Abstract:
In this talk we discuss instantons on asymptotically conical Spin(7)-manifolds where the instanton is asymptotic to a fixed nearly G2-instanton at infinity. After discussing the preliminary notions of holonomy groups, G2 & Spin(7)-manifolds, asymptotically conical manifolds, and Yang-Mills equations & instantons in 4-dimensions, we mainly focus on the deformation theory of instantons on 8-dimensional the asymptotically conical Spin(7)-manifold.
As examples, we consider two important Spin(7) manifolds: $mathbb{R}^8$, where $mathbb{R}^8$ is considered to be an asymptotically conical manifold asymptotic to the cone over the round 7-sphere, and Bryant-Salamon manifold - the negative spinor bundle over 4-sphere, asymptotic to the cone over the squashed 7-sphere. We apply the deformation theory to describe deformations of Fairlie-Nuyts-Fubini-Nicolai (FNFN) Spin(7)-instantons on $mathbb{R}^8$, and the Clarke-Oliviera instanton on the negative spinor bundle over 4-sphere. We also calculate the virtual dimensions of the moduli spaces using Atiyah-Patodi-Singer index theorem and the spectrum of the twisted Dirac operators.
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