Fusion transformations in Liouville theory

Abstract

We study the fusion kernel for nondegenerate conformal blocks in the Liouville theory as a solution of difference equations originating from the pentagon identity. We propose an approach for solving these equations based on a “nonperturbative” series expansion that allows calculating the fusion kernel iteratively. We also find exact solutions for the special central charge values c = 1+6(b − b⁻¹)², b ∈ ℕ. For c = 1, the obtained result reproduces the formula previously obtained from analytic properties of a solution of a Painlev´e equation, but our solution has a significantly simplified form.