In the modular linear differential equation (MLDE) approach to classifying rational conformal field theories (RCFTs) both the MLDE and the RCFT are identified by a pair of non-negative integers [n,l]. n is the number of characters of the RCFT as well as the order of the MLDE that the characters solve and l, the Wronskian index, is associated to the structure of the zeroes of the Wronskian of the characters. In this paper, we study [3,0] and [3,2] MLDEs in order to classify the corresponding CFTs. We reduce the problem to a “finite” problem: to classify CFTs with central charge 0 < c ≤ 96, we need to perform 6, 720 computations for the former and 20, 160 for the latter. Each computation involves (i) first finding a simultaneous solution to a pair of Diophantine equations and (ii) computing Fourier coefficients to a high order and checking for positivity.In the [3,0] case, for 0 < c ≤ 96, we obtain many character-like solutions: two infinite classes and a discrete set of 303. After accounting for various categories of known solutions, including Virasoro minimal models, WZW CFTs, Franc-Mason vertex operator algebras and Gaberdiel-Hampapura-Mukhi novel coset CFTs, we seem to have seven hitherto unknown character-like solutions which could potentially give new CFTs. We also classify [3,2] CFTs for 0 < c ≤ 96: each CFT in this case is obtained by adjoining a constant character to a [2,0] CFT, whose classification was achieved by Mathur-Mukhi-Sen three decades ago.
The classification scheme for rational conformal field theories, given by the Mathur-Mukhi-Sen (MMS) program, identifies a rational conformal field theory by two numbers: (n, l). n is the number of characters of the rational conformal field theory. The characters form linearly independent solutions to a modular linear differential equation (which is also labelled by (n, l)); the Wronskian index l is a non-negative integer associated to the structure of zeroes of the Wronskian.In this paper, we compute the (n, l) values for three classes of well-known CFTs viz. the WZW CFTs, the Virasoro minimal models and the $$ \mathcal{N} $$ N = 1 super-Virasoro minimal models. For the latter two, we obtain exact formulae for the Wronskian indices. For WZW CFTs, we get exact formulae for small ranks (upto 2) and all levels and for all ranks and small levels (upto 2) and for the rest we compute using a computer program. We find that any WZW CFT at level 1 has a vanishing Wronskian index as does the $$ {\hat{\mathbf{A}}}_{\mathbf{1}} $$ A ̂ 1 CFT at all levels. We find intriguing coincidences such as: (i) for the same level CFTs with $$ {\hat{\mathbf{A}}}_{\mathbf{2}} $$ A ̂ 2 and $$ {\hat{\mathbf{G}}}_{\mathbf{2}} $$ G ̂ 2 have the same (n, l) values, (ii) for the same level CFTs with $$ {\hat{\mathbf{B}}}_{\mathbf{r}} $$ B ̂ r and $$ {\hat{\mathbf{D}}}_{\mathbf{r}} $$ D ̂ r have the same (n, l) values for all r ≥ 5.Classifying all rational conformal field theories for a given (n, l) is one of the aims of the MMS program. We can use our computations to provide partial classifications. For the famous (2, 0) case, our partial classification turns out to be the full classification (achieved by MMS three decades ago). For the (3, 0) case, our partial classification includes two infinite series of CFTs as well as fifteen “discrete” CFTs; except three all others have Kac-Moody symmetry.
In recent years it has been understood that new rational CFTs can be discovered by applying the coset construction to meromorphic CFTs. Here we turn this approach around and show that the coset construction, together with the classification of meromorphic CFT with c ≤ 24, can be used to predict the existence of new meromorphic CFTs with c ≥ 32 whose Kac-Moody algebras are non-simply-laced and/or at levels greater than 1. This implies they are non-lattice theories. Using three-character coset relations, we propose 34 infinite series of meromorphic theories with arbitrarily large central charge, as well as 46 theories at c = 32 and c = 40.
We have shown analytically that radiation from a collapsing shell which leads to a charged black hole, whose exterior is described by the RN (Reissner-Nordström) metric (and hence the background spacetime is non-globally hyperbolic), is processed with a unitary evolution. For the analysis, we have used the Wheeler-deWitt formalism which in turn gave rise to a Schrödinger-like wave equation. We showed the existence of unitarity by proving that the trace of the squared density matrix of the outgoing radiation, from a quantized massless scalar field, is unity and that the conservation of probability holds for the wave function of the system.
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.
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