Connected components of Prym eigenform loci in genus five

Moduli spaces of holomorphic differentials on genus g Riemann surfaces admit a natural $GL_2(\mathbb{R})$ action. The known examples of orbifolds, which are unions of orbit closures for the $GL_2(\mathbb{R})$ action, are Prym eigenform loci, which are non-empty for the Riemann surfaces of genus no greater than 5, and in this talk I plan to tell about the number of the connected components of Prym eigenform loci for genus five surfaces.