T-duality relations between hyperkähler and bi-hypercomplex structures

Abstract

Abstract We investigate the T-duality relations between hyperkähler and bi-hypercomplex structures using the doubled formalism. In generalized geometry, both the hyperkähler and bi-hypercomplex structures are embedded in generalized hyperkähler structures that satisfy the split-bi-quaternion algebra. We write down the analogue of the Buscher rule, which is the T-duality transformation of the hyperkähler and bi-hypercomplex structures. As a practical example, we construct the bi-hypercomplex structure of the 5 2 2 -brane, known as a T-fold, from the hyperkähler structure of the Taub-NUT space using the T-duality transformation. The bi-hypercomplex structures of the T-fold have non-trivial monodromies. This results in the fact that the worldsheet instantons on the T-fold are multi-valued. We comment on the resolution of this issue using the Born sigma model.