Spin(7) Instantons on asymptotically conical Calabi-Yau Fourfolds

The present thesis examines the role of Hermitian Yang-Mills (HYM) connections in Spin(7) instanton moduli spaces over asymptotically conical (AC) Calabi-Yau (CY) fourfolds. The starting point is Lewis' energy estimate: Lewis: if a principal G-bundle P over a closed CY fourfold X8} admits HYM solutions, then all Spin(7) instantons on P are HYM. The question we are interested in is whether this result persists in the AC setting. A first-order approach is to pass to the (Sasaki-Einstein) asymptotic link, which is not Ricci flat and could thus be homogeneous. The role of the Spin(7) instanton equation is assumed by the G2 instanton equation and that of the HYM equation by the contact instanton equation. It is easier to explore the relationship between these 7-dimensional systems instead. Imposing equivariance reduces the PDEs to representation theory. This allows us to exhibit an explicit non-contact G2 instanton on S7. This example agrees with the limiting connection of the standard octonionic instanton of Fubini and Nicolai. Prior to the results of this thesis, this was the only known non-HYM Spin(7) instanton on a CY fourfold. It originally appeared in the physics literature. We provide an alternative construction, in line with the modern framework for equivariant gauge theory. Because its limiting connection is not contact, its moduli space cannot contain HYM connections. Consequently, it does not help resolve the question we set out to answer. We extend Lewis's theorem to the AC setting, conditioning on decay rates. We construct the moduli space of SO(5)-invariant Spin(7) instantons with structure group SO(3) on the Stenzel space. These new instantons sit exactly at the slow-rate cut-off point of our extension of Lewis's theorem. They provide a negative answer to the question we set out to answer: the moduli space is one-dimensional and contains precisely two HYM connections. One of these is the epicentre of a removable singularity/ bubbling phenomenon and the development of a corresponding Fueter section. We compute this explicitly and verify (after suitable modifications) an infinite-energy version of Tian's energy conservation identity. This phenomenon hints at a possible relationship between the AC Spin(7) instanton and HYM systems.