Double Field Theory: A Pedagogical Review

TL;DR

This work surveys Double Field Theory, a T-duality–invariant reformulation of the massless NS-NS sector of string theory through a doubled spacetime with an $O(D,D)$ structure. It develops the generalized diffeomorphism symmetry, the strong (section) constraint, and an invariant action expressed in terms of a generalized metric and dilaton, then connects these constructions to Scherk–Schwarz compactifications and gauged supergravities, including non-geometric fluxes. The review extends to U-duality via Extended Field Theory (EFT), worldsheet perspectives, and a broad set of developments such as non-geometry, finite gauge transformations, and potential implications for moduli stabilization and de Sitter vacua. Collectively, it frames DFT/ EFT as a string-inspired geometric paradigm that unifies geometric and non-geometric backgrounds and opens pathways beyond conventional supergravity. The material highlights both established results and open questions about relaxation of constraints, higher-derivative corrections, and the precise string-theoretic embedding of extended geometries.

Abstract

Double Field Theory (DFT) is a proposal to incorporate T-duality, a distinctive symmetry of string theory, as a symmetry of a field theory defined on a double configuration space. The aim of this review is to provide a pedagogical presentation of DFT and its applications. We first introduce some basic ideas on T-duality and supergravity in order to proceed to the construction of generalized diffeomorphisms and an invariant action on the double space. Steps towards the construction of a geometry on the double space are discussed. We then address generalized Scherk-Schwarz compactifications of DFT and their connection to gauged supergravity and flux compactifications. We also discuss U-duality extensions, and present a brief parcours on world-sheet approaches to DFT. Finally, we provide a summary of other developments and applications that are not discussed in detail in the review.

Figures (2)

• Figure 1: We picture the logic of DFT compactifications Aldazabal:2011nj. While standard SS reductions from supergravity in $D = d + n$ dimensions (solid line) give rise to gauged supergravity involving only geometric fluxes in $d$ dimensions, invoking duality arguments at the level of the effective action one can conjecture the need for dual fluxes stwacfi to complete all the deformations of gauged supergravity (waved line). More fundamentally, DFT is the $O(D,D)$-covariantization of supergravity (dashed line), and generalized SS compactifications of DFT give rise to gauged supergravities with all possible deformations (dotted line).
• Figure 2: We picture the space of gaugings (or fluxes) in half-maximal supergravities in $d=7,8$Dibitetto:2012rk. A point in this diagram corresponds to a given configuration. If two points lie in the same diagonal line (orbit) they are related by a duality transformation. Different theories are classified by orbits (lines) rather than configurations (points). The configuration space splits in a subgroup of geometric (i.e. only involving fluxes like $H_{abc}$ and $\omega_{ab}{}^{c}$) and non-geometric (involving fluxes $Q_a{}^{bc}$ and $R^{abc}$) configurations. The space of orbits then splits in two: (1) non-geometric orbits (truly half-maximal) and (2) geometric orbits (basically maximal) that intersect the geometric space (between A and B).