The Quantum Group Structure of 2D Gravity and Minimal Models II: The Genus-Zero Chiral Bootstrap

TL;DR

This paper derives explicit formulas for the F and B matrices in 2D gravity and minimal models using quantum-group symmetry, clarifying their algebraic structure and connections to 3D gravity.

Contribution

It provides new explicit formulas for the F and B matrices involving quantum-group symmetries and clarifies their relationship with different chiral operator-algebras.

Findings

Derived closed-form F and B matrices involving additional coupling constants.

Clarified the relationship between IRF-type and vertex-type quantum-group-invariant algebras.

Connected the operator algebra to a generalized 3D gravity model with tetrahedra.

Abstract

The F and B matrices associated with Virasoro null vectors are derived in closed form by making use of the operator-approach suggested by the Liouville theory, where the quantum-group symmetry is explicit. It is found that the entries of the fusing and braiding matrices are not simply equal to quantum-group symbols, but involve additional coupling constants whose derivation is one aim of the present work. Our explicit formulae are new, to our knowledge, in spite of the numerous studies of this problem. The relationship between the quantum-group-invariant (of IRF type) and quantum-group-covariant (of vertex type) chiral operator-algebras is fully clarified, and connected with the transition to the shadow world for quantum-group symbols. The corresponding 3-j-symbol dressing is shown to reduce to the simpler transformation of Babelon and one of the author (J.-L. G.) in a suitable infinite limit defined by analytic continuation. The above two types of operators are found to coincide when applied to states with Liouville momenta going to $\infty$ in a suitable way. The introduction of quantum-group-covariant operators in the three dimensional picture gives a generalisation of the quantum-group version of discrete three-dimensional gravity that includes tetrahedra associated with 3-j symbols and universal R-matrix elements. Altogether the present work gives the concrete realization of Moore and Seiberg's scheme that describes the chiral operator-algebra of two-dimensional gravity and minimal models.