Bootstrapping Non-Invertible Symmetries

TL;DR

This paper uses numerical bootstrap methods to establish universal bounds on operators in 1+1d conformal field theories with non-invertible symmetries described by fusion categories, impacting the understanding of gapless phases.

Contribution

It introduces a novel bootstrap approach to constrain 1+1d CFTs with non-invertible symmetries, providing rigorous bounds on operator dimensions related to fusion categories.

Findings

Derived upper bounds on the lightest $ ext{C}$-preserving scalar operators.

Established bounds on the lightest $ ext{C}$-violating local operators.

Connected bootstrap equations to a slab construction involving 2+1d TQFTs.

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.