In this work, which lead to my undergraduate thesis and written jointly with my undergraduate advisor, I studied Cayley 4-submanifolds of 8-dimensional Euclidean space. These are certain calibrated submanifolds, and play a role in string theory and its generalizations. The main technique was to use 3-dimensional subgroup of Spin(7) to simplify and solve the PDEs defining these Cayley 4-manifolds.
We determine several families of so-called Cayley 4-dimensional manifolds in the real Euclidean 8-space. Such manifolds are of interest because Cayley 4-manifolds and Cayley 4-cycles in Calabi-Yau 4-folds and Spin(7) holonomy manifolds are supersymmetric cycles that are candidates for representations of fundamental particles in String Theory. Moreover, some of the examples of Cayley manifolds discovered in this paper may be modified to construct explicit examples in our current search for new holomorphic invariants for Calabi-Yau 4-folds and for the further development of mirror symmetry.
We apply the classic results of Harvey and Lawson to find Cayley manifolds which are graphs of functions from the set of quaternions to itself. We consider graphs which are invariant under the action of three dimensional subgroups of Spin(7) which fix the quaternions as a subgroup of the Cayley numbers. Spin(7) is a subgroup of SO(8) which preserves the Cayley form. Systems of ODEs and PDEs are derived and solved, some special cases of a classic theorem of Harvey and Lawson are investigated, and theorems aiding in the classification of all such manifolds described here are proven. Several families of interesting Cayley 4-dimensional manifolds are discovered. Some of them are novel.