Anti-Holomorphic Involutions and Langlands Duality on the Moduli Space of Principal G2-Bundles

Let X be a compact Riemann surface of genus g≥2 equipped with an anti-holomorphic involution τ, and let M(G2) denote the moduli space of stable principal G2-bundles over X. We study the fixed-point loci M(G2)σ of the involutions σ=θ∗∘τ∗ on M(G2), where θ is a Cartan involution of G2 corresponding to one of its two real forms: the compact form G2c and the split form G22,2. We prove that M(G2)σc is connected with Euler characteristic 1, and that M(G2)σs has exactly 2c connected components, where c is the number of connected components of the fixed-point set Xτ. Each component is shown to be a compact, real-analytic, totally real submanifold of M(G2) of real dimension 14(g−1). For the split form, we establish that χ(M(G2)σs)=3g−1 when c=0, where the factor 3=|W(G2)|/|W(SO(4))| arises from the Borel–Hirzebruch formula applied to the symmetric space G22,2/SO(4). We further analyse the monodromy element k0∈SO(4) associated to the non-trivial topological type over the ovals, proving that its centraliser in G22,2 is exactly SO(4) and that its conjugacy class coincides with G22,2/SO(4), with e(O(k0))=3. This supports a conjecture asserting χC(w1,…,wc)=3g−1 for each component when c≥1. Finally, exploiting the Langlands self-duality G2∨≅G2, we show that M(G2)σ is a (B,B,B)-brane in the hyperkähler manifold T∗M(G2), and that it is self-specular: it is preserved by the derived Fourier–Mukai autoequivalence Φ:DbCoh(M(G2))→∼DbCoh(M(G2)) induced by the Langlands self-duality, which restricts to a self-equivalence of DbCoh(M(G2)σ).