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Journal of Mathematical Physics

Primary operators of coset conformal field theories

Hyung‐Joon Shin

Published Feb 1, 1992• 0 citations
Publisher

Abstract

The nature of primary operators of coset conformal field theories is discussed. Some primary states of typical coset conformal field theory are explicitly calculated that have correct conformal dimensions and dynamics. The nature of modular invariant partition functions from the diagonal sum of absolute squares of characters of coset primary states is studied.

DOI
10.1063/1.529780

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The nature of primary operators of coset conformal field theories is discussed. Some primary states of typical coset conformal field theory are explicitly calculated that have correct conformal dimensions and dynamics. The nature of modular invariant partition functions from the diagonal sum of absolute squares of characters of coset primary states is studied.
We prove using invariance under the modular SSS- and STSTST-transformations that every unitary two-dimensional conformal field theory (CFT) of only even-spin operators (with no extended chiral algebra and with central charges c,c~>1c,\tilde{c}>1c,c~>1) contains a primary operator with dimension Δ1Δ_1Δ1​ satisfying 0<Δ1<(c+c~)/24+0.09280...0 < Δ_1 < (c+\tilde{c})/24 + 0.09280...0<Δ1​<(c+c~)/24+0.09280... After deriving both analytical and numerical bounds, we discuss how to extend our methods to bound higher conformal dimensions before deriving lower and upper bounds on the number of primary operators in a given energy range. Using the AdS3_33​/CFT2_22​ dictionary, the bound on Δ1Δ_1Δ1​ proves the lightest massive excitation in appropriate theories of 3D matter and gravity with cosmological constant Λ<0Λ< 0Λ<0 can be no heavier than 1/(8GN)+O(−Λ)1/(8G_N)+O(\sqrt{-Λ})1/(8GN​)+O(−Λ​); the bounds on the number operators are related via AdS/CFT to the entropy of states in the dual gravitational theory. In the flat-space approximation, the limiting mass is exactly that of the lightest BTZ black hole.
We develop a general method of solving non-trivial conformal field theories by coset imbeddings into free boson models. In particular, we consider ZN parafermion models represented by the coset model constructions su(N1)⊕su(N)1/su(N)2, and bosonize the su(N)1 models. The calculation of correlation functions of these parafermion theories is formulated so that detailed information from the diagonal su(N)2 theory is not required. A differential equation for the four-point functionof order operators σ1 is derived as an example. We also derive differential equations with respect to the modular parameter τ for the parafermion characters in terms of the su(N)1 characters only. In particular, the equations for the Z2 and Z3 characters are first and second order respectively.
A (1 + 1)D unitary bosonic rational conformal field theory (RCFT) may be organized according to its genus, a tuple (c,C) consisting of its central charge c and a unitary modular tensor category C which describes the (2 + 1)D topological quantum field theory for which its maximally extended chiral algebra forms a holomorphic boundary condition. We establish a number of results pertaining to RCFTs in “small” genera, by which we informally mean genera with the central charge c and the number of primary operators rank(C) both not too large. We start by completely solving the modular bootstrap problem for theories with at most four primary operators. In particular, we characterize, and provide an algorithm which efficiently computes, the function spaces to which the partition function of any bosonic RCFT with rank(C)≤4 must belong. Using this result, and leveraging relationships between RCFTs and holomorphic vertex operator algebras which come from “gluing” and cosets, we rigorously enumerate all bosonic theories in 95 of the 105 genera (c,C) with c ≤ 24 and rank(C)≤4. This includes as (new) special cases the classification of chiral algebras with three primaries and c < 120/7 ∼ 17.14, and the classification of chiral algebras with four primaries and c < 62/3 ∼ 20.67. We then study two applications of our classification. First, by making use of chiral versions of bosonization and fermionization, we obtain the complete list of purely left-moving fermionic RCFTs with c < 23 as a corollary of the results of the previous paragraph. Second, using a (conjectural) concept which we call “symmetry/subalgebra duality,” we precisely relate our bosonic classification to the problem of determining certain generalized global symmetries of holomorphic vertex operator algebras.