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Anti-holomorphic involutions of the moduli spaces of Higgs bundles. (Involutions anti-holomorphes des espaces de modules de fibrés de Higgs.) (English. French summary) Zbl 1333.14032

Consider a complex semisimple Lie group \(G\). Using the notions of pseudo-real \(G\)-Higgs bundles, introduced in [I. Biswas et al., Ann. Inst. Fourier 64, No. 6, 2527–2562 (2014; Zbl 1329.14106)], the authors study the fixed point locus of certain anti-holomorphic involutions on the moduli space \(\mathcal{M}(G)\) of polystable \(G\)-Higgs bundles over a compact Riemann surface \(X\) of genus at least two.
Let \(\alpha:X\to X\) be an anti-holomorphic involution of \(X\) and \(\sigma: G\to G\) be also an anti-holomorphic involution of \(G\). Denote by \(Z^\sigma\) the fixed point locus of \(\sigma\) in the center of \(G\). Choose an element \(c\) belonging to the subgroup of \(Z^\sigma\) generated by the elements of order two.
A holomorphic principal \(G\)-bundle \(E\) over \(X\) is said to be \((\alpha,\sigma,c)\)-pseudo-real if \(E\) is equipped with an anti-holomorphic map \(\tilde\alpha:E\to E\) which covers \(\alpha\) and such that \(\tilde\alpha(eg)=\tilde\alpha(e)\sigma(g)\), for \(e\in E\) and \(g\in G\) and, moreover, \(\tilde\alpha^2(e)=ec\), for \(e\in E\). A \(G\)-Higgs bundle \((E,\varphi)\) is \((\alpha,\sigma,c,\pm)\)-pseudo-real if \(E\) is \((\alpha,\sigma,c)\)-pseudo-real and \(\tilde\alpha(\varphi)=\pm\varphi\), where here \(\tilde\alpha:E(\mathfrak{g})\otimes K\to E(\mathfrak{g})\otimes K\) is induced by \(\tilde\alpha:E\to E\) and by \(\alpha:X\to X\). If \(c=1\), \((E,\varphi)\) is \((\alpha,\sigma,\pm)\)-real. Denote by \(\mathcal{M}(\alpha,\sigma,c,\pm)\) the moduli space of polystable \((\alpha,\sigma,c,\pm)\)-pseudo-real \(G\)-Higgs bundles over \(X\).
In [loc. cit.], it was proved that a \((\alpha,\sigma,c,\pm)\)-pseudo-real \(G\)-Higgs bundle \((E,\varphi)\) is polystable if and only if the underlying holomorphic \(G\)-Higgs bundle is polystable, so that there is a map \(\mathcal{M}(\alpha,\sigma,c,\pm)\to\mathcal{M}(G)\) that forgets the pseudo-real structure.
The main result of the paper gives a description of the fixed point locus of two natural anti-holomorphic involutions of \(\mathcal{M}(G)\) in terms of the images of the maps \(\mathcal{M}(\alpha,\sigma,c,\pm)\to\mathcal{M}(G)\), when \(c\) varies. The involutions are denoted by \(\iota(\alpha,\sigma)^\pm\) and are defined in Section 4.1 using \(\alpha\) and \(\sigma\).
The interpretations of these locus in \(\mathcal{M}(G)\) in terms of representations of the orbifold fundamental group of \(X\) are studied, using again results of the authors and J. Hurtubise in [Zbl 1329.14106].
Finally, considering the hyperkähler structure on (the smooth locus of) \(\mathcal{M}(G)\), found by N. J. Hitchin in [Proc. Lond. Math. Soc. (3) 55, 59–126 (1987; Zbl 0634.53045)], given by complex structures \(J_1,J_2,J_3\) (which is reviewed in Section 2.4) it is proved that the fixed point locus of \(\iota(\alpha,\sigma)^+\) is an \((A,A,B)\)-brane, while the one of \(\iota(\alpha,\sigma)^+\) is an \((A,B,A)\)-brane, generalizing previous work of D. Baraglia and L. P. Schaposnik in [Commun. Math. Phys. 331, No. 3, 1271–1300 (2014; Zbl 1311.53058)] and in [“Real structures on moduli spaces of Higgs bundles”, arXiv:1309.1195, to appear in Adv. Theor. Math. Phys].

MSC:

14H60 Vector bundles on curves and their moduli
57R57 Applications of global analysis to structures on manifolds
58D29 Moduli problems for topological structures
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