In search of conformal theories

TL;DR

This paper reformulates the conformal crossing equation to clarify conformal symmetry and explores its solutions for various Lie groups, revealing infinitely many solutions, including for the conformal group.

Contribution

It introduces a new formulation of the crossing equation that makes conformal symmetry more transparent and generalizes it to arbitrary Lie groups.

Findings

Infinite solutions to the G-crossing equation for SU(2).

Infinite solutions to the conformal crossing equation for SO(d+1,1).

Enhanced understanding of conformal data constraints.

Abstract

The conformal crossing equation puts very stringent constraints on the conformal data. We formulate it in way that makes the conformal symmetry more transparent. This allows for generalization of the crossing equation to arbitrary Lie group G. Using the crossing equation for SU(2) as a toy model, we find infinitely many solutions to the G-crossing equation. In particular, when G is specialized to the conformal group SO(d+1,1), we get infinitely many solutions to the conformal crossing equation.