A geometric approach to orthogonal Higgs bundles. (English) Zbl 1405.14035
This paper studies the geometry of the moduli space of \(\mathrm{SO}(p+1,p)\)-Higgs bundles in the moduli space of \(\mathrm{SO}(2p+1,\mathbb{C})\)-Higgs bundles on a Riemann surface of genus greater than or equal to \(2\).
In particular, using the correspondence with \(\mathrm{Sp}(2p,\mathbb{C})\)-Higgs bundles due to Langlands duality, the author obtains that the intersection of the moduli space of \(\mathrm{SO}(p+1,p)\)-Higgs bundles with the regular fibres of \(\mathrm{SO}(2p+1,\mathbb{C})\)-Hitchin fibration is given by two copies of a quotient of \(\mathrm{Prym}(S,S/\sigma)[2]\), where \(S\) is the associated spectral curve over \(\Sigma\) and \(\sigma\) is a natural involution.
Another significant result in the paper, obtained by the use of KO-theory, is the computation of the Stiefel-Whitney classes of an \(\mathrm{SO}(p+1,p)\)-Higgs bundle in terms of its spectral data.
The article gives also a natural stratification of the smooth loci of the moduli space of \(\mathrm{SO}(p+1,p)\)-Higgs bundles.
In particular, using the correspondence with \(\mathrm{Sp}(2p,\mathbb{C})\)-Higgs bundles due to Langlands duality, the author obtains that the intersection of the moduli space of \(\mathrm{SO}(p+1,p)\)-Higgs bundles with the regular fibres of \(\mathrm{SO}(2p+1,\mathbb{C})\)-Hitchin fibration is given by two copies of a quotient of \(\mathrm{Prym}(S,S/\sigma)[2]\), where \(S\) is the associated spectral curve over \(\Sigma\) and \(\sigma\) is a natural involution.
Another significant result in the paper, obtained by the use of KO-theory, is the computation of the Stiefel-Whitney classes of an \(\mathrm{SO}(p+1,p)\)-Higgs bundle in terms of its spectral data.
The article gives also a natural stratification of the smooth loci of the moduli space of \(\mathrm{SO}(p+1,p)\)-Higgs bundles.
Reviewer: Michele Savarese (Roma)
MSC:
| 14D20 | Algebraic moduli problems, moduli of vector bundles |
| 14D21 | Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) |
| 14H70 | Relationships between algebraic curves and integrable systems |
| 14P25 | Topology of real algebraic varieties |
| 20C33 | Representations of finite groups of Lie type |
| 53C07 | Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills) |
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