Symmetries and Higher-Form Connections in Derived Differential Geometry
Abstract
We introduce a general definition of higher-form connections on principal ∞-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∞-algebroid of a principal ∞-bundle and establish its global sections as the L∞-algebra of the derived higher symmetry group of the bundle. We define the space of p-form connections on an ∞-bundle as the space of order p splittings of its Atiyah L∞-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 Čech-Deligne differential cocycles. We further relate the L∞-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.
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BibTeX
@misc{bunk2026symmetries,
title = {{Symmetries and Higher-Form Connections in Derived Differential Geometry}},
author = {Severin Bunk and Lukas Müller and Joost Nuiten and Richard J. Szabo},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2602.03441},
eprint = {2602.03441},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05737889v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Symmetries and Higher-Form Connections in Derived Differential Geometry
AU - Severin Bunk
AU - Lukas Müller
AU - Joost Nuiten
AU - Richard J. Szabo
PY - 2026
DA - 2026/09/03
PB - HAL
DO - 10.48550/arxiv.2602.03441
UR - https://hal.science/hal-05737889v1
AB - We introduce a general definition of higher-form connections on principal ∞-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_∞-algebroid of a principal ∞-bundle and establish its global sections as the L_∞-algebra of the derived higher symmetry group of the bundle. We define the space of p-form connections on an ∞-bundle as the space of order p splittings of its Atiyah L_∞-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 Čech-Deligne differential cocycles. We further relate the L_∞-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.
ER -