I want to understand the paper which belongs to Ludwig (I put it below). I do not understand why exactly he got the new space $U(m+n)/U(m) \times U(n)$. My understanding from Grassmannian Manifold is that I think because we have m positive eigenvalues which are identified and n minus eigenvalues which are identified also we should mod out these two. Maybe it is wrong!
Source: Topological phases: Classification of topological insulators and superconductors of non-interacting fermions, and beyond Below equation 38