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$S^1$-quotient of $Spin(7)$-structures

Udhav Fowdar

Abstract

If a $Spin(7)$ manifold $N^8$ admits a free $S^1$ action preserving the fundamental $4$-form then the quotient space $M^7$ is naturally endowed with a $G_2$-structure. We derive equations relating the intrinsic torsion of the $Spin(7)$-structure to that of the $G_2$-structure together with the additional data of a Higgs field and the curvature of the $S^1$-bundle; this can be interpreted as a Gibbons-Hawking-type ansatz for $Spin(7)$-structures. We focus on the three $Spin(7)$ torsion classes: torsion-free, locally conformally parallel and balanced. In particular we show that if $N$ is a $Spin(7)$ manifold then $M$ cannot have holonomy contained in $G_2$ unless $N$ is in fact a Calabi-Yau $4$-fold and $M$ is the product of a Calabi-Yau $3$-fold and an interval. We also derive a new formula for the Ricci curvature of $Spin(7)$-structures in terms of the torsion forms. We then describe this $S^1$-quotient construction in detail for the Bryant-Salamon $Spin(7)$ metric on the spinor bundle of $S^4$ and for the flat metric on $\mathbb{R}^8$.

$S^1$-quotient of $Spin(7)$-structures

Abstract

If a manifold admits a free action preserving the fundamental -form then the quotient space is naturally endowed with a -structure. We derive equations relating the intrinsic torsion of the -structure to that of the -structure together with the additional data of a Higgs field and the curvature of the -bundle; this can be interpreted as a Gibbons-Hawking-type ansatz for -structures. We focus on the three torsion classes: torsion-free, locally conformally parallel and balanced. In particular we show that if is a manifold then cannot have holonomy contained in unless is in fact a Calabi-Yau -fold and is the product of a Calabi-Yau -fold and an interval. We also derive a new formula for the Ricci curvature of -structures in terms of the torsion forms. We then describe this -quotient construction in detail for the Bryant-Salamon metric on the spinor bundle of and for the flat metric on .

Paper Structure

This paper contains 14 sections, 14 theorems, 128 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The quotient construction
  4. The torsion free quotient
  5. The locally conformally parallel quotient
  6. The balanced quotient
  7. Ricci and Scalar curvatures
  8. $S^1$-quotient of the Spinor bundle of $S^4$
  9. The Gibbons-Hawking ansatz
  10. $S^1$-quotient of a fixed fibre of the spinor bundle
  11. $S^1$-quotient of the Bryant-Salamon cone metric
  12. Remarks on the induced $SU(3)$-structure on $\mathbb{C} \mathbb{P}^3$
  13. $S^1$-quotient of flat $Spin(7)$ metric
  14. Appendix

Key Result

Theorem 2.1

$\nabla^{g_\varphi}\varphi=0$ if and only if $d\varphi=0$ and $d*_\varphi \varphi=0.$

Theorems & Definitions (30)

  • Theorem 2.1: Fernandez1982
  • Lemma 2.2: Bryant06someremarks
  • Definition 2.3
  • Theorem 2.4
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Remark 3.3
  • Theorem 3.4
  • ...and 20 more