Deformations of Asymptotically Cylindrical Cayley Submanifolds

We study the deformations of an asymptotically cylindrical Cayley submanifold inside an asymptotically cylindrical Spin(7)-manifold. We prove an index formula for the operator of Dirac type that arises as the linearisation of the deformation map and show that if the Spin(7)-structure is generic, then there are no obstructions, and hence the moduli space is a smooth finite-dimensional manifold whose dimension is equal to the index of the operator of Dirac type. We further construct examples of asymptotically cylindrical Cayley submanifolds inside the asymptotically cylindrical Riemannian 8-manifolds with holonomy Spin(7) constructed by Kovalev.