Grassmannians,Calibrations and Five-Brane Intersections

TL;DR

This paper constructs new hyper-Kahler and Kahler geometries with torsion using quaternionic calibrations, explores their moduli spaces, and applies these geometries to supergravity solutions and supersymmetric sigma models.

Contribution

It introduces a novel geometric construction of hyper-Kahler and Kahler manifolds with torsion via quaternionic calibrations and analyzes their applications in string theory and supergravity.

Findings

New hyper-Kahler manifolds with torsion constructed

Found moduli spaces using quaternionic line bundles

Presented intersecting NS-5-brane solutions in supergravity

Abstract

We present a geometric construction of a new class of hyper-Kahler manifolds with torsion. This involves the superposition of the four-dimensional hyper-Kahler geometry with torsion associated with the NS-5-brane along quaternionic planes in quaternionic k-space, $\bH^k$. 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 $\bH^k$. 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-Kahler geometry with torsion along complex planes in $\bC^{2k}$. We find that the resulting geometry is Kahler 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, 1/8, 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.