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Higgs bundles for the Lorentz group. (English) Zbl 1273.14026

Let \(X\) be a compact Riemann surface of genus \(g\geq 2\), and \(G\) be a real reductive Lie group. Let \(H\subseteq G\) be a maximal compact subgroup and \(H^\mathbb{C}\subseteq G^\mathbb{C}\) their complexifications. Let \(\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{m}\) be a Cartan decomposition of \(\mathfrak{g}\), where \(\mathfrak{m}\) is the complement of \(\mathfrak{h}\) with respect to the non-degenerate \(\text{Ad}(G)\)-invariant bilinear form on \(\mathfrak{g}\). If \(\theta:\mathfrak{g}\to\mathfrak{g}\) is the corresponding Cartan involution then \(\mathfrak{h}\) and \(\mathfrak{m}\) are its \(+1\)-eigenspace and \(-1\)-eigenspace, respectively. Complexifying, we have the decomposition \(\mathfrak{g}^{\mathbb{C}}=\mathfrak{h}^{\mathbb{C}}\oplus\mathfrak{m}^{\mathbb{C}}\) and \(\mathfrak{m}^{\mathbb{C}}\) is a representation of \(H^\mathbb{C}\) through the so-called isotropy representation, \(\iota:H^\mathbb{C}\longrightarrow\operatorname{Aut}(\mathfrak{m}^{\mathbb{C}})\), which is induced by the adjoint representation of \(G^\mathbb{C}\) on \(\mathfrak{g}^{\mathbb{C}}\). If \(E_{H^\mathbb{C}}\) is a principal \(H^{\mathbb{C}}\)-bundle over \(X\), we denote by \(E_{H^\mathbb{C}}(\mathfrak{m}^{\mathbb{C}})=E_{H^\mathbb{C}}\times_{H^{\mathbb{C}}}\mathfrak{m}^{\mathbb{C}}\) the vector bundle, with fiber \(\mathfrak{m}^{\mathbb{C}}\), associated to the isotropy representation.
Let \(K=T^*X^{1,0}\) be the canonical line bundle of \(X\). A \(G\)-Higgs bundle over \(X\) is a pair \((E_{H^\mathbb{C}},\varphi)\) where \(E_{H^\mathbb{C}}\) is a principal holomorphic \(H^\mathbb{C}\)-bundle over \(X\) and \(\varphi\) is a global holomorphic section of \(E_{H^\mathbb{C}}(\mathfrak{m}^{\mathbb{C}})\otimes K\), called the Higgs field.
Higgs bundles on compact Riemann surfaces were introduced by N. J. Hitchin in [Proc. Lond. Math. Soc., III. Ser. 55, 59–126 (1987; Zbl 0634.53045)]. There is a notion of polystability for \(G\)-Higgs bundles which allows the construction of the corresponding moduli spaces \(\mathcal{M}(G)\).
In the present paper, the group which is considered is \(G=\mathrm{SO}_0(1,n)\) — the connected component containing the identity of \(\mathrm{SO}(1,n)\). This is the Lorentz group of special relativity, and its adjoint form is the groups of isometries of real hyperbolic space. The authors prove that if \(n>1\) is odd, then \(\mathcal{M}(\mathrm{SO}_0(1,n))\) has two connected components.
The theorem is obtained through the Morse-theoretic approach pioneered by Hitchin in the above mentioned paper, and which has already been applied for several other groups. This method reduces the problem to the study of connectedness of certain subvarieties of \(\mathcal{M}(\mathrm{SO}_0(1,n))\). For that, the authors obtain first a detailed description of smooth points of the moduli space \(\mathcal{M}(\mathrm{SO}_0(1,n))\), and also give an explicit description of those Higgs bundles which represent singular points.
For any real reductive Lie group, non-abelian Hodge theory on \(X\) establishes a homeomorphism between \(\mathcal{M}(G)\) and the moduli space of reductive representations of \(\pi_1X\) in \(G\) (see N. J. Hitchin [Proc. Lond. Math. Soc., III. 55, 59–126 (1987; Zbl 0634.53045)]; C. T. Simpson [Publ. Math., Inst. Hautes Étud. Sci. 79, 47–129 (1994; Zbl 0891.14005); Inst. Hautes Étud. Sci. 80, 5–79 (1995; Zbl 0891.14006)]; S. K. Donaldson [Proc. Lond. Math. Soc., III. 55, 127–131 (1987; Zbl 0634.53046)] and K. Corlette [J. Differ. Geom. 28, No. 3, 361–382 (1988; Zbl 0676.58007)]). Hence, a direct consequence of the main result of the paper is that, for \(n>1\) odd, the moduli space of reductive representations of \(\pi_1X\) in \(\mathrm{SO}_0(1,n)\) has two connected components.
It should also be mentioned that a large part of this study was already carried out for the case of \(\mathrm{SO}_0(p,q)\) — the connected component containing the identity of \(\mathrm{SO}(p,q)\). However, in general, this case is far more involved technically than for the other groups studied in the literature, so a general solution for the count of connected components of \(\mathcal{M}(\mathrm{SO}_0(p,q))\) has not yet been achieved.

MSC:

14D20 Algebraic moduli problems, moduli of vector bundles
14F45 Topological properties in algebraic geometry
14H60 Vector bundles on curves and their moduli
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References:

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