Higgs bundles, branes and Langlands duality

Given a compact Riemann surface $X$ and a complex reductive Lie group $G$ equipped with real structures, we define antiholomorphic involutions on the moduli space of $G$-Higgs bundles over $X$. We investigate how the various components of the fixed point locus match up, as one passes from $G$ to its Langlands dual ${}^{L}G$. Special attention is given to the case $G=SL(2,\text{C})$ and ${}^{L}G=PGL(2,\text{C})$.