Coassociative SubmanifoldEdit

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Coassociative Submanifold

Coassociative submanifolds are a class of calibrated 4-dimensional submanifolds that arise in the geometry of seven-dimensional manifolds endowed with a G2-structure. These objects sit at the intersection of differential geometry, topology, and mathematical physics, playing a central role in the study of manifolds with exceptional holonomy and in the physics of M-theory. The existence and deformation theory of coassociatives are governed by the underlying G2-structure and its torsion properties, with deep links to calibrated geometry and minimal submanifolds.

Definition and calibration

Let M be a smooth 7-manifold equipped with a G2-structure, encoded by a differential 3-form φ that determines a Riemannian metric g and an orientation. Associated to φ is a 4-form ψ = *φ, where * is the Hodge star with respect to g. A 4-dimensional submanifold L ⊂ M is called coassociative if the restriction of φ to L vanishes, φ|_L = 0. Equivalently, L is calibrated by ψ in the sense of calibrated geometry, meaning that the volume form on L agrees with ψ|_L and L minimizes volume in its homology class whenever the G2-structure is torsion-free (i.e., φ is closed and co-closed).

In formulas, L is coassociative ⇔ φ|_L = 0 ⇔ ψ|_L = vol_L.
The G2-structure thus provides a natural notion of calibrated, and hence volume-minimizing, 4-dimensional submanifolds in M.

The concept dualizes the associative submanifolds, which are 3-dimensional and calibrated by φ. For readers, see G2-manifold and calibration for broader context on these calibrated subobjects and their geometric significance.

Normal bundle and tangent structure

There is a natural identification between the normal bundle N_L of a coassociative submanifold and the bundle of self-dual 2-forms on L. Concretely, the map that sends a normal vector field to the interior product with φ identifies N_L ≅ Λ^2_+(T^*L), where Λ^2_+ denotes self-dual 2-forms on L. This identification is central to the deformation theory of coassociatives because deformations of L within M correspond to sections of N_L, which, under the above isomorphism, correspond to self-dual 2-forms on L.

The interplay between the tangent bundle of L and the ambient G2-structure also manifests through the octonionic cross product associated to φ. Coassociative submanifolds can be characterized by how their tangent spaces interact with this cross product, a viewpoint that is especially useful in flat-model analyses such as in R^7 with the standard G2-structure.

For more on the ambient geometry, see G2-manifold and related discussions of how φ defines cross products and calibrations.

Deformation theory

A fundamental result in the study of coassociative submanifolds is McLean’s deformation theory. If L is a compact coassociative submanifold of a torsion-free G2-manifold (i.e., φ is closed and co-closed), then the moduli space of deformations of L as a coassociative submanifold is finite-dimensional and smooth near L. The tangent space to this moduli space is naturally identified with the space of harmonic self-dual 2-forms on L, i.e.,

T_L Mod ≅ H^2_+(L),

where H^2_+(L) denotes the space of harmonic self-dual 2-forms on L. Consequently, the dimension of the local deformation space equals the second Betti number's positive part, b^2_+(L). This result provides a precise link between the topology of L and its geometric flexibility within a G2-manifold.

The key reference is McLean's theorem (1998), which develops the operator-theoretic framework that identifies infinitesimal deformations with harmonic self-dual 2-forms and shows the smoothness of the moduli space under the torsion-free hypothesis.
The isomorphism N_L ≅ Λ^2_+(T^*L) and the role of harmonic forms on L are central to the deformation analysis.

Examples and existence results

Coassociative submanifolds appear in several standard contexts:

Flat model: In R^7 equipped with the standard torsion-free G2-structure, there exist linear coassociative 4-planes. These flat models provide a laboratory for explicit calculations and for testing deformation theories.
Compact G2-manifolds: In the constructions of compact G2-manifolds, such as those due to Dominic Joyce (and related orbifold resolutions), coassociative submanifolds arise as natural objects to study within the global holonomy context. Joyce and collaborators developed techniques for producing and analyzing coassociatives in these highly structured spaces.
Gluing and singularities: Beyond smooth examples, a line of work investigates how coassociative submanifolds behave under gluing constructions and how singularities may arise and be resolved. Such analyses intersect with nonsingular deformation theory and with the study of moduli spaces at the boundary of smooth structures.

In the literature, coassociative submanifolds are also discussed in connection with M-theory, where M5-branes can wrap calibrated 4-cycles in a G2-compactification, linking the mathematics of coassociatives to physical models. See M-theory for broader physical context and calibrated geometry for a broader mathematical framework.

Intersections and singularities

The study of coassociative submanifolds often involves their intersections with other calibrated submanifolds, singularities that may occur in families of coassociatives, and questions about stability under perturbations of the ambient G2-structure. The deformation theory provides constraints on how L can move within a family of coassociatives, and singularities frequently appear as limits of smooth deformations. Analyzing these phenomena requires tools from elliptic partial differential equations, geometric measure theory, and topology, with key input from the calibration framework.