Bootstrapping noninvertible symmetries

Using the numerical modular bootstrap, we constrain the space of 1+1d conformal field theories (CFTs) with a finite noninvertible global symmetry described by a fusion category C. We derive universal and rigorous upper bounds on the lightest 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 C-violating local operator. Our bootstrap equations naturally arise from a ``slab construction, '' where the CFT is coupled to the 2+1d Turaev-Viro topological quantum field theory, also known as the symmetry TFT.