Double Field Theory

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.

51 pages. v2: Corrected typo in eqn. (2.48) and very minor typos elsewhere. Added ref. [9] to M. Van Raamsdonk

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• Weaving the Exotic Web
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• From Exceptional Field Theory to Heterotic Double Field Theory via K3
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• The Classical Double Copy for Half-Maximal Supergravities and T-duality
• Symmetries and Covering Maps for the Minimal Tension String on $\mathbf{Ad}{\mathbf{S}}_3\times {\mathbf{S}}^3\times {\mathbf{T}}^4$
• Introductory Lectures on Extended Supergravities and Gaugings
• Exotic Branes in Exceptional Field Theory: $E_{7(7)}$ and Beyond
• Unwinding strings in semi-flatland
• The Worldsheet Perspective of T-duality Symmetry in String Theory
• Noncommutative gauge theories on D-branes in non-geometric backgrounds
• Point Particle Motion in DFT and a Singularity-Free Cosmological Solution
• T-duality equivalences beyond string theory
• Cosmological Implications from O(D,D)
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• Quantum Gravity Phenomenology in the Infrared
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• A note on large gauge transformations in double field theory
• Solution of the Size and Horizon Problems from Classical String Geometry
• Ten-dimensional origin of Minkowski vacua in $\mathcal{N}=8$ supergravity
• Yang-Baxter deformations and generalized supergravity -- A short summary
• Higher Gauge Structures in Double and Exceptional Field Theory
• Doubled Aspects of Vaisman Algebroid and Gauge Symmetry in Double Field Theory
• $L_{\infty }$ algebras and Tensor Hierarchies in Exceptional Field Theory and Gauged Supergravity
• Aspects of the Doubled Worldsheet
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• Twistors and supertwistors for exceptional field theory
• Deformed Weitzenböck Connections and Double Field Theory
• The different faces of branes in Double Field Theory
• Surface terms in effective action of O-plane at order ${\alpha }^{\prime 2}$
• T-duality constraint on R-R couplings
• 2D Black Holes, Bianchi I Cosmologies, and ${\alpha }^{\prime }$
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• Twisted C-brackets
• The Doubled Geometry of Nilmanifold Reductions
• Causal structures and dynamics of black-hole-like solutions in string theory
• A note on Faddeev--Popov action for doubled-yet-gauged particle and graded Poisson geometry
• On a class of conformal $\mathcal{E}$-models and their chiral Poisson algebras
• String Excitation by Initial Singularity of Inflation
• Courant bracket twisted both by a 2-form $B$ and by a bi-vector $\theta$
• A Simple Model of Double Dynamics on Lie Groups
• On section conditions of E7(7) exceptional field theory and superparticle in N=8 central charge superspace
• Long Strings and Quasinormal Winding Modes
• Gauged Supergravities and the Physics of Extra Dimensions
• Geometric Low-Energy Effective Action in a Doubled Spacetime
• Symmetry enhancement interpolation, non-commutativity and Double Field Theory
• Reviving 3D $\mathcal{N}=8$ superconformal field theories
• Geometry of ${\mathbb{R}}^{+}\times E_{3(3)}$ Exceptional Field Theory and F-theory
• Rainbow Spacetime from a Nonlocal Gravitational Uncertainty Principle
• Towards an extended/higher correspondence -- Generalised geometry, bundle gerbes and global Double Field Theory
• Metric Algebroid and Poisson-Lie T-duality in DFT
• The teleparallel complex
• Higher Spin Double Field Theory : A Proposal
• Extended Drinfel'd algebras and non-Abelian duality
• The Algebroid Structure of Double Field Theory
• Loop Corrections in Double Field Theory: Non-trivial Dilaton Potentials
• From geometry to non-geometry via T-duality
• An invitation to Poisson-Lie T-duality in Double Field Theory and its applications
• Higher-derivative Heterotic Kerr-Sen Black Holes
• $L_{\infty }$ algebras for extended geometry
• Quantum Klein Space and Superspace
• Duality Constraints on String Theory: Instantons and spectral networks
• Poisson-Lie T-plurality for WZW backgrounds
• Classical worldvolumes as generalised geodesics
• T-folds as Poisson-Lie plurals
• Solution-generating transformations in duality-invariant theories and the fluid/gravity correspondence
• Poisson-Lie T-plurality for dressing cosets
• Jacobi-Lie Models and Supergravity Equations
• Duality invariant string beta functions at two loops
• Notes on Ramond-Ramond spinors and bispinors in double field theory
• Perturbations in $\mathbf{O}(D,D)$ string cosmology from double field theory
• $Ad{S}_3\times S^3$ Background from Poisson-Lie T-Duality
• Equivalence of open/closed strings
• The puzzle of global Double Field Theory: open problems and the case for a Higher Kaluza-Klein perspective
• Extended Doubled Structures of Algebroids for Gauged Double Field Theory
• Generalised Geometry and Flux Vacua
• A class of non-geometric M-theory compactification backgrounds
• Gravitational Core of Double Field Theory: Lecture Notes
• Towards quantum Mechanics on the curved cotangent bundle
• Stability Analysis of the Cosmological Dynamics of $O(D,D)$-complete Stringy Gravity
• Double Field Theory on Group Manifolds in a Nutshell
• Late-time Cosmology without Dark Sector but with Closed String Massless Sector
• Perturbative quantum gravity in double field theory
• Dilatonic Effect in Double Field Theory Cosmology
• Gauge Five-brane Solutions of Co-dimension Two in Heterotic Supergravity
• World-volume Effective Theories of Locally Non-geometric Branes
• Open-String Non-Associativity in an R-flux Background
• Exotic Branes from del Pezzo Surfaces
• T-duality of D-brane versus O-plane actions
• C-spaces, generalized geometry and double field theory
• Renormalization of a Lorentz invariant doubled worldsheet theory
• Pin(d,d) covariance of pure spinor equations for supersymmetric vacua and Non-Abelian T-duality
• Supergravity solution-generating techniques and canonical transformations of $\sigma$-models from $O(D,D)$
• Double Field Theory and Pseudo-Supersymmetry
• $O(d,d;\mathbb{Z})$ invariant Fierz-Pauli massive gravity
• Generalized symmetries as homotopy Lie algebras
• $\mathcal{N}=2$ ${\text{CFT}}_3\text{’s}$ from $\mathcal{N}=4$ gauged supergravity
• Aspects of the T-duality construction for the Supermembrane theory
• Black hole solutions in double field theory
• Plane-parallel waves as Jacobi-Lie models
• Quantum Theory, Gravity and Second Order Geometry
• Integrable deformed H${}_{{}_4}$ WZW models and their non-Abelian duals as solutions of generalized supergravity equations
• Dualising consistent truncations
• Lagrangian formulation for electric charge in a magnetic monopole distribution
• Quantum Gravity Corrections to the Mean Field Theory of Nucleons
• The Duality Symmetric String at Two-loops
• Pre-NQ Manifolds and Correspondence Spaces: the Nilmanifold Example
• Path Integral in Modular Space
• Born Geometry in a Nutshell
• Flux Backgrounds and Exceptional Generalised Geometry
• Duality Covariant Solutions in Extended Field Theories
• Higher time derivatives in the microcanonical ensemble describe dynamics of flux-coupled classical and quantum oscillators
• Dimensional Reductions of DFT and Mirror Symmetry for Calabi-Yau Three-folds and $K3\times T^2$
• Double Field Theory for Generalized $\lambda$-deformation
• Interactions of strings on a T-fold
• $L_{\infty }$-algebras and membrane sigma models
• Exotic Aspects of Extended Field Theories
• On global aspects of duality invariant theories: M2-brane vs DFT
• A T-Fold Black Hole in Doubled Type IIB
• Lagrangian and orthogonal splittings, quasitriangular Lie bialgebras and almost complex product structures
• Non-invertible symmetries of two-dimensional Non-Linear Sigma Models
• The Two-Parameter Brane Sigma-Model and Extended Solutions of M-Theory
• Gaugings and other aspects in supergravity
• On spectator dependence of Jacobi-Lie T-plurality
• T-duality on Almost Hermitian Spaces
• Strings and D-branes in curved space-time
• Backgrounds in Boundary String Field Theory
• Gauged Double Field Theory as an $L_{\infty }$ algebra
• NS-NS Spacetimes from Amplitudes
• Luis Santaló and classical field theory
• Doubled space and extended supersymmetry
• Half-maximal Extended Drinfel'd Algebras
• Localized Kaluza-Klein 6-brane
• Extended Field Theories as higher Kaluza-Klein theories
• Duality rules for more mixed-symmetry potentials
• Higher dimensional Lie algebroid sigma model with WZ term
• Low Energy Consequences of Loop Quantum Gravity
• T-symmetry in String Geometry Theory
• Algebraic structures in exceptional geometry
• Geometric Approaches to Quantum Fields and Strings at Strong Couplings
• Duality covariant field redefinitions
• More Stringy Effects in Target Space from Double Field Theory