Symmetries and Higher-Form Connections in Derived Differential Geometry

TL;DR

This paper develops a comprehensive framework for higher-form connections on principal infinity-bundles using derived differential geometry, unifying known concepts and extending to higher gauge theories.

Contribution

It introduces a new definition of higher-form connections via derived stacks and Atiyah L-infinity algebroids, connecting to higher gauge theories and supergravity.

Findings

Recovered known higher U(1)-bundle connections via derived geometry

Established the Atiyah L-infinity algebroid as a key structure

Related higher symmetries to higher Courant algebroids

Abstract

We introduce a general definition of higher-form connections on principal $\infty$-bundles in differential geometry. This is achieved by developing the formal differentiation and integration of maps from smooth manifolds to derived stacks with sufficient deformation theory. That allows us to introduce the Atiyah $L_\infty$-algebroid of a principal $\infty$-bundle and establish its global sections as the $L_\infty$-algebra of the derived higher symmetry group of the bundle. We define the space of $p$-form connections on an $\infty$-bundle as the space of order $p$ splittings of its Atiyah $L_\infty$-algebroid. This can be cast equivalently as lifting the classifying map of a bundle on a manifold to the order $p$ truncation of the de Rham stack of the manifold. We demonstrate that our new concept of derived geometric $p$-form connections recovers the known notion of connections on higher U(1)-bundles defined via \v{C}ech-Deligne differential cocycles. We further relate the $L_\infty$-algebras of derived higher symmetries of higher U(1)-bundles and higher Courant algebroids. Some applications in higher gauge theory and in supergravity are mentioned.