Real Slices of $\mathrm{S}\mathrm{L}(r,\mathbb{C})$-Opers

Through the action of an anti-holomorphic involution $\sigma$ (a real structure) on a Riemann surface $X$, we consider the induced actions on $\mathrm{S}\mathrm{L}(r,\mathbb{C})$-opers and study the real slices fixed by such actions. By constructing this involution for different descriptions of the space of $\mathrm{S}\mathrm{L}(r,\mathbb{C})$-opers, we are able to give a natural parametrization of the fixed point locus via differentials on the Riemann surface, which in turn allows us to study their geometric properties.