Skyrmion

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In theoretical physics, a skyrmion, conceived by Tony Skyrme, is a homotopically non-trivial classical solution of a nonlinear sigma model with a non-trivial target manifold topology i.e. a particular case of a topological soliton. It arises, for example, in chiral models of mesons where the target manifold is a homogeneous space of

$SU(N)_L \times SU(N)_R \,$

(the structure group),

$\left(\frac{SU(N)_L\times SU(N)_R}{SU(N)_{diag}}\right)$

where

SU(N)_L and SU(N)_R are the left and right copies respectively

SU(N)_diag is the diagonal subgroup

If spacetime has the topology S³×R (for space and time respectively), then classical configurations are classified by an integral winding number because the third homotopy group,

$\pi_3\left(\frac{SU(N)_L\times SU(N)_R}{SU(N)_{diag}}\cong SU(N)\right)=\mathbb{Z}$

(the congruence sign here refers to homeomorphism, not isomorphism).

It is possible to add a topological term to the chiral lagrangian whose integral only depends upon the homotopy class. This results in superselection sectors in the quantized model.

Skyrmions have been used to model baryons.