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Stratifying monopoles and rational maps. (English) Zbl 0693.32012
The author presents a stratification of the space of monopoles $$(A,\Phi)$$ in the following way. Let $$\Phi^{\infty}:=\lim_{R\to \infty} \Phi | S^ 2(\infty): S^ 2(\infty)\to SU(N).$$ Then $$\Phi^{\infty}(S^ 2(\infty))$$ is contained in some adjoint orbit $$F(n)$$ of $$SU(N)$$. We call $$(A,\Phi)$$ to be framed by $$(\mu_ 1,...,\mu_ q)$$ iff $$\Phi^{\infty}| \{positive\quad z- axis\}=(\mu_ 1^{\oplus n_ 1}\oplus...\oplus \mu_ q^{\oplus n_ q})$$ (diagonal). On the other hand, using the action of $$\Phi^{\infty}(S^ 2(\infty))$$ on $$S^ 2\times {\mathbb{C}}^ N$$, $$S^ 2\times {\mathbb{C}}^ N$$ split as $$S^ 2\times {\mathbb{C}}^ N=W_ 1\oplus,...,\oplus W_ q,$$ where $$W_ j$$ is the eigenspace of $$\Phi^{\infty}(S^ 2(\infty))$$ with the eigenvalue $$i\mu_ j$$. We call the integer $$m_ j:=c_ 1(W_ 1\oplus,...,\oplus W_ q)$$ $$(c_ 1$$ denotes the first Chern class) the magnetic monopole of (A,$$\Phi)$$. Due to Grothendieck’s splitting theorem, one obtains $W_ j\simeq {\mathcal O}(k_{n_ 1+,...,+n_{j-1}+1})\oplus,...,\oplus {\mathcal O}(k_{n_ 1+,...,+n_ j}).$ We call $$K_ i:=\{k_{n_{i- 1}+1},...,k_{n_ i}\}$$ a holomorphic charge of (A,$$\Phi)$$. Now the spaces of monopoles of magnetic charge $$m=(m_ 1,...,m_ q):\quad {\mathcal M}(m,F(n))$$ has a stratification $${\mathcal M}(m,F(n))=\cup_{k}{\mathcal M}_ k(m,F(n))$$ by the space of monopoles of holomorphic charge $$k=(K_ 1,...,K_ q)$$. On the other hand, for each $$(A,\Phi)\in {\mathcal M}(m,F(n)),$$ one can define a rational map $${\mathbb{P}}^ 1\to F(n)$$ of “magnetic energy” m using the twistor theory and the above filtration. Therefore we have a map from $${\mathcal M}(m,F(n))$$ to $${\mathcal R}(m,F(n)):=\{the$$ space of rational maps from $${\mathbb{P}}^ 1$$ to F(n) of magnetic energy $$m\}$$. He also shows that $${\mathcal R}(m,F(n))$$ can be stratified by “holomorphic charge” as before and the map defined above preserves the filtration. As an example, in the case of fundamental monopoles, he explicitly describes $${\mathcal M}(m,F(n))$$ and also shows that the map is a diffeomorphism.
Reviewer: K.Sugiyama

##### MSC:
 32G13 Complex-analytic moduli problems 58J90 Applications of PDEs on manifolds 14E05 Rational and birational maps 55R10 Fiber bundles in algebraic topology 14M99 Special varieties 81T08 Constructive quantum field theory
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##### References:
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