Frame-like Geometry of Double Field Theory (1011.4101v2)

Abstract: We relate two formulations of the recently constructed double field theory to a frame-like geometrical formalism developed by Siegel. A self-contained presentation of this formalism is given, including a discussion of the constraints and its solutions, and of the resulting Riemann tensor, Ricci tensor and curvature scalar. This curvature scalar can be used to define an action, and it is shown that this action is equivalent to that of double field theory.

• The paper presents a geometrical frame formalism that integrates double field theory with Siegel's approach, emphasizing T-duality and gauge symmetry.
• It establishes a novel action principle analogous to Einstein-Hilbert gravity by utilizing a GL(D)×GL(D)-covariant scalar curvature.
• The study compares E- and H-formulations, offering deeper insights into generalized metric constructions and the consistency of duality symmetries.

Frame-like Geometry of Double Field Theory

The paper "Frame-like Geometry of Double Field Theory" by Olaf Hohm and Seung Ki Kwak presents a rich interplay between the double field theory (DFT) and frame-like geometrical formalisms, specifically those developed by Siegel. The authors integrate DFT, a construct that naturally extends conventional field theories to embody T-duality—a symmetry prominent in string theory—with Siegel's formalism, thereby providing deeper insights into its geometric underpinnings.

Overview

At the heart of the paper is T-duality, a fundamental concept within string theory that connects different physical regimes by transforming the string’s momentum and winding modes through the non-compact O(D,D) duality group. This duality is made manifest in the DFT framework by incorporating doubled coordinates for both world-sheet and space-time structures. The resulting theory provides a unified description of gravitational and non-gravitational fields, such as the metric and the Kalb-Ramond field, using an O(D,D) covariant action.

The paper explores the link between two formulations of DFT and Siegel's geometrical formalism, which is framed in terms of GL(D)×GL(D) symmetry. This connection is elucidated by expressing the DFT action using a scalar curvature that aligns with those in Riemannian geometry.

Main Contributions

1. Geometrical Frame Formalism: The authors review Siegel’s frame-like formalism characterized by frame fields and GL(D)×GL(D) connections. A key aspect is the introduction of generalized Lie derivatives that maintain covariance under gauge transformations parameterized by specific vector fields. This provides a robust framework for exploring the algebra and constraints of DFT.
2. Action Principle and Gauge Invariance: The paper proposes an action principle akin to the Einstein-Hilbert action in Riemannian geometry but adapted to DFT's doubled structures. Via scalar curvature derived from the GL(D)×GL(D) curvature tensor, an action is constructed, maintaining gauge invariance under the novel DFT gauge transformations. The resulting field equations are correlated with the motion equations presented in former works, substantiating their equivalence.
3. H-Formulation: The discussion extends to the H-formulation of DFT, translating the frame geometry into expressions involving the generalized metric HMN. The gauge fixing and subsequent derivations elucidate the symmetry properties and invariant structures within this formalism, demonstrating equivalence to the earlier E-formulation of the double field theory.

Numerical Results and Claims

The paper does not focus on numerical results but presents strong theoretical constructs by asserting that the geometrical reformulation of DFT via Siegel's formalism offers crucial clarifications of DFT’s structure. The consistency between different formulations—typically intricate due to the theory's doubled nature—is meticulously demonstrated.

Implications and Future Directions

Practically, this research enriches our geometric comprehension of DFT, equipping theorists with better tools to handle dualities in string theory. Theoretically, this work lays a foundation for further exploration into generalizations of DFT, potentially unlocking insights into more expansive frameworks like type II string theories or M-theory.

Looking forward, a promising avenue involves extending these geometrical methods to incorporate generalized geometrical structures. This may not only enhance our understanding of existing theories but also pave the way toward a genuinely doubled formulation of field theory, overcoming current constraints by allowing non-trivial dependency on both momentum and winding coordinates.

In conclusion, Hohm and Kwak’s paper makes a significant contribution to theoretical physics by articulating a geometric interpretation of DFT, bridging intricacies within duality symmetries, and proffering a fertile ground for future theoretical developments in the field of string theory.