Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Parallel torsion and $G_2, Spin(7)$ instantons

Stefan Ivanov, Alexander Petkov, Luis Ugarte

TL;DR

The paper analyzes how skew-torsion connections associated with G_2 and Spin(7) structures yield instanton conditions, and when these instanton properties force the torsion to be parallel. By deriving and leveraging precise identities among dT, d^∇T, σ^T, and the Lee form, it establishes equivalences between ∇T=0 and curvature-instanton conditions under integrable, Gauduchon, and closed-torsion settings, including corresponding results for compact and noncompact cases. The results extend prior work by removing Killing-torsion assumptions and provide a comprehensive framework connecting instanton geometry to parallel torsion, with implications for Hull-Strominger-type systems and conformal deformations (Gauduchon structures) in both G_2 and Spin(7) contexts. The work also clarifies how Ricci and scalar invariants behave when torsion is parallel and when instanton conditions hold, offering a rigorous foundation for applications in special holonomy with skew torsion in mathematical physics.

Abstract

Instanton properties of the characteristic connection $\nabla$ on an integrable $G_2$ manifold as well as instanton condition of the torsion connection $\nabla$ on a $Spin(7)$ manifold are investigated. It is shown that for an integrable $G_2$ manifold with $\nabla$-parallel Lee form the curvature of the characteristic connection is a $G_2$ instanton exactly when the torsion 3-form is $\nabla$-parallel. It is observed that on a compact $Spin(7)$ manifold with $\nabla$ closed torsion 3-form the torsion connection is a $Spin(7)$ instanton if and only if the torsion 3-form is parallel with respect to the torsion connection.

Parallel torsion and $G_2, Spin(7)$ instantons

TL;DR

The paper analyzes how skew-torsion connections associated with G_2 and Spin(7) structures yield instanton conditions, and when these instanton properties force the torsion to be parallel. By deriving and leveraging precise identities among dT, d^∇T, σ^T, and the Lee form, it establishes equivalences between ∇T=0 and curvature-instanton conditions under integrable, Gauduchon, and closed-torsion settings, including corresponding results for compact and noncompact cases. The results extend prior work by removing Killing-torsion assumptions and provide a comprehensive framework connecting instanton geometry to parallel torsion, with implications for Hull-Strominger-type systems and conformal deformations (Gauduchon structures) in both G_2 and Spin(7) contexts. The work also clarifies how Ricci and scalar invariants behave when torsion is parallel and when instanton conditions hold, offering a rigorous foundation for applications in special holonomy with skew torsion in mathematical physics.

Abstract

Instanton properties of the characteristic connection on an integrable manifold as well as instanton condition of the torsion connection on a manifold are investigated. It is shown that for an integrable manifold with -parallel Lee form the curvature of the characteristic connection is a instanton exactly when the torsion 3-form is -parallel. It is observed that on a compact manifold with closed torsion 3-form the torsion connection is a instanton if and only if the torsion 3-form is parallel with respect to the torsion connection.

Paper Structure

This paper contains 19 sections, 33 theorems, 134 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $G_2$ structure
  4. The $G_2$-connection with skew-symmetric torsion
  5. The torsion and the Ricci tensor of the characteristic connection
  6. $G_2$-instanton. Proof of Theorem \ref{['instm']}, Theorem \ref{['instqjm']} and Theorem \ref{['cmpga']}
  7. Proof of Theorem \ref{['instm']}
  8. Proof of Theorem \ref{['instqjm']}
  9. Compact Gauduchon $G_2$ manifolds. Proof of Theorem \ref{['cmpga']}
  10. Closed torsion
  11. $Spin(7)$-structure
  12. Decomposition of the space of forms
  13. The $Spin(7)$-connection with skew-symmetric torsion
  14. $Spin(7)$-instanton. Proof of Theorem \ref{['main2']}, Theorem \ref{['mainsp0']} and Theorem \ref{['mainsp']}
  15. Proof of Theorem \ref{['main2']}
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $(M,{\varphi})$ be an integrable $G_2$ manifold with ${\nabla}$-parallel Lee form and the curvature of the characteristic connection ${\nabla}$ is a $G_2$-instanton, i.e. Then the torsion 3-form is parallel with respect to the characteristic connection, ${\nabla} T=0$. In particular, the $G_2$ manifold is of constant type, the characteristic Ricci tensor is symmetric, ${\nabla}$-parallel and

Theorems & Definitions (62)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Remark 1.7
  • Theorem 1.8
  • Corollary 1.9
  • Theorem 1.10
  • ...and 52 more