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On the relation between the modular double of U_q(sl(2,R)) and the quantum Teichmueller theory

I. Nidaiev, J. Teschner

TL;DR

The work establishes a direct bridge between the modular double of ${\mathcal{U}}_q({\mathfrak sl}(2,\mathbb{R}))$ and quantum Teichmüller theory, revealing that Casimir-length observables, Clebsch–Gordan maps, and braiding in the modular double correspond to geometric-length operators, cutting, and braiding moves in Teichmüller theory. By constructing explicit factorized Clebsch–Gordan maps and their $b$-6j/$b$-CG kernels, the authors derive a streamlined path to the principal-series Clebsch–Gordan decomposition and connect fusion and braiding to Teichmüller moves. The paper also develops the quantum Teichmüller framework for pants decompositions and the Moore–Seiberg groupoid, providing explicit operator realizations (A,B,Z) in genus 0 that mirror the modular double structure. Overall, these results yield a cohesive, geometrically flavored realization of noncompact quantum group representation theory within quantum Teichmüller theory, with potential applications to conformal field theory and integrable models.

Abstract

We exhibit direct relations between the modular double of U_q(sl(2,R)) and the quantum Teichmueller theory. Explicit representations for the fusion- and braiding operations of the quantum Teichmueller theory are immediate consequences. Our results include a simplified derivation of the Clebsch-Gordan decomposition for the principal series of representation of the modular double of U_q(sl(2,R)).

On the relation between the modular double of U_q(sl(2,R)) and the quantum Teichmueller theory

TL;DR

The work establishes a direct bridge between the modular double of and quantum Teichmüller theory, revealing that Casimir-length observables, Clebsch–Gordan maps, and braiding in the modular double correspond to geometric-length operators, cutting, and braiding moves in Teichmüller theory. By constructing explicit factorized Clebsch–Gordan maps and their -6j/-CG kernels, the authors derive a streamlined path to the principal-series Clebsch–Gordan decomposition and connect fusion and braiding to Teichmüller moves. The paper also develops the quantum Teichmüller framework for pants decompositions and the Moore–Seiberg groupoid, providing explicit operator realizations (A,B,Z) in genus 0 that mirror the modular double structure. Overall, these results yield a cohesive, geometrically flavored realization of noncompact quantum group representation theory within quantum Teichmüller theory, with potential applications to conformal field theory and integrable models.

Abstract

We exhibit direct relations between the modular double of U_q(sl(2,R)) and the quantum Teichmueller theory. Explicit representations for the fusion- and braiding operations of the quantum Teichmueller theory are immediate consequences. Our results include a simplified derivation of the Clebsch-Gordan decomposition for the principal series of representation of the modular double of U_q(sl(2,R)).

Paper Structure

This paper contains 43 sections, 11 theorems, 139 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Some notations and conventions
  3. Modular double
  4. Prinicpal series representations of ${\mathcal{U}}_q({\mathfrak s}{\mathfrak l}(2,{\mathbb R}))$
  5. Modular duality
  6. The Whittaker model for ${\mathcal{D}}{\mathcal{U}}_q({\mathfrak s}{\mathfrak l}_2)$
  7. The model space
  8. The R-operator
  9. The Clebsch-Gordan maps of the modular double
  10. Definition of Clebsch-Gordan maps
  11. Factorization of Clebsch-Gordan maps
  12. Construction of the Clebsch-Gordan maps
  13. Verification of intertwining property
  14. The b-Clebsch-Gordan coefficients
  15. The fusion operation
  16. ...and 28 more sections

Key Result

Theorem 1

The operator ${\mathsf R}$ has the following properties:

Figures (6)

  • Figure 1: Graphical representation of the vertex $v$ dual to a triangle $t$. The marked corner defines a corresponding numbering of the edges that emanate at $v$.
  • Figure 2: The move $B_v:\sigma\to B_v\sigma$
  • Figure 3: The move $Z_v:\sigma\to Z_v\sigma$
  • Figure 4: The move $F_e:\sigma_s\to \sigma_t\equiv F_e\sigma_s$
  • Figure 5: Graphical representation for Equation \ref{['SFprform']}.
  • ...and 1 more figures

Theorems & Definitions (15)

  • Theorem 1
  • Lemma 1
  • Proposition 1
  • Proposition 2
  • Lemma 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • proof
  • Proposition 6
  • ...and 5 more