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

Florian Girelli, Christopher Pollack, Aldo Riello

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 $σ$-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 $σ$-models, specifically the A-model and KS model.

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

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) -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 -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 -models, specifically the A-model and KS model.

Paper Structure

This paper contains 28 sections, 15 theorems, 281 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. From Noether to Poisson-Lie symmetries: A spacetime covariant approach
  3. Spaces of fields & locality
  4. The covariant phase space
  5. Symmetries: Noether & Poisson-Lie
  6. Poisson-Lie groups, their actions, & their doubles
  7. Poisson-Lie symmetry for a 0+1D field theory
  8. The (generalized) spinning top
  9. The deformed (generalized) spinning top
  10. Setup: Action principle
  11. Aside: The variational action principle for $\Psi(\gamma)$
  12. Symmetries of the deformed (generalized) spinning top
  13. Poisson-Lie symmetry for a 1+1D field theory
  14. Formulating the covariant Klimčík & Ševera open string
  15. Warm up: Noetherian symmetries of the principal chiral model
  16. ...and 13 more sections

Key Result

Theorem 2.3

Lagrangian symmetries preserve the equations of motion and thus (assuming smoothness of ${\mathcal{F}}_\text{EL}\hookrightarrow{\mathcal{F}}$) descends to an action on the covariant phase space. Furthermore, if ${\partial} C = \emptyset$, this action is Noetherian, i.e. it is Hamiltonian with conserved (i.e. $C$-independent) local generators which are given by the integral of the conserved Noeth

Figures (1)

  • Figure 1: We illustrate how the ribbon variables are associated to a triangulation and its dual complex. In particular, the expression of the constraint makes clear the action of the symmetry transformation on the phase space variables.

Theorems & Definitions (38)

  • Definition 2.1: Noetherian symmetry
  • Definition 2.2: Lagrangian symmetry
  • Theorem 2.3
  • Definition 2.4
  • Definition 2.5: Poisson-Lie symmetries
  • Definition 2.6: Poisson-Lie Group
  • Example 2.7
  • Definition 2.8: Poisson Action
  • Theorem 2.9: Lu and Weinstein lu_poisson_1990
  • Theorem 2.10: Lu lu_multiplicative_nodate, Babelon and Bernard babelon_dressing_1992
  • ...and 28 more