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Recent progress on Liouville Field Theory

Benedicte Ponsot

TL;DR

Liouville field theory with a noncompact spectrum challenges the conformal bootstrap program. The paper builds the Liouville fusion monodromy from Racah–Wigner data of the continuous-series representations of U_q(sl(2,R)), yielding an explicit fusion matrix and a proof of crossing symmetry in the weak-coupling regime with analytic continuation to strong coupling. It also provides a complete construction of boundary Liouville correlators, including the boundary three-point function and the normalization of boundary operators, expressed through the same quantum-group data. Together, these results reveal a concrete quantum-group backbone for Liouville CFT and supply exact, nonperturbative data for bulk and boundary correlators in noncompact conformal field theories.

Abstract

An explicit construction for the monodromy of the Liouville conformal blocks in terms of Racah-Wigner coefficients of the quantum group U_q(sl(2,R)) is proposed. As a consequence, crossing-symmetry for four point functions is analytically proven, and the expression for the correlator of three boundary operators is obtained.

Recent progress on Liouville Field Theory

TL;DR

Liouville field theory with a noncompact spectrum challenges the conformal bootstrap program. The paper builds the Liouville fusion monodromy from Racah–Wigner data of the continuous-series representations of U_q(sl(2,R)), yielding an explicit fusion matrix and a proof of crossing symmetry in the weak-coupling regime with analytic continuation to strong coupling. It also provides a complete construction of boundary Liouville correlators, including the boundary three-point function and the normalization of boundary operators, expressed through the same quantum-group data. Together, these results reveal a concrete quantum-group backbone for Liouville CFT and supply exact, nonperturbative data for bulk and boundary correlators in noncompact conformal field theories.

Abstract

An explicit construction for the monodromy of the Liouville conformal blocks in terms of Racah-Wigner coefficients of the quantum group U_q(sl(2,R)) is proposed. As a consequence, crossing-symmetry for four point functions is analytically proven, and the expression for the correlator of three boundary operators is obtained.

Paper Structure

This paper contains 16 sections, 80 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Three point function DornZZ
  4. Crossing symmetry
  5. Properties of the fusion matrix
  6. Construction of the fusion matrix: strategy PT1
  7. Clebsch-Gordan coefficients and Racah-Wigner symbol for a continuous series of representations of $\mathcal{U}_q(\mathfrak{sl}(2,\mathbb{R}))$ PT1PT2
  8. A class of representations of $\mathcal{U}_q(\mathfrak{sl}(2,\mathbb{R}))$
  9. Clebsch-Gordan decomposition of tensor products
  10. Racah-Wigner coefficients for $\mathcal{U}_q(\mathfrak{sl}(2,\mathbb{R}))$
  11. Explicit computation of the Racah-Wigner coefficients
  12. Expression of the fusion matrix
  13. Application: Boundary three point function PT3
  14. Associativity condition
  15. Normalization of the boundary operator
  16. ...and 1 more sections