Grassmannians , Calibrations and Five - Brane Intersections

We present a geometric construction of a new class of hyper-Kähler manifolds with torsion. This involves the superposition of the four-dimensional hyper-Kähler geometry with torsion associated with the NS-5-brane along quaternionic planes in H k. We find the moduli space of these geometries and show that it can be constructed using the bundle space of the canonical quaternionic line bundle over a quaternionic projective space. We also investigate several special cases which are associated with certain classes of quaternionic planes in H k. We then show that the eight-dimensional geometries we have found can be constructed using quaternionic calibrations. We generalize our construction to superpose the same four-dimensional hyper-Kähler geometry with torsion along complex planes in C 2k. We find that the resulting geometry is Kähler with torsion. The moduli space of these geometries is also investigated. In addition the applications of these new geometries to M-theory and sigma models are presented. In particular, we find new solutions of IIA supergravity with the interpretation of intersecting NS-5-branes at Sp(2)-angles on a string and show that they preserve 3/32, 5/32 and 3/16 of supersymmetry. We also show that two-dimensional sigma models with target spaces the above manifolds have (p,q) extended supersymmetry.