A Note on the Symplectic Structure on the Space of G-monopoles
نویسنده
چکیده
1.1. Let G be a semisimple complex Lie group with the Cartan datum (I, ·) and the root datum (Y,X, . . . ). Let H ⊂ B = B+,B− ⊂ G be a Cartan subgroup and a pair of opposite Borel subgroups respectively. Let X = G/B be the flag manifold of G. Let C = P ∋ ∞ be the projective line. Let α = ∑ i∈I aii ∈ N[I] ⊂ H2(X,Z). The moduli space of G-monopoles of topological charge α (see e.g. [4]) is naturally identified with the space Mb(X, α) of based maps from (C,∞) 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 Mb(X, α). It generalizes the well known formula for G = SL2 [1]. 1.2. Recall that for G = SL2 we have (X,B+) = (P ,∞). Recall the natural local coordinates on Mb(P , a) (see [1]). We fix a coordinate z on C such that z(∞) = ∞. Then a based map φ : (C,∞) → (P,∞) of degree a is a rational function p(z) q(z) where
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تاریخ انتشار 1999