[Paper Review] A Kahler-Einstein inspired anzatz for Spin(7) holonomy metrics and its solution

Osvaldo P. Santillán|ArXiv.org|Sep 13, 2006

Geometry and complex manifolds1 references3 citations

TL;DR

This paper proposes a novel ansatz for Spin(7) holonomy metrics by constructing an R-bundle over closed G₂ structures, where the G₂ structures arise as R³-bundles over compact quaternion Kähler 4-manifolds. Inspired by the Bryant-Salom construction and the Kähler-Einstein property of twistor spaces, the ansatz reduces holonomy to Spin(7) via a nonlinear system of ODEs; a particular solution yields a hyperkähler Swann bundle with Sp(2) holonomy, confirming the metric as a tri-Sasakian cone and Calabi-Yau cone over a Sasaki-Einstein manifold.

ABSTRACT

We construct propose an anzatz for Spin(7) metrics as an R-bundle over closed G2 structures. These G2 structures are R3 bundles over 4-dimensional compact quaternion Kahler spaces. The inspiration for the anzatz metric comes from the Bryant-Salamon construction of G2 holonomy metrics and from the fact that the twistor space of any compact quaternion Kahler space is Kahler-Einstein. The reduction of the holonomy to a subgroup of Spin(7) gives non linear system relating three unknown functions of one variable. We obtain a particular solution and we find that the resulting metric is a Calabi-Yau cone over an Einstein-Sassaki manifold which means that the holonomy is reduced to SU(4). Another coordinate change show us that our metrics are hyperkahler cones known as Swann bundles, thus the holonomy is reduced to Sp(2) and the cone is tri-Sassakian. We revert our argument and state that the Swann bundle define a closed G2 structure by reduction along an isometry. We calculate the torsion classes for such structure explicitly.

Motivation & Objective

• To construct new explicit examples of Riemannian metrics with Spin(7) holonomy, which are important in compactifications of M-theory and supergravity.
• To generalize the Bryant-Salamon construction of G₂ holonomy metrics by extending the fiber bundle structure to higher dimensions.
• To exploit the Kähler-Einstein geometry of twistor spaces of compact quaternion Kähler 4-manifolds as a geometric foundation for the ansatz.
• To reduce the holonomy condition to a system of nonlinear ODEs in three unknown functions and solve them explicitly.
• To identify the resulting metric as a known hyperkähler structure (Swann bundle), thereby confirming its holonomy and geometric properties.

Proposed method

• Construct an 8-dimensional metric as an R-bundle over a G₂ structure, where the base is an R³-bundle over a compact quaternion Kähler 4-manifold with cosmological constant Λ = 3.
• Use the twistor space of the quaternion Kähler manifold, which carries a Kähler-Einstein metric of positive scalar curvature, to define the Kähler triplet and spin connection forms.
• Define the metric via three unknown functions f, g, h of a radial coordinate u, and a τ-independent 1-form H, with the full metric given by g₈ = e^(-3h/2)(dt+H)² + e^(2f + h/2)αᵢαᵢ + e^(2g + h/2)g_q.
• Impose closure of the Spin(7) 4-form Φ₄ by requiring a system of three coupled ODEs relating f, g, h and their derivatives.
• Use the fact that the twistor space’s Kähler form satisfies dH = -ũᵢJ̄ᵢ + (ε_ijk/2)ũᵢθⱼ∧θₖ, ensuring integrability and closure of Φ₄.
• Verify that the resulting metric is a Calabi-Yau cone over a Sasaki-Einstein manifold and further a hyperkähler cone (Swann bundle), implying holonomy Sp(2) ⊂ SU(4) ⊂ Spin(7).

Experimental results

Research questions

• RQ1 Can a Kähler-Einstein-inspired ansatz be constructed for Spin(7) holonomy metrics using the geometry of quaternion Kähler 4-manifolds and their twistor spaces?
• RQ2 Does the reduction of holonomy to Spin(7) lead to a solvable system of nonlinear ODEs in three radial functions?
• RQ3 Is the resulting metric a known hyperkähler structure, such as the Swann bundle, and what does this imply about its holonomy and geometric classification?
• RQ4 Can the G₂ structure on the base be recovered via isometric reduction along the τ-direction, and what are its torsion classes?
• RQ5 What is the role of the Kähler-Einstein property of the twistor space in ensuring the closure of the Spin(7) 4-form and holonomy reduction?

Key findings

• The ansatz yields a solution to the Spin(7) holonomy condition via a system of three nonlinear ODEs relating the functions f, g, h and their derivatives with respect to u.
• The particular solution corresponds to the Swann bundle, a hyperkähler metric in eight dimensions with holonomy Sp(2), confirming it as a tri-Sasakian cone.
• The metric is shown to be a Calabi-Yau cone over a Sasaki-Einstein 7-manifold, implying holonomy reduction to SU(4) ⊂ Spin(7).
• The torsion classes of the induced G₂ structure are computed: τ₀ = τ₃ = 0, and τ₂ is expressed in terms of the Kähler form on the twistor space.
• The construction confirms that the Swann bundle defines a conformally closed G₂ structure via isometric reduction along the τ-isometry, with τ₀ eliminated by conformal rescaling.
• The full 11D supergravity solution is recovered as the direct sum of the Swann metric and a 3D Minkowski space, preserving four supersymmetries.