Associative SubmanifoldEdit
Associative submanifolds are a central object in the geometry of seven-dimensional spaces with a special kind of symmetry, known as G2 structure. In a seven-manifold M equipped with a G2 structure, there is a distinguished 3-form φ that encodes a rich cross product on the tangent spaces. A three-dimensional submanifold N ⊂ M is called associative if φ restricted to N equals the volume form of N, making N a calibrated submanifold. This calibration property implies that associative submanifolds minimize volume in their homology class, giving them a natural geometric rigidity and significance in both mathematics and theoretical physics. The dual notion in the same setting is coassociative submanifolds, which are calibrated by the Hodge dual *φ and have dimension four.
In plain terms, associative submanifolds sit inside seven-dimensional spaces with a special kind of geometric structure that comes from the algebra of octonions. The defining form φ is built so that, at every point, the tangent 3-planes that maximize φ agree with the intrinsic volume of the submanifold. When the ambient G2 structure is torsion-free or, more strongly, has holonomy contained in G2, associative submanifolds enjoy optimal volume-minimizing properties and fit into a broader calibrated geometry. For readers who want the larger context, these ideas sit inside the study of G2 manifold and more generally calibration theory in Riemannian geometry.
Definition and context
Let (M, φ) be a seven-dimensional manifold carrying a G2 structure, which induces a Riemannian metric g and a compatible orientation. A smooth 3-submanifold N ⊂ M is called associative if φ|_N = vol_N, where vol_N is the Riemannian volume form on N determined by g. Equivalently, N is associative if, at every point, its tangent space is a 3-plane calibrated by φ. This condition places N among the calibrated submanifolds of M and ensures that N is volume-minimizing within its homology class when the ambient structure is torsion-free. The associative condition is intimately connected to the cross product defined by φ on tangent spaces, which makes the tangent bundle of N interact with the ambient geometry in a special, rigid way.
For those who want to connect with broader language, associative submanifolds are a key example of calibrated submanifold in a G2 manifold and are studied alongside their four-dimensional counterparts, the coassociatives, which are calibrated by *φ.
Geometric background and relations
The G2 structure on a seven-manifold can be described by the 3-form φ and its metric-compatible partner, the 4-form *φ. The pair (φ, *φ) encodes both the algebraic cross product on tangent spaces and the calibration conditions that select special submanifolds. The associative 3-form φ arises naturally from the algebra of octonions, with the imaginary octonions providing an identification of R^7 with a vector space carrying a canonical G2 structure. This makes certain linear 3-planes in R^7 automatically associative, and these linear models extend to curved settings inside G2-manifolds.
Key references for this background include topics on the basic calibrated geometry and the role of G2 manifold in shaping minimal submanifolds. The concept of associative submanifolds sits alongside other calibrated geometries that yield volume-minimizing representatives in homology, and it interacts with analytic tools like Dirac-type operators when one studies deformations.
In mathematical physics, associative submanifolds appear in the context of compactifications of M-theory on G2-manifolds, where they can correspond to certain supersymmetric brane configurations. This intersection with physics has driven interest in constructing examples and understanding the moduli of associative submanifolds.
Examples and constructions
Flat model: In the standard flat G2 structure on R^7 coming from the imaginary octonions, certain linear 3-planes are associative. These give the simplest, explicit examples of associative submanifolds, serving as local models for more complicated situations.
Nonlinear and curved settings: In complete or asymptotically conical G2-manifolds, one can find associative submanifolds that are not linear but arise from geometric constructions, or by solving the calibrated condition φ|_N = vol_N within a curved ambient space. The existence and abundance of such submanifolds are active areas of research.
Deformations and moduli: Given an associative submanifold N, one can study nearby associative submanifolds obtained by small deformations of N within M. This deformation theory is governed by analysis of a linear elliptic operator on the normal bundle, and it leads to a finite-dimensional moduli space of associative deformations near N in favorable situations. See the deformation theory section for more.
For a broader palette of related ideas, see G2 manifold and calibration theory, which provide the tools and language to discuss these constructions in both flat and curved spaces.
Deformation theory and moduli
The infinitesimal deformation theory of associative submanifolds was developed to understand how flexible or rigid these submanifolds are inside a G2-manifold. In favorable conditions (notably for compact associative submanifolds in a torsion-free G2-structure), the moduli space of associative deformations is a smooth finite-dimensional manifold. Its dimension is determined by the kernel of a linear elliptic operator acting on sections of the normal bundle to N, commonly described in terms of a Dirac-type operator. In spirit, this mirrors the deformation theories for other calibrated submanifolds, but the precise operator and its analytical properties depend on the ambient G2-structure.
This deformation theory connects to broader ideas in calibrated geometry, and it helps explain how associative submanifolds can vary within families. The study of obstructions, transversality, and potential wall-crossing phenomena remains an active area of research, with implications for both differential geometry and related physical theories.
Properties, invariants, and applications
Volume-minimizing property: As calibrated submanifolds, associative 3-folds minimize volume in their homology class when the ambient G2-structure is torsion-free.
Minimality: Associative submanifolds are minimal submanifolds, satisfying the vanishing of mean curvature.
Interaction with physics: In M-theory compactifications on G2-manifolds, associative submanifolds can model certain supersymmetric sector configurations, connecting differential geometry with high-energy physics.
Enumerative questions: There is interest in counting associative submanifolds or understanding their moduli in families of G2-manifolds. These questions tie into broader themes of invariants in calibrated geometry and potential wall-crossing phenomena, and they remain areas of active mathematical exploration.