Let \(X\) be a compact Riemann surface and \(G\) a complex connected semisimple Lie group. A \(G\)-Higgs bundle over \(X\) is a pair \((E, \varphi)\) where \(E\) is a principal \(G\)-bundle over \(X\) and \(\varphi\) a section of the vector bundle \(E( \mathfrak{g} ) \otimes K\) associated to \(E\) via the adjoint representation of \(G\) on \(T_e G =: \mathfrak{g}\).
The moduli space \({\mathcal M} (G)\) of polystable \(G\)-Higgs bundles over \(X\) is an algebraic variety over \({\mathbb C}\), and has a hyperkähler structure. It is moreover homeomorphic to the space \({\mathcal R} (G)\) of reductive representations of \(\pi_1 (X)\) in \(G\) modulo conjugation. See for example [O. García-Prada, Lond. Math. Soc. Lect. Note Ser. 359, 265–310 (2009; Zbl 1187.14037)] for an overview.
In the present paper, the authors study the fixed loci of certain finite order automorphisms of \({\mathcal M} (G)\), describing them in terms of decorated principal bundles. For involutions, the fixed loci are also interpreted in terms of representations of \(\pi_1 (X)\) in certain real forms of \(G\), leading to a description as hyperkähler or Lagrangian submanifolds of \({\mathcal M} (G)\).
To give some more detail: Suppose \(\theta \in \mathrm{Aut} (G)\) has order \(n\), and write \(\zeta_k = \exp \left( 2\pi \sqrt{-1} \frac{k}{n} \right)\). Then the map \((E, \varphi) \mapsto ( \theta(E), \zeta_k \cdot \theta (\varphi) )\) defines an order \(n\) automorphism \((\theta, \zeta_k)\) of \({\mathcal M} (G)\), which in fact only depends on the class of \(\theta\) in \(\mathrm{Out} (G)\). A main result (Theorem 6.3) is that if \((E, \varphi)\) is a polystable and simple fixed point of \(( \theta , \zeta_k)\), then the structure group of \(E\) can be reduced to the fixed subgroup \(G^{\theta'} \subset G\) for a certain conjugate \(\theta'\) of \(\theta\); and \(\varphi\) belongs to the \(\zeta_k\)-eigenspace for the action of \(\theta'\) on \(\mathfrak{g}\). Moreover, the fixed locus depends only on the class of \(\theta\) in \(\mathrm{Out} (G)\).
Generalising further; let \(\alpha \in H^1 ( X, Z )\) be a principal \(Z\)-bundle, where \(Z\) is the centre of \(G\). Via the multiplication \(G \times Z \to G\), one can define a \(G\)-bundle \(E \otimes \alpha\), generalising the notion of tensor product of a vector bundle by a line bundle. For \((\alpha, a )\) an element of order \(n\) in \(H^1 ( X, Z ) \rtimes \mathrm{Out} (G)\), the authors consider automorphisms \((\alpha, a, \zeta_k)\) of \({\mathcal M} (G)\) given by \[ (E, \varphi) \ \mapsto \ ( a(E) \otimes \alpha , \zeta_k \cdot a(\varphi) ) . \] Let \(G_\theta \subset G\) be the subgroup \(\{ s \in G : \theta(s) \cdot s \ \in \ Z \}\), which the authors show in fact coincides with \(N_G ( G^\theta )\). In Theorem 6.10, the authors describe the fixed locus \({\mathcal M} (G)^{(\alpha, \theta, \zeta_k )}\) in terms of \((G_\theta, \zeta_k )\)-Higgs bundles, directly generalising Theorem 6.3.
To prove Theorems 6.3 and 6.10, in Sections 2–5, technical tools are developed. In section 2, required statements on automorphisms and real forms of Lie algebras and Lie groups are obtained. To mention just one result of independent interest; following an approach in [J. Adams and O. Taïbi, Duke Math. J. 167, No. 6, 1057–1097 (2018; Zbl 1410.11036)], real forms of \(G\) are classified by the cohomology set \(H^1 ( \mathbb{Z}/2 , \mathrm{Aut}(G) )\).
In Section 3, the relation is studied between (possibly twisted) automorphisms of a principal \(G\)-bundle and reductions of structure group of the bundle. The main statements are Propositions 3.3. and 3.9, whose ingenious methods of proof may well be further applicable.
Section 4 contains a discussion of degree, stability, polystability, semistability and simplicity for \(G\)-Higgs bundles, and the relation of polystability with the Hitchin equations (see [N. J. Hitchin, Proc. Lond. Math. Soc. (3) 55, 59–126 (1987; Zbl 0634.53045)], [C. T. Simpson, J. Am. Math. Soc. 1, No. 4, 867–918 (1988; Zbl 0669.58008); Publ. Math., Inst. Hautes Étud. Sci. 75, 5–95 (1992; Zbl 0814.32003)]).
Section 5 introduces \((G^\theta , \zeta_k)\)-bundles and \((G_\theta , \zeta_k)\)-bundles, a generalisation of \(G\)-Higgs bundles. The Hitchin equations are a useful tool for relating \(\alpha\)-polystability of \(( G^\theta, \zeta_k )\)- or \(( G_\theta, \zeta_k )\)-Higgs bundles with polystability of \(G\)-Higgs bundles.
Section 6 contains the main results on the fixed loci outlined above.
In Sections 7–8, the authors study the images of the fixed loci under the homeomorphism between \({\mathcal M} (G)\) and the moduli space \({\mathcal R} (G)\) of reductive representations of \(\pi_1 (X)\), focusing primarily on involutions. In this case, the fixed loci of \((\theta, 1)\) correspond to representations of \(\pi_1 ( X )\) in \(G^\theta\), and those of \((\theta, -1)\) to representations of \(\pi_1 (X)\) in certain real forms of \(G\). Similar statements are obtained for the involutions \((\alpha, a , \pm 1 )\). In Theorem 8.10, it is shown how the two types of fixed loci give rise to hyperkähler and Lagrangian submanifolds of \({\mathcal M} (G)\) respectively; and thereby to \((B, B, B)\)-branes and \((B, A, A)\)-branes respectively.
The authors conclude with a detailed discussion of the cases \(G = \mathrm{SL} (n , {\mathbb C} )\) and \(G = \mathrm{Spin} ( 8, {\mathbb C} )\) to illustrate the general theory. Of particular interest for \(\mathrm{SL} (2, {\mathbb C})\) is the relation with Prym varieties. Also, an example is given showing that the nonsimple part of the fixed locus may contain points which are not defined by \((G_\theta, \zeta_k)\)-Higgs bundles (compare with Theorem 6.3). The case of \(\mathrm{Spin} ( 8, {\mathbb C} )\) is notable because, uniquely among simply connected complex simple Lie groups, \(\mathrm{Spin} (8, \mathbb{C})\) has an outer automorphism group which is nonabelian and also contains elements of order greater than two, namely \(S^3\).
This is a highly interesting and stimulating paper. Moreover, the authors have made a considerable effort to make the work as self-contained and accessible as possible, being generous with explanations and remarks.