Hyper-K\"ahler with Torsion, T-duality, and Defect (p,q) Five-branes

We investigate a five-branes interpretation of hyper-K"{a}hler geometry with torsion (HKT). This geometry is obtained by conformal transformation of the Taub-NUT space which represents a Kaluza-Klein five-brane. This HKT would represent an NS5-brane on the Taub-NUT space. In order to explore the HKT further, we compactify one transverse direction, and study the $O(2,2;{\mathbb Z}) = SL(2,{\mathbb Z}) \times SL(2,{\mathbb Z})$ monodromy structure associated with two-torus. Performing the conjugate transformation, we obtain a new solution whose physical interpretation is a defect $(p,q)$ five-brane on the ALG space. Throughout this analysis, we understand that the HKT represents a coexistent state of two kinds of five-branes. This situation is different from composite states such as $(p,q)$ five-branes or $(p,q)$ seven-branes in type IIB theory. We also study the T-dualized system of the HKT. We again find a new solution which also indicates another defect $(p,q)$ five-brane on the ALG space.