## Geometry of hyper-Kähler connections with torsion.(English)Zbl 0993.53016

A hypercomplex structure $$H = (J_\alpha)$$ on a manifold $$M$$ is given by 3 complex structures $$J_1, J_2$$ and $$J_3$$ such that $$J_3 = J_1J_2$$. The pair $$(M,H)$$ is called a hypercomplex manifold. A hyper-Hermitian manifold $$(M,H,g)$$ is a hypercomplex manifold $$(M,H)$$ together with a Riemannian metric $$g$$ which is Hermitian with respect to the 3 complex structures $$J_\alpha$$. It is called an HKT-manifold (hyper-Kähler manifold with torsion) if there exists an HKT-connection on $$M$$: a connection $$\nabla$$ preserving the hyper-Hermitian structure and which has totally skew-symmetric torsion $$T$$, more precisely, $$g(T(\cdot , \cdot), \cdot)$$ is a 3-form. The special case $$T=0$$ corresponds to ordinary hyper-Kähler manifolds (HK-manifolds). In that case $$\nabla$$ is the Levi-Civita connection of $$g$$.
HKT-manifolds occur in physics, e.g. as allowed targets for N=4 nonlinear $$\sigma$$-models with Wess-Zumino-Witten-term in 2 and 3 spacetime dimensions.
The mathematical paper under review is a systematic study of HKT-manifolds. It was proven by Gauduchon that every Hermitian manifold $$(M,J,g)$$ admits a unique Hermitian connection with totally skew-symmetric torsion (KT-connection) and that the torsion can be easily computed in terms of $$J$$ and $$g$$. Using this fact the authors first observe that the HKT-connection on an HKT-manifold is unique and give a necessary and sufficient condition for the existence of an HKT-connection on a hyper-Hermitian manifold. Then they study the problem whether a given hypercomplex manifold admits a compatible HKT-structure and derive some sufficient condition in terms of the existence of an appropriate $$\partial$$-closed $$(2,0)$$-form (In the HK-case the form is $$d$$-closed and hence holomorphic). Using this condition Joyce’s twist construction of hypercomplex manifolds is implemented for HKT-manifolds. Also, based on work of Howe/Papadopoulos, the following general constructions of hypercomplex and HK-geometry are shown to have HKT-versions: twistor construction (S. Salamon, Hitchin/Karlhede/Lindström/Roček, Joyce), reduction (HKLR, Joyce) and the notion of HK-potential (HKLR). The twistor construction is non-holomorphic in the sense that it involves the canonical non-integrable almost complex structure on the twistor space. Potentials and reduction are well suited for the construction of examples, as demonstrated in the paper.
The following examples are discussed: HKT-structure on Joyce’s homogeneous hypercomplex manifolds associated to compact semisimple groups (Opfermann/Papadopoulos), on compact hypercomplex nilmanifolds constructed by Barberis/Dotti-Miatello, on Pedersen/Poon’s (generically inhomogeneous) hypercomplex manifolds $$S^1\times S^{4n-1}$$, on $$S^1\times SU(3)/U(1)$$ and, finally, on the Swann bundle (considered only as hypercomplex manifold) of any quaternionic Kähler manifold. It was proven by Pedersen/Poon/Swann that the Swann bundle of any quaternionic manifold (without assuming a quaternionic Kähler metric) is a hypercomplex manifold. One should expect that it has a compatible HKT-structure. However the authors’ construction does not apply since there is no potential available. As pointed out by the authors, the problem of existence of a compatible HKT-structure on an arbitrary hypercomplex manifold remains open.

### MSC:

 53C26 Hyper-Kähler and quaternionic Kähler geometry, “special” geometry 81T60 Supersymmetric field theories in quantum mechanics 53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills) 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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