Weak Lie 3-groups, 2-gerbes over torus fibrations of type F1, and T-Duality

Abstract

We construct a weak Lie 3-group $T_2{\mathbb{B}}_n^{F_1}$ from the 2-category of gerbes over ${\mathbb{R}}^n/{\mathbb{Z}}^n$ and the ${\mathbb{R}}^n/{\mathbb{Z}}^n$-action on it by pullback along translations. We also construct a homotopy equivalence between $T_2{\mathbb{B}}_n^{F_1}$ and a different Lie 3-group $T_2{\mathbb{D}}_n^{F_1}$, which admits a dimensional reduction to the homotopy equivalence of Lie 2-groups introduced by Nikolaus and Waldorf to model half-geometric T-duality. This is a first step towards establishing a higher form of T-duality for 2-gerbes over torus fibrations, relevant to supergravity and M-theory.

40 pages