From geometry to non-geometry via T-duality

Abstract

Reconsideration of the T-duality of the open string allows us to introduce some geometric features in non-geometric theories. First, we have found what symmetry is T-dual to the local gauge transformations. It includes transformations of background fields but does not include transformations of the coordinates. According to this we have introduced a new, up to now missing term, with additional gauge field $A^D_i$ (D denotes components with Dirichlet boundary conditions). It compensates non-fulfilment of the invariance under such transformations on the end-points of an open string, and the standard gauge field $A^N_a$ (N denotes components with Neumann boundary conditions) compensates non-fulfilment of the gauge invariance. Using a generalized procedure we will perform T-duality of vector fields linear in coordinates. We show that gauge fields $A^N_a$ and $A^D_i$ are T-dual to ${}^\star A_D^a$ and ${}^\star A_N^i$ respectively. We introduce the field strength of T-dual non-geometric theories as derivatives of T-dual gauge fields along both T-dual variable $y_μ$ and its double ${\tilde y}_μ$. This definition allows us to obtain gauge transformation of non-geometric theories which leaves the T-dual field strength invariant. Therefore, we introduce some new features of non-geometric theories where field strength has both antisymmetric and symmetric parts. This allows us to define new kinds of truly non-geometric theories.