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Cohomogeneity One Manifolds of Spin(7) and G(2) Holonomy

M. Cvetic, G. W. Gibbons, H. Lu, C. N. Pope

TL;DR

<3-5 sentence high-level summary>This work advances the construction of non-compact metrics with exceptional holonomy by developing new cohomogeneity-one ans"atze for Spin(7) in eight dimensions and G_2 in seven, exploring principal orbits including S^7 viewed as a triaxially squashed S^3 bundle over S^4 and the Aloff-Wallach spaces N(k,ell). It derives comprehensive first-order systems from a superpotential, employs analytic reductions and numerical analysis to uncover new ALC Spin(7) metrics C_8 on line bundles over CP^3 and AC/ALC families for all N(k,ell), as well as explicit G_2 constructions on R^3-bundles over CP^2 and CP^3-related settings, with perturbative AC limits around known solutions. The results include an explicit global Spin(7) solution for all N(k,ell) and a rich array of AC/ALC metrics for various principal orbits, some with orbifold structures on CP^2 bolts. These findings enrich the catalog of explicit exceptional-holonomy geometries and offer new pathways for holography, M-theory compactifications, and geometric analysis.

Abstract

In this paper, we look for metrics of cohomogeneity one in D=8 and D=7 dimensions with Spin(7) and G_2 holonomy respectively. In D=8, we first consider the case of principal orbits that are S^7, viewed as an S^3 bundle over S^4 with triaxial squashing of the S^3 fibres. This gives a more general system of first-order equations for Spin(7) holonomy than has been solved previously. Using numerical methods, we establish the existence of new non-singular asymptotically locally conical (ALC) Spin(7) metrics on line bundles over \CP^3, with a non-trivial parameter that characterises the homogeneous squashing of CP^3. We then consider the case where the principal orbits are the Aloff-Wallach spaces N(k,\ell)=SU(3)/U(1), where the integers k and \ell characterise the embedding of U(1). We find new ALC and AC metrics of Spin(7) holonomy, as solutions of the first-order equations that we obtained previously in hep-th/0102185. These include certain explicit ALC metrics for all N(k,\ell), and numerical and perturbative results for ALC families with AC limits. We then study D=7 metrics of $G_2$ holonomy, and find new explicit examples, which, however, are singular, where the principal orbits are the flag manifold SU(3)/(U(1)\times U(1)). We also obtain numerical results for new non-singular metrics with principal orbits that are S^3\times S^3. Additional topics include a detailed and explicit discussion of the Einstein metrics on N(k,\ell), and an explicit parameterisation of SU(3).

Cohomogeneity One Manifolds of Spin(7) and G(2) Holonomy

TL;DR

<3-5 sentence high-level summary>This work advances the construction of non-compact metrics with exceptional holonomy by developing new cohomogeneity-one ans"atze for Spin(7) in eight dimensions and G_2 in seven, exploring principal orbits including S^7 viewed as a triaxially squashed S^3 bundle over S^4 and the Aloff-Wallach spaces N(k,ell). It derives comprehensive first-order systems from a superpotential, employs analytic reductions and numerical analysis to uncover new ALC Spin(7) metrics C_8 on line bundles over CP^3 and AC/ALC families for all N(k,ell), as well as explicit G_2 constructions on R^3-bundles over CP^2 and CP^3-related settings, with perturbative AC limits around known solutions. The results include an explicit global Spin(7) solution for all N(k,ell) and a rich array of AC/ALC metrics for various principal orbits, some with orbifold structures on CP^2 bolts. These findings enrich the catalog of explicit exceptional-holonomy geometries and offer new pathways for holography, M-theory compactifications, and geometric analysis.

Abstract

In this paper, we look for metrics of cohomogeneity one in D=8 and D=7 dimensions with Spin(7) and G_2 holonomy respectively. In D=8, we first consider the case of principal orbits that are S^7, viewed as an S^3 bundle over S^4 with triaxial squashing of the S^3 fibres. This gives a more general system of first-order equations for Spin(7) holonomy than has been solved previously. Using numerical methods, we establish the existence of new non-singular asymptotically locally conical (ALC) Spin(7) metrics on line bundles over \CP^3, with a non-trivial parameter that characterises the homogeneous squashing of CP^3. We then consider the case where the principal orbits are the Aloff-Wallach spaces N(k,\ell)=SU(3)/U(1), where the integers k and \ell characterise the embedding of U(1). We find new ALC and AC metrics of Spin(7) holonomy, as solutions of the first-order equations that we obtained previously in hep-th/0102185. These include certain explicit ALC metrics for all N(k,\ell), and numerical and perturbative results for ALC families with AC limits. We then study D=7 metrics of holonomy, and find new explicit examples, which, however, are singular, where the principal orbits are the flag manifold SU(3)/(U(1)\times U(1)). We also obtain numerical results for new non-singular metrics with principal orbits that are S^3\times S^3. Additional topics include a detailed and explicit discussion of the Einstein metrics on N(k,\ell), and an explicit parameterisation of SU(3).

Paper Structure

This paper contains 33 sections, 175 equations, 4 figures.

Table of Contents

  1. Introduction
  2. New Spin(7) metrics with triaxial $S^3$ bundle over $S^4$
  3. Ansatz and first-order equations
  4. Some properties of the equations
  5. Truncations to simpler systems
  6. Some observations about the Spin(7) system
  7. Numerical analysis
  8. Numerical analysis for ${{\Bbb C}{\Bbb P}}^3$ bolts: New Spin(7) metrics ${{\Bbb C}}_8$
  9. Numerical analysis for $S^4$ bolts: The ${{\Bbb B}}_8$, ${{\Bbb B}}_8^+$ and ${{\Bbb B}}_8^-$ examples recovered
  10. Analysis for NUTs: The ${{\Bbb A}}_8$ example recovered
  11. Spin(7) metrics with $SU(3)/U(1)$ principal orbits
  12. The principal orbits: Aloff-Wallach spaces
  13. Einstein metrics on $N(k,\ell)$ from first-order equations
  14. Einstein metrics on $N(k,\ell)$ from second-order equations
  15. Global considerations
  16. ...and 18 more sections

Figures (4)

  • Figure 1: The new non-singular Spin(7) metrics ${{\Bbb C}}_8$ as a function of $\lambda$
  • Figure 2: The metric functions for a typical non-singular ALC solution
  • Figure 3: The non-singular Spin(7) metrics ${{\Bbb B}}_8$ and ${{\Bbb B}}_8^\pm$ as a function of $q_2/q_1$
  • Figure 4: The non-singular $G_2$ metrics ${{\Bbb B}}_7$ and ${{\Bbb B}}_7^\pm$ as a function of $q_2/q_1$