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Wronskian indices and rational conformal field theories

Das, Arpit and Gowdigere, Chethan N. and Santara, Jagannath (2021) 'Wronskian indices and rational conformal field theories.', Journal of High Energy Physics, 2021 (4). p. 294.


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 A^1 CFT at all levels. We find intriguing coincidences such as: (i) for the same level CFTs with A^2 and G^2 have the same (n, l) values, (ii) for the same level CFTs with B^r and 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.

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Publisher statement:Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Date accepted:04 April 2021
Date deposited:28 July 2021
Date of first online publication:30 April 2021
Date first made open access:28 July 2021

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