Implications of an arithmetical symmetry of the commutant for modular invariants

Abstract

We point out the existence of an arithmetical symmetry for the commutant of the modular matrices S and T . This symmetry holds for all affine simple Lie algebras at all levels and implies the equality of certain coefficients in any modular invariant. Particularizing to SU (3) k , we classify the modular invariant partition functions when k + 3 is an integer coprime with 6 and when it is a power of either 2 or 3. Our results imply that no detailed knowledge of the commutant is needed to undertake a classification of all modular invariants.

On parity functions in conformal field theories

“…These rather trivial solutions can lead to non-trivial couplings in terms of the weights, and it turns out that many apparently non-trivial couplings are in fact trivial in this sense. For instance in su(5), it was found in [6], and checked the hard way, that the identity p = (1, 1, 1, 1) can couple, for even n, to the following three weights p ′ = (1,…”

“…In the case of su(3), the parity equation is considerably more complex, and it is only recently that the general solution has been given [7], though in a totally different context. As noticed in [6], the su(3) parity plays a fundamental role in the description of the Jacobian varieties of the complex Fermat curves, and it is in this geometric setting that, in disguise, the equation for su(3) was solved in all generality (see [8] for a review of the connections between the two problems). The su(3) solution yields, as a special case, the solution for the su(2) case.…”

“…An algorithm to compute the parity of an arbitrary weight can be given, that requires evaluating congruences on Dynkin labels and determinants of permutations (see [6] for G = A ℓ ). It is not our purpose to describe that algorithm in the general case since, as we shall soon see, G parities can be reduced to the much simpler su(2) parities, which we now make explicit.…”

Ramond sector characters and N = 2 Landau-Ginzburg models

“…Consider the SU(3) Kazama-Suzuki theory with modular invariant arising from the SU (3) 9 exceptional modular invariant (see for instance [20], and [22] for a complete classification) which reads…”

Modular Data: The Algebraic Combinatorics of Conformal Field Theory

“…No. However, a remarkable connection [101,12] has been observed between the A (1) 2 level k modular invariants, and the Jacobian of the Fermat curve x k+3 + y k+3 + z k+3 = 0. In particular, the A…”