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: [$\sigma$], [$\epsilon$], and $[1]$. The [$\sigma$] sector consists of all ${\mathbb{Z}}_2$-odd operators. The ${\mathbb{Z}}_2$-even operators are further divided into the [$\epsilon$] 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 [$\epsilon$] sector in the $[\epsilon ]\times [\epsilon ]$ operator product expansion. By applying this constraint to the lowest-lying relevant scalars, we derive bounds on their conformal dimensions $({\mathrm{\Delta }}_{\sigma },{\mathrm{\Delta }}_{\epsilon })$ 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 nontrivial constraints in $d=3$ (${\mathrm{\Delta }}_{\sigma }\gtrsim 0.85$), relevant for theories like ${\mathrm{QED}}_3$. This work opens a new avenue for nonperturbatively probing the landscape of CFTs constrained by fusion rules.

Close

PrevNext

Reuse & PermissionsView figure in article

• 

• Conformal field theory
• Conformal symmetry
• Continuous symmetries
• Discrete symmetries
• Symmetries