Cayley 4 FoldEdit

Cayley 4-fold is a central object in the study of higher-dimensional calibrated geometry, describing a special kind of four-dimensional submanifold inside eight-dimensional space endowed with a Spin(7) structure. Named for its roots in octonionic geometry, the Cayley 4-fold is defined by a particular 4-form, the Cayley form, which singles out volume-minimizing submanifolds. In plain terms, a Cayley 4-fold is a 4-dimensional surface that sits inside an 8-manifold in such a way that its four-dimensional volume is as small as possible within its homology class, a property guaranteed by calibration theory.

In the simplest setting, R^8 with its standard Spin(7) structure provides a flat model for Cayley 4-folds. More generally, Cayley submanifolds can be studied inside any eight-manifold equipped with a Spin(7) structure, where the Cayley 4-form Φ plays a central role in defining the calibration. A submanifold N^4 is called Cayley if the restriction of Φ to N equals the volume form on N, denoted vol_N. This condition implies that N is area-minimizing in its homology class and has vanishing mean curvature.

Definition and basic properties

The Cayley 4-form: In an eight-manifold M with Spin(7) structure, there exists a parallel, closed 4-form Φ, known as the Cayley form. A 4-submanifold N ⊂ M is Cayley precisely when Φ|_N = vol_N. This makes N a calibrated submanifold in the sense of Calibrated geometry.
Calibration and minimality: Because Φ calibrates N, the volume of N is minimal among all submanifolds in its homology class. Equivalently, Cayley submanifolds are minimal submanifolds in the ambient metric.
Local and global perspectives: The Cayley condition can be examined both locally (as a first-order PDE constraint on the embedding) and globally (in terms of the topology of N and its embedding in M). In the flat model R^8 with the standard Spin(7) structure, researchers study both linear (plane) and nonlinear (curved) Cayley submanifolds.

Construction and examples

Linear Cayley 4-planes: A basic source of examples consists of linear Cayley 4-planes in R^8. These serve as simple local models for more complicated Cayley submanifolds and help illuminate the geometry of the calibration.
Complex submanifolds as Cayley: In the standard embedding of C^4 ≅ R^8, any complex two-dimensional submanifold is Cayley with respect to the canonical Spin(7) structure. This follows from the identity that combines the Kähler form and the holomorphic volume form to reproduce the Cayley form; thus, many familiar complex-analytic submanifolds provide explicit Cayley examples.
Product and bundle constructions: Various constructions take simpler geometric ingredients (such as special Lagrangian submanifolds in lower dimensions, complex submanifolds, or certain bundle-sum operations) and assemble them into Cayley 4-folds under suitable Spin(7) structures. These methods give a bridge between familiar complex or symplectic geometry and the four-dimensional Cayley world.
Gluing and deformation techniques: Modern constructions employ gluing techniques, where pieces of Cayley submanifolds are joined along controlled neck regions to create new examples. This sits alongside deformation theory, which studies how a given Cayley submanifold can be varied within the ambient Spin(7) structure while remaining Cayley.

Deformation theory and moduli

McLean-type deformations: For calibrated submanifolds, infinitesimal deformations are governed by elliptic differential systems. In the Cayley setting, the deformation theory can be described in terms of harmonic forms on the submanifold, often yielding a moduli space whose dimension is governed by topological data such as the first Betti number b^1(N). In favorable situations the deformations are unobstructed, and the moduli space is a smooth manifold of dimension equal to the relevant cohomological count.
Global implications: The deformation theory of Cayley submanifolds connects local geometric control to global topology, influencing the structure of the space of Cayley submanifolds within a given Spin(7) manifold and informing stability questions under perturbations of the ambient structure.

Topology, singularities, and physics

Singular Cayley submanifolds: Just as with other calibrated geometries, Cayley submanifolds can develop singularities. Studying singular Cayley cones and their desingularizations is an active area, with connections to singularity theory and geometric analysis.
Applications in physics: Cayley 4-folds appear in the context of M-theory and string theory, where calibrated submanifolds correspond to supersymmetric cycles. In particular, Cayley submanifolds model certain BPS states and play a role in compactifications of higher-dimensional theories on spaces with exceptional holonomy such as Spin(7) manifolds.

Related concepts and context

Spin(7) structure: The Cayley 4-fold is defined relative to a Spin(7) structure on an eight-manifold, a special holonomy group reducing the frame bundle and yielding the parallel Cayley form Φ.
Octonions and exceptional geometry: The appearance of the Cayley form is intimately linked to octonionic geometry, and the term “Cayley” reflects that heritage. The octonion algebra Octonions provides a natural arena for the underlying algebraic structure.
Calibrated geometry and minimal submanifolds: Cayley submanifolds are a key instance of calibrated submanifolds, a concept developed to identify volume-minimizing submanifolds through differential forms, with wide-ranging consequences in global analysis and geometry Calibrated geometry.
Complex submanifolds and calibrations: In the flat model, complex two-dimensional submanifolds of C^4 are Cayley, illustrating the close ties between complex geometry and the Cayley calibration. This connection sits alongside broader calibration theory and special holonomy.