Deformations of Asymptotically Cylindrical Cayley Submanifolds

TL;DR

This paper investigates the deformation theory of asymptotically cylindrical Cayley submanifolds within Spin(7)-manifolds, establishing an index formula and conditions for smooth moduli spaces, with explicit examples constructed in specific geometries.

Contribution

It provides a new index formula for the deformation operator and demonstrates that generic Spin(7)-structures yield unobstructed, smooth moduli spaces of Cayley submanifolds.

Findings

Index formula for the Dirac-type operator derived.

Moduli space is smooth and finite-dimensional under generic conditions.

Explicit examples constructed in Kovalev's asymptotically cylindrical Spin(7) manifolds.

Abstract

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.