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The internal space of a N=4 supersymmetric model with Wess-Zumino term has a connection with totally skew-symmetric torsion and holonomy in Sp(n). We study the mathematical background of this type of connections. In particular, we relate it to classical Hermitian geometry, construct homogeneous as well as inhomogeneous examples, characterize it in terms of holomorphic data, develop its potential theory and reduction theory.

In this paper we provide examples of hypercomplex manifolds which do not carry HKT structure, thus answering a question in [16]. We also prove that the existence of HKT structure is not stable under small deformations. Similarly we provide examples of compact complex manifolds with vanishing first Chern class which do not admit a Hermitian structure with restricted holonomy of its Bismut connection in SU(n), thus providing a counter-example of the conjecture in [18]. Again we prove that such property is not stable under small deformations.

We find sufficient conditions for principal toric bundles over compact Kähler manifolds to admit Calabi-Yau connections with torsion as well as conditions to admit strong Kähler connections with torsion. With the aid of a topological classification, we construct such geometry on (k − 1)(S 2 × S 4 )#k(S 3 × S 3 ) for all k ≥ 1.

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