An analytic invariant of G₂ manifolds

We prove that the moduli space of holonomy G2-metrics on a closed 7-manifold can be disconnected by presenting a number of explicit examples. We detect different connected components of the G2-moduli space by defining an analytic refinement ν¯(M,g)∈Z of the defect invariant ν(M,φ)∈Z/48 of G2-structures φ on a closed 7-manifold M introduced by the first and third authors. The ν¯-invariant is defined using η-invariants and Mathai-Quillen currents on M and we compute it for twisted connected sums à la Kovalev, Corti-Haskins-Nordström-Pacini and extra-twisted connected sums as constructed by the second and third authors. In particular, we find examples of G2-holonomy metrics in different components of the moduli space where the associated G2-structures are homotopic and other examples where they are not.