HKT metric - Knowlpedia

Definition

Let $(M,I,J,K,g)$ be a hyper-Hermitian manifold, with fundamental two-forms $\omega_I,\omega_J,\omega_K$ . Set, in the left-action convention,

\Omega_I=\omega_J+i\omega_K\in\Omega_I^{2,0}(M).

The metric $g$ is an HKT metric—hyperkähler with torsion—if

\partial_I\Omega_I=0.

Connection characterization

Equivalently, the Hermitian Bismut connections of $(g,I)$ , $(g,J)$ , and $(g,K)$ coincide. Their common connection preserves the hypercomplex triple and has totally skew-symmetric torsion. This explains the name.

Relation to hyperkähler geometry

Every hyperkähler metric is HKT with zero torsion. An HKT metric need not be hyperkähler: the individual fundamental forms need not be closed, and the Levi-Civita connection need not preserve $I,J,K$ . Thus HKT is an intermediate differential condition on a hyper-Hermitian metric, not an alias for either neighboring structure.

Potentials

Every HKT metric admits local potentials. More precisely, locally there is a smooth strictly quaternionic plurisubharmonic function $u$ for which the metric is obtained from $\partial\partial_Ju$ ; conversely, such a function produces an HKT metric.

Sign convention

With the right-action convention of Alesker–Verbitsky, one writes $\Omega=\omega_J-i\omega_K$ . The sign changes with the action and fundamental-form conventions; the condition must be read together with the definition of $\Omega$ .

References

P. S. Howe and G. Papadopoulos, “Twistor spaces for hyper-Kähler manifolds with torsion,” Physics Letters B 379 (1996), 80–86. DOI record00442-8).
Semyon Alesker and Misha Verbitsky, “Plurisubharmonic functions on hypercomplex manifolds and HKT-geometry,” Journal of Geometric Analysis 16 (2006), 375–399. arXiv record.