Perspectives on the moduli space of torsion-free G2-structures

Abstract

The moduli space of torsion-free G₂-structures for a compact 7-manifold forms a non-singular smooth manifold. This was originally proved by Joyce. In this thesis, we present the details of this proof, modifying some of the arguments using new techniques. Next, we consider the action of gauge transformations on the space of torsion-free G₂-structures. This gives us a new framework to study the moduli space.

We show that the torsion-free condition under the action of gauge transformations almost exactly corresponds to a particular 3-form, which arises naturally from the G₂-structure and the gauge transformation, being harmonic when we add a "gauge-fixing" condition. This may be the first step in giving an alternate proof of the fact that the moduli space forms a manifold in our framework of gauge transformations.