Beyond Noether: A Covariant Study of Poisson-Lie Symmetries in Low Dimensional Field Theory

TL;DR

This paper investigates Poisson-Lie symmetries in low-dimensional field theories using a covariant phase space approach, highlighting structural challenges and illustrating with models like the non-linear sigma model and 2+1D gravity.

Contribution

It provides a covariant framework for understanding Poisson-Lie symmetries in field theories, emphasizing non-locality and non-Abelian momentum maps, with explicit low-dimensional examples.

Findings

Identifies structural challenges in implementing Poisson-Lie symmetries in field theories.

Demonstrates non-locality and non-Abelian momentum maps in explicit models.

Connects low-dimensional models to 2D sigma models like the A-model and KS model.

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 Klim\v{c}\'ik and \v{S}evera (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.