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 $\varphi :(M,g)\to (N,h)$ between Riemannian manifolds is restricted harmonic (RH) 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 RH maps in the BPS limit. It is shown that $\varphi$ is RH if and only if ${\varphi }^{\ast }h$ has exact divergence, and a linear stability theory of RH maps is developed, whence it follows that all weakly conformal maps, for example, are stable RH. Examples of RH maps in every degree class $R^3\to SU(2)$ and $R^2\to S^2$ 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 RH, so each such field can be deformed along $SDiff(R^3)$ to yield BPS skyrmions with lower $E_2$, casting doubt on the predictions of such studies. The problem of minimizing $E_2$ for $\varphi :R^k\to N$ 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 $(M,g)\to (N,h)$ 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 $R^3\to S^3\equiv SU(2)$ is stable restricted $F$-critical for every such $F$.

arXiv comment26 pages, 0 figures, updated bibliography, published version

References in corpus (13)

• A Skyrme-type proposal for baryonic matter
• A BPS Skyrme model and baryons at large Nc
• Skyrmions in a truncated BPS theory
• Nuclei as near BPS-Skyrmions
• Compactons and semi-compactons in the extreme baby Skyrme model
• Near-BPS Skyrmions: Non-shell configurations and Coulomb effects
• The gauged BPS baby Skyrme model
• Baryonic sphere: a spherical domain wall carrying baryon number
• Scaling Identities for Solitons beyond Derrick's Theorem
• Solitons on tori and soliton crystals
• Rotational-vibrational coupling in the BPS Skyrme model of baryons
• Perfect fluids from high power sigma-models
• The vector BPS Skyrme model

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• Generalized Skyrmions and hairy black holes in asymptotically AdS${}_4$ spacetime
• Radial vibrations of BPS skyrmions
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• Bogomolny equation for the BPS Skyrme model from the strong necessary conditions
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• On the spin excitation energy of the nucleon in the Skyrme model
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• Near-BPS baby Skyrmions with Gaussian tails
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