Let
X be a compact Riemann surface of genus
,
G be a complex semisimple Lie group, and
be the moduli space of stable principal
G-bundles. This paper studies the fixed point set of
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Let
X be a compact Riemann surface of genus
,
G be a complex semisimple Lie group, and
be the moduli space of stable principal
G-bundles. This paper studies the fixed point set of involutions on
induced by an anti-holomorphic involution
on
X and a Cartan involution
of
G, producing an involution
. These fixed points are shown to correspond to stable
-bundles over the real curve
, where
is the real form associated with
. The fixed point set
consists of exactly
connected components, each a smooth complex manifold of dimension
, where
r is the rank of the fundamental group of the compact form of
G. A cohomological obstruction in
characterizes which bundles are fixed. A key result establishes a derived equivalence between coherent sheaves on
and on the fixed point set of the dual involution on the moduli space of
-local systems, where
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
as a
-brane, mirror to an
-brane in the Hitchin system, revealing new links between real structures, quantization, and mirror symmetry.
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