Overview
A mathematical framework models T-duality relating general geometric and non-geometric backgrounds using higher groupoid bundles equipped with connections. The formulation establishes manifest covariance under the full T-duality group $\mathsf{GO}(n,n;\mathbb{Z})$ while recovering established examples documented across theoretical literature.
Research Context
The formulation directly builds upon and broadens earlier geometric insights established by Nikolaus and Waldorf. In that prior work, higher geometry was demonstrated to characterize the topological properties governing both geometric and half-geometric T-dualities. The current proposal extends these topological treatments into a broader differential and algebraic framework capable of addressing non-geometric backgrounds.
Approach
The construction introduces two specific structural modifications to Nikolaus and Waldorf's original higher-geometric formulation:
- Adjusted Connections: The higher geometric spaces are endowed with adjusted connections. This geometric enrichment facilitates the derivation and analysis of explicit formulas for both the target space metric and the Kalb-Ramond field corresponding to a given T-background.
- Principal 2-Groupoid Bundles: The authors generalize standard principal 2-bundles to principal 2-groupoid bundles. This structural enlargement accommodates scalar fields generated during dualities executed along two directions, while simultaneously accounting for the presence of non-geometric $Q$-fluxes.
Findings
The framework reproduces established reference configurations and key examples found in existing duality literature. Furthermore, the higher groupoid bundle formulation maintains manifest covariance under the entire discrete orthogonal structure group $\mathsf{GO}(n,n;\mathbb{Z})$. The authors also examine and provide commentary on the behavior of T-duality when scalar fluxes are present.
Why This Matters
According to the authors, realizing non-geometric and geometric T-dualities through higher groupoid bundles with connections carries notable physical and mathematical implications, specifically by unifying metric components, Kalb-Ramond fields, scalar fields, and non-geometric fluxes within a unified, duality-covariant higher-gauge geometric framework.