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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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