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**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.

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.

Reviewer: Vicente Cortés (Bonn)

### 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.) |