Cayley submanifolds in Spin(7)-manifolds

In 1955, Marcel Berger classified the possible holonomy groups of Riemannian metrics g on a manifold M (satisfying some basic conditions, giving the list SO(n), U(m), SU(m), Sp(m), G2 and Spin(7) of possible holonomy groups. If the holonomy group is not SO(n) then g is compatible with additional geometric structures on M -- this makes it special and interesting. Metrics with holonomy SU(m), Sp(m), G2 and Spin(7) are Ricci-flat, and are important in Physics, especially in String Theory and M-Theory. G2 (in 7 dimensions) and Spin(7) (in 8 dimensions) are called the exceptional holonomy groups. Professor Joyce constructed the first examples of compact manifolds with holonomy G2 and Spin(7) in 1993-5. G2-manifolds are especially important in M-Theory, as ingredients to build the universe from.
Calibrated geometry is a natural companion subject to Riemannian holonomy groups. For any special holonomy group, it defines intersting classes of special minimal submanifolds called calibrated submanifolds. In Calabi-Yau manifolds (with holonomy SU(m)) these are special Lagrangian submanifolds (SL m-folds). In G2 manifolds there are associative 3-folds and coassociative 4-folds. In Spin(7) manifolds there are Cayley 4-folds.
Calibrated submanifolds are important in String Theory and M Theory, as the classical geometry underlying 'branes'.
The SYZ Conjecture in 1995 explained Mirror Symmetry in terms of dual fibrations of a Calabi-Yau manifold by special Lagrangian submanifolds, including singular fibres. Ever since then, it has been an important open question to construct examples of fibrations of compact manifolds with special holonomy by calibrated submanifolds, including singular fibres. The interesting possibilities are Calabi-Yau manifolds fibred by special Lagrangians, G2 manifolds fibred by coassociative 4-folds, and Spin(7) manifolds fibred by Cayley 4-folds.
We propose to study Cayley 4-folds in compact and Spin(7) manifolds, including singular Cayley 4-folds (probably with 'isolated conical singularities'). We should look at their deformation theory, and resolution of singularities.
A long term goal, which may or may not be achieved in the PhD, would be to construct examples of compact Spin(7) manifolds with fibrations by compact Cayley 4-folds, including singular fibres. Before we get there, we need to develop technology to deal with families of Cayley 4-folds, including singularities, and construction of families in examples.
There are several notions of 'fibration', of varying strength:
a) The strongest is that exactly one fibre passes though each point.
b) A weaker notion is that one has a family of Cayley 4-folds parametrized by a compact 4-manifold, such that one fibre passes through each point counted with signs.
c) One could also ask for an open set comprising 99% of the volume of the manifold, with a fibration of the open set by Cayley 4-folds; or maybe for a family of manifolds, open sets and fibrations such that the proportion of volume in the fibration (e.g. 99%) tends to 100% in a limit.
Studying singular Cayley 4-folds and resolutions of singularities using gluing may yield a fibration of type b) (if we work hard). It is not yet clear to us whether the fibration will also satisfy a). One question we could investigate is whether fibrations of type a), with given singular models, are stable under small perturbations, as this would probably help ensure a) holds in examples.
If Y is a G2 manifold then X = Y x S1 is a (degenerate) Spin(7) manifold. A fibration of Y by coassociative 4-folds yields a fibration of X by Cayley 4-folds. Conversely, an S1-invariant fibration of X by Cayley 4-folds descends to a fibration of Y by coassociative 4-folds.
This project falls within the EPSRC's Geometry and Topology area.