A Geometry for Non-Geometric String Backgrounds

TL;DR

This work develops a duality-invariant description of string backgrounds that are not globally geometric by doubling the internal torus to a $T^{2n}$ fiber and enforcing a self-duality constraint, so T-duality acts as a geometric transformation on the enlarged space. A polarization selects a physical $T^n$ within $T^{2n}$, and T-duality corresponds to changing this polarization, thereby relating geometrically distinct descriptions of the same CFT; geometric backgrounds admit a global polarization while non-geometric backgrounds (T-folds) do not. The formalism is illustrated in the $n=1$ case, shows Buscher T-duality arises from the duality rules, and is extended to open strings and D-branes, where D-brane configurations transform nontrivially under dualities. The framework provides a concrete, manifestly duality-covariant method to analyze non-geometric fluxes, duality twists, and the emergence of spacetime from a doubled geometry, with potential extensions to U-duality and M-theory.

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.