On parity functions in conformal field theories

Abstract

We examine general aspects of parity functions arising in rational conformal field theories, as a result of Galois theoretic properties of modular transformations. We focus more specifically on parity functions associated with affine Lie algebras, for which we give two efficient formulas. We investigate the consequences of these for the modular invariance problem.a Chercheur Qualifié FNRS

Modular Data: The Algebraic Combinatorics of Conformal Field Theory

“…. , k} of the affine algebra A (1) 1 at even level k (see (3.5) below). Also, the 'classifying algebra' in boundary conformal field theory [17] can be a generalised fusion ring.…”

“…The source of some of the most interesting modular data are the affine nontwisted KacMoody algebras X (1) r . The simplest way to construct affine algebras is to let X r be any finite-dimensional simple (more generally, reductive) Lie algebra.…”

“…This is a Lie algebra, using the obvious bracket, and is infinite-dimensional. The affine algebra X (1) r is simply a certain central extension of the loop algebra. (As usual, the central extension is taken in order to get a rich supply of representations.…”

“…The representation theory of X (1) r is analogous to that of X r . We are interested in the so-called integral highest weight representations.…”

“…Thanks mostly to the structure and action of the affine Weyl group on the Cartan subalgebra of X (1) r , the character χ λ is essentially a lattice theta function, and so transforms nicely under the modular group SL 2 (Z). In fact, for fixed algebra X (1) r and level k ∈ Z ≥ , these χ λ define a representation of SL 2 (Z), exactly as in (1.2) above, and the matrices S and T constitute modular data. The 'identity' is 0 = k 0 , and the set of 'primaries' is the highest weights…”