Higgs bundles for the non-compact dual of the special orthogonal group. (English) Zbl 1314.14060
The authors studies the moduli of \(G-\)Higgs bundles for the case \(G = \mathrm{SO}^*(2n)\) over Riemann surface \(X\). A topological invariant \(d\) is attached to a representation of \(\pi_1(X)\) in \(\mathrm{SO}^*(2n)\) called the Toledo invariant. The authors show that this invariant coincides with the first Chern class of a reduction to a \(\mathrm{U}(n)\)-bundle of the flat \(\mathrm{SO}^*(2n)\)-bundle associated to that representation. They give a bound of this invariant such that the moduli space of \(\mathcal{M}_d(\mathrm{SO}^*(2n))\) of polystable \(\mathrm{SO}^*(2n)-\)Higgs bundles \((E, \varphi)\) with first Chern class \(c(E)=d\) become nonempty and study the moduli space for the extreme cases.
In this extreme situation when \(n = 2m\), to every \(\mathrm{SO}*(2n)-\)Higgs bundle they associate a \(K^2-\)twisted \(\mathrm{U}^*(2m)\)-Higgs pair called Cayley partner and show that the moduli space of \(\mathcal{M}_d(\mathrm{SO}^*(4m))\) of polystable \(\mathrm{SO}^*(4m)\)-Higgs bundles is isomorphic to \(\mathcal{M}_{K^2}(\mathrm{U}^*(2m))\), the moduli space of polystable \(K^2\)-twisted \(\mathrm{U}^*(2m)\)-Higgs pairs and for \(n=2m+1\), it is isomorphic to \(\mathcal{M}_d(\mathrm{SO}^*(4m)) \times \text{Jac}(X)\). Using this and Morse theory in the moduli space of Higgs bundles, they show that the moduli space is connected in the maximal Toledo case.
The paper ends with extensive study of \(\mathrm{SO}^*(2n)\)-Higgs bundles for low values of \(n\).
In this extreme situation when \(n = 2m\), to every \(\mathrm{SO}*(2n)-\)Higgs bundle they associate a \(K^2-\)twisted \(\mathrm{U}^*(2m)\)-Higgs pair called Cayley partner and show that the moduli space of \(\mathcal{M}_d(\mathrm{SO}^*(4m))\) of polystable \(\mathrm{SO}^*(4m)\)-Higgs bundles is isomorphic to \(\mathcal{M}_{K^2}(\mathrm{U}^*(2m))\), the moduli space of polystable \(K^2\)-twisted \(\mathrm{U}^*(2m)\)-Higgs pairs and for \(n=2m+1\), it is isomorphic to \(\mathcal{M}_d(\mathrm{SO}^*(4m)) \times \text{Jac}(X)\). Using this and Morse theory in the moduli space of Higgs bundles, they show that the moduli space is connected in the maximal Toledo case.
The paper ends with extensive study of \(\mathrm{SO}^*(2n)\)-Higgs bundles for low values of \(n\).
Reviewer: Sarbeswar Pal (Kelambakkam)
MSC:
| 14H60 | Vector bundles on curves and their moduli |
| 53C07 | Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills) |
| 58D29 | Moduli problems for topological structures |
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