Gravitational Core of Double Field Theory: Lecture Notes
Jeong-Hyuck Park
TL;DR
This work presents Double Field Theory (DFT) as an O$(D,D)$-symmetric framework for gravity rooted in string theory, introducing doubled coordinates, the section condition, and a generalized diffeomorphism structure. It builds the theory from a minimal set of geometric ingredients—the generalized metric ${\cal H}_{AB}$ and the dilaton $d$—and introduces a master derivative with compatible Christoffel and spin connections, followed by a projection-based route to full covariance. The paper derives the Einstein Double Field Equation $G_{AB}=T_{AB}$, discusses matter couplings in the string frame, and analyzes key solutions: a three-parameter spherically symmetric vacuum generalizing Schwarzschild and an open-Universe cosmology that aligns with late-time cosmological data while avoiding de Sitter. It demonstrates how non-Riemannian geometries, open-string effects, and the dilaton flux contribute rich, testable phenomenology, including solar-system consistency under suitable conditions and cosmological acceleration driven by the dilaton in string frame. Overall, DFT provides a rigorous, falsifiable gravity framework emerging from string theory with clear predictions and avenues for observational and theoretical exploration.
Abstract
Double Field Theory (DFT) has emerged as a comprehensive framework for gravity, presenting a testable and robust alternative to General Relativity (GR), rooted in the $\mathbf{O}(D,D)$ symmetry principle of string theory. These lecture notes aim to provide an accessible introduction to DFT, structured in a manner similar to traditional GR courses. Key topics include doubled-yet-gauged coordinates, Riemannian versus non-Riemannian parametrisations of fundamental fields, covariant derivatives, curvatures, and the $\mathbf{O}(D,D)$-symmetric augmentation of the Einstein field equation, identified as the unified field equation for the closed string massless sector. By offering a novel perspective, DFT addresses unresolved questions in GR and enables the exploration of diverse physical phenomena, paving the way for significant future research.