Spin7Edit
Spin(7) is a compact, simply connected Lie group of dimension 21, and it serves as a key example in the study of exceptional structures in geometry. It is the universal covering group of SO(7) and can be realized as a subgroup of SO(8) that preserves a particular 4-form on eight-dimensional real space. Concretely, Spin(7) is the stabilizer of a distinguished 4-form known as the Cayley form on R^8, and this stabilizer description provides a direct link between abstract group theory and concrete geometric objects. The center of Spin(7) is a cyclic group of order two, and the Lie algebra associated with Spin(7) is so(7), which in the Cartan classification sits in the B3 family. For representation theory, Spin(7) admits an 8-dimensional real spin representation that plays a central role in how the group acts on geometry and topology in eight dimensions. These features place Spin(7) at the crossroads of Lie theory, differential geometry, and mathematical physics.
Algebraic and Lie-theoretic structure
Spin(7) as a Lie group: Spin(7) is the simply connected cover of SO(7), realized as a subgroup of SO(8) that preserves a fixed 4-form. Its Lie algebra so(7) has dimension 21, and the root system is of type B3. Its center is isomorphic to Z/2Z. The eight-dimensional real spin representation provides a faithful action of Spin(7) on R^8, which is central to interpretations in geometry and physics. See Lie group and Spin group for broader context.
Embedding into GL(8,R) and the Cayley form: Spin(7) can be described as the stabilizer inside GL(8,R) of the Cayley 4-form Φ on R^8. This invariant object defines a Spin(7)-structure on eight-dimensional spaces and establishes Spin(7) as a subgroup of SO(8) that preserves Φ. See Cayley form for the explicit geometric object and holonomy for how such invariants influence geometry.
Representations and geometry: The 8-dimensional spin representation of Spin(7) is fundamental in understanding how the group acts on eight-dimensional spaces. This representation underpins constructions in differential geometry and is related to the special algebraic structures that Spin(7) preserves. See Representation theory and Spin representation for related topics.
Geometry and holonomy
Spin(7) structures and holonomy: An 8-manifold can carry a Spin(7) structure if its frame bundle reduces to Spin(7) via a globally defined 4-form Φ that is stabilized by Spin(7). Such a structure is called torsion-free when Φ is closed and co-closed (dΦ = 0 and d*Φ = 0), in which case the Levi-Civita connection has holonomy contained in Spin(7). When the holonomy is exactly Spin(7), the manifold is Ricci-flat and exhibits exceptional geometric properties. See holonomy and Spin(7) structure for broader context.
Calibrated geometry and Cayley submanifolds: The Cayley 4-form Φ calibrates certain 4-dimensional submanifolds, known as Cayley submanifolds, which minimize volume in their homology class. This calibrated geometry is a central feature of Spin(7) manifolds and connects to broader ideas in calibrated geometry, minimal submanifolds, and geometric measure theory. See Calibrated geometry and Cayley form for related material.
Examples of Spin(7) metrics: Bryant and Salamon constructed complete Riemannian metrics with Spin(7) holonomy on the total spaces of certain vector bundles over S^4, illustrating how these structures can be realized on noncompact spaces. Joyce later produced compact examples of Spin(7) manifolds by resolving orbifold quotients of tori with carefully chosen group actions, demonstrating that the exceptional holonomy can occur in compact settings as well. See Bryant–Salamon and Joyce for detailed constructions and context.
Compact versus noncompact cases: Noncompact Spin(7) manifolds often arise from bundle constructions and explicit cohomogeneity-one ansatzes, while compact Spin(7) manifolds require delicate global analysis and resolution techniques. These developments together illustrate the flexibility of Spin(7) geometry across different global topologies. See Joyce and Bryant–Salamon for representative examples.
Connections and applications
Interplay with other exceptional holonomy groups: Spin(7) sits alongside G2 as part of the broader landscape of special holonomy, with distinct geometric signatures and calibrated geometries. The stabilizer viewpoint in eight dimensions contrasts with G2 holonomy, which occurs in seven dimensions. See G2 and Holonomy for a comparative perspective.
Physical relevance: In theoretical physics, spaces with Spin(7) holonomy appear in compactifications of higher-dimensional theories, including certain formulations of M-theory and string theory, where the reduced holonomy constrains the amount of preserved supersymmetry and shapes the low-energy effective theory. These connections highlight how deep geometric structures inform physical models. See String theory and M-theory for broader physical context.
Topological and geometric invariants: The study of Spin(7) manifolds engages tools from differential geometry, topology, and global analysis, including the examination of cohomology, index theory, and moduli of Spin(7) structures. This area remains a vibrant intersection of pure mathematics and mathematical physics. See Index theory and Cohomology for foundational topics relevant to this field.