In search of conformal theories

TL;DR

The paper reframes the conformal crossing equation in a group-theoretic language based on $SO(d+1,1)$ and harmonic analysis, enabling a generalization to arbitrary groups $G$. It identifies tempered principal-series representations $[\Delta,\ell]$ with $\Delta=\frac{d}{2}+ic$ and uses the Racah coefficient $W$ and the pentagon identity to construct an infinite family of analytic solutions to the generalized $G$-crossing equation, including a conformal 6j-symbol construction. A residue-contour approach links these principal-series solutions to the physical conformal spectrum, providing a bridge between group theory and the bootstrap. The work suggests deep connections between representation theory, boundary CFT, and AdS/CFT-type dualities, with potential applications to higher-dimensional theories and nontrivial boundary conditions.

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.

Figures (6)

• Figure 1: The integral over the principal series representations is represented by the contour A. It can be expressed as a sum over physical representations corresponding to the poles on the real axis given that the contour can be deformed to B without picking any additional complex poles.
• Figure 2: Converting the principal series representations to physical representations.
• Figure 3: Graphical representation of the operators and their product algebra.
• Figure 4: Geometric interpretation of the $6j$-symbol as a tetrahedron.
• Figure 5: Two triangulations of octahedron corresponding to the two sides of the pentagon identity.