[Paper Review] Instantons in G2 manifolds from J-holomorphic curves in coassociative submanifolds

Naichung Conan Leung, Xiaowei Wang+1 more|arXiv (Cornell University)|Mar 27, 2013

Geometry and complex manifolds24 references3 citations

TL;DR

This paper establishes a correspondence between instantons—3-dimensional associative submanifolds—in G₂ manifolds and J-holomorphic curves in coassociative submanifolds, using a deformation method in the C^{1,α} setting to overcome nonlinearities and weak coupling in the linearized instanton equation. The key result is a construction of new instantons via perturbations of complex structures on coassociative submanifolds, linking their count to Seiberg-Witten invariants via Taubes' GW=SW correspondence.

ABSTRACT

In G2 manifolds, 3-dimensional associative submanifolds (instantons) play a role similar to J-holomorphic curves in symplectic geometry. In [21], instantons in G2 manifolds were constructed from regular J-holomorphic curves in coassociative submanifolds. In this exposition paper, after reviewing the background of G2 geometry, we explain the main ingredients in the proofs in [21]. We also construct new examples of instantons.

Motivation & Objective

• To establish a correspondence between associative submanifolds (instantons) in G2 manifolds and J-holomorphic curves in coassociative submanifolds.
• To overcome analytical challenges in the instanton equation, including nonlinearity and weak coupling, by using the Schauder (C^{1,α}) setting instead of W^{1,p}.
• To construct new examples of instantons by deforming complex structures on coassociative submanifolds in Calabi-Yau threefolds.
• To extend the correspondence beyond non-degenerate symplectic forms, including cases with degenerate or vanishing normal vector fields.
• To explore the validity of the correspondence in the almost G2 setting, where the G2 structure is closed but not necessarily parallel.

Proposed method

• A deformation method is used to construct instantons in a G2 manifold M by deforming submanifolds rather than maps, relying on normal frames in the ambient space.
• The instanton equation is treated in the C^{1,α} setting due to the presence of cubic derivative terms that prevent L^p estimates in W^{1,p} spaces.
• A right inverse bound is established for the linearized instanton operator using Schauder estimates, crucial for applying the implicit function theorem.
• The construction involves a family of coassociative submanifolds C_t in M, with boundaries on C_0 and C_ε, and a perturbed almost complex structure J_n induced by a normal vector field n.
• The implicit function theorem is applied to solve F_ε(V) = 0, where F_ε is the instanton operator, using error estimates and quadratic estimates in the C^{1,α} norm.
• The method relies on Fredholm regularity of the J_n-holomorphic curve Σ_n in the coassociative submanifold C_0, ensuring persistence under small complex structure perturbations.

Experimental results

Research questions

• RQ1 Can associative submanifolds (instantons) in G2 manifolds be systematically constructed from J-holomorphic curves in coassociative submanifolds?
• RQ2 How can the nonlinear and weakly coupled nature of the instanton equation be handled analytically when standard L^p methods fail?
• RQ3 What is the role of the Seiberg-Witten invariant in counting instantons arising from J-holomorphic curves in coassociative submanifolds?
• RQ4 Does the correspondence between instantons and J-holomorphic curves persist when the normal vector field to the coassociative family has zeros or degeneracies?
• RQ5 Can the correspondence be extended to the almost G2 setting, where the G2 structure is closed but not parallel?

Key findings

• A new family of instantons is constructed in G2 manifolds by perturbing the complex structure on a coassociative submanifold, using a J_n-holomorphic curve near a Fredholm regular J_0-holomorphic curve.
• The correspondence between instantons and J-holomorphic curves holds even when the normal vector field to the coassociative family has zeros, provided the curve avoids the zero locus.
• The analysis in the C^{1,α} setting successfully handles the cubic nonlinearity in the instanton equation, which obstructs standard W^{1,p} estimates.
• The implicit function theorem is applied to solve the instanton equation F_ε(V) = 0, with error estimates showing ||F_ε(0)||_{C^{α}} ≤ Cε^{1−α}, enabling the solution to exist for small ε.
• The construction yields non-trivial instantons not diffeomorphic to the product Σ×[0,ε], distinguishing them from trivial cylindrical solutions.
• The method suggests a potential generalization to almost G2 structures, as the gluing analysis depends primarily on Fredholm regularity of the linearized operator.