An analytic invariant of G_2 manifolds

We prove that the moduli space of holonomy G_2-metrics on a closed 7-manifold is in general disconnected by presenting a number of explicit examples. We detect different connected components of the G_2-moduli space by defining an integer-valued analytic refinement of the nu-invariant, a Z/48-valued defect invariant of G_2-structures on a closed 7-manifold introduced by the first and third authors. The refined invariant is defined using eta invariants and Mathai-Quillen currents on the 7-manifold and we compute it for twisted connected sums \`a la Kovalev, Corti-Haskins-Nordstr\"om-Pacini and extra-twisted connected sums as constructed by the second and third authors. In particular, we find examples of G_2-holonomy metrics in different components of the moduli space where the associated G_2-structures are homotopic and other examples where they are not.