[Paper Review] Homogeneous HKT and QKT manifolds

A. Opfermann, George Papadopoulos|ArXiv.org|Jul 24, 1998

Geometry and complex manifolds6 references19 citations

TL;DR

This paper constructs a large class of homogeneous KT, HKT, and QKT manifolds using invariant metrics and the canonical connection on homogeneous spaces $G/K$, leveraging a decomposition of the Lie algebra $\mathfrak{g}$ via colorings of Dynkin diagrams. The key contribution is a systematic classification of such spaces, including the discovery that the twistor spaces of homogeneous QKT manifolds admit KT structures.

ABSTRACT

We present the construction of a large class of homogeneous KT, HKT and QKT manifolds, $G/K$, using an invariant metric on $G$ and the canonical connection. For this a decomposition of the Lie algebra of $G$ is employed, which is most easily described in terms of colourings of Dynkin diagrams of simple Lie algebras. KT structures on homogeneous spaces are associated with different colourings of Dynkin diagrams. The colourings which give rise to HKT structures are found using extended Dynkin diagrams. We also construct homogeneous QKT manifolds from homogeneous HKT manifolds and show that their twistor spaces admit a KT structure. Many examples of homogeneous KT, HKT and QKT spaces are given.

Motivation & Objective

• To systematically construct homogeneous KT, HKT, and QKT manifolds using invariant geometry on $G/K$.
• To classify these structures via colorings of Dynkin diagrams of simple Lie algebras.
• To establish a correspondence between specific colorings and the existence of HKT or QKT structures.
• To demonstrate that the twistor spaces of homogeneous QKT manifolds admit KT structures.
• To extend known results on group manifolds to broader classes of homogeneous spaces with torsion-compatible connections.

Proposed method

• Utilize an invariant metric on a semi-simple, compact Lie group $G$ to define a canonical connection on $G/K$.
• Employ a decomposition of the Lie algebra $\mathfrak{g}$ into subspaces associated with root spaces, guided by root system data.
• Use colorings of Dynkin diagrams to encode the decomposition and classify possible KT, HKT, and QKT structures.
• Apply extended Dynkin diagrams to identify colorings that yield HKT structures.
• Construct QKT structures from HKT manifolds via a fibration involving $\Phi(U(2))$ and $U(1)$-twisting.
• Analyze the exterior derivative of the torsion form $H$ to determine closure properties and relate them to curvature traces.

Experimental results

Research questions

• RQ1 Which colorings of Dynkin diagrams correspond to KT, HKT, or QKT structures on $G/K$?
• RQ2 How can the canonical connection and invariant metric on $G$ be used to construct homogeneous HKT and QKT manifolds?
• RQ3 What is the structure of the twistor space of a homogeneous QKT manifold, and does it admit a KT structure?
• RQ4 Under what conditions is the torsion $H$ closed (strong structure) or non-closed (weak structure) in these geometries?
• RQ5 Can the fibration structure of QKT manifolds be generalized to non-homogeneous cases, and what are the implications for string theory?

Key findings

• A large class of homogeneous KT manifolds is constructed from complex homogeneous spaces via Dynkin diagram colorings.
• HKT structures on $G/K$ arise precisely from colorings of extended Dynkin diagrams, providing a complete classification method.
• The twistor space of a homogeneous QKT manifold admits a KT structure, generalizing the twistor construction for QK manifolds.
• For four-dimensional QKT manifolds, the torsion $H$ is closed only if the curvature is proportional to the $Sp(1)$-connection, implying vanishing torsion unless $d=1$.
• The exterior derivative of the torsion $\mathrm{d}H$ is shown to be proportional to the trace of the square of the canonical connection's curvature.
• Eight-dimensional homogeneous HKT manifolds are often of the form $M_{(7)} \times U(1)$, where $M_{(7)}$ are Freud-Rubin spaces, linking them to known Einstein manifolds.