Skyrmion

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In theoretical physics, a skyrmion, conceived by Tony Skyrme, is a mathematical model used to model baryons (a subatomic particle). A skyrmion is a homotopically non-trivial classical solution of a nonlinear sigma model with a non-trivial target manifold topology: 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),

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

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,

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

(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. It has been predicted that they could be created in a multi-component Bose-Einstein condensate^[1].