Involutive Symmetries and Langlands Duality in Moduli Spaces of Principal G-Bundles

Let X be a compact Riemann surface of genus g≥2, G be a complex semisimple Lie group, and MG(X) be the moduli space of stable principal G-bundles. This paper studies the fixed point set of involutions on MG(X) induced by an anti-holomorphic involution τ on X and a Cartan involution θ of G, producing an involution σ=θ∗∘τ∗. These fixed points are shown to correspond to stable GR-bundles over the real curve (Xτ,τ), where GR is the real form associated with θ. The fixed point set MG(X)σ consists of exactly 2r connected components, each a smooth complex manifold of dimension (g−1)dimG2, where r is the rank of the fundamental group of the compact form of G. A cohomological obstruction in H2(Xτ,π1(GR)) characterizes which bundles are fixed. A key result establishes a derived equivalence between coherent sheaves on MG(X)σ and on the fixed point set of the dual involution on the moduli space of G∨-local systems, where G∨ denotes the Langlands dual of G. This provides an extension of the Geometric Langlands Correspondence to settings with involutions. An application to the Chern–Simons theory on real curves interprets MG(X)σ as a (B,B,B)-brane, mirror to an (A,A,A)-brane in the Hitchin system, revealing new links between real structures, quantization, and mirror symmetry.