Asymptotic density of states in 2d CFTs with non-invertible symmetries

TL;DR

This work generalizes Cardy’s universal density of states to 2d CFTs with finite, non-invertible symmetries described by a fusion category $\mathcal{C}$. It develops two complementary frameworks—the 2d tube-algebra view and the 3d bulk TV$_{\mathcal{C}}$ perspective with Drinfeld center $\mathcal{Z}(\mathcal{C})$—to classify irreducible sectors as simple objects of $\mathcal{Z}(\mathcal{C})$ and to derive their asymptotic counts. The main result states that the density in the $a$-twisted sector transforming in the center-label $\mu$ scales as $\langle \mu,a\rangle\dim\mu$, up to the universal Cardy growth $e^{(L/T)\pi c/6}$; in the finite-group limit this reproduces Pal–Sun’s $(\dim\rho)^2$ formula and extends naturally to modular and Haagerup fusion-category symmetries. The findings provide a unified description of state densities and selection rules across a broad class of fusion-category symmetries, with concrete illustrations for finite groups, diagonal RCFTs, and the Haagerup case.

Abstract

It is known that the asymptotic density of states of a 2d CFT in an irreducible representation $ρ$ of a finite symmetry group $G$ is proportional to $(\dimρ)^2$. We show how this statement can be generalized when the symmetry can be non-invertible and is described by a fusion category $\mathcal{C}$. Along the way, we explain what plays the role of a representation of a group in the case of a fusion category symmetry; the answer to this question is already available in the broader mathematical physics literature but not yet widely known in hep-th. This understanding immediately implies a selection rule on the correlation functions, and also allows us to derive the asymptotic density.

Figures (5)

• Figure 1: States in the sector $V_c\otimes\overline{V_d}\subset\mathcal{H}_a$ can be realized in the 3d TQFT description as in this figure. Associated with each end $T^2$ are holomorphic and antiholomorphic boundary conditions. Take operators $\mathcal{O}_c$ and $\overline{\mathcal{O}_d}$ on the boundaries, corresponding to states in $V_c$ and $\overline{V_d}$, respectively. They are connected by topological lines labeled by $c$ and $d$ meeting in the middle. Choose $x_{cd}^a\in\mathrm{Hom}(a\otimes d,c)$ to fuse $c$ and $d$ into $a$. Then $\mathcal{O}_{cd}^{a;x}=(\mathcal{O}_c,\mathcal{O}_d,x_{cd}^a)$ is the defect operator corresponding to a state in the sector $V_c\otimes\overline{V_d}\subset\mathcal{H}_a$ (See footnote \ref{['folding']}). The degree of freedom in choosing $x_{cd}^a$ is $\dim \mathrm{Hom}(a\otimes d,c)=N_{ad}^c$, which leads to the multiplicity of the sector $V_c\otimes\overline{V_d}$ in $\mathcal{H}_a$.
• Figure 2: The 2d theory $Q$ on $\Sigma$ is obtained by putting the 3d TQFT $\mathrm{TV}_\mathcal{C}$ on $\Sigma \times I$. We have the non-topological boundary condition $\mathbb{B}_Q$ on the left and the (Dirichlet) topological boundary condition $\mathbb{D}$ on the right. The topological lines $a,b,c,\dots \in \mathcal{C}$ on the $\mathbb{D}$ boundary implement the $\mathcal{C}$ symmetry of $Q$.
• Figure 3: A defect operator $\mathcal{O}\in\mathcal{H}_a$ of the 2d theory $Q$ is composed of a bulk anyon $\mu\in\mathcal{Z}(\mathcal{C})$ and a pair of boundary (defect) operators. Via the state-operator correspondence, $\widetilde{\mathcal{O}}$ belongs to the $\mu$-punctured disk Hilbert space of $\mathrm{TV}_\mathcal{C}$ with boundary condition $\mathbb{B}_Q$, and $x$ belongs to $\mathrm{Hom}_\mathcal{C}(F(\mu),a)$ where $F:\mathcal{Z}(\mathcal{C}) \to \mathcal{C}$ describes the fusion of bulk anyons with the topological boundary $\mathbb{D}$.
• Figure 4: The partition function of $\mathrm{TV}_\mathcal{C}$ on the solid torus $D_L^2 \times S_T^1$ with $\mathbb{B}_Q$ boundary condition and anyon $\mu$ in the middle. $D_L^2$ is the horizontal disk of circumference $L$, and $S^1_T$ is the vertical circle of size $T$. Since the boundary condition $\mathbb{B}_Q$ is conformal, this partition function depends only on $q=e^{-2\pi(T/L)}$.
• Figure 5: Boundary crossing relation in 2+1d TQFT. The blue dashed lines sketch the surfaces of the gapped boundaries, such that the regions outside the slabs and inside the cylinders are empty. Top: A special case that involves the half-linking matrices $\Psi_{a(\mu,xy)}$. Bottom: The general case.