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.Comment: 16 pages, typos removed incl. important one in (48
