Principal 3-Bundles with Adjusted Connections

TL;DR

This paper develops the theory of adjusted connections for principal 3-bundles using $L_$-algebras, providing explicit local descriptions, symmetries, and applications in high-energy physics, especially string and M-theory.

Contribution

It introduces the concept of adjusted connections for principal 3-bundles, deriving explicit forms, symmetries, and their integration into higher groupoids and differential cohomology.

Findings

Explicit form of adjustment datum for 3-term $L_$-algebras

Description of local adjusted connections and symmetries

Applications in gauged supergravity and string/M-theory

Abstract

We explore the notion of an adjusted connection for principal 3-bundles. We first derive the explicit form of an adjustment datum for 3-term $L_\infty$-algebras, which allows us to give a local description of such adjusted connections and their infinitesimal symmetries. We then integrate the corresponding action Lie 3-algebroid to an action Lie 3-groupoid, encoding local connection forms with finite (higher) symmetries. This also yields the notion of an adjusted 2-crossed module of Lie groups. Stackifying the action Lie 3-groupoid then gives us the explicit description of principal 3-bundles with adjusted connections in terms of differential cohomology. These connections appear in a number of contexts within high-energy physics, and we list local examples arising in gauged supergravity as well as a global example arising in various contexts in string/M-theory. Our primary motivation, however, stems from U-duality, and we also define a notion of categorified torus that forms an adjusted 2-crossed module, which we hope to be useful in lifting T-duality to M-theory.