Bootstrapping Non-Invertible Symmetries

TL;DR

This work develops a numerical modular bootstrap for 1+1d CFTs with finite non-invertible symmetries described by fusion categories (Fibonacci, Ising, su(2)_2) by employing a slab construction that couples to a 2+1d Turaev-Viro TQFT. By analyzing modular properties of twisted partition functions and using the linear functional method, the authors derive universal bounds on the lightest C-preserving scalar and on C-preserving/violating operators across central charges, revealing no C-protected gapless phases in key ranges and saturations by known RCFTs. The framework leverages the Drinfeld center, anyon sectors, and defect Hilbert spaces to translate 2+1d modular data into 1+1d operator bounds, with implications for robust gapless phases in lattice models like anyonic chains. The results illustrate how non-invertible symmetries impose strong, model-independent constraints on the infrared behavior of quantum systems and point to broader generalizations to other fusion categories and higher-dimensional bulk theories.

Abstract

Using the numerical modular bootstrap, we constrain the space of 1+1d CFTs with a finite non-invertible global symmetry described by a fusion category $\mathcal{C}$. We derive universal and rigorous upper bounds on the lightest $\mathcal{C}$-preserving scalar local operator for fusion categories such as the Ising and Fibonacci categories. These numerical bounds constrain the possible robust gapless phases protected by a non-invertible global symmetry, which commonly arise from microscopic lattice models such as the anyonic chains. We also derive bounds on the lightest $\mathcal{C}$-violating local operator. Our bootstrap equations naturally arise from a "slab construction", where the CFT is coupled to the 2+1d Turaev-Viro TQFT, also known as the Symmetry TFT.

Figures (9)

• Figure 1: A topological defect line $\cal L$ wrapped around the spatial circle of a cylinder leads to an action $\widehat{{\mathcal{L}}}$ on the Hilbert space $\cal H$.
• Figure 2: A local operator $\phi(x)$ is said to commute with a topological line $\cal L$ if we can sweep the line past the operator without changing any correlation function. In this case we say that $\phi$ is symmetric under $\cal L$.
• Figure 3: By quantizing the system on a spatial circle with a topological defect line $\cal L$ inserted at a point in space, we define a defect Hilbert space ${\cal H}_{\cal L}$. Via the operator-state correspondence, the states in ${\cal H}_{\cal L}$ are mapped to operators living at the end of $\cal L$.
• Figure 4: A defect operator $\mathcal{O}\in\mathcal{H}_a$ of the 2d theory $Q$ is composed of a bulk anyon $\mu\in\mathcal{Z(C)}$ and a pair of boundary (defect) operators. (Figure reproduced from Lin:2022dhv.)
• Figure 5: The 2+1d TQFT $\mathrm{TV}_{\mathcal{C}}$ quantized on a spatial disk with boundary condition $\mathbb{B}_Q$ and the anyon line $\mu$ inserted at the origin. We denote the Hilbert space of this 2+1d system on a disk by ${\cal V}_\mu$. We denote the partition function over this Hilbert space by $Z^\text{3d}_\mu(\tau,\bar{\tau})$. (Figure reproduced from Lin:2022dhv.)