Orthogonal Higgs bundles with singular spectral curves (Q2227306)

In the paper under review, the authors carry out a detailed study of the Higgs bundles for noncompact real forms of \(\mathrm{SL}(2,\mathbb C)\times \mathrm{SL}(2,\mathbb C)\) and the isogenous group \(\mathrm{SO}(4,\mathbb C)\), on a compact Riemann surface \(\Sigma\) with genus \(g\geq 2\). Let \(\mathcal I_2:\mathrm{SL}(2,\mathbb C)\times \mathrm{SL}(2,\mathbb C)\rightarrow \mathrm{SO}(4,\mathcal C)\) be the isogeny defined by the tensor product. There are only two isogenous pairs of real forms, which are neither compact nor split, i.e: \begin{itemize} \item[1.] \(\mathrm{SL}(2,\mathbb C)\), \(\mathrm{SO}(1,3)\); \item[2.] \(\mathrm{SU}(2)\times \mathrm{SL}(2,\mathbb R)\), \(\mathrm{SO}^*(4)\). \end{itemize} The authors give a concrete description of the map induced by the \(\mathcal I_2\), between Higgs bundles for \(\mathrm{SL}(2,\mathbb C)\) and \(\mathrm{SO}(1,3)\) (resp. \(\mathrm{SU}(2)\times \mathrm{SL}(2,\mathbb R)\) and \(\mathrm{SO}^*(4)\)). Also, a geometric description of the induced map on spectral datas is given in this paper. The main construction used in this paper is: build an \(\mathrm{SO}(4,\mathbb C)\)-spectral curve by the fiber product \(S_1\times_\Sigma S_2\), where \(S_1\) and \(S_2\) are \(\mathrm{SL}(2,\mathbb C)\)-spectral curves. Using this construction, the locus of \(\mathrm{SO}_0(1,3)\)-Higgs bundles inside the Hitchin fiber over some special spectral curve is dual to a Prym variety, where \(\mathrm{SO}_0(1,3)\) is the connected component of identity. Similarly, there is a description for \(\mathrm{SO}^*(4)\)-Higgs bundles. The authors also give some comments on the relations between the support of the (B,B,B) brane and the moduli space of real Higgs bundles.