Generalized metric formulation of double field theory (1006.4823v2)

Abstract: The generalized metric is a T-duality covariant symmetric matrix constructed from the metric and two-form gauge field and arises in generalized geometry. We view it here as a metric on the doubled spacetime and use it to give a simple formulation with manifest T-duality of the double field theory that describes the massless sector of closed strings. The gauge transformations are written in terms of a generalized Lie derivative whose commutator algebra is defined by a double field theory extension of the Courant bracket.

• The paper introduces a generalized metric that recasts double field theory to achieve manifest O(D,D) symmetry.
• The authors extend the Courant bracket to a C bracket, ensuring algebraic closure of gauge transformations in a doubled coordinate space.
• The study compares vielbein formulations, highlighting the equivalence of metric representations and potential enhancements for higher-dimensional theories.

Overview of "Generalized Metric Formulation of Double Field Theory"

The paper authored by Olaf Hohm, Chris Hull, and Barton Zwiebach presents a reformulation of Double Field Theory (DFT) through the introduction of a generalized metric. The research provides a framework with manifest T-duality for the massless sector of closed string theories, leveraging the metric’s integration with T-duality transformations. Central to the pursuit of this paper is the incorporation of O(D,D) symmetry and the Courant bracket, expanded to the field of double field theories.

Generalized Metric and O(D,D) Symmetry

The authors introduce a generalized metric, $H_{MN}$, which is a 2D by 2D symmetric matrix constructed from traditional D-dimensional spacetime metrics and antisymmetric tensor fields. This metric retains its O(D,D) covariance, a critical component for ensuring the theory’s invariance under T-duality. Through this framework, the paper elegantly delineates the complex non-linear transformations of traditional variables $g_{ij}$ and $b_{ij}$ into a simpler linear form in terms of the generalized metric.

Gauge Algebra and Courant Brackets

Hohm, Hull, and Zwiebach extend the Courant bracket to accommodate doubled field configurations, introducing the C bracket. This extension allows the formulation of the gauge algebra in double field theory to operate effectively within a doubled coordinate space. Notably, the authors elucidate on the algebraic closure of gauge transformations, demonstrating that these transformations align with the C bracket's operation.

Vielbein Formulations and Frame Fields

To further realize the advantages of the generalized metric formalism, the paper explores different vielbein formulations. The system is articulated through frame fields with a GL(D,R) x GL(D,R) symmetry. These formulations provide valuable insight into the equivalence between formulations centered around $H_{MN}$ and those employing the metric $E_{ij}$. Moreover, this exposition aligns with prior work by Siegel while providing a different methodological approach to the geometric structures foundational to the theory.

Implications and Potential Developments

The implications of this paper are significant for theoretical physics, particularly in string theory and related aspects of quantum gravity. By furnishing a cleaner mathematical formulation in terms of tensors and generalized geometry, the paper advances toward resolving the constraints double field theory faces. The elaboration of gauge invariance shows potential for further generalizations, possibly contributing to the structure of higher-dimensional theories.

Future Directions in AI and Theoretical Physics

Central challenges remain in extending these results beyond the confines of the strong constraint, allowing true utilization of all components of the doubled space. Future research needs to develop a deeper understanding, potentially leading to a formally complete double field theory, which may implicate higher derivative theories and string field theory’s quantum landscape. As a speculative endeavor, AI-driven symbolic computation tools, backed by natural language processing, could facilitate these advances by uncovering relationships within complex algebraic structures beyond human computational capacity.

In conclusion, this paper provides an innovative perspective on double field theory's mathematical underpinnings, establishing a cornerstone for pursuing new theoretical insights into the fabric of spacetime and string theory. By seamlessly integrating T-duality and a sophisticated gauge framework, the work progresses toward a holistic understanding of these profound theoretical constructs.