Einstein-Cartan and Palatini-Cartan Gravity from Graded Poisson Geometry
Abstract
Using notions from graded Poisson geometry, we develop a framework that allows for concise formulations of field theories such as Einstein-Cartan and Palatini-Cartan gravity. We show that Einstein-Cartan gravity is equivalent to a variant of Palatini-Cartan gravity in which the vielbein fields are constrained to satisfy appropriate symmetry relations, which we refer to as constrained vielbein gravity. To that end, we relate these theories via a diffeomorphism of graded manifolds. We then show how to obtain the full Palatini-Cartan theory by removing the symmetry constraint. Finally, we show how our graded geometric framework allows for a direct incorporation of the minimal BV extension of the graded constrained vielbein and Palatini-Cartan theories.
Related
- Holomorphic Conformal Cartan Geometry: Conformal Gravity, Twistors, and Ambitwistors
- A Cartan geometry from a correspondence space in Riemannian geometry
- Cartan Determinants in algebraic geometry
- Quasi-Einstein metrics from Horava gravity
- An outline of shifted Poisson structures and deformation quantisation in derived differential geometry
BibTeX
@misc{moerland2026einsteincartan,
title = {{Einstein-Cartan and Palatini-Cartan Gravity from Graded Poisson Geometry}},
author = {Johannes Moerland and Peter Schupp},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.11844},
eprint = {2610.11844},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.11844},
}
RIS
TY - UNPB
TI - Einstein-Cartan and Palatini-Cartan Gravity from Graded Poisson Geometry
AU - Johannes Moerland
AU - Peter Schupp
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.11844
UR - https://arxiv.org/abs/2610.11844
AB - Using notions from graded Poisson geometry, we develop a framework that allows for concise formulations of field theories such as Einstein-Cartan and Palatini-Cartan gravity. We show that Einstein-Cartan gravity is equivalent to a variant of Palatini-Cartan gravity in which the vielbein fields are constrained to satisfy appropriate symmetry relations, which we refer to as constrained vielbein gravity. To that end, we relate these theories via a diffeomorphism of graded manifolds. We then show how to obtain the full Palatini-Cartan theory by removing the symmetry constraint. Finally, we show how our graded geometric framework allows for a direct incorporation of the minimal BV extension of the graded constrained vielbein and Palatini-Cartan theories.
ER -