We prove that the monodromy of a cohomologically rigid integrable connection (E,∇) on a smooth complex projective variety X is integral. This answers positively a special case of a conjecture by Carlos Simpson. To this aim, we prove that the mod p reduction of a rigid integrable connection (E,∇) has the structure of an isocrystal with Frobenius structure. We also prove that rigid integrable connections with vanishing p- curvatures are unitary. This allows one to prove new cases of Grothendieck’s p-curvature conjecture. Joint with Michael Groechenig.