The Conformal Characters

TL;DR

This paper provides a comprehensive mathematical framework for understanding conformal multiplets across dimensions, focusing on highest weight representations, their characters, and unitarity conditions, with explicit classifications and computational tools.

Contribution

It offers a unified, mathematically rigorous treatment of conformal multiplet characters and unitarity, including explicit classifications and computational methods up to rank four.

Findings

Classification of unitary highest weight representations.

Explicit character formulas using Kazhdan-Lusztig polynomials.

Unified treatment applicable to all dimensions.

Abstract

We revisit the study of the multiplets of the conformal algebra in any dimension. The theory of highest weight representations is reviewed in the context of the Bernstein-Gelfand-Gelfand category of modules. The Kazhdan-Lusztig polynomials code the relation between the Verma modules and the irreducible modules in the category and are the key to the characters of the conformal multiplets (whether finite dimensional, infinite dimensional, unitary or non-unitary). We discuss the representation theory and review in full generality which representations are unitarizable. The mathematical theory that allows for both the general treatment of characters and the full analysis of unitarity is made accessible. A good understanding of the mathematics of conformal multiplets renders the treatment of all highest weight representations in any dimension uniform, and provides an overarching comprehension of case-by-case results. Unitary highest weight representations and their characters are classified and computed in terms of data associated to cosets of the Weyl group of the conformal algebra. An executive summary is provided, as well as look-up tables up to and including rank four.