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Liouville bootstrap via harmonic analysis on a noncompact quantum group

B. Ponsot, J. Teschner

TL;DR

This work advances Liouville theory by formulating the Liouville bootstrap as a problem of harmonic analysis on a noncompact, self-dual quantum group. By positing invertible fusion transformations, the authors derive a closed system of functional equations for the fusion coefficients, showing a unique solution for irrational ${c>25}$, constructed from Racah coefficients of a continuous series of ${\mathcal U}_q({\mathfrak{sl}}(2,\mathbb{R}))$ representations and their Clebsch-Gordan data; the same Racah data satisfy the Moore-Seiberg pentagon equations and extend analytically to the strong coupling region ${1<c<25}$ via ${b\leftrightarrow b^{-1}}$ duality. The main technical achievements include an explicit Racah-coefficient formula via overlaps of specially constructed eigenfunctions, a complete harmonic analysis framework on ${SL_q^+(2,\mathbb{R})}$, and a demonstration that these coefficients solve the full fusion equations, thereby confirming crossing symmetry and locality of Liouville correlators within the stated assumptions. This work also points to concrete applications to boundary Liouville theory and D-brane physics, while highlighting a rare example of a nonclassical, self-dual quantum group underpinning a noncompact CFT bootstrap, with connections to Teichmüller space quantization.

Abstract

The purpose of this short note is to announce results that amount to a verification of the bootstrap for Liouville theory in the generic case under certain assumptions concerning existence and properties of fusion transformations. Under these assumptions one may characterize the fusion and braiding coefficients as solutions of a system of functional equations that follows from the combination of consistency requirements and known results. This system of equations has a unique solution for irrational central charge c>25. The solution is constructed by solving the Clebsch-Gordan problem for a certain continuous series of quantum group representations and constructing the associated Racah-coefficients. This gives an explicit expression for the fusion coefficients. Moreover, the expressions can be continued into the strong coupling region 1<c<25, providing a solution of the bootstrap also for this region.

Liouville bootstrap via harmonic analysis on a noncompact quantum group

TL;DR

This work advances Liouville theory by formulating the Liouville bootstrap as a problem of harmonic analysis on a noncompact, self-dual quantum group. By positing invertible fusion transformations, the authors derive a closed system of functional equations for the fusion coefficients, showing a unique solution for irrational , constructed from Racah coefficients of a continuous series of representations and their Clebsch-Gordan data; the same Racah data satisfy the Moore-Seiberg pentagon equations and extend analytically to the strong coupling region via duality. The main technical achievements include an explicit Racah-coefficient formula via overlaps of specially constructed eigenfunctions, a complete harmonic analysis framework on , and a demonstration that these coefficients solve the full fusion equations, thereby confirming crossing symmetry and locality of Liouville correlators within the stated assumptions. This work also points to concrete applications to boundary Liouville theory and D-brane physics, while highlighting a rare example of a nonclassical, self-dual quantum group underpinning a noncompact CFT bootstrap, with connections to Teichmüller space quantization.

Abstract

The purpose of this short note is to announce results that amount to a verification of the bootstrap for Liouville theory in the generic case under certain assumptions concerning existence and properties of fusion transformations. Under these assumptions one may characterize the fusion and braiding coefficients as solutions of a system of functional equations that follows from the combination of consistency requirements and known results. This system of equations has a unique solution for irrational central charge c>25. The solution is constructed by solving the Clebsch-Gordan problem for a certain continuous series of quantum group representations and constructing the associated Racah-coefficients. This gives an explicit expression for the fusion coefficients. Moreover, the expressions can be continued into the strong coupling region 1<c<25, providing a solution of the bootstrap also for this region.

Paper Structure

This paper contains 9 sections, 3 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Bootstrap for Liouville theory
  3. Difference equations for the fusion coefficients
  4. A tensor category of quantum group representations
  5. Space of functions on the quantum group
  6. Calculation of Racah coefficients
  7. Functional equations for Racah-Wigner coefficients
  8. Discussion
  9. Appendix: Special functions

Key Result

Theorem 1

The ${\mathcal{U}}_{q}({\mathfrak s}{\mathfrak l}(2,{\mathbb R}))$-representation ${\mathcal{P}}_{\alpha_2}\otimes{\mathcal{P}}_{\alpha_1}$ defined on $L^2({\mathbb R}^2)$ by means of $\Delta$ decomposes as follows

Theorems & Definitions (11)

  • Remark 1
  • Conjecture 1
  • Remark 2
  • Conjecture 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Theorem 1
  • Theorem 2
  • Remark 6
  • ...and 1 more