An analytic invariant of G₂ manifolds

The first and third authors have constructed a defect invariant nu in Z/48 for G_2-structures on a closed 7-manifold. We describe the nu-invariant using eta-invariants and Mathai-Quillen currents on M and show that it can be refined to an integer-valued invariant bar nu for G_2-holonomy metrics. As an example, we determine the bar nu invariants of twisted and extra twisted connected sums a la Kovalev, Corti-Haskins-Nordström-Pacini, and Nordström. In particular, we exhibit examples of 7-manifolds where the moduli space of G_2-holonomy metrics has at least two connected components. In one of these examples, the underlying G_2-structures are homotopic, in another one, they are not.