# Crossing Symmetry in Alpha Space

## TL;DR

This paper introduces a novel approach to the conformal bootstrap in one-dimensional CFTs by using Sturm-Liouville theory and alpha space, transforming crossing symmetry into an eigenvalue problem for an integral operator.

## Contribution

It develops a systematic method for conformal block decomposition and analyzes crossing symmetry as an eigenvalue problem in alpha space, connecting to Wilson transform eigenfunctions.

## Key findings

- Eigenfunctions of the integral operator K can be found in closed form.
- The method provides a new perspective on crossing symmetry as an eigenvalue problem.
- The approach simplifies conformal bootstrap computations in 1D CFTs.

## Abstract

We initiate the study of the conformal bootstrap using Sturm-Liouville theory, specializing to four-point functions in one-dimensional CFTs. We do so by decomposing conformal correlators using a basis of eigenfunctions of the Casimir which are labeled by a complex number alpha. This leads to a systematic method for computing conformal block decompositions. Analyzing bootstrap equations in alpha space turns crossing symmetry into an eigenvalue problem for an integral operator K. The operator K is closely related to the Wilson transform, and some of its eigenfunctions can be found in closed form.