Classification of Unitary RCFTs with Two Primaries and Central Charge Less Than 25

Meromorphic cosets and the classification of three-character CFT

“…Returning to the three-character case, the restriction on Wronskian index makes this in one sense a weaker classification than that of [39] for two primaries, where there was no restriction on the Wronskian index, but in one sense also stronger since the present work has no restriction on the central charge. This should finally bring closure to a programme for the "simplest" three-character theories (those with vanishing Wronskian index) that was initiated over three decades ago in [3].…”

“…While much of the literature cited above has focused on classifying admissible characters, from the CFT point of view the result should be interpreted as a "superset" of candidates of which actual CFTs form a subset. The problem of identifying this subset has been addressed in varying degrees of detail, for small numbers of characters, in [3,17,18,20,24,27,28,[35][36][37][38][39]. In the present work we take this goal forward by completing the classification of three-character CFT with vanishing Wronskian index (explained below) for any central charge, but excluding central charge = 8, 16 where the classification of admissible characters is itself problematic [21,[33][34][35].…”

“…Apart from the fact that we restrict the Wronskian index but not the central charge, the approach in the present work has some other important differences from [39]. Here we start from a given finite set of admissible characters, then look for bilinear coset-type relations for them based on their q-expansion.…”

“…This technical point is of importance because it seems to have been missed in much of the previous literature, starting with [17] that only considered a special sub-class of cosets where each n i ≥ 2. More general cosets of meromorphic theories were studied recently, and apparently for the first time, in [39] in the context of theories with exactly two primaries. The values of n i have considerable significance for the structure of the bilinear pair, which we now explain.…”

“…Suppose a bilinear relation holds between CFTs C, C with Kac-Moody algebras h, h, and they pair up to a CFT H with Kac-Moody algebra g. Then h must be the commutant of h in g. Now suppose that for any of i = 1, 2 we have n i = 1. This means that some spin-1 currents in the theory H arise as "composites" of primaries in C, C. This in turn means the total number of spin-1 currents of C, C is strictly smaller than that of H, in other words dim h + dim h < dim g, so the embedding of h in g is a non-trivial onetypically h is embedded into a simple factor of g. Such cases were discussed for the case of two primaries in [39]. On the other hand whenever all n i ≥ 2, no currents of H can arise as composites of primaries of C, C. Therefore we have dim h + dim h = dim g. This can only happen if g is non-simple and h corresponds to one or more of its simple factors.…”