Yang-Mills theory for multiplicative Ehresmann connections
Abstract
We develop a twofold generalization of classical Yang-Mills theory, extending it from principal bundles to the setting of possibly non-transitive and non-integrable Lie algebroids. The classical theory is recovered when one considers the Atiyah algebroid of a principal bundle. In our framework, principal bundle connections are replaced by the more general notion of (infinitesimal) multiplicative Ehresmann connections. An action functional for such connections is constructed, now including a curvature 3-form contribution, alongside the usual curvature 2-form term, and the resulting variational problem is naturally constrained by a cohomological condition. We derive the associated Euler-Lagrange equations, and define a class of self-dual solutions (instantons) in both 4 and 5 dimensions. We also show that the solution space is invariant under gauge transformations, and compute its tangent space at a solution. As an important example, we show that our framework produces a Yang-Mills theory for connections on bundle gerbes.
Related
- Eliminating the Higgs field and perturbing Yang-Mills connections in cone Yang-Mills theory
- Decay estimates for Yang-Mills connections and Yang-Mills-Higgs pairs in higher dimensions
- Algebraic limits of Hermitian-Yang-Mills connections on Kähler surfaces
- Nonclosure of stationary Yang-Mills connections under weak convergence
- Non-Hermitian Yang--Mills Connections with Vanishing Chern Classes
BibTeX
@misc{grad2026yangmills,
title = {{Yang-Mills theory for multiplicative Ehresmann connections}},
author = {Žan Grad},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.10904},
eprint = {2607.10904},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.10904},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Yang-Mills theory for multiplicative Ehresmann connections
AU - Žan Grad
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2607.10904
UR - https://arxiv.org/abs/2607.10904
AB - We develop a twofold generalization of classical Yang-Mills theory, extending it from principal bundles to the setting of possibly non-transitive and non-integrable Lie algebroids. The classical theory is recovered when one considers the Atiyah algebroid of a principal bundle. In our framework, principal bundle connections are replaced by the more general notion of (infinitesimal) multiplicative Ehresmann connections. An action functional for such connections is constructed, now including a curvature 3-form contribution, alongside the usual curvature 2-form term, and the resulting variational problem is naturally constrained by a cohomological condition. We derive the associated Euler-Lagrange equations, and define a class of self-dual solutions (instantons) in both 4 and 5 dimensions. We also show that the solution space is invariant under gauge transformations, and compute its tangent space at a solution. As an important example, we show that our framework produces a Yang-Mills theory for connections on bundle gerbes.
ER -