Faisal Romshoo
Perspectives on the moduli space of torsion-free $\textrm{G}_2$-structures
Joyce showed that the moduli space of torsion-free $\textrm{G}_2$-structures for a compact $7$-manifold forms a non-singular smooth manifold. In this talk, we consider the action of gauge transformations of the form $e^{tA}$ where $A$ is a $2$-tensor, on the space of torsion-free $\textrm{G}_2$-structures. This gives us a new framework to study the moduli space.
We will see that a $\textrm{G}_2$-structure $\widetilde{\varphi} = P^*\varphi$ acted upon by a gauge transformation $P = e^{tA}$ is infinitesimally torsion-free if and only if $A \diamond \varphi$ is harmonic and if $A$ satisfies a ``gauge-fixing" condition, where $A \diamond \varphi$ is a special type of $3$-form. 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.
MC 5501