Beyond Noether: A covariant study of Poisson-Lie symmetries in low dimensional field theory, by Florian Girelli, Christopher Pollack, Aldo Riello

SciPost Physics Home Authoring Refereeing Submit a manuscript About Beyond Noether: A covariant study of Poisson-Lie symmetries in low dimensional field theory Florian Girelli, Christopher Pollack, Aldo Riello SciPost Phys. 20, 057 (2026) · published 24 February 2026 doi: 10.21468/SciPostPhys.20.2.057 pdf BiBTeX RIS Submissions/Reports Abstract We explore global Poisson-Lie (PL) symmetries using a Lagrangian, or "covariant phase space" approach, that manifestly preserves spacetime covariance. PL symmetries are the classical analog of quantum-group symmetries. In the Noetherian framework symmetries leave the Lagrangian invariant up to boundary terms and necessarily yield (on closed manifolds) $\mathfrak{g}^*$-valued conserved charges which serve as Hamiltonian generators of the symmetry itself. Non-trivial PL symmetries transcend this framework by failing to be symplectomorphisms and by admitting (conserved) non-Abelian group-valued momentum maps. In this paper we discuss various structural and conceptual challenges associated with the implementation of PL symmetries in field theory, focusing in particular on non-locality. We examine these issues through explicit examples of low-dimensional field theories with non-trivial PL symmetries: the deformed spinning top (or, the particle with curved momentum and configuration space) in 0+1D; the non-linear $\sigma$-model by Klimčík and Ševera (KS) in 1+1D; and gravity with a cosmological constant in 2+1D. Although these examples touch on systems of different dimensionality, they are all ultimately underpinned by 2D $\sigma$-models, specifically the A-model and KS model. × TY - JOURPB - SciPost FoundationDO - 10.21468/SciPostPhys.20.2.057TI - Beyond Noether: A covariant study of Poisson-Lie symmetries in low dimensional field theoryPY - 2026/02/24UR - https://scipost.org/SciPostPhys.20.2.057JF - SciPost PhysicsJA - SciPost Phys.VL - 20IS - 2SP - 057A1 - Girelli, FlorianAU - Pollack, ChristopherAU - Riello, AldoAB - We explore global Poisson-Lie (PL) symmetries using a Lagrangian, or "covariant phase space" approach, that manifestly preserves spacetime covariance. PL symmetries are the classical analog of quantum-group symmetries. In the Noetherian framework symmetries leave the Lagrangian invariant up to boundary terms and necessarily yield (on closed manifolds) $\mathfrak{g}^*$-valued conserved charges which serve as Hamiltonian generators of the symmetry itself. Non-trivial PL symmetries transcend this framework by failing to be symplectomorphisms and by admitting (conserved) non-Abelian group-valued momentum maps. In this paper we discuss various structural and conceptual challenges associated with the implementation of PL symmetries in field theory, focusing in particular on non-locality. We examine these issues through explicit examples of low-dimensional field theories with non-trivial PL symmetries: the deformed spinning top (or, the particle with curved momentum and configuration space) in 0+1D; the non-linear $\sigma$-model by Klimčík and Ševera (KS) in 1+1D; and gravity with a cosmological constant in 2+1D. Although these examples touch on systems of different dimensionality, they are all ultimately underpinned by 2D $\sigma$-models, specifically the A-model and KS model.ER - × @Article{10.21468/SciPostPhys.20.2.057, title={{Beyond Noether: A covariant study of Poisson-Lie symmetries in low dimensional field theory}}, author={Florian Girelli and Christopher Pollack and Aldo Riello}, journal={SciPost Phys.}, volume={20}, pages={057}, year={2026}, publisher={SciPost}, doi={10.21468/SciPostPhys.20.2.057}, url={https://scipost.org/10.21468/SciPostPhys.20.2.057},} Ontology / Topics See full Ontology or Topics database. Global symmetries Integrability/integrable models Lie algebras and groups Non-abelian symmetries Quantum gravity Sigma models Authors / Affiliations: mappings to Contributors and Organizations See all Organizations. 1 Florian Girelli, 1 2 Christopher Pollack, 1 2 Aldo Riello 1 University of Waterloo [UW] 2 Institut Périmètre de physique théorique / Perimeter Institute [PI] Funder for the research work leading to this publication Conseil de Recherches en Sciences Naturelles et en Génie / Natural Sciences and Engineering Research Council [NSERC / CRSNG]