Bosonic Rational Conformal Field Theories in Small Genera, Chiral Fermionization, and Symmetry/Subalgebra Duality

TL;DR

This work advances a comprehensive, genus-based classification program for bosonic and fermionic RCFTs in small genera by combining modular bootstrap (via Bantay–Gannon vector-valued modular forms) with coset/gleaning techniques from holomorphic VOAs. It achieves a near-complete enumeration of bosonic chiral algebras for $0<c\le 24$ and $\mathrm{rank}(\mathcal{C})\le 4$, and uses chiral fermionization to derive a complete list of left-moving chiral fermionic RCFTs with $c<23$, tying the results to generalized global symmetries through a proposed symmetry/subalgebra duality. The paper also develops a gluing framework to construct holomorphic VOAs from seeds and cosets, and demonstrates how symmetry data and Drinfeld-center structures govern possible extensions and dualities, with explicit examples such as rank-2 symmetry actions on $E_8$-theory and interplays between Verlinde lines and topological boundary conditions. Collectively, these results offer a powerful, scalable approach to understanding 1+1D RCFT landscapes, inform symmetry structures in holomorphic VOAs, and suggest avenues toward higher-rank, higher-$c$ classifications and connections to moonshine and TMF. $${\ }$$

Abstract

A (1+1)D unitary bosonic rational conformal field theory (RCFT) may be organized according to its genus, a tuple $(c,\mathscr{C})$ consisting of its central charge $c$ and a unitary modular tensor category $\mathscr{C}$ which describes the (2+1)D topological quantum field theory (TQFT) 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$(\mathscr{C})$ 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$(\mathscr{C})\leq 4$ 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 $(c,\mathscr{C})$ with $c\leq 24$ and rank$(\mathscr{C})\leq 4$. This includes as (new) special cases the classification of chiral algebras with three primaries and $c<120/7\sim 17.14$, and the classification of chiral algebras with four primaries and $c<62/3\sim 20.67$. 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.

Figures (2)

• Figure 1: A (1+1)D RCFT on a Riemann surface $\Sigma$ may be thought of as a (2+1)D TQFT $\mathcal{C}$ on $\Sigma\times [-1,1]$ with a boundary condition corresponding to the chiral algebra $\mathcal{V}$ (resp. $\overline{\mathcal{V}}$) placed on $\Sigma\times \{-1\}$ (resp. $\Sigma \times \{+1\})$. The choice of a modular invariant corresponds to the choice of a topological surface operator (TSO) inserted at $\Sigma \times \{0\}$. Alternatively, this setup may be folded so that the topological surface operator corresponds to a topological boundary condition (TBC) in the doubled TQFT $\mathcal{C}\boxtimes \overline{\mathcal{C}}$.
• Figure 2: A chiral CFT/holomorphic VOA $\mathcal{A}$ with a fusion category symmetry $\mathcal{F}$ may be coupled to a (2+1)D topological $\mathcal{F}$ gauge theory $\mathcal{Z}(\mathcal{F})$ to obtain a conformal subalgebra $\mathcal{A}^{\mathcal{F}}$. Conversely, given a conformal subalgebra $\mathcal{W}$ of a chiral CFT $\mathcal{A}$, there exists a topological boundary condition $\mathcal{A}_{\mathrm{top}}$ such that sandwiching the (2+1)D TQFT $\textsl{Rep}(\mathcal{W})$ with the $\mathcal{A}_{\mathrm{top}}$ and $\mathcal{W}$ boundary conditions recovers $\mathcal{A}$; the fusion category $\textsl{Rep}(\mathcal{W})_{\mathcal{A}}$ of topological line operators supported on $\mathcal{A}_{\mathrm{top}}$ then acts on $\mathcal{A}$ by symmetries which commute with $\mathcal{W}$.

Theorems & Definitions (129)

• Conjecture 1.1
• Definition B.1: Linear category
• Definition B.2: Deligne's tensor product
• Definition B.3: Unitary structure
• Definition B.4: Abelian category
• Definition B.5
• Definition B.6: Monoidal category
• Definition B.7: Rigidity
• Definition B.8: Trace and dimension
• Definition B.9: Braiding