2026/07/30 by Spencer Whitehead
Mathematics · #math.DG #msc:53C07
20 pages
arxiv created 2026/07/30 · arxiv updated 2026/07/31
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 T4 and the moduli space of ASD instantons on the dual torus T4 parametrising flat line bundles on T4. 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 ≫ 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 \mathfraku(1)4, and thus not Spin(7).