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Chern-Weil Maslov index and its orbifold analogue

We give Chern-Weil definitions of the Maslov indices of bundle pairs over a Riemann surface Σwith boundary, which consists of symplectic vector bundle on Σand a Lagrangian subbundle on \partialΣ as well as its generalization for transversely intersecting Lagrangian boundary conditions. We discuss their properties and relations to the known topological definitions. As a main application, we extend Maslov index to the case with orbifold interior singularites, via curvature integral, and find also an analogous topological definition in these cases.