Lagrangians from Equations: A Skyrmion Case Study

Abstract

Atiyah and Manton^[1] have shown recently that they could approximate the configuration of the actual Skyrme model^[2] with an ansatz derived from a SU(2) Yang-Mills instanton field A _μ in four dimensions (x,t) using the holonomy

$$ U\left( x \right) = T\exp {\mkern 1mu} \left( { - \int_{ - \infty }^\infty {{A_t}} \left( {x,t} \right)dt} \right), $$

((1.1))

when static solutions with spherical symmetry are considered, by identifying this with the unitary field of the model^[2]

$$ U\left( x \right) = \exp \left( {f\left( r \right){{\hat x}^2}{\sigma _2}} \right) $$

((1.2))

where r = (x _i x _i )¹/² is the distance to the origin in space and ${{\hat x}^2} = {x^2}/r$ is the unit vector in the direction of x ⁱ.