Global symmetry and conformal bootstrap in the two-dimensional $Q$-state Potts model

TL;DR

This paper numerically analyzes the four-point functions of the two-dimensional Potts conformal field theory across complex Q values, revealing solutions consistent with symmetry constraints and uncovering additional solutions with unknown significance.

Contribution

It provides the first numerical solutions to crossing-symmetry equations for the Potts CFT for complex Q, detailing spectra, fusion rules, and symmetry properties.

Findings

Found crossing-symmetry solutions consistent with $S_Q$ symmetry.

Determined spectra and fusion rules for several four-point functions.

Discovered extra solutions with unknown interpretations.

Abstract

The Potts conformal field theory is an analytic continuation in the central charge of conformal field theory describing the critical two-dimensional $Q$-state Potts model. Four-point functions of the Potts conformal field theory are dictated by two constraints: the crossing-symmetry equation and $S_Q$ symmetry. We numerically solve the crossing-symmetry equation for several four-point functions of the Potts conformal field theory for $Q\in\mathbb{C}$. In all examples, we find crossing-symmetry solutions that are consistent with $S_Q$ symmetry of the Potts conformal field theory. In particular, we have determined their numbers of crossing-symmetry solutions, their exact spectra, and a few corresponding fusion rules. In contrast to our results for the $O(n)$ model, in most of examples, there are extra crossing-symmetry solutions whose interpretations are still unknown.