Double Field Theory

TL;DR

Double Field Theory provides a gauge-invariant framework for massless closed string fields on a doubled torus by treating momentum- and winding-related zero modes on equal footing. The authors derive a quadratic action from closed string field theory and extend it to cubic order, revealing a novel doubled-diffeomorphism symmetry and a central constraint $oxed{ riangle=0}$ that enforces level matching while preserving T-duality as a geometric action of $O(d,d;Z)$. A key result is that the massless sector necessarily contains the metric, Kalb-Ramond field, and dilaton, and their interactions exhibit intricate couplings through dual derivatives, while the action remains consistent under generalized Buscher-type dualities and a discrete $B o -B$ symmetry. The paper argues that the doubled geometry is physical, not merely auxiliary, and outlines how to relate the doubled framework to conventional and dual backgrounds via field redefinitions and projection operators, though a full nonlinear theory remains challenging. Overall, this work establishes a concrete, two-derivative, gauge-invariant starting point for a doubled-field description of stringy geometry with promising ties to generalized geometry and future nonperturbative insights.

Abstract

The zero modes of closed strings on a torus --the torus coordinates plus dual coordinates conjugate to winding number-- parameterize a doubled torus. In closed string field theory, the string field depends on all zero-modes and so can be expanded to give an infinite set of fields on the doubled torus. We use the string field theory to construct a theory of massless fields on the doubled torus. Key to the consistency is a constraint on fields and gauge parameters that arises from the L_0 - \bar L_0=0 condition in closed string theory. The symmetry of this double field theory includes usual and 'dual diffeomorphisms', together with a T-duality acting on fields that have explicit dependence on the torus coordinates and the dual coordinates. We find that, along with gravity, a Kalb-Ramond field and a dilaton must be added to support both usual and dual diffeomorphisms. We construct a fully consistent and gauge invariant action on the doubled torus to cubic order in the fields. We discuss the challenges involved in the construction of the full nonlinear theory. We emphasize that the doubled geometry is physical and the dual dimensions should not be viewed as an auxiliary structure or a gauge artifact.