Abstract

There are two main methods of constructing compact manifolds with holonomy G2, viz. resolution of singularities (first applied by Joyce) and twisted connect sum (first applied by Kovalev). In the second case, there is a known, computable invariant (the \nu-invariant, introduced by Crowley–Goette–Nordström) which can be used to distinguish between different examples. However no such invariant is known for the first construction.

In this talk, I will introduce two new invariants of G2-manifolds, termed \mu-invariants, and explain why these promise to be well-suited to, and computable on, Joyce’s examples of G2-manifolds. These invariants are related to \eta- and \zeta-invariants and should be regarded as the Morse indices of a G2-manifold when it is viewed as a critical point of certain Hitchin functionals. I shall also explain how to compute the \mu-invariants in closed form on the orbifolds used in Joyce’s construction, using Epstein \zeta-functions.