Liouville bootstrap via harmonic analysis on a noncompact quantum group

TL;DR

This paper verifies the bootstrap for Liouville theory at irrational central charge c>25 by solving functional equations derived from quantum group representations, providing explicit fusion coefficients and extending results into the strong coupling region.

Contribution

It introduces a method to explicitly solve the Liouville bootstrap using harmonic analysis on a noncompact quantum group, covering both weak and strong coupling regions.

Findings

Explicit fusion coefficients for irrational c>25

Solution extended into the strong coupling region 1<c<25

Verification of bootstrap consistency under certain assumptions

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.