7 problems

Hypercomplex HKT or bi-HKT conjecture. Any hypercomplex manifold is either HKT or bi-HKT.

Let $(M^{4n},I,J,K,\Omega)$ be a compact HKT-manifold, where $I,J,K$ are the quaternionic complex structures and $\Omega$ is its HKT-form. Let $F\in C^\infty(M,\mathbb{R})$ be a sm…

The model is an ${\cal N}=4$ HKT sigma model formulated in harmonic superspace in terms of original $q^{+}$ superfields. A linear harmonic constraint is a harmonic-superspace const…

A hyper-Kähler with torsion (HKT) metric is a Riemannian metric with the geometric structure described in the paper. The harmonic action with the constraints … produces an HKT geom…

Nonlinear multiplet conjecture. Models described by nonlinear multiplets describe bi-HKT and bi-Kähler manifolds, but with nonzero Obata connections.

Converse conjecture. Every Obata-flat HKT manifold can be described with linear $({\bf 4},{\bf 4},{\bf 0})$ multiplets.

Quaternionic Calabi–Yau conjecture. On a compact HKT manifold with $SL(n,\mathbb H)$-holonomy of the Obata connection, there exists a balanced HKT metric; if it exists, it is uniqu…