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Dimension Reduction of Generalized ASD Instantons

Dylan Galt, Langte Ma

TL;DR

The paper develops a comprehensive dimension-reduction theory for generalized ASD instantons on product manifolds $(M=Y\times X)$ equipped with a parallel $(n-4)$-form $\theta$. By introducing the $\alpha$-null energy condition and an integrability framework for families of connections, it proves a canonical isomorphism between irreducible $\theta$-instantons on $M$ and fiber data consisting of flat holonomy on $Y$ and $\beta$-instantons on $X$, with an abelian variant arising from the Picard torus. It also constructs a well-behaved compactification of the moduli space when $X$ is 4-dimensional and demonstrates that a broad class of calibrated fibrations (coassociative and Cayley) reduce to hyperkähler product-type descriptions, including G2 and Spin(7) cases. The results yield explicit moduli-space descriptions in several key geometries (e.g., Hermitian Yang–Mills, G2-, Spin(7)-instantons) and provide a framework for adiabatic-limit problems and potential applications to calibrated bubbling phenomena.

Abstract

We study generalized anti-self-dual instantons defined over Riemannian manifolds equipped with a parallel codimension-$4$ differential form. In particular, for product Riemannian manifolds possessing such a form, we study dimension reduction phenomena, finding a topological criterion for bundles which, when satisfied, allows for a complete characterization of dimension reduction for the corresponding moduli space of generalized ASD instantons. By establishing an integrability result for families of connections, we then deduce explicit descriptions for these moduli spaces, including those of Hermitian Yang--Mills connections, $G_2$-, and $\Spin(7)$-instantons. When one factor in the product is a $4$-manifold, we establish well-behaved compactifications for these moduli spaces.

Dimension Reduction of Generalized ASD Instantons

TL;DR

The paper develops a comprehensive dimension-reduction theory for generalized ASD instantons on product manifolds equipped with a parallel -form . By introducing the -null energy condition and an integrability framework for families of connections, it proves a canonical isomorphism between irreducible -instantons on and fiber data consisting of flat holonomy on and -instantons on , with an abelian variant arising from the Picard torus. It also constructs a well-behaved compactification of the moduli space when is 4-dimensional and demonstrates that a broad class of calibrated fibrations (coassociative and Cayley) reduce to hyperkähler product-type descriptions, including G2 and Spin(7) cases. The results yield explicit moduli-space descriptions in several key geometries (e.g., Hermitian Yang–Mills, G2-, Spin(7)-instantons) and provide a framework for adiabatic-limit problems and potential applications to calibrated bubbling phenomena.

Abstract

We study generalized anti-self-dual instantons defined over Riemannian manifolds equipped with a parallel codimension- differential form. In particular, for product Riemannian manifolds possessing such a form, we study dimension reduction phenomena, finding a topological criterion for bundles which, when satisfied, allows for a complete characterization of dimension reduction for the corresponding moduli space of generalized ASD instantons. By establishing an integrability result for families of connections, we then deduce explicit descriptions for these moduli spaces, including those of Hermitian Yang--Mills connections, -, and -instantons. When one factor in the product is a -manifold, we establish well-behaved compactifications for these moduli spaces.
Paper Structure (25 sections, 31 theorems, 230 equations, 1 table)

This paper contains 25 sections, 31 theorems, 230 equations, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Main Results
  4. Generalized Anti-Self-Dual Instantons
  5. Anti-Self-Dual $\theta$-Instantons
  6. Anti-Self-Dual Connections on $4$-Manifolds
  7. Hermitian Yang--Mills Connections
  8. $G_2$-Instantons
  9. $\mathop{\mathrm{Spin}}\nolimits(7)$-Instantons
  10. Curvature and Parallelism
  11. Bundles over Product Manifolds
  12. Stabilization
  13. Integrability
  14. Parallelism
  15. Dimension Reduction Phenomena
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $M = Y \times X$ be the product of two closed Riemannian manifolds and $\theta = \mathop{\mathrm{vol}}\nolimits_Y \wedge \beta + \alpha$ an admissible $(n-4)$-form compatible with the product structure on $M$. Suppose $\pi: P \to M$ is an $\alpha$-null principal $SU(r)$-bundle such that $\mathca where $\mathcal{N}^*_{\beta}(X, P_X)$ denotes the moduli space of irreducible $\beta$-instantons ov

Theorems & Definitions (80)

  • Definition 1
  • Definition 2
  • Theorem 1.1
  • Theorem 1.2
  • Definition 3
  • Theorem 1.3
  • Proposition 1
  • Theorem 1.4
  • Lemma 1
  • proof
  • ...and 70 more