Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Complete non-compact $\operatorname{Spin}(7)$-manifolds from $T^2$-bundles over AC Calabi Yau manifolds

Nicolò Cavalleri

TL;DR

This work constructs new complete non-compact $8$-manifolds with holonomy $Spin(7)$ by equipping total spaces of principal $T^2$-bundles over asymptotically conical Calabi–Yau $3$-folds with torsion-free $Spin(7)$ structures. The authors reduce the torsion-free condition to a tractable system in the adiabatic limit (shrinking fibres) and solve it via an implicit function theorem, yielding a one-parameter analytic curve of Spin($7$) metrics that collapse to the base CY geometry while preserving full holonomy. They also introduce and analyze asymptotically $T^2$-fibred conical ($AT^2C$) geometry in higher dimensions, establish topological constraints, and generalize to orbifolds. The construction produces infinitely many diffeomorphism types and the first complete toric Spin($7$) manifolds, with further toric families accessible via orbifold techniques, thereby expanding the landscape of non-compact Spin($7$) geometries and linking them to $G_2$ and CY geometry through a unifying adiabatic-IFT approach.

Abstract

We develop a new construction of complete non-compact 8-manifolds with Riemannian holonomy equal to $\operatorname{Spin}(7)$. As a consequence of the holonomy reduction, these manifolds are Ricci-flat. These metrics are built on the total spaces of principal $T^2$-bundles over asymptotically conical Calabi Yau manifolds, and the result is generalized to orbifolds. The resulting metrics have a new geometry at infinity that we call asymptotically $T^2$-fibred conical ($AT^2C$) and which generalizes to higher dimensions the ALG metrics of 4-dimensional hyperkähler geometry, analogously to how ALC metrics generalize ALF metrics. As an application of this construction, we produce infinitely many diffeomorphism types of $AT^2C$ $\operatorname{Spin}(7)$-manifolds and the first known examples of complete toric $\operatorname{Spin}(7)$-manifold.

Complete non-compact $\operatorname{Spin}(7)$-manifolds from $T^2$-bundles over AC Calabi Yau manifolds

TL;DR

This work constructs new complete non-compact -manifolds with holonomy by equipping total spaces of principal -bundles over asymptotically conical Calabi–Yau -folds with torsion-free structures. The authors reduce the torsion-free condition to a tractable system in the adiabatic limit (shrinking fibres) and solve it via an implicit function theorem, yielding a one-parameter analytic curve of Spin() metrics that collapse to the base CY geometry while preserving full holonomy. They also introduce and analyze asymptotically -fibred conical () geometry in higher dimensions, establish topological constraints, and generalize to orbifolds. The construction produces infinitely many diffeomorphism types and the first complete toric Spin() manifolds, with further toric families accessible via orbifold techniques, thereby expanding the landscape of non-compact Spin() geometries and linking them to and CY geometry through a unifying adiabatic-IFT approach.

Abstract

We develop a new construction of complete non-compact 8-manifolds with Riemannian holonomy equal to . As a consequence of the holonomy reduction, these manifolds are Ricci-flat. These metrics are built on the total spaces of principal -bundles over asymptotically conical Calabi Yau manifolds, and the result is generalized to orbifolds. The resulting metrics have a new geometry at infinity that we call asymptotically -fibred conical () and which generalizes to higher dimensions the ALG metrics of 4-dimensional hyperkähler geometry, analogously to how ALC metrics generalize ALF metrics. As an application of this construction, we produce infinitely many diffeomorphism types of -manifolds and the first known examples of complete toric -manifold.
Paper Structure (43 sections, 62 theorems, 136 equations)

This paper contains 43 sections, 62 theorems, 136 equations.

Table of Contents

  1. Introduction
  2. Generalizations of theorem \ref{['thm:main_theorem']}
  3. Connections to physics
  4. Organization
  5. Acknowledgments
  6. Preliminaries
  7. $\operatorname{SU}(3)$-structures
  8. Calabi-Yau manifolds
  9. $\operatorname{Spin}(7)$ structures
  10. Spin geometry
  11. Connections on $T^2$-bundles
  12. Asymptotic geometries
  13. Calabi Yau cones
  14. Asymptotically Conical Calabi Yau 3-folds
  15. Asymptotically $T^k$-fibred conical manifolds
  16. ...and 28 more sections

Key Result

Theorem 1.1

Let $\left(B, g_B, \omega_0, \Omega_0\right)$ be an asymptotically conical Calabi Yau 3-fold asymptotic with rate $\nu \in \mathbb{R}^-$ to the Calabi-Yau cone $\left(\mathrm{C}(\Sigma), g_{\mathrm{C}}, \omega_{\mathrm{C}}, \Omega_{\mathrm{C}}\right)$ over a smooth Sasaki-Einstein 5-manifold $\Sigma Then the 8-manifold $M$ carries an analytic curve of $T^2$-invariant torsion-free $\operatorname{Sp

Theorems & Definitions (129)

  • Theorem 1.1
  • Definition 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • Theorem 2.6
  • Proposition 2.7
  • ...and 119 more