Non-geometric backgrounds in string theory (1811.11203v3)

Abstract: This review provides an introduction to non-geometric backgrounds in string theory. Starting from a discussion of T-duality, geometric and non-geometric torus-fibrations are reviewed, generalised geometry and its relation to non-geometric backgrounds are explained and compactifications of string theory with geometric and non-geometric fluxes are discussed. Furthermore covered are doubled geometry as well as non-commutative and non-associative structures in the context of non-geometric backgrounds.

• The paper introduces non-geometric backgrounds in string theory by revealing how T-duality redefines flux configurations and challenges traditional geometric frameworks.
• It employs generalized and doubled geometry techniques to integrate diffeomorphisms and gauge transformations, thereby broadening the conceptual landscape.
• The analysis of torus fibrations with H-, f-, Q-, and speculative R-flux informs new approaches to moduli stabilization and the realization of supergravity vacua.

Essay: An Introduction to Non-Geometric Backgrounds in String Theory

The paper "Non-geometric backgrounds in string theory" by Erik Plauschinn offers a comprehensive review of the concept of non-geometric backgrounds in string theory, exploring a broad range of topics including T-duality, geometric and non-geometric torus fibrations, generalized geometry, and doubled geometry. The document aims to provide a lucid introduction to non-geometric backgrounds by weaving together various complex ideas inherent in string theory and offering potential applications.

String theory, as discussed, fundamentally extends beyond point-particle theories by considering one-dimensional extended objects, or strings, typically described by a two-dimensional conformal field theory. An interesting realization in string theory is the existence of T-duality, a phenomenon wherein strings compactified on a circle of radius $R$ are indistinguishable from those compactified at a radius $\alpha'/R$. This duality plays a significant role in investigating geometric and non-geometric backgrounds, particularly influencing the observed spectrum of possible string configurations on toroidal and manifold backgrounds.

One essential focus of the paper is on the paradigm shift from conventional geometric descriptions to non-geometric backgrounds, spaces that defy traditional Riemannian frameworks. The hallmark of non-geometric spaces is the need for T-duality (geometric spaces use diffeomorphism as transition functions) to define the background—or a situation where a space remains inherently non-geometric even after employing dualities. Plauschinn illustrates these properties using the example of a three-torus with $H$-flux, presenting subsequent twists and T-folds characterized by geometric and non-geometric fluxes. Through this discussion, the relationship between $H$-flux, $f$-flux (geometric), and $Q$-flux (non-geometric) is elucidated, while postulating the existence of $R$-flux in fully non-geometric settings that remain elusive to description exclusively in point-particle or string theory.

Additionally, the paper discusses generalized geometry, which expands the notion of space in string theory to encompass both vector and covector fields, unifying elements like diffeomorphisms and $B$-field gauge transformations. This fusion results in the generalization of symmetries within the model, accommodating T-duality transformations as part of an $O(D,D,\mathbb{Z})$ duality group. This combination offers a powerful approach for characterizing non-geometric backgrounds, allowing for non-trivial fibration structures of torus bases and extended spaces.

Notably, generalized geometry provides an anchor for theoretical constructs that capture the complexities of non-geometric string vacua and elaborate on the intertwining nature of geometric and non-geometric facets of cohesive string landscapes. The utilization of Courant brackets further aids in representing the integrability conditions pertinent to efficient symmetry algebra operations on generalized tangent bundles.

The application of generalized geometry extends beyond theoretical musings, holding promise for string phenomenology where non-geometric fluxes contribute to moduli stabilization and metamorphosize into viable supergravity vacua. It lays the foundation for more nuanced explorations in quantum gravity, establishing bridges between string theory and low-energy physics models, such as effective field theories and moduli space compactifications.

In summary, the document by Plauschinn serves as a meticulous introduction to a captivating area in string theory, challenging the preponderant geometric intuition by reconstructing landscape explorations through T-duality, flux compactifications, and string theory compactifications. While the notions of fluxes like $R$-flux remain speculative, they tantalizingly point towards advanced non-geometric string landscapes, possibly reshaping the paradigms of physics beyond current theoretical frameworks. This paper represents a pivotal reference for researchers exploring both the intricacies and prospects of non-geometric backgrounds.