Background independent action for double field theory

TL;DR

This work presents a nonlinear, background‑independent action for double field theory on a doubled space with coordinates $(x^i,\tilde{x}_i)$ that is invariant under $O(D,D)$ and realizes the Courant (C‑bracket) gauge algebra. The action uses a background‑independent field ${\cal E}_{ij}$ and the dilaton $d$, together with calligraphic derivatives ${\cal D}_i$ and $\bar{\cal D}_i$, and it decomposes into standard NS‑NS, dual, and mixed sectors, reducing to the conventional NS‑NS theory when tilde derivatives are absent. An $O(D,D)$ geometric structure is developed via covariant derivatives and a curvature scalar ${\cal R}({\cal E},d)$, showing the action can be written as $S' = \int dxd\tilde{x}\, e^{-2d}{\cal R}$ and that the dilaton equation is ${\cal R}=0$. The strong constraint implies fields live on a maximal isotropic subspace, and, in toroidal reductions, some $O(d,d)$ dualities arise from gauge symmetry, connecting duality with geometry and hinting at extensions toward U‑duality and RR sectors. This framework advances a consistent, fully nonlinear, duality‑invariant formulation of the NS‑NS sector and provides a pathway toward a broader doubled‑geometry understanding of string theory.

Abstract

Double field theory describes a massless subsector of closed string theory with both momentum and winding excitations. The gauge algebra is governed by the Courant bracket in certain subsectors of this double field theory. We construct the associated nonlinear background-independent action that is T-duality invariant and realizes the Courant gauge algebra. The action is the sum of a standard action for gravity, antisymmetric tensor, and dilaton fields written with ordinary derivatives, a similar action for dual fields with dual derivatives, and a mixed term that is needed for gauge invariance.