Obstructions to Spin(7) Nahm transforms on tori

The Nahm transform for 4-dimensional flat hyperkahler tori is an isometry between the moduli space of anti-self-dual (ASD) instantons on a torus $T^4$ and the moduli space of ASD instantons on the dual torus $\hat{T^4}$ parametrising flat line bundles on $T^4$. This paper studies a generalised Nahm transform on an 8-dimensional torus with a Spin(7) structure. I construct instanton bundles with Dirac kernels respectively in positive and negative chiralities, demonstrating that the usual Nahm transform is not well-defined. I then define a notion of asymptotic holonomy for instantons twisted by a high power $k \gg 1$ of an instanton line bundle, and I show that this asymptotic holonomy reduces to Spin(7) to second order in $k$. Finally, I provide examples for which the asymptotic holonomy is $\mathfrak{u}(1)^4$, and thus not Spin(7).