The Spacetime of Double Field Theory: Review, Remarks, and Outlook (1309.2977v3)

Abstract: We review double field theory (DFT) with emphasis on the doubled spacetime and its generalized coordinate transformations, which unify diffeomorphisms and b-field gauge transformations. We illustrate how the composition of generalized coordinate transformations fails to associate. Moreover, in dimensional reduction, the O(d,d) T-duality transformations of fields can be obtained as generalized diffeomorphisms. Restricted to a half-dimensional subspace, DFT includes `generalized geometry', but is more general in that local patches of the doubled space may be glued together with generalized coordinate transformations. Indeed, we show that for certain T-fold backgrounds with non-geometric fluxes, there are generalized coordinate transformations that induce, as gauge symmetries of DFT, the requisite O(d,d;Z) monodromy transformations. Finally we review recent results on the \alpha' extension of DFT which, reduced to the half-dimensional subspace, yields intriguing modifications of the basic structures of generalized geometry.

• The paper introduces a doubled spacetime framework that unifies diffeomorphisms and b-field gauge transformations via generalized coordinate transformations.
• It details how DFT addresses non-geometric backgrounds using T-duality and discusses possible relaxations of the strong constraint for broader applications.
• The review examines the inclusion of α′ corrections and their impact on refining the generalized geometric structure and advancing string theory compactifications.

The Spacetime of Double Field Theory: Review, Remarks, and Outlook

The paper provides a comprehensive overview of Double Field Theory (DFT), focusing on its underlying geometry, symmetries, and implications for theoretical and practical developments in string theory. DFT is an extension of conventional field theories that incorporates the additional winding modes of string theory, leading to a doubled formulation of spacetime. At its core, DFT unifies diffeomorphisms and $b$-field gauge transformations into the framework of generalized coordinate transformations. This approach reflects the inherent symmetry of string theory's target space, most notably the $O(d,d)$ T-duality group that emerges on toroidal backgrounds.

Overview of Key Concepts in DFT

DFT extends the notion of ordinary spacetime to a doubled space of dimension $2D$, where each spatial dimension is accompanied by a dual winding coordinate. The basic dynamical entities include the generalized metric ${\cal H}_{MN}$ and the dilaton $d$, which are invariant under the $O(D,D)$ T-duality group transformations. The theory is constructed such that ${\cal H}_{MN}$ encapsulates both the traditional metric $g_{ij}$ and the antisymmetric $b$-field, linking them through Buscher's T-duality transformations.

Generalized Coordinate Transformations: These transformations are at the heart of DFT, serving as symmetries that blend diffeomorphisms with $b$-field gauge invariance. The closure of these transformations is governed by the Courant bracket, which, however, does not possess the explicit associativity seen in classical geometry, leading to non-associative coordinate transformations under successive applications. This unusual structure emphasizes the departure of DFT from classical field theories and the necessity of a generalized geometric framework.

Major Results and Implications

Non-Geometric Backgrounds: One of the primary applications of DFT is in addressing 'non-geometric' backgrounds, which are not fully describable by traditional geometry. These include T-folds, which require patching by generalized coordinate transformations including T-duality flips. These transformations ensure global consistency in spacetime descriptions, leading to new insights into the landscape of string compactifications.

Relaxation of the Strong Constraint: While DFT traditionally enforces strong constraints—nullifying $\partial^M \Phi \partial_M \Psi$ for all fields $\Phi$ and $\Psi$—it has been hypothesized that these constraints can be partially relaxed, particularly in the context of low-energy effective actions, or in string field theory where level-matching constraints suffice. This opens avenues to explore more general backgrounds like non-geometric fluxes in string compactifications, enriching the framework's applicability.

Inclusion of $\alpha'$ Corrections: Recent extensions of DFT have incorporated stringy $\alpha'$ corrections, providing a more complete formulation that remains $O(D,D)$ covariant. These corrections manifest as modifications to generalized geometric structures, suggesting profound effects on the underlying geometry. The inclusion of $\alpha'$ corrections also leads to a "doubled $\alpha'$-geometry," refining the traditional spacetime perspectives and potentially impacting duality symmetries.

Speculations on Future Developments

The research opens several potential directions for advancing both the theoretical formalisms of string theory and their practical applications. DFT's framework could be a seminal stepping stone for exploring the full gamut of non-geometric backgrounds that are predicted by string dualities. Moreover, its ability to gracefully integrate $\alpha'$ corrections promises insights into quantum aspects of gravity and manifolds with inherent non-commutative features.

The possibility of further relaxing the strong constraint, especially in generalized reduction schemes like those involving Scherk-Schwarz reductions, suggests a unitarily complete treatment of supergravity theories akin to the consistent gauged supergravities. This could lead to an enhanced understanding of U-duality symmetries and the compactification of higher-dimensional theories such as M-theory.

In summary, DFT's introduction of doubled geometry, enriched symmetry structures, and unification of field theories marks a significant advancement in exploring the rich tapestry of string and M-theory landscapes. Future research on DFT could potentially unravel further layers of the structure of spacetime, offering new horizons for high-energy theory and its unification with quantum mechanics.