Disconnecting the moduli space of G_2-metrics via U(4)-coboundary defects

Abstract

We exhibit examples of closed Riemannian 7-manifolds with holonomy G_2 such that the underlying manifolds are diffeomorphic but whose associated G_2-structures are not homotopic. This is achieved by defining two invariants of certain U(3)-structures. We show that these agree with the invariants of G_2-structures defined by Crowley and Nordstr\"om. We construct a suitable coboundary for G_2 manifolds obtained via the Twisted Connected Sum method that allows the invariants to be computed in terms of the input data of the construction. We find explicit examples where the invariants detect different connected components of the moduli of G_2-metrics.