Journal
JOURNAL OF HIGH ENERGY PHYSICS
Volume -, Issue 4, Pages -Publisher
SPRINGER
DOI: 10.1007/JHEP04(2021)294
Keywords
Conformal and W Symmetry; Conformal Field Theory
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The paper discusses the classification scheme for rational conformal field theories, which identifies them by two numbers (n, l). It computes the (n, l) values for three classes of well-known CFTs and finds interesting patterns and coincidences in the Wronskian indices of certain CFTs. The goal is to classify all rational conformal field theories for a given (n, l), with partial classifications provided using the computations.
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 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 A1 CFT at all levels. We find intriguing coincidences such as: (i) for the same level CFTs with A2 and G2 have the same (n, l) values, (ii) for the same level CFTs with Br and Dr 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.
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