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.

10 pages, 3 figures. Published in Physical Review D

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