Bootstrapping noninvertible symmetries
Ying-Hsuan Lin, Shu-Heng Shao
Abstract
Using the numerical modular bootstrap, we constrain the space of $1+1\mathrm{d}$ conformal field theories (CFTs) with a finite noninvertible 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 noninvertible 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 $2+1\mathrm{d}$ Turaev-Viro topological quantum field theory , also known as the symmetry TFT.