Conformal Bootstrap with Duality-Inspired Fusion Rule

Abstract

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 $(\Delta_\sigma, \Delta_\epsilon)$ 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 non-trivial constraints in $d=3$ ($\Delta_\sigma \gtrsim 0.85$), relevant for theories like QED$_3$. This work opens a new avenue for non-perturbatively probing the landscape of CFTs constrained by fusion rules.