On classification of extremal non-holomorphic conformal field theories

TL;DR

The paper develops a program to classify unitary rational chiral CFTs by their genus $(\mathcal{C},c)$, introducing extremal non-holomorphic VOAs as a tractable proxy by maximizing the sum of minimal conformal weights within a genus. It advances a computational framework based on vector-valued modular forms, using exponent matrices $\Lambda$ and the fundamental matrix $\Xi$ (via the Bantay–Gannon method) to derive possible VOA characters from fixed genus data, resolving ambiguities numerically and enforcing integrality and positivity of coefficients. The authors prove that for rank $\mathcal{C}\le 3$ and admissible $c$, the extremal VOA character vector is uniquely determined by the minimal energies $h_i$, yielding finitely many candidates; they provide explicit tables of candidate characters for rank-2 genera with $c\le 72$ and rank-3 genera with $c\le 48$, encompassing familiar theories like WZW models, Ising, and related cosets. This work concretizes a tractable path toward realizing genera with extremal constraints, informs the realizability of small-genus VOAs, and connects VOA character theory with topological phases via modular tensor categories.

Abstract

Rational chiral conformal field theories are organized according to their genus, which consists of a modular tensor category $\mathcal{C}$ and a central charge $c$. A long-term goal is to classify unitary rational conformal field theories based on a classification of unitary modular tensor categories. We conjecture that for any unitary modular tensor category $\mathcal{C}$, there exists a unitary chiral conformal field theory $V$ so that its modular tensor category $\mathcal{C}_V$ is $\mathcal{C}$. In this paper, we initiate a mathematical program in and around this conjecture. We define a class of extremal vertex operator algebras with minimal conformal dimensions as large as possible for their central charge, and non-trivial representation theory. We show that there are finitely many different characters of extremal vertex operator algebras V possessing at most three different irreducible modules. Moreover, we list all of the possible characters for such vertex operator algebras with $c$ at most 48.

Theorems & Definitions (11)

• Theorem
• Definition 2.1
• Conjecture 2.2
• Conjecture 2.3
• Theorem 2.4
• Definition 2.5
• Remark 2.6
• Conjecture 2.7
• Theorem 3.1
• proof