On 2d Conformal Field Theories with Two Characters

TL;DR

<3-5 sentence high-level summary>This work develops a modular-invariant differential-equation framework to classify two-character rational conformal field theories (RCFTs) by the Wronskian zero count $\\ell \\in\\ {0,2,3,4}$. It systematically analyzes each $\\ell$, finding a complete set of viable ℓ = 0 theories and identifying nine potentially consistent two-character theories for ℓ = 2 with central charges in $16<c<24$, while ruling out new two-character theories for ℓ = 3 and exploring ℓ = 4 with connections to tensor-product constructions. It further investigates the chiral algebras underlying the ℓ = 2 candidates, showing that their current algebras are exotic level-1 combinations rather than straightforward WZW models, and outlines a path toward rigorous consistency checks via Zhu theory and vector-valued modular forms. The results reveal surprising links between ℓ = 0 and ℓ = 2 spectra and set the stage for deeper structural understanding of two-character RCFTs and their holographic/monodromy properties.

Abstract

Rational CFT's are classified by an integer $\ell$, the number of zeroes of the Wronskian of their characters in moduli space. For $\ell=0$ they satisfy non-singular modular-invariant differential equations, while for $\ell>0$ the corresponding equations have singularities. We survey CFT's with two characters and $\ell=0,2,3,4$ and verify the consistency, at the level of characters, of some candidate theories with $\ell\ne 0$. For $\ell=2$ there are seven consistents sets of characters. We identify specific combinations of level-1 current algebras that are potential symmetries of the corresponding CFT's.