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6j symbols for the modular double, quantum hyperbolic geometry, and supersymmetric gauge theories

J. Teschner, G. S. Vartanov

TL;DR

The paper defines and analyzes the $b$-$6j$ symbols for the modular double of ${\mathcal U}_q({\mathfrak sl}(2,{\mathbb R}))$, identifying natural normalization choices and deriving new integral representations that reveal structural parallels with Liouville theory and Teichmüller theory.A central result is that the semiclassical limit $b\to0$ of the $b$-$6j$ symbols reproduces the hyperbolic volume of non-ideal tetrahedra, with the semiclassical action serving as the generating function for Fenchel-Nielsen coordinates.The work also establishes a deep connection to ${\rm Diff}(S^1)$ via unitary normalization, showing that the $6j$ symbols coincide with corresponding invariants in ${\rm Diff}(S^1)$ tensor products and with Liouville fusion kernels.Furthermore, the authors link these algebraic objects to two-dimensional quantum hyperbolic geometry and propose interpretations as partition functions of three-dimensional ${\mathcal N}=2$ gauge theories, suggesting avenues for geometric constructions of 3d theories from non-ideal tetrahedral decompositions.

Abstract

We revisit the definition of the 6j-symbols from the modular double of U_q(sl(2,R)), referred to as b-6j symbols. Our new results are (i) the identification of particularly natural normalization conditions, and (ii) new integral representations for this object. This is used to briefly discuss possible applications to quantum hyperbolic geometry, and to the study of certain supersymmetric gauge theories. We show, in particular, that the b-6j symbol has leading semiclassical asymptotics given by the volume of a non-ideal tetrahedron. We furthermore observe a close relation with the problem to quantize natural Darboux coordinates for moduli spaces of flat connections on Riemann surfaces related to the Fenchel-Nielsen coordinates. Our new integral representations finally indicate a possible interpretation of the b-6j symbols as partition functions of three-dimensional N=2 supersymmetric gauge theories.

6j symbols for the modular double, quantum hyperbolic geometry, and supersymmetric gauge theories

TL;DR

The paper defines and analyzes the $b$-$6j$ symbols for the modular double of ${\mathcal U}_q({\mathfrak sl}(2,{\mathbb R}))$, identifying natural normalization choices and deriving new integral representations that reveal structural parallels with Liouville theory and Teichmüller theory.A central result is that the semiclassical limit $b\to0$ of the $b$-$6j$ symbols reproduces the hyperbolic volume of non-ideal tetrahedra, with the semiclassical action serving as the generating function for Fenchel-Nielsen coordinates.The work also establishes a deep connection to ${\rm Diff}(S^1)$ via unitary normalization, showing that the $6j$ symbols coincide with corresponding invariants in ${\rm Diff}(S^1)$ tensor products and with Liouville fusion kernels.Furthermore, the authors link these algebraic objects to two-dimensional quantum hyperbolic geometry and propose interpretations as partition functions of three-dimensional ${\mathcal N}=2$ gauge theories, suggesting avenues for geometric constructions of 3d theories from non-ideal tetrahedral decompositions.

Abstract

We revisit the definition of the 6j-symbols from the modular double of U_q(sl(2,R)), referred to as b-6j symbols. Our new results are (i) the identification of particularly natural normalization conditions, and (ii) new integral representations for this object. This is used to briefly discuss possible applications to quantum hyperbolic geometry, and to the study of certain supersymmetric gauge theories. We show, in particular, that the b-6j symbol has leading semiclassical asymptotics given by the volume of a non-ideal tetrahedron. We furthermore observe a close relation with the problem to quantize natural Darboux coordinates for moduli spaces of flat connections on Riemann surfaces related to the Fenchel-Nielsen coordinates. Our new integral representations finally indicate a possible interpretation of the b-6j symbols as partition functions of three-dimensional N=2 supersymmetric gauge theories.

Paper Structure

This paper contains 33 sections, 5 theorems, 127 equations.

Table of Contents

  1. Introduction
  2. Racah-Wigner 6j symbols for the modular double
  3. Self-dual representations of ${\mathcal{U}}_q({\mathfrak s}{\mathfrak l}(2,{\mathbb R}))$ and the modular double
  4. Normalized Clebsch-Gordan coefficients for the modular double
  5. Normalized b-6j symbols for the modular double
  6. 3j symbols for the modular double
  7. A new integral formula for the b-6j symbols
  8. Relations to three-dimensional hyperbolic geometry
  9. Volumes of non-ideal tetrahedra
  10. Semiclassical limit
  11. Relation to Liouville theory and the representation theory of ${\rm Diff(S^1)}$
  12. Fusion kernel
  13. Unitary normalization
  14. 6j symbols of ${\rm Diff}(S^1)$
  15. Application to two-dimensional quantum hyperbolic geometry
  16. ...and 18 more sections

Key Result

Theorem 1

S2 where and

Theorems & Definitions (5)

  • Theorem 1
  • Corollary 1
  • Corollary 2
  • Corollary 3
  • Corollary 4