Solving the Ward Identities of Irrational Conformal Field Theory

TL;DR

This paper analyzes the affine-Virasoro Ward identities in irrational conformal field theory, providing solutions for correlators, an algebraic formulation for global solutions, and insights into braiding and fusion rules.

Contribution

It introduces a method to solve the Ward identities for all affine-Virasoro constructions, including irrational cases, and explores their correlators and algebraic structure.

Findings

Solved correlators for affine-Sugawara nests

Developed an algebraic formulation for global solutions

Identified braiding and fusion rule behaviors in correlators

Abstract

The affine-Virasoro Ward identities are a system of non-linear differential equations which describe the correlators of all affine-Virasoro constructions, including rational and irrational conformal field theory. We study the Ward identities in some detail, with several central results. First, we solve for the correlators of the affine-Sugawara nests, which are associated to the nested subgroups $g\supset h_1 \supset \ldots \supset h_n$. We also find an equivalent algebraic formulation which allows us to find global solutions across the set of all affine-Virasoro constructions. A particular global solution is discussed which gives the correct nest correlators, exhibits braiding for all affine-Virasoro correlators, and shows good physical behavior, at least for four-point correlators at high level on simple $g$. In rational and irrational conformal field theory, the high-level fusion rules of the broken affine modules follow the Clebsch-Gordan coefficients of the representations.