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Beyond Noether: A Covariant Study of Poisson-Lie Symmetries in Low Dimensional Field Theory
by Florian Girelli, Christopher Pollack, Aldo Riello
Submission summary
| Authors (as registered SciPost users): | Christopher Pollack |
| Submission information | |
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| Preprint Link: | scipost_202508_00012v2 (pdf) |
| Date submitted: | Dec. 12, 2025, 10:15 p.m. |
| Submitted by: | Christopher Pollack |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
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| Approach: | Theoretical |
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) $\fg^*$-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 \klimcik{} and \severa{} (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.
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List of changes
Change 2: Corrected the following minor grammatical typos as per request of referee and editor in charge:
p. 6, l. 185: "On the space $M\times \mathcal{F}$, one can defined the bi-complex" -> define
p. 32, l. 978: “transformation of the (off-shell) worldine $\xi(t)$” -> worldline
p. 32, l. 989: "the dynamics $\mathcal{H} = c_{1}{\rm Tr}(h)-c_{2}$ for e.g., $G = SU(2)$" -> "$\mathcal{H} = c_{1}{\rm Tr}(h)-c_{2}$ e.g."
p. 63, l. 1917: “interplay between between Poisson-Lie symmetric field theories” -> “interplay between Poisson-Lie”
p. 37, l. 1137: "where on-shell, neither $q$ nor $\omega$ dependss on the choice of $C\to \Sigma$. -> depends