Double Field Theory (0904.4664v2)

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.

• The paper introduces a doubled torus framework that explicitly incorporates T-duality as a fundamental symmetry in string theory.
• It employs a string field theory approach to integrate gravity, Kalb-Ramond fields, and dilatons within a consistent doubled geometry with key constraints.
• The work reveals a homotopy Lie algebra structure in gauge symmetries, setting the stage for future studies on non-linear extensions and non-geometric backgrounds.

A Study of Double Field Theory: Symmetries, Constraints, and T-Duality

The paper entitled "Double Field Theory" by Chris Hull and Barton Zwiebach provides a rigorous exploration into the formulation of a field theory that incorporates T-duality as an explicit symmetry. This work is fundamentally built upon the framework of string field theory applied to toroidal compactifications and investigates the geometric and algebraic structures facilitating T-duality in this context.

Key Elements and Contributions

The authors develop a double field theory (DFT) which involves augmenting the dimensionality of the torus under consideration. Here, a doubled torus is introduced, accommodating both the momenta and winding states of the closed strings. The field theory is constructed to maintain non-trivial dependence on these additional winding coordinates. This work expands on the string field theory results for closed strings on toroidal backgrounds, emphasizing the inclusion of all zero-modes, thus leading to an infinite set of fields on the doubled torus. The key constraint that emerges from the string theory is the $\bar{L}_0 - L_0 = 0$ condition, which translates into a field-theoretical constraint that is pivotal for achieving consistency within the double field theory.

A significant contribution is the authors' identification of the appropriate field content, which includes gravity, a Kalb-Ramond field, and a dilaton. These fields are shown to be necessary for maintaining the symmetry and structure of the theory, notably in the presence of doubled diffeomorphisms which explicitly act on the doubled coordinate set.

Symmetric and Constraint Structures

The symmetry structure in DFT is non-trivial. The paper outlines the presence of standard and "dual" diffeomorphisms alongside T-duality operations, the latter operating on an expanded set of coordinates. An intriguing feature is the algebraic structure of the gauge symmetries, suggesting that they form a homotopy Lie algebra, an observation underscoring the absence of a straightforward Lie algebra structure. This result aligns with previous observations within string field theory and suggests a deeper algebraic interconnection beyond traditional gauge symmetries.

The work emphasizes the physicality of the doubled geometry, arguing against interpreting the extra dimensions merely as auxiliary artifacts. This interpretation suggests new ways of engaging with string theories' higher-dimensional facets, where the dual dimensions provide essential physical information rather than supplementary constraints or gauge artifacts.

Implications and Future Directions

From a practical standpoint, the constructed double field theory presents valuable insights into symmetries observed in string theory, particularly the realization of T-duality as a manifest symmetry. The theoretical formalism potentially applies to cases involving non-geometric backgrounds where reliance on conventional geometric intuition is insufficient.

The paper speculates on future explorations into the full non-linear structure of DFT and the challenges with non-trivial projection and constraint management, as discussed in Section 5. The presence of cocycles and sign factors in products introduces subtleties that are anticipated to play critical roles in future non-linear explorations and extensions of the theory.

Conclusion

Hull and Zwiebach's "Double Field Theory" paper provides a systematic and rigorous account of how string field theory's inherent structures inform and empower a broader geometric and algebraic framework, presenting a sophisticated approach to engaging with T-duality symmetries. The doubled torus construction challenges conventional views on increased dimensionality in theoretical physics and underscores the potential of investigating non-geometric string backgrounds with field-theoretic approaches. As this work sets a foundation for treating these symmetries and structures, it offers a valuable stepping stone for subsequent research efforts aiming to unravel deeper facets of string dualities and their implications within physics.