Moduli spaces of anti-invariant vector bundles over a curve

Let $X$ be a smooth irreducible projective curve with an involution $\sigma$. A vector bundle $E$ over $X$ is called anti-invariant if there exists an isomorphism ${\sigma }^{\ast }E\to E^{\ast }$. In this paper, we give a construction of the moduli spaces of anti-invariant vector bundles over $X$.