Conformal Bootstrap with Duality-Inspired Fusion Rule

We present the first systematic exploration of conformal field theories (CFTs) possessing fusion rules inspired by categorical (non-invertible) symmetries, using the conformal bootstrap. Specifically, we impose a selection rule motivated by Kramers-Wannier duality and derive bounds on the conformal dimensions $(Δ_σ, Δ_ε)$ of the lowest-lying $\mathbb{Z}_2$-odd scalar $σ$ and $\mathbb{Z}_2$-even scalar $ε$ in dimensions $d=2$ through $d=7$. Our bounds correctly allow the $d=2$ Ising model while excluding the $d=3$ Ising model, demonstrating the effectiveness of the imposed condition. Furthermore, we observe a distinct feature in $d=2$ corresponding to the $\mathcal{M}(8,7)$ minimal model and find non-trivial constraints in $d=3$ ($Δ_σ\gtrsim 0.85$), relevant for theories like QED$_3$. This work opens a new avenue for non-perturbatively probing the vast landscape of CFTs constrained by non-invertible symmetries.