Geometry of degenerate quantum states

The geometric information contained in a single non-degenerate quantum state is known to reduce to the Fubini-Study (quantum) metric and symplectic (Berry curvature) form. I will explain how to reduce the geometry of degenerate states to the non-Abelian (Wilczek-Zee) curvature and a matrix-valued metric tensor. First, I describe all (functionally-)independent invariants associated with a collection of $m$-dimensional subspaces of a Hilbert space $\mathbb{C}^n$. The invariants are characterized from the perspective of linear algebra as well as in terms of local invariant tensors on the Grassmanian $Gr_{m,n}$. For two subspaces, the configuration is described by a set of $m$ principle angles that generalize the notion of quantum distance. For more subspaces, there are additional invariants assoicated with triples of subspaces. Some of them generalize the Berry-Pancharatnam phase and some do no have analogues for 1-dimensional spaces. We present a procedure for calculating global invariants as integrals of invariants tensors over $Gr_{m,n}$ submanifolds constructed from geodesics. At the technical level, we find it convenient to represent the subspaces via the corresponding orthogonal projectors.