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Description

All known examples of instantons on (non-trivial) asymptotically conical $G_2$ manifolds asymptote to nearly K\"ahler instantons which live on the link of the cone. I will use this observation to develop a framework for studying the moduli space of such a $G_2$ instanton, showing that the expected dimension of this space is the index of a twisted Dirac operator on a weighted Sobolev space. I will focus on the example of $\mathbb R^7$ where the link is the homogeneous space $G_2/\mathrm{SU}(3)$ and show how one can use representation theoretic methods to calculate the expected dimension of the moduli space.