The gauge algebra of double field theory and Courant brackets

Abstract: We investigate the symmetry algebra of the recently proposed field theory on a doubled torus that describes closed string modes on a torus with both momentum and winding. The gauge parameters are constrained fields on the doubled space and transform as vectors under T-duality. The gauge algebra defines a T-duality covariant bracket. For the case in which the parameters and fields are T-dual to ones that have momentum but no winding, we find the gauge transformations to all orders and show that the gauge algebra reduces to one obtained by Siegel. We show that the bracket for such restricted parameters is the Courant bracket. We explain how these algebras are realised as symmetries despite the failure of the Jacobi identity.

• The paper presents a T-duality covariant gauge bracket that links string theory symmetries with the Courant bracket in generalized geometry.
• It formulates non-linear, O(D,D) invariant gauge transformations for DFT that reduce to known cases under specific restrictions.
• It establishes a bridge between mathematical structures and physical symmetries, paving the way for advancements in understanding non-geometric string backgrounds.

Overview of "The Gauge Algebra of Double Field Theory and Courant Brackets"

The paper investigates the symmetry algebra of a recently proposed double field theory (DFT), focusing on the dynamics of closed string modes on a doubled torus. The authors, Chris Hull and Barton Zwiebach, explore a formulation that seeks to capture both momentum and winding modes on a toroidal background, considering the symmetries that arise from T-duality. This investigation is pivotal as it positions the DFT within a framework where both the geometry and duality symmetries are treated uniformly.

Key Contributions

1. T-Duality Covariant Bracket: The paper presents a detailed exploration of the gauge algebra's symmetry structure corresponding to T-duality. A significant outcome of this study is the establishment of a T-duality covariant gauge bracket. The authors demonstrate that, under specific constraints, this bracket coincides with the Courant bracket, prominent in the context of generalized geometry.
2. Gauge Transformation and Algebra: The authors achieve a formulation of non-linear gauge transformations suitable for restricted fields in DFT. Interestingly, these transformations are shown to uphold the established O(D,D) symmetry inherent to the theory and reduce to known transformations under specific configurations—demonstrating the robustness of the DFT framework.
3. Courant and C-Bracket Relations: A novel aspect of this work is the analysis of the Courant bracket within the gauge algebra. The research establishes that restricted field applications of the T-duality invariant bracket (C-bracket) indeed yield the Courant bracket, establishing a concrete bridge between mathematical structures in generalized geometry and physical symmetries in string theory.

Theoretical Implications

The implications of this research are multifaceted, both enhancing the theoretical foundation of double field theories and drawing broader connections with existing mathematical structures like Lie bialgebroids and Courant algebroids. The equivalence between the C-bracket and the Courant bracket under certain restrictions aligns DFT more closely with generalized geometry, paving the way for deeper insights into the geometric nature of string dualities.

The authors address the algebraic challenges linked to the failure of the Jacobi identity, showing that the symmetries and gauge transformations can still be consistently realized within this framework. This resolution implies a degree of flexibility in DFT, allowing it to encapsulate more complex geometric and duality situations than traditional field theories.

Practical Implications and Future Directions

Practically, the work sets the stage for further advancements in constructing a fully-fledged double field theory encompassing all modes—momentum and winding—without reliance on specific space reductions like the null subspace. This could provide a strong tool for tackling non-geometric backgrounds that naturally appear in advanced string-theoretical settings.

The research invites future developments in two primary directions: First, achieving an all-orders construction of the double field theory, building on the symmetry principles outlined. Secondly, further connection with generalized complex geometry could yield new insights into string compactifications and possibly unify aspects of string field theory and M-theory under the double field framework.

In conclusion, by addressing both the algebraic structures and symmetry properties inherent in double field theory, this paper not only enhances our understanding of DFT's physical implications but also strengthens its mathematical foundation, positioning it as a vital tool for exploring string theory's full geometric and symmetric spectrum.