Instantons on Asymptotically Conical Spin(7)-Manifolds

We develop the deformation theory of instantons on asymptotically conical Spin(7)-manifolds where the instanton is asymptotic to a fixed nearly G2-instanton at infinity. By relating the deformation complex with spinors, we identify the space of infinitesimal deformations with the kernel of the twisted negative Dirac operator on the asymptotically conical Spin(7)-manifold.

We apply this theory to describe the deformations of the Fairlie-Nuyts-Fubini-Nicolai (FNFN) Spin(7)-instantons on R8, where R8 is considered to be an asymptotically conical Spin(7)-manifold asymptotic to the cone over S7. We calculate the virtual dimension of the moduli space using the Atiyah-Patodi-Singer index theorem and the spectrum of the twisted Dirac operator.

We then apply the deformation theory to compute the deformations of Clarke--Oliveira's instanton on the Bryant-Salamon Spin(7)-Manifold. The Bryant-Salamon Spin(7)-manifold: negative spinor bundle over 4-sphere is an asymptotically conical manifold where the link is the squashed sphere Sp(2) × Sp(1)/Sp(1) × Sp(1).

Finally, we show that with gauge groups U(1) and SU(2), no irreducible Sp(2) × U(1)-invariant asymptotically conical instantons on R8 exist. Using this result, we prove that any asymptotically conical U(1)- or SU(2)-instanton on R8 asymptotic to the flat connection on S7 satisfying certain conditions is obstructed.