Cayley SubmanifoldEdit

Cayley submanifolds are a distinguished class of four-dimensional submanifolds living inside eight-dimensional space endowed with a Spin(7) structure. They are defined via a geometric object called the Cayley 4-form, which calibrates certain submanifolds and makes them volume-minimizing within their homology class. The subject sits at the intersection of differential geometry, topology, and mathematical physics, and it has become a central example in the broader study of calibrated geometry and exceptional holonomy. For readers familiar with the landscape of geometric analysis, Cayley submanifolds provide a concrete arena where ideas about minimality, deformation theory, and nonlinear partial differential equations meet the special symmetry encoded by Spin(7).

The Cayley construction sits within the broader framework of holonomy and calibration. Spin(7) is one of the special holonomy groups identified in the Berger classification, and it stabilizes a distinguished 4-form on R^8 known as the Cayley form. When a submanifold M^4 ⊂ R^8 is such that the pullback of this 4-form equals the intrinsic volume form on M, M is called a Cayley submanifold. This condition implies that M is volume-minimizing in its homology class, a hallmark of calibrated submanifolds studied in calibrated geometry. The Cayley form and its associated geometry have natural expressions in terms of octonions, linking the analytic theory to algebraic structures with non-associative products. Cayley submanifolds thus sit at the crossroads of octonionic algebra, exceptional holonomy, and minimal surface theory, with implications for mathematical physics in contexts such as compactifications that preserve a portion of supersymmetry. See calibration, Spin(7) holonomy, and octonions for foundational context, as well as Harvey-Lawson for the origin of calibration ideas.

Definition and basic properties

A four-dimensional oriented submanifold M^4 ⊂ R^8 is called Cayley if the Cayley 4-form Φ satisfies Φ|_M = vol_M, where vol_M is the volume form induced on M. Equivalently, M is calibrated by Φ, hence minimizes volume in its homology class.
The Cayley 4-form Φ is the Spin(7)-invariant calibration on R^8. If a submanifold is Cayley, its tangent spaces are constrained so that the restriction of Φ to each tangent 4-plane equals the volume form on that plane.
Because Φ is closed and Spin(7)-invariant, Cayley submanifolds inherit a stable, volume-minimizing character. This connects to broader ideas in calibration theory and places Cayley submanifolds among the canonical examples of minimal submanifolds with extra symmetry.
The standard model for intuition is to view R^8 as equipped with a Spin(7) structure that preserves Φ. In that setting, linear Cayley 4-planes provide the simplest nontrivial examples, and more generally one looks for nonlinear submanifolds whose tangent spaces satisfy the Cayley condition along every point.

Geometry and examples

Linear Cayley subspaces: The most basic examples are flat 4-planes in R^8 that are calibrated by Φ. These serve as the local models around which nonlinear Cayley submanifolds are studied.
Cones and singularities: There are Cayley cones arising as cones over certain 3-manifolds, which illuminate how singularities can form in calibrated geometries and how local models guide desingularization and deformation analyses.
Constructive schemes: One fruitful approach to producing Cayley submanifolds uses the octonionic viewpoint, where certain subspaces compatible with the octonion multiplication give rise to Cayley structures. This ties into the algebraic side of the story via octonions and the interplay between non-associativity and exceptional symmetry.
Complex and hyperkähler links: In particular ambient geometries with extra structure, Cayley submanifolds can appear as special representatives among other calibrated classes, such as those related to complex or hyperkähler geometries, when viewed through the prism of Spin(7) symmetry. See also G2 holonomy for parallel calibration themes in related holonomy settings.

Deformation theory and moduli

McLean’s framework for calibrated submanifolds gives a general method to study deformations of Cayley submanifolds. Small deformations are governed by elliptic equations, and under favorable circumstances the moduli space of Cayley deformations is finite-dimensional and smooth.
The dimension and smoothness of the moduli space depend on the geometry of the underlying M and the ambient Spin(7) structure. In many situations one obtains a well-behaved deformation theory with a finite number of parameters describing nearby Cayley submanifolds, while in other cases obstructions may arise.
The deformation picture for Cayley submanifolds connects to broader questions about stability, obstructions, and the structure of moduli spaces in geometric analysis. See McLean (mathematician) for foundational results on deformations of calibrated submanifolds, and calibration for the general framework.

In physics

Calibrated submanifolds, including Cayley submanifolds, appear in theoretical physics as stable, supersymmetric cycles in M-theory compactifications on manifolds with Spin(7) holonomy. Their volume-minimizing property ensures stability of the corresponding brane configurations, making them relevant in the study of non-perturbative effects and the landscape of possible compactifications. See M-theory and Calibrated geometry for the physical background, and Spin(7) holonomy to connect to the geometric structure underlying these phenomena.

Controversies and debates

Existence and construction of compact Spin(7) manifolds: A central challenge in the field is producing explicit, compact manifolds with Spin(7) holonomy. The existence of Cayley submanifolds within such ambient spaces depends on delicate geometric and analytic constructions, and different gluing and perturbation techniques have been proposed. Debates in this area often focus on the most robust methods for controlling singularities and achieving global consistency, rather than on foundational principles of calibrated geometry.
Regularity versus singularities: In calibrated geometry, one encounters submanifolds that are smooth away from a set of singularities. The extent to which singular Cayley submanifolds can be understood, classified, or desingularized remains an active area of research, with viewpoints ranging from analytic regularity results to topological and algebro-geometric constructions inspired by octonionic and Spin(7) symmetry.
Role of octonions: The octonionic model provides a powerful heuristic and constructive tool, but non-associativity can complicate attempts to translate algebraic pictures into global geometric constructions. Some researchers emphasize the clarity of the octonionic approach for intuition and explicit examples, while others push toward more intrinsic, coordinate-free descriptions that avoid reliance on non-associative algebra. See octonions and Bryant-Salamon for the standard geometric context and explicit examples in Spin(7) settings.
Interplay with other calibrated geometries: Cayley submanifolds sit alongside associative and coassociative submanifolds in G2 geometry, as well as special Lagrangian submanifolds in Calabi–Yau geometries. Debates often center on which methods transfer most effectively from one calibration context to another, and how insights from one holonomy group inform the understanding of another. See G2 holonomy and Calibrated geometry for broader comparisons.