Quasi-Characters for three-character Rational Conformal Field Theories

TL;DR

This work advances the classification of three-character RCFTs by unifying $(3,0)$ characters and quasi-characters through a uniform ${}_3F_2$ hypergeometric description that encodes elliptic monodromy; it leverages Bantay–Gannon duality to map $(3,0)$ to $(3,3)$ and extract modular data such as the S-matrix and fusion rules. The authors show that only a subset of the known $(3,3)$ solutions yields admissible fusion, and they develop a matrix MLDE–based quasi-character construction to generate $(3,6)$ and $(3,9)$ families, revealing a polytope structure in the space of admissible theories. A duality-driven procedure then identifies RCFTs among these higher-ℓ solutions and provides a blueprint for constructing large families of admissible theories with higher Wronskian index, while preserving a clear link to existing RCFT data and modular data. The results illuminate a geometric and algebraic framework that can extend to four- and five-character theories through similar dualities and quasi-character constructions, with potential applications to modular tensor category classifications and coset constructions.

Abstract

We revisit (3,0) and (3,3) admissible solutions obtained using the MLDE method. We show that all $(3,0)$ solutions can be written in terms of a universal formula involving the ${}_3F_2$ hypergeometric function that takes into account the monodromy at the elliptic points. We construct $(3,3)$ admissible solutions from (3,0) CFTs using a duality due to Bantay and Gannon. This enables us to compute their modular properties such as the S-matrix and the fusion rules. We find that only 7 of the 15 known (3,3) admissible solutions have proper fusion rules. Using the theory of matrix MLDE, starting with a known (3,0) and (3,3) solutions, we construct two other solutions, that are typically quasi-characters that share the same multiplier as the original solution. We then construct linear combinations that lead to new admissible solutions. We observe that admissible solutions arise as integer points that lie on a polytope. We construct all possible (3,6) and (3,9) admissible solutions that arise in this fashion. In some cases, we identify RCFT that arise from our (3,6) admissible solutions. In addition, we obtain a large family of admissible solutions with higher Wronskian index.

Figures (2)

• Figure 1: Example of type $U$ admissible characters starting with a $c = 23$, $h_1= 3/2$, $h_2=15/8$ RCFT. They occur as integral points on the interior and boundary of a quadrilateral. We see two other $(3,0)$ theories at the corners $(b_1,b_2)=(46,0)$ and $(b_1,b_2)=(0,575)$. These are shown as red dots and connected by green dots which are $(3,6)$ admissible characters. We show the $(3,12)$ points that lie on the boundaries as blue dots. All integral interior points are $(3,12)$ admissible characters and are not shown in the figure to avoid clutter.
• Figure 2: In this type $\mathbb{W}_{1,1}$ example, we obtain $(3,6)$ theories on integral points that lie on the line $\frac{1}{46}(c_1-8742)$. All integral points inside the polytope and on the $c_1=0$ line are $(3,12)$ admissible solutions. There are no $(3,0)$ solutions.

Theorems & Definitions (2)

• Proposition 3.1
• proof : Proof