[Paper Review] Elementary deformations and the hyperKähler-quaternionic Kähler correspondence

Óscar Maciá, Andrew Swann|arXiv (Cornell University)|Apr 4, 2014

Geometry and complex manifolds8 references3 citations

TL;DR

This paper establishes a uniqueness result for the hyperKähler-quaternionic Kähler correspondence by interpreting it as a combination of the twist construction and elementary deformations. It shows that this correspondence has only one degree of freedom, and applies the framework to realize c-map constructions globally, verifying known examples like those of Ferrara and Sabharwal via Lie group structures on the twisted space.

ABSTRACT

The hyperKähler-quaternionic Kähler correspondence constructs quaternionic Kähler metrics from hyperKähler metrics with a rotating circle symmetry. We discuss how this may be interpreted as a combination of the twist construction with the concept of elementary deformation, surveying results of our forthcoming paper. We outline how this leads to a uniqueness statement for the above correspondence and indicate how basic examples of c-map constructions may be realised in this context.

Motivation & Objective

• To clarify the geometric mechanism behind the hyperKähler-quaternionic Kähler correspondence using the twist construction and elementary deformations.
• To prove that the correspondence has only one degree of freedom, establishing a uniqueness statement.
• To provide a global, computational framework for realizing c-map constructions in the context of hyperKähler metrics with rotating circle symmetry.
• To verify known examples of quaternionic Kähler metrics—particularly those from the rigid c-map—by constructing them via the twist and deformation procedure.
• To demonstrate that the resulting twisted metrics are complete and admit left-invariant structures on solvable Lie groups.

Proposed method

• The twist construction is applied to hyperKähler manifolds with a rotating circle symmetry, using twist data (M, X, F, a) where X generates the circle action, F is an X-invariant integral 2-form, and a satisfies da = -X⌟F.
• Elementary deformations are introduced by adjusting the connection form and curvature to preserve geometric structures under the twist, particularly ensuring the Kähler forms remain closed after twisting.
• The method uses H-related tensors to transfer invariant structures from the original manifold to the twist, with key formulas governing the exterior derivative of forms on the twisted space.
• The framework leverages the fact that the curvature F is of type (1,1) for the complex structure I, ensuring integrability of the twisted almost complex structure.
• For the c-map example, the construction starts from a projective special Kähler manifold (e.g., ℝH(2)), lifts to a cone metric with flat symplectic connection, and constructs a hyperKähler metric on the cotangent bundle.
• The twist is performed using the lifted rotating symmetry, with the twist function chosen as -t²/2 + c, leading to a globally defined, complete quaternionic Kähler metric on the twisted space.

Experimental results

Research questions

• RQ1 What is the minimal set of data required to define the hyperKähler-quaternionic Kähler correspondence, and is it unique?
• RQ2 How can the twist construction be combined with elementary deformations to preserve the geometric structures needed for the correspondence?
• RQ3 Can the c-map construction be fully realized and verified using the twist and deformation framework, particularly in the case of non-compact symmetric spaces?
• RQ4 What is the global structure of the resulting quaternionic Kähler metric after the twist, and does it admit a Lie group structure?
• RQ5 Under what conditions does the twist produce a complete, positive-definite metric, and how can this be confirmed via structure constants?

Key findings

• The hyperKähler-quaternionic Kähler correspondence is shown to have only one degree of freedom, establishing a uniqueness result for the construction.
• The twist of a hyperKähler manifold with rotating circle symmetry, combined with elementary deformation, yields a quaternionic Kähler metric, and this construction is unique up to the choice of twist function.
• For the rigid c-map example with λ² = 4, the resulting twisted metric is isometric to the symmetric space Gr₂(ℂ²²), realized as a left-invariant metric on a solvable Lie group.
• For λ² = 4/3, the universal cover of the twisted space is identified as G₂*/SO(4), confirming the metric structure of the corresponding symmetric space.
• The resulting quaternionic Kähler metric on the twisted space is complete, positive definite, and has constant coefficients in a coframe H-related to the original hyperKähler structure.
• The structure functions of the H-related coframe are constant, confirming that the twisted space carries a Lie group structure with a left-invariant metric.