Harmonic, Holomorphic and Rational Maps from Self-Duality

TL;DR

The paper extends the rational map ansatz to a broad class of compact simple Lie groups by exploiting Hermitian symmetric cosets $G/\widehat{K}\otimes U(1)$. It builds a self-dual, first-order framework that links harmonic holomorphic/anti-holomorphic maps from $S^2$ to these cosets with a three-dimensional ansatz in $\mathbb{R}^3$, via a radial profile $f(r)$ and a principal variable $g(z,\bar z)$. This yields explicit constructions for $CP^1$, $CP^N$, and SU$(p+q)/$SU$(p)\times$SU$(q)\times U(1)$, as well as Sp$(N)$, and provides a practical route to approximate topological solitons in nonlinear sigma models and the SU$(N)$ Skyrme model. The results show how holomorphic structure governs the soliton content, with concrete energy computations indicating near-optimal scalings and a viable path to studying Skyrmions for larger symmetry groups.

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 ansätze 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 $E=\int_{S^2} \mid dX\mid^2 dμ$. 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 ansätze in $\mathbb{R}^3$. We apply our results to construct approximate Skyrme solutions for the $SU(N)$ Skyrme model.

Figures (1)

• Figure 1: The Dynkin diagrams of the simple Lie algebras. The $\alpha_a$'s below the spots label the simple roots, and the numbers above correspond to the integers $m_a$ in the expansion of the highest root $\psi=\sum_{a=1}^r m_a\,\alpha_a$. The black spots correspond to $m_a=1$. Note that for $E_8$, $F_4$ and $G_2$, none of the $m_a$'s equal unity, and so such groups do not lead to Hermitian symmetric spaces.