On homogeneous HKT manifolds and the Einstein condition

Abstract: We consider homogeneous hypercomplex manifolds with a transitive action of a compact Lie group and we give a characterization of invariant HKT metrics on them. On every such hypercomplex manifold we prove the existence of an invariant HKT-Einstein metric, which is unique up to scaling. Furthermore, we determine for which invariant HKT metrics the torsion and the curvature of the Bismut connection are Bismut-parallel, showing that invariant strong HKT metrics have this property.

• The paper establishes a complete characterization of invariant HKT metrics via Lie algebra decompositions on Joyce hypercomplex manifolds.
• It proves the existence and uniqueness (up to scaling) of G-invariant HKT-Einstein metrics by linking scaling factors to Lie-theoretic invariants.
• It demonstrates that strong HKT metrics exhibit Bismut-parallel torsion and curvature, which restricts the geometry to specific homogeneous structures.

Homogeneous HKT Manifolds and the Einstein Condition

Introduction and Motivation

The paper "On homogeneous HKT manifolds and the Einstein condition" (2604.22404) investigates the geometry of hypercomplex manifolds with transitive compact Lie group actions, specifically focusing on invariant hyperkähler metrics with torsion (HKT metrics). The principal aim is to systematically characterize invariant HKT metrics on Joyce hypercomplex manifolds, to establish existence and uniqueness results for HKT-Einstein metrics in this category, and to analyze when invariant HKT metrics exhibit Bismut-parallel torsion and curvature. The work leverages advanced Lie-theoretic constructions to extend results analogous to classical Kahler-Einstein theory to the hypercomplex, torsion-including setting.

Structure and Properties of Joyce Hypercomplex Manifolds

The paper considers manifolds $M = G/L$, where $G$ is a compact Lie group, $L$ is a closed subgroup, and $M$ carries a $G$-invariant hypercomplex structure constructed following Joyce's framework. The authors employ a decomposition of the Lie algebra $\mathfrak{g}$, parameterized by a set of strongly orthogonal roots, to build hypercomplex structures on $M$:

$$\mathfrak{g} = \mathfrak{l} \oplus \mathfrak{m},\quad\mathfrak{m} = \bigoplus_{j=1}^m \mathfrak{m}_j$$

Each component $\mathfrak{m}_j$ admits an independent hypercomplex structure, and invariant HKT metrics correspond to scalar products that are diagonal on this decomposition.

Characterization and Construction of Invariant HKT Metrics

The authors provide a complete characterization of all $G$-invariant HKT metrics on Joyce hypercomplex manifolds: every such metric arises from a scalar product

$G$0

for some positive real numbers $G$1. The distinguished metric $G$2 is crafted by modifying the opposite of the Killing form in a manner compatible with the hypercomplex structure. This construction is general; invariant HKT metrics need not be naturally reductive, nor do they require restrictions on hypercomplex structures at each coset layer.

Existence and Uniqueness of HKT-Einstein Metrics

For Joyce hypercomplex manifolds, the HKT-Einstein condition is formulated via the Chern-Ricci form of the Hermitian pair $G$3:

$G$4

with $G$5 a smooth function. The paper rigorously proves that for every such manifold, there exists a unique (up to scaling) $G$6-invariant HKT-Einstein metric. This result parallels the classical uniqueness of invariant Kähler-Einstein metrics on homogeneous simply connected compact Kähler manifolds.

The explicit computation shows that the Einstein metric corresponds to selecting coefficients $G$7 equated with positive Lie-theoretic invariants $G$8 associated with strongly orthogonal roots, reproducing and generalizing earlier results for the Lie group case.

Bismut Connection: Parallelism of Torsion and Curvature

The paper conducts a meticulous analysis of the Bismut connection associated with invariant HKT metrics, focusing on the properties of parallel torsion (Bismut-Torsion-Parallel, BTP) and curvature. The main claims are:

• An invariant HKT metric exhibits Bismut-parallel torsion if and only if its restriction to each root system component is $G$9-invariant, which is equivalent to having equal scaling parameters $L$0 amongst components belonging to the same irreducible root system.
• If an HKT metric is strong (pluriclosed or SKT in all complex structures), then it is necessarily BTP and, further, exhibits parallel Bismut curvature.
• For simply connected Joyce hypercomplex manifolds admitting invariant strong HKT metrics, the only possibilities are products of odd-rank special unitary groups with bi-invariant metrics.

Strong HKT-Einstein metrics are rare and exist only when all scaling coefficients are equal, which by the Lie theoretic structure restricts possible manifolds severely.

Implications and Future Developments

The theoretical advancement presented clarifies the landscape of canonical metrics on homogeneous hypercomplex manifolds and exposes a robust parallel with classical homogeneous Kähler geometry, but in the presence of torsion and with richer Lie algebraic structure. The explicit algebraic recipe for constructing HKT-Einstein metrics provides immediate computational feasibility for invariant examples, and the characterization results may guide the search for new special metrics in string theory, mathematical physics, and Geometric Analysis.

On the side of geometric flows, the uniqueness and existence of invariant HKT-Einstein metrics situate them as stationary points of relevant geometric flows (cf. quaternionic Calabi-Yau). Moreover, the precise criteria for Bismut-parallel structures lay groundwork for classification and rigidity results in both geometric analysis and algebraic holonomy theory.

Future work could extend the classification to non-compact settings, explore moduli space structures of HKT metrics in more general homogeneous spaces, and probe the interactions with topological and holonomy aspects, particularly in relation to supersymmetric sigma models and their geometric backgrounds.

Conclusion

This paper establishes a comprehensive Lie-theoretic framework for homogeneous HKT geometry, elucidates the construction and uniqueness of invariant HKT-Einstein metrics, and provides explicit, verifiable criteria for Bismut-parallel torsion and curvature. These results constitute a substantial step forward in understanding canonical metrics in hypercomplex geometry, with concrete implications for differential geometry and mathematical physics, and serve as a foundation for further investigation into geometric flows, holonomy, and classification problems.