A Geometry for Non-Geometric String Backgrounds

A geometric string solution has background fields in overlapping coordinate patches related by diffeomorphisms and gauge transformations, while for a non-geometric background this is generalised to allow transition functions involving duality transformations. Non-geometric string backgrounds arise from T-duals and mirrors of flux compactifications, from reductions with duality twists and from asymmetric orbifolds. Strings in ‘T-fold’ backgrounds with a local n-torus fibration and T-duality transition functions in O(n, n;Z) are formulated in an enlarged space with a T 2n fibration which is geometric, with spacetime emerging locally from a choice of a T n submanifold of each T 2n fibre, so that it is a subspace or brane embedded in the enlarged space. T-duality acts by changing to a different T n subspace of T 2n. For a geometric background, the local choices of T n fit together to give a spacetime which is a T n bundle, while for non-geometric string backgrounds they do not fit together to form a manifold. In such cases spacetime geometry only makes sense locally, and the global structure involves the doubled geometry. For open strings, generalised D-branes wrap a T n subspace of each T 2n fibre and the physical D-brane is the part of the part of the physical space lying in the generalised D-brane subspace.