Resonant JPZ algebras and 3D Chern--Simons gravity
Abstract
In this work we construct three-dimensional Chern--Simons gravity theories for the six resonant JPZ algebras generated by the Lorentz generators Ja, translations Pa, and an additional Lorentz-like generator Za. The resonance condition together with the Jacobi identities leaves precisely six configurations arising from the semigroup expansions. Our main result is that three of them, B4, C̃4, and C4, are isomorphic to iso(2,1)⊕ so(2,1), although they yield different gravitational realizations when we keep the original JPZ field identification fixed. For each case JPZ, we determine the invariant tensors, gauge curvatures, Chern--Simons action, field equations, and the corresponding maximal-rank coupling sectors. The field equations can be written in the compact form ΩσF=0, which makes explicit how the additional field ha contributes to the curvature, torsion, and the remaining gauge-curvature components. We use the direct-sum isomorphisms to classify the local metric sectors and to relate them to known global and asymptotic solutions, while keeping the gravitational interpretation tied to the original (ω,e,h) fields.
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BibTeX
@misc{durka2026resonant,
title = {{Resonant JPZ algebras and 3D Chern--Simons gravity}},
author = {Remigiusz Durka and Nelson Merino and D. M. Peñafiel},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17480},
eprint = {2610.17480},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17480},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Resonant JPZ algebras and 3D Chern--Simons gravity
AU - Remigiusz Durka
AU - Nelson Merino
AU - D. M. Peñafiel
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.17480
UR - https://arxiv.org/abs/2610.17480
AB - In this work we construct three-dimensional Chern--Simons gravity theories for the six resonant JPZ algebras generated by the Lorentz generators J_a, translations P_a, and an additional Lorentz-like generator Z_a. The resonance condition together with the Jacobi identities leaves precisely six configurations arising from the semigroup expansions. Our main result is that three of them, B_4, C̃_4, and C_4, are isomorphic to iso(2,1)⊕ so(2,1), although they yield different gravitational realizations when we keep the original JPZ field identification fixed. For each case JPZ, we determine the invariant tensors, gauge curvatures, Chern--Simons action, field equations, and the corresponding maximal-rank coupling sectors. The field equations can be written in the compact form Ω_σF=0, which makes explicit how the additional field h^a contributes to the curvature, torsion, and the remaining gauge-curvature components. We use the direct-sum isomorphisms to classify the local metric sectors and to relate them to known global and asymptotic solutions, while keeping the gravitational interpretation tied to the original (ω,e,h) fields.
ER -