Conformal Bootstrap with Duality-Inspired Fusion Rule

TL;DR

This work develops a conformal bootstrap framework constrained by duality-inspired, non-invertible fusion rules to bound CFT data non-perturbatively. By analyzing mixed correlators of two relevant scalars σ and ε under the fusion rules $[σ]×[σ]∼[1]+[ε]$, $[ε]×[ε]∼[1]$, and $[σ]×[ε]∼[σ]$, the authors obtain bounds on $(Δ_{σ},Δ_{ε})$ for $d=2$–$7$; notably, the 2D Ising point is allowed while the 3D Ising point is excluded, and a kink matching the $M(8,7)$ minimal model appears in $d=2$. They discuss candidate CFTs consistent with the bounds, including a free boson in $d=2$, QED$_3$ monopole operators in $d=3$, and conformal gauge theories in higher $d$, illustrating the utility of categorical-symmetry constraints. This work opens a new non-perturbative avenue to explore the landscape of CFTs governed by non-invertible symmetries and could guide searches for theories with duality-driven fine-tuning protection.

Abstract

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.

Figures (2)

• Figure 1: Bootstrap bounds on the $(\Delta_{\sigma}, \Delta_{\epsilon})$ plane imposing the duality-inspired fusion rule for (a) $d=2$ and (b) $d=3$. Blue dots represent the not excluded points, while red crosses are excluded. The purple dash-dotted lines are candidate CFTs discussed in the main text.
• Figure 2: Bootstrap bounds on the $(\Delta_{\sigma}, \Delta_{\epsilon})$ plane imposing the duality-inspired fusion rule for higher dimensions: (a) $d=4$, (b) $d=5$, (c) $d=6$, and (d) $d=7$. Blue dots represent the not excluded points, while red crosses are excluded.