A Geometry for Non-Geometric String Backgrounds

TL;DR

This paper introduces a geometric framework for understanding non-geometric string backgrounds using doubled geometry, where T-duality transitions are incorporated into an enlarged space, revealing how spacetime emerges locally from this structure.

Contribution

It develops a geometric description of non-geometric backgrounds via doubled space, extending traditional geometry to include T-duality as a transition function.

Findings

Non-geometric backgrounds arise from T-duals and flux compactifications.

Spacetime in non-geometric backgrounds is locally defined within a doubled space.

Generalized D-branes wrap subspaces in the doubled geometry.

Abstract

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.