Harmonic, Holomorphic and Rational Maps from Self-Duality

Abstract

We propose a generalization of the so-called rational map ansatz on the Euclidean space $\mathbb{R}^3$, for any compact simple Lie group $G$ such that $G/{\widehat K}\otimes U(1)$ is an Hermitian symmetric space, for some subgroup ${\widehat K}$ of $G$. It generalizes the rational maps on the two-sphere $SU(2)/U(1)$, and also on $CP^N=SU(N+1)/SU(N)\otimes U(1)$, and opens up the way for applications of such ans\"atze on non-linear sigma models, Skyrme theory and magnetic monopoles in Yang-Mills-Higgs theories. Our construction is based on a well known mathematical result stating that stable harmonic maps $X$ from the two-sphere $S^2$ to compact Hermitian symmetric spaces $G/{\widehat K}\otimes U(1)$ are holomorphic or anti-holomorphic. We derive such a mathematical result using ideas involving the concept of self-duality, in a way that makes it more accessible to theoretical physicists. Using a topological (homotopic) charge that admits an integral representation, we construct first order partial differential self-duality equations such that their solutions also solve the (second order) Euler-Lagrange associated to the harmonic map energy $E=\int_{S^2} \mid dX\mid^2 d\mu$. We show that such solutions saturate a lower bound on the energy $E$, and that the self-duality equations constitute the Cauchy-Riemann equations for the maps $X$. Therefore, they constitute harmonic and (anti)holomorphic maps, and lead to the generalization of the rational map ans\"atze in $\mathbb{R}^3$. We apply our results to construct approximate Skyrme solutions for the $SU(N)$ Skyrme model.