Meromorphic cosets and the classification of three-character CFT

Abstract

We investigate the admissible vector-valued modular forms having three independent characters and vanishing Wronskian index and determine which ones correspond to genuine 2d conformal field theories. This is done by finding bilinear coset-type relations that pair them into meromorphic characters with central charges 8, 16, 24, 32 and 40. Such pairings allow us to identify some characters with definite CFTs and rule out others. As a key result we classify all unitary three-character CFT with vanishing Wronskian index, excluding c = 8, 16. The complete list has two infinite affine series Br,1, Dr,1 and 45 additional theories. As a by-product, at higher values of the total central charge we also find constraints on the existence or otherwise of meromorphic theories. We separately list several cases that potentially correspond to Intermediate Vertex Operator Algebras.

Classifying three-character RCFTs with Wronskian index equalling 0 or 2

“…Combining the two facts (a) all solutions to the [3, 3] MLDE lie on the plane (58), (b) the MLDE on the plane reduces to a two-parameter rigid MLDE, we can arrive at the following statement : the [3,3] MLDE is in some sense a two-parameter rigid MLDE in disguise. A better theory of MLDEs and their parameter spaces and solutions, incorporating considerations coming from seriesexpansions around other cusps (beyond the τ = i∞, q = 0 considered in this paper), has appeared in [50]. Applying that theory to the [3,3] MLDE would properly explain this statement 14 .…”

“…CNG thanks Arpit Das and Sunil Mukhi for collaborations in [14,30,50]; he is especially indebted to the latter for providing understanding and insights into contemporary RCFT theory. CNG also thanks Iosif Bena and gratefully acknowledges the hospitality of CEA Saclay where some part of this work was done.…”

“…Initially, admissible character solutions to the non-rigid [2,6] MLDE were obtained in [20], not by directly solving the MLDE, but in an indirect manner via the theory of quasi-characters [18]. A very recent paper [50] gives a general theory of non-rigid MLDEs and their solutions. Other studies of two-character…”

“…See also appendix of [50]. 6 Non-rigid MLDEs and non-rigid parameters display some phenomenon which are absent in MLDEs with only rigid parameters.…”

“…7 Although there are cogent arguments why this should be so, there is no proof for the general case. The recent work [50] proves, starting from the MLDE, that every Frobenius solution of the [2,6] study based on modular data and SL(2, Z) representation theory while the latter two directly solve the two-parameter MLDE. The stage (III) of the analysis that identifies genuine CFT amongst the large number (303, excluding an infinite class) of admissible character solutions was done in [30].…”