Beyond the circle: current algebras, geometry and higher gauge structures
Abstract
Replacing the circle by a manifold leads to current Lie algebras and their central and derivation extensions. We separate the coefficient algebra, harmonic analysis and geometric data. Differentials modulo exact forms describe central extensions; divergence-free vector fields give explicit cocycles, whose compatible derivations satisfy a stabilizer equation. Compact examples include the three-sphere and its current--Virasoro extension. We also explain the analytic qualifications for SL(2,R) and its symmetric space and the role of transported multiplication in soft geometries. The recent harmonic constructions of Campoamor-Stursberg, Marrani and Rausch de Traubenberg are compared with the cohomological work of Coquereaux, Frappat, Ragoucy and Sorba and Castellani's loop geometry and infinite-dimensional free differential algebras. A coefficient-algebra construction connects currents to higher brackets and clarifies the additional requirements of a dynamical theory. For Lie algebras of order F, central extensions obey coupled binary and symmetric F-ary conditions, with explicit examples and obstructions.
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BibTeX
@misc{campoamorstursberg2026beyond,
title = {{Beyond the circle: current algebras, geometry and higher gauge structures}},
author = {R. Campoamor-Stursberg and A. Marrani and M. Rausch de Traubenberg},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17781},
eprint = {2610.17781},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17781},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Beyond the circle: current algebras, geometry and higher gauge structures
AU - R. Campoamor-Stursberg
AU - A. Marrani
AU - M. Rausch de Traubenberg
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.17781
UR - https://arxiv.org/abs/2610.17781
AB - Replacing the circle by a manifold leads to current Lie algebras and their central and derivation extensions. We separate the coefficient algebra, harmonic analysis and geometric data. Differentials modulo exact forms describe central extensions; divergence-free vector fields give explicit cocycles, whose compatible derivations satisfy a stabilizer equation. Compact examples include the three-sphere and its current--Virasoro extension. We also explain the analytic qualifications for SL(2,R) and its symmetric space and the role of transported multiplication in soft geometries. The recent harmonic constructions of Campoamor-Stursberg, Marrani and Rausch de Traubenberg are compared with the cohomological work of Coquereaux, Frappat, Ragoucy and Sorba and Castellani's loop geometry and infinite-dimensional free differential algebras. A coefficient-algebra construction connects currents to higher brackets and clarifies the additional requirements of a dynamical theory. For Lie algebras of order F, central extensions obey coupled binary and symmetric F-ary conditions, with explicit examples and obstructions.
ER -