Modular Data: The Algebraic Combinatorics of Conformal Field Theory

Abstract

Abstract. This paper is primarily intended as an introduction for mathematicians to some of the rich algebraic combinatorics arising in for instance conformal field theory (CFT). It tries to refine, modernise, and bridge the gap between papers [6] and [55]. Our paper is essentially self-contained, apart from some of the background motivation (Section 1) and examples (Section 3) which are included to give the reader a sense of the context. Detailed proofs will appear elsewhere. The theory is still a work-in-progress, and emphasis is given here to several open questions and problems.

Nonsemisimple Fusion Algebras and the Verlinde Formula

“…We also note the following complications that are already apparent in nonunitary semisimple fusion (see the relevant remarks in [16]), but become more acute for nonsemisimple fusion:…”

“…We restrict our attention to the particular case where ν A ≤ 2 because of the lack of instructive examples of higher-"rank" logarithmic theories; the definitions may need to be refined as further examples are worked out. When the set of idempotents with ν A = 2 is empty, we recover the semisimple case [16] (we do not impose conditions F2 and F3 in [16] because they imply semisimplicity of the fusion algebra).…”

Higher-Level Appell Functions, Modular Transformations, and Characters

“…As noted in [16], (κ( · , a; q), ϑ(q, · )) is a section of a rank-2 vector bundle over the torus. For K ℓ with ℓ > 1, we unify K ℓ and the theta functions arising in the open quasiperiodicity formula for K ℓ into a vector K ℓ representing 3 We are grateful to V.I. Ritus for the elegant derivation in Eqs.…”

“…Strictly speaking, this applies to rational conformal field theories, where modular properties of the characters and the structure of the Verlinde formula are known, at least in principle (for a discussion of the modular transformation properties of characters and other quantities and for further references, see [3,4,5,6,7]). …”

On character generators for simple Lie algebras

“…The Demazure character formulas [6] are valid for all simple Lie algebras (and others), and lead directly to non-negative expressions for the characters. Gaskell [9] wrote a general formula for the character generator as a Demazure operator acting on the highest-weight generating function (see (97) below), and calculated several examples. 3 In the second part of this paper, we will push Gaskell's methods, and apply Demazure character formulas to the character generator.…”