Signature, Toledo and eta invariants for surface group representations in the real symplectic group

In this paper, by using the Atiyah–Patodi–Singer index theorem, we obtain a formula for the signature of a flat symplectic vector bundle over a surface with boundary, which is related to the Toledo invariant of a surface group representation in the real symplectic group and ρ invariant on the boundary. As an application, we obtain a Milnor–Wood type inequality for the signature. In particular, we give a new proof of the Milnor–Wood inequality for the Toledo invariant in the case of closed surfaces and obtain some modified inequalities for surfaces with boundary.