Conformal bootstrap with duality-inspired fusion rule

We present a systematic exploration of conformal field theories (CFTs) constrained by duality-inspired fusion rules using the conformal bootstrap. We classify the operator spectrum into three sectors: [ σ ], [ ε ], and [ 1 ] . The [ σ ] sector consists of all Z 2 -odd operators. The Z 2 -even operators are further divided into the [ ε ] sector, which contains only the operators that change sign under duality, and the [ 1 ] sector, which encompasses all remaining operators. We impose a selection rule motivated by Kramers-Wannier duality, specifically forbidding the appearance of the [ ε ] sector in the [ ε ] × [ ε ] operator product expansion. By applying this constraint to the lowest-lying relevant scalars, we derive bounds on their conformal dimensions ( Δ σ , Δ 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 M ( 8 , 7 ) minimal model and find nontrivial constraints in d = 3 ( Δ σ ≳ 0.85 ), relevant for theories like QED 3 . This work opens a new avenue for nonperturbatively probing the landscape of CFTs constrained by fusion rules.