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International Journal of Modern Physics A

MODULAR INVARIANCE AND FUSION RULE ALGEBRAS: CLASSIFICATION OF THE THREE-OPERATOR CASE

Michele Caselle, G. Ponzano

Published Dec 20, 1990• 3 citations
Publisher

Abstract

We show how to handle together constraints coming from associativity, modular invariance and analyticity in RCFT's. In particular, looking at the theories with three operators we classify fusion rule algebras, central charges and conformal weights fulfilling such constraints. We also show that all allowed theories with three operators are algebraic.

DOI
10.1142/s0217751x90002002

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We show how to handle together constraints coming from associativity, modular invariance and analyticity in RCFT's. In particular, looking at the theories with three operators we classify fusion rule algebras, central charges and conformal weights fulfilling such constraints. We also show that all allowed theories with three operators are algebraic.
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.
We study constraints coming from the modular invariance of the partition function of two-dimensional conformal field theories. We constrain the spectrum of CFTs in the presence of holomorphic and anti-holomorphic currents using the semi-definite programming. In particular, we find the bounds on the twist gap for the non-current primaries depend dramatically on the presence of holomorphic currents, showing numerous kinks and peaks. Various rational CFTs are realized at the numerical boundary of the twist gap, saturating the upper limits on the degeneracies. Such theories include Wess-Zumino-Witten models for the Deligne’s exceptional series, the Monster CFT and the Baby Monster CFT. We also study modular constraints imposed by W \mathcal{W} W -algebras of various type and observe that the bounds on the gap depend on the choice of W \mathcal{W} W -algebra in the small central charge region.