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In this paper, by using Atiyah-Patodi-Singer index theorem, we obtain aformula for the signature of a flat symplectic vector bundle over a surfacewith boundary, which is related to the Toledo invariant of a surface grouprepresentation in the real symplectic group and the Rho invariant on theboundary. As an application, we obtain a Milnor-Wood type inequality for thesignature. In particular, we give a new proof of the Milnor-Wood inequality forthe Toledo invariant in the case of closed surfaces and obtain some modifiedinequalities for the surface with boundary.

Signature, Toledo invariant and surface group representations in the real symplectic group