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.
Highlights
Quantum field theory is one of the most successful tools of theoretical physics
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
We review how the multiplicities of the irreducible modules in the Verma modules are given by the evaluation of Kazhdan-Lusztig polynomials at argument equal to one, and how the inversion of the decomposition fixes the irreducible characters
Summary
Quantum field theory is one of the most successful tools of theoretical physics. It is ubiquitous in our understanding of physical phenomena from the smallest to the largest scales. We observe that the expression of the character of the irreducible module depends on the relative position of the weight λ with respect to (minus) the Weyl vector −ρ = −1 All these observations generalize to other semisimple Lie algebras.. The root system Φ[λ] determines the integrality class of λ In this low rank case, the integer coefficients bw,w again simplify to a sign depending on the length of the elements in the group W[λ]. The character formula takes the form (2.7), but where the sum is restricted to the Weyl group elements W[λ] and the length function is inside this group In this manner, we have found the characters of all irreducible highest weight representations of the B2 algebra
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