We consider a class of Dirac-type operators, on the universal covering of a closed manifold, that are invariant under the projective action associated to a two-cocycle of the fundamental group.
These operators give interesting invariants analogous to those studied in L^2-index theory for covering spaces. The key property of this setting is that the twist by a two-cocycle naturally yields an (infinite dimensional) bundle of arbitrary small curvature.
We shall describe the construction of eta and rho invariants, and discuss some of the geometric properties that follow from an Atiyah—Patodi—Singer index formula in this setting. This is based on joint work with Charlotte Wahl.