NEW EXCEPTIONAL $(A_1^{(1)})^{\oplus r} $ INVARIANTS AND THE ASSOCIATED GEPNER MODELS

Abstract

New modular invariant partition functions for tensor products of [Formula: see text] affine Lie algebras are presented. These exceptional modular invariants can be understood in terms of automorphisms of the fusion rules of the affine algebra or of its extensions. There are three isolated cases, as well as an infinite series of new invariants. As an application, the new modular invariants are employed to produce new Gepner type compactifications of the heterotic string.

ON THE LANDAU–GINZBURG DESCRIPTION OF $(A_1^{(1)})^{\oplus N}$ INVARIANTS

“…Note that this requires the use of appropriate discrete torsions, as explained at the end of section 4.2. Actually some of the orbifold results have been reproduced incorrectly in table 3 of [14]. Therefore we also list, in table 2, the correct spectra for these theories.…”

“…Ten of these correspond to A-D-E-type potentials, namely C (1,1) [10], C (1,2,4) [10], C (1,1,2,2) [5], C (1,5) [30], C (1,10,10) [30], C (2,5,14) [30], C (1,5,5,7) [15], C (5,6,10) [30], C (3,5,5,5) [15], and C (4,5,5) [20], while the remaining ones cannot be obtained from tensor products of minimal models: C (1,2) [15], C (1,4) [25], C (2,5) [35], C (4,5) [45], C (2,7,19) [40], C (2,16,17) [50], C (3,19,20) [60], C (7,10,25) [60], C (10,16,37) [90], and C (3,4,5,6) [15]. All but the first three of these configurations lead to undesirable states in the untwisted sector or to asymmetric chiral states in the twisted sectors.…”

“…The corresponding numbers for the invariants (9) and (10) have been calculated in [12] and [14], respectively. In the case of the modular invariant (9), we have checked that none of the orbifolds is able to reproduce these numbers in all cases, which is another proof that this invariant cannot be described by an orbifold of a non-degenerate Landau-Ginzburg model with respect to a manifest symmetry of its potential.…”

“…Let us remark that only for N = 2 is the invariant defined by (17) 'fundamental' in the sense [22] that it cannot be obtained by forming sums and/or products of the invariant tensors corresponding to simpler invariants; in that case this invariant was found in [13,14]. Also note the structural difference between even and odd values of N which shows up when describing the left-right symmetric primary fields: for these, either I 1 = ∅, or else N 0 ∈ 2Z Z + N + 1 and l i = k i /2 for all i ∈ I 0 .…”

“…In the case of the invariant (25), as well as for (15) with N = 2 and k 1 , k 2 ∈ 4Z Z, these spectra have already been listed in [14]. The spectra for a few other models employing invariants of the type (15) are listed in table 1, namely for some models with …”