关于G-单极子空间辛结构的一个注记
A note on the symplectic structure on the space of G-monopoles
摘要：
令$G$是具有Borel子群$B$的半单复李群。设$X=G/B$为$G$的旗流形。设$C=P^1\ni\infty$为投影线。令$α\in H_2(X,{\Bbb Z})$。拓扑电荷$α$的$G$ -单极子的模空间（例如参见[Jarvis]）自然地与从$(C,\infty)$到度$α$的$(X,B)$的基映射的空间$M_b(X,α)$相同。$G$ -单极子的模空间具有自然的超凯勒结构，因此是全纯辛结构。我们给出了$M_b(X,α)$上辛结构的一个简单的显式公式。它推广了$G=SL_2$的公知公式（参见例如[Atiyah-Hitchin]）。设$P\supset B$是抛物子群。在$M_b(X,α)$上构造Poisson结构，将Poisson结构完全推广到基映射空间$M=M_b(G/P,β)$。在大多数情况下，相应的映射$T^*M\to TM$不是同构，即$M$分裂成非平凡的辛叶。这些叶子被明确地描述了。
英文摘要：
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 $α\in H_2(X,{\Bbb Z})$. The moduli space of $G$-monopoles of topological charge $α$ (see e.g. [Jarvis]) is naturally identified with the space $M_b(X,α)$ of based maps from $(C,\infty)$ to $(X,B)$ of degree $α$. The moduli space of $G$-monopoles carries a natural hyperkähler structure, and hence a holomorphic symplectic structure. We propose a simple explicit formula for the symplectic structure on $M_b(X,α)$. 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 $M_b(X,α)$ generalizes verbatim to the space of based maps $M=M_b(G/P,β)$. 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.
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