T-duality, Generalized Geometry and Non-Geometric Backgrounds (0807.4527v1)

Abstract: We discuss the action of O(d,d), and in particular T-duality, in the context of generalized geometry, focusing on the description of so-called non-geometric backgrounds. We derive local expressions for the pure spinors descibing the generalized geometry dual to an SU(3) structure background, and show that the equations for N=1 vacua are invariant under T-duality. We also propose a local generalized geometrical definition of the charges f, H, Q and R appearing in effective four-dimensional theories, using the Courant bracket. We then address certain global aspects, in particular whether the local non-geometric charges can be gauged away in, for instance, backgrounds admitting a torus action, as well as the structure of generalized parallelizable backgrounds.

• The paper uses generalized geometry to provide a framework for T-duality in non-geometric backgrounds, emphasizing the O(d,d) symmetry.
• It defines generalized charges (f, H, Q, R) for non-geometric backgrounds using the Courant bracket and spin connection within this generalized geometry framework.
• The authors verify that supersymmetry equations for N=1 backgrounds with SU(3) x SU(3) structures remain invariant under T-duality transformations.

Overview of T-duality, Generalized Geometry, and Non-Geometric Backgrounds

The paper titled "T-duality, Generalized Geometry and Non-Geometric Backgrounds" by Graña, Minasian, Petrini, and Waldram explores the intricate relationships between T-duality and generalized geometry with a focus on non-geometric string backgrounds. This work situates itself at the crossroads of modern string theory, aiming to elucidate the complexities that arise when conventional geometrical intuitions are challenged by phenomena unique to string theory, such as non-geometric flux backgrounds.

The investigation explores the peculiar action of the O(d,d) symmetry group, emphasizing T-duality's impact within the construct of generalized geometry, which provides a robust framework for considering geometric and non-geometric configurations. The authors extend the formalism to explore SU(3) × SU(3) structures, integral to N = 1 supersymmetric vacua, and demonstrate the T-duality invariance of the associated equations.

Generalized Geometry and T-duality

Generalized geometry, bringing synergy between the tangent and cotangent bundles of a manifold through the O(d,d) structure, offers a powerful paradigm for addressing string theory compactifications. The authors articulate various constructs essential for these descriptions - generalized metric, generalized vielbeins, and O(d) × O(d) structures. Specifically, generalized spinors reveal the potential for embedding string backgrounds comprising metric and B-field interchanges under T-duality. By refining the formalism to encapsulate O(d,d) spinors, the work adroitly accommodates flux compactifications.

Non-Geometric Backgrounds and Charges

The paper proposes a coherent characterization of the generalized charges (f, H, Q, R) inherent to non-geometric backgrounds. Employing the Courant bracket, the authors offer a local description of these charges, rooting them in the spin connection and generalized geometry framework. This approach offers a nuanced insight into the distinction between geometric and non-geometric vacua, pertinent to understanding the lifting of solutions in gauged supergravity to consistent string compactifications.

Supersymmetry Preserving Transformations

In a compelling exploration of N = 1 supersymmetric backgrounds, the authors verify the invariance of supersymmetry equations under the cloak of T-duality. By working through explicit T-duality examples on toroidal manifolds, exhibiting phenomena such as H-flux and mirror symmetry, they establish the transformation dynamics of SU(3) × SU(3) structures. This invariance is crucial for affirming the role of T-duality as a legitimate symmetry in extended supersymmetric settings.

Implications and Future Directions

The paper's formulation of the T-duality operations within generalized geometry directs us toward broader implications for understanding string compactifications with non-geometric fluxes. The delineation of global properties, including potential obstructions, underscores the complexity in achieving a universally applicable description. The identification of generalized parallelizable backgrounds further extends these ideas, offering pathways to conceptualize more intricate algebraic structures aligned with string dualities.

The discourse on T-duality transformation within such unified geometric settings might unfold novel methodologies for addressing unsolved problems in string theory. As an academic community, continuing to explore these foundational aspects could unveil further insights into string dualities and their incumbent algebraic and geometric interpretations. Future endeavors might address the echo of these transformations in broader physical scenarios or other forms of string-theoretic dualities, thereby enriching our global understanding of high-dimensional string backgrounds.