϕ : (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 ϕ is restricted harmonic if and only if ϕ * 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 ϕ : 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 F -criticality 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 ≡ SU (2) is stable restricted F -critical for every such F .

Magnetothermodynamics of BPS baby skyrmions

“…It is also known that some crystal structures usually emerge [53]. However, if we assume that the main contribution to the energy comes from the BPS part of the full model, i.e., the Dirichlet part is multiplied by a small parameter , we are still in a near BPS regime with only softly broken APD symmetries (for a recent investigation of this issue, see [54]). Hence, one may wonder whether, for sufficiently small , we would continue to have liquid (plastic) ferromagnetic matter, as found for the BPS limit.…”

Volume of a vortex and the Bradlow bound

“…We act with an infinitesimal scale transformation x i → ð1 þ λÞx i , for infinitesimal λ, on all fields that appear in the constrained energy functional (but not on the metric). Then we assume that all terms in the energy density are "geometrically natural" [14] such that a coordinate transformation acting on the fields may be traded for the inverse coordinate transformation of the metric, δ ij → ð1 − 2λÞδ ij . That is to say, we act with the transformation…”

Radial vibrations of BPS skyrmions

“…Classical binding energies are, therefore, zero, and small nonzero binding energies may be generated, either already for the BPS submodel, by including further energy contributions (spin and isospin excitations, Coulomb energy [18]), or by adding more terms to the lagrangian [19] - [21], leading to the general lagrangian (I.1).…”