Periods of harmonic Prym differentials on a compact Riemann surface

We study the period classes of closed, harmonic, and holomorphic Prym differentials on a compact Riemann surface of any genus g greater than or equal to 2 for arbitrary characters of its fundamental group. We prove that the harmonic Prym vector bundle of harmonic Prym differentials and the Gunning cohomology bundle are real-analytically isomorphic over the base of nontrivial normalized characters for every compact Riemann surface of genus g greater than or equal to 2.