Cayley 4 FormEdit
The Cayley 4-form is a distinguished geometric object in eight dimensions that plays a central role in the study of exceptional holonomy and calibrated geometry. It is a particular differential 4-form on ℝ^8 (and more generally on 8-manifolds) that is stabilized by the Lie group Spin(7). The form is intimately tied to the octonions octonions and provides a canonical way to endow eight-dimensional spaces with a rich geometric structure. In flat space its stabilizer in GL(8,ℝ) is exactly Spin(7), which makes the form a natural gateway to Spin(7) geometry and to the broader theory of special holonomy.
The Cayley 4-form is named after Arthur Cayley and is sometimes referred to simply as the Cayley form. It emerges from the algebra of octonions and the associated cross-product-like structures that exist in eight dimensions. One of its key features is that it defines a calibration: submanifolds on which the form attains its maximal possible value are volume-minimizing in their homology class. This leads to the notion of Cayley submanifolds, which are the calibrated 4-dimensional submanifolds singled out by the form. The framework carrying this idea is part of the larger subject of calibrations, developed in part by Harvey and Lawson, which provides tools for identifying minimal submanifolds in Riemannian manifolds Calibrated geometry.
Construction and definition
The most standard setting identifies ℝ^8 with the octonions, denoted by O. Using the octonionic multiplication and the natural inner product, one can define a specific 4-form φ on ℝ^8 that is invariant under the action of Spin(7). The form φ is the unique (up to scaling) Spin(7)-invariant 4-form whose stabilizer inside GL(8,ℝ) is Spin(7). In this sense, φ encodes a reduction of the frame bundle from SO(8) to Spin(7) and thereby determines a Riemannian metric and orientation on the underlying space. While the explicit coordinate expression for φ is lengthy, its geometric content is transparent: φ records how eight-dimensional frames interact under the octonionic structure, and it singles out a distinguished class of 4-planes called Cayley 4-planes.
One can also view φ as the cornerstone of a Spin(7) structure on an 8-manifold M. Such a structure is given by a nowhere-vanishing 4-form φ whose value at each point matches the standard Cayley form under a suitable identification of the tangent space with ℝ^8. This pointwise condition induces a metric g and an orientation on M. When φ is parallel with respect to the Levi-Civita connection (equivalently, when dφ = 0 and d*φ = 0), the holonomy group of the metric is contained in Spin(7), yielding a manifold of exceptional holonomy.
Geometry of Spin(7) structures
Spin(7) is a simply connected Lie group that sits inside SO(8) as the stabilizer of the Cayley 4-form. The eight-dimensional geometry determined by φ is thus encoded by a reduction of the structure group of the tangent bundle from SO(8) to Spin(7). This reduction imposes strong geometric constraints and leads to special curvature properties. In particular, a torsion-free Spin(7) structure—one for which φ is parallel—produces a Ricci-flat metric, making Spin(7) holonomy manifolds analogues to other exceptional holonomy spaces (such as those with G2 holonomy in seven dimensions) within the broader framework of special holonomyHolonomy.
The deformation theory of torsion-free Spin(7) structures is a topic of active study. On a compact 8-manifold, the space of such structures is governed by differential forms and the topology of the underlying manifold, and its dimension is controlled by certain cohomology groups. The interplay between φ, the metric, and the topology of M gives rise to rich moduli spaces that have been explored in both pure geometry and mathematical physics.
Calibrations and Cayley submanifolds
Calibrated geometry provides a mechanism for identifying volume-minimizing submanifolds. The Cayley 4-form φ calibrates a special class of 4-dimensional submanifolds in M, known as Cayley submanifolds. These submanifolds minimize volume within their homology class and enjoy stability properties that are useful in variational problems. The concept of calibration was developed to give intrinsic criteria, in terms of differential forms, for minimal submanifolds without requiring the equations of motion to be solved directly.
Cayley submanifolds are among the key objects studied in calibrated geometry, alongside other calibrated submanifolds associated with different holonomy groups (for example, special Lagrangian submanifolds in Calabi–Yau geometry and associative/coassociative submanifolds in G2 geometry). The Cayley calibration thus sits at a crossroads of differential geometry and geometric analysis, with implications for the study of minimal surfaces, moduli spaces, and geometric flows in eight dimensions Calibrated geometry.
Existence, examples, and applications
Flat space provides the canonical example: ℝ^8 equipped with the standard Cayley 4-form φ yields the basic model of Spin(7) geometry. Beyond the flat case, the construction of global examples with Spin(7) holonomy is more delicate. A major breakthrough came with the work of Dominic Joyce, who produced compact examples of manifolds with Spin(7) holonomy by resolving singularities in suitable orbifolds and endowing them with torsion-free Spin(7) structures. These manifolds are Ricci-flat and exhibit rich geometric and topological features that illuminate the landscape of exceptional holonomyDominic Joyce.
Spin(7) geometry intersects with theoretical physics, particularly in string theory and M-theory, where exceptional holonomy manifolds can furnish backgrounds with reduced supersymmetry and interesting compactification properties. The Cayley 4-form thus serves not only as a mathematical object of intrinsic interest but also as a bridge to physical theories that rely on special holonomy and calibrated branes String theory, M-theory.
In addition to Joyce’s compact examples, there are noncompact or asymptotically cylindrical constructions due to Bryant and Salamon and further developments that explore various asymptotic geometries and singular limits. These examples help illustrate the range of phenomena associated with Spin(7) structures and the behavior of Cayley submanifolds in different geometric contexts Bryant–Salamon.