A B\"acklund transformation for elliptic four-point conformal blocks

TL;DR

This paper introduces an integral transformation technique to generate a lattice of conformal blocks in Liouville theory, providing new integral representations for elliptic four-point blocks with potential applications in conformal field theory.

Contribution

It presents a novel integral transformation method that systematically constructs elliptic four-point conformal blocks in Liouville theory, expanding the toolkit for analyzing conformal correlators.

Findings

Derived integral representations for elliptic conformal blocks.

Established a lattice of conformal dimensions via repeated transformations.

Applied the method specifically to elliptic blocks from previous work.

Abstract

We apply an integral transformation to solutions of a partial differential equation for five-point correlation functions in Liouville theory on a sphere with one degenerate field $V_{-\frac{1}{2b}}$. By repeating this transformation, we can reach a whole lattice of values for the conformal dimensions of the four other operators. Factorizing out the degenerate field leads to integral representations of the corresponding four-point conformal blocks. We illustrate this procedure on the elliptic conformal blocks discovered in a previous publication.