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Classical dynamical $r$-matrices for the Chern-Simons formulation of generalised 3d gravity

Juan Carlos Morales Parra, Bernd Schroers

Abstract

Classical dynamical $r$-matrices arise naturally in the combinatorial description of the phase space of Chern-Simons theories, either through the inclusion of dynamical sources or through a gauge-fixing procedure involving two punctures. Here we consider classical dynamical $r$-matrices for the family of Lie algebras which arise in the Chern-Simons formulation of 3d gravity, for any value of the cosmological constant. We derive differential equations for classical dynamical $r$-matrices in this case, and show that they can be viewed as generalised complexifications, in a sense which we define, of the equations governing dynamical $r$-matrices for $\mathfrak{su}(2)$ and $\mathfrak{sl}(2,\mathbb{R})$. We obtain explicit families of solutions and relate them, via Weierstrass factorisation, to solutions found by Feher, Gabor, Marshall, Palla and Pusztai in the context of chiral WZWN models.

Classical dynamical $r$-matrices for the Chern-Simons formulation of generalised 3d gravity

Abstract

Classical dynamical -matrices arise naturally in the combinatorial description of the phase space of Chern-Simons theories, either through the inclusion of dynamical sources or through a gauge-fixing procedure involving two punctures. Here we consider classical dynamical -matrices for the family of Lie algebras which arise in the Chern-Simons formulation of 3d gravity, for any value of the cosmological constant. We derive differential equations for classical dynamical -matrices in this case, and show that they can be viewed as generalised complexifications, in a sense which we define, of the equations governing dynamical -matrices for and . We obtain explicit families of solutions and relate them, via Weierstrass factorisation, to solutions found by Feher, Gabor, Marshall, Palla and Pusztai in the context of chiral WZWN models.
Paper Structure (13 sections, 6 theorems, 236 equations, 2 tables)

This paper contains 13 sections, 6 theorems, 236 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Motivation
  3. Definitions
  4. Example
  5. Chern-Simons formulation of 3d gravity
  6. Structure of the CDYBE for
  7. The Lie algebras as generalised complexifications
  8. Casimirs and the CDYBE for
  9. Classical dynamical -matrices for
  10. Classical dynamical -matrices
  11. Dynamical generalizations of classical -matrices for
  12. The FGMPP dynamical -matrices for
  13. Conclusion and outlook

Key Result

Lemma 1

Meusburger:2006fn. For $\lambda \in \mathbb{R}$ we have the following isomorphisms

Theorems & Definitions (15)

  • Definition 1
  • Lemma 1
  • Remark 1
  • Theorem 1
  • proof
  • Lemma 2
  • proof
  • Theorem 2
  • proof
  • Lemma 3
  • ...and 5 more