A Note on a Symplectic Structure on the Space of G -Monopoles

Abstract

Let $G$ be a semisimple complex Lie group with a Borel subgroup $B$. Let $X=G/B$ be the flag manifold of $G$. Let $C=P^1\ni\infty$ be the projective line. Let $\alpha\in H_2(X,{\Bbb Z})$. The moduli space of $G$-monopoles of topological charge $\alpha$ (see e.g. [Jarvis]) is naturally identified with the space $M_b(X,\alpha)$ of based maps from $(C,\infty)$ to $(X,B)$ of degree $\alpha$. The moduli space of $G$-monopoles carries a natural hyperk\"ahler structure, and hence a holomorphic symplectic structure. We propose a simple explicit formula for the symplectic structure on $M_b(X,\alpha)$. It generalizes the well known formula for $G=SL_2$ (see e.g. [Atiyah-Hitchin]). Let $P\supset B$ be a parabolic subgroup. The construction of the Poisson structure on $M_b(X,\alpha)$ generalizes verbatim to the space of based maps $M=M_b(G/P,\beta)$. In most cases the corresponding map $T^*M\to TM$ is not an isomorphism, i.e. $M$ splits into nontrivial symplectic leaves. These leaves are explicilty described.