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 _i ^D(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 _a ^N(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 _a ^Nand A _i ^Dare T-dual to ^★ A _D ^aand ^★ A _N ⁱrespectively. We introduce the field strength of T-dual non-geometric theories as derivatives of T-dual gauge fields along both T-dual variable y _μand its double ȳ _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.