Abstract

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} = 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}} CFT at all levels. We find intriguing coincidences such as: (i) for the same level CFTs with {hat{mathbf{A}}}_{mathbf{2}} and {hat{mathbf{G}}}_{mathbf{2}} have the same (n, l) values, (ii) for the same level CFTs with {hat{mathbf{B}}}_{mathbf{r}} and {hat{mathbf{D}}}_{mathbf{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.

Highlights

  • Two dimensional conformal field theory (CFT) is a very important subject since it bears relevance to many areas in physics and mathematics; [3,4,5,6] are a partial list of references for CFTs

  • One could perhaps ask the question: are there any new rational conformal field theories (RCFTs) with a given (n, l) besides the known RCFTs such as the Virasoro minimal models, the WZW CFTs and others, and various tensor products of these theories? To ask this question, we would need to know the (n, l) values of known RCFTs and this is one of the things we study in this paper: we compute the number of characters and the Wronskian-indices for a large class of RCFTs

  • We use the computer program to obtain the number of primary fields as well as their scaling dimensions and compute the Wronskian index of the CFT; the results are in table 5

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Summary

Introduction

Two dimensional conformal field theory (CFT) is a very important subject since it bears relevance to many areas in physics and mathematics; [3,4,5,6] are a partial list of references for CFTs. We would need to know the (n, l) values of known RCFTs and this is one of the things we study in this paper: we compute the number of characters and the Wronskian-indices for a large class of RCFTs. We study the WZW CFTs for all Lie algebras and all levels and obtain exact formulae for many cases and for others via computer programs. In 3.9, we summarise the results and provide various conjectures about WZW CFTs. In section 4, we first study Virasoro minimal models in 4.1 and are able to perform an exact computation for the infinite series of RCFTs and remarkably, they all have a vanishing Wronskian index.

MLDE approach to RCFTs
The Wronskian W
The valence formula and the Wronskian index
Generalities
A series
B series
C series
D series
Exceptional series
All Lie algebras at level one
All Lie algebras at level two
Summary of results for WZW CFTs
Conclusion and future directions
Full Text
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