Non-Geometric T-Duality as Higher Groupoid Bundles with Connections

We propose a description of T-duality between general geometric and non-geometric backgrounds as higher groupoid bundles with connections. Our description extends the previous observation by Nikolaus and Waldorf that the topological aspects of geometric and half-geometric T-dualities can be described in terms of higher geometry. We extend their construction in two ways. First, we endow the higher geometries with adjusted connections, which allow us to discuss explicit formulas for the metric and the Kalb-Ramond field of a T-background. Second, we extend the principal 2-bundles to principal 2-groupoid bundles, which accommodate the scalar fields arising in T-dualities along two directions as well as $Q$-fluxes. Our proposals reproduce key examples from the literature. They are manifestly covariant under the full T-duality group $\mathsf{GO}(n,n;\mathbb{Z})$ and have interesting physical and mathematical implications. Eventually, we also comment on the case of T-duality in the presence of scalar fluxes.