Charting the space of 3D CFTs with a continuous global symmetry

TL;DR

This work implements a nonperturbative bootstrap program for 3D CFTs with a continuous global symmetry by analyzing the four-point function of a conserved current $J_$ and exploiting current conservation to derive a minimal set of crossing equations. It develops a spin-1 conformal-block recurrence, computes blocks via a radial-parameter approach, and applies semidefinite programming to bound light operator dimensions and the central charge $C_T$; the results reproduce the conformal collider bounds and place stringent constraints on the critical $O(2)$ model. The analysis reveals that both parity-even and parity-odd scalars must appear in the $J imes J$ OPE, and it uncovers a kink near the $O(2)$ model on the boundary of the allowed region, suggesting a potential UV-complete realization. By incorporating spectrum gaps, the authors obtain tighter central-charge bounds and demonstrate how a large spin-4 gap drives $C_T$ to infinity, consistent with holographic expectations; the work sets the stage for mixed scalar-current bootstrap and extensions to larger symmetry groups or higher dimensions.

Abstract

We study correlation functions of a conserved spin-1 current $J_μ$ in three dimensional Conformal Field Theories (CFTs). We investigate the constraints imposed by permutation symmetry and current conservation on the form of three point functions $\langle J_μJ_ν\mathcal O_{Δ,\ell}\rangle $ and the four point function $\langle J_μJ_νJ_ρJ_σ\rangle $ and identify the minimal set of independent crossing symmetry conditions. We obtain a recurrence relation for conformal blocks for generic spin-1 operators in three dimensions. In the process, we improve several technical points, facilitating the use of recurrence relations. By applying the machinery of the numerical conformal bootstrap we obtain universal bounds on the dimensions of certain light operators as well as the central charge. Highlights of our results include numerical evidence for the conformal collider bound and new constraints on the parameters of the critical $O(2)$ model. The results obtained in this work apply to any unitary, parity-preserving three dimensional CFT.

Figures (18)

• Figure 1: Exclusion plot in the plane $(\Delta^+_{0},\Delta^-_{0})$ corresponding to the lightest parity even and parity odd scalars appearing in the OPE of two equal conserved currents. The shaded region is allowed.
• Figure 2: Lower bound on the central charge normalized to the one of a free complex scalar as a function of $\gamma$. The vertical dashed lines corresponds to the conformal collider bound $-1\leq 12\gamma\leq 1$. We impose that the first spin-2 operator after $T_{\mu\nu}$ has dimension larger than 3.5 (see section \ref{['sec:results']} for explanation).
• Figure 3: Change of coordinates from the usual cross ratio $u$ and $v$ to the new ones $y=u+v-1/2$ and $t=u-v$. The conservation equation can be used to evolve 12 functions (6 even and 6 odd under the action of $\mathtt{\left[\,P\,\right]}$) from the line $t=0$ to the full plane. We can further evolve 2 functions (1 even and 1 odd) from the point $(t,y)=(0,0)$, to the line $t=0$.
• Figure 4: The picture represents the conformal multiplet ${\mathcal{H}}_{\mathcal{O}}$ of an operator ${\mathcal{O}}$ with dimension $\Delta$ and spin $\ell$. The arrows represent the descendant operators and the horizontal and vertical axis are labeled respectively by the conformal dimensions and the spin. When the primary operator ${\mathcal{O}}$ has dimension $\Delta=\Delta_A^\star$, its descendant ${\mathcal{O}}_A$ (with dimension $\Delta_A=\Delta^\star_A+n_A$ and spin $\ell_A$) becomes primary. The state ${\mathcal{O}}_A$ and all its descendants form a conformal multiplet of null states ${\mathcal{H}}_{{\mathcal{O}}_A}$.
• Figure 5: Allowed region consistent with crossing symmetry assuming that all the parity-even scalars appearing in the OPE $J\times J$ have dimension larger than $\Delta_0^+$ and all parity-odd scalars have dimensions larger than $\Delta_0^-$. The orange shaded region is allowed. Marks correspond to known CFTs: free complex boson $((\Delta_0^+,\Delta_0^-)=(1,7))$, free Dirac fermion $((\Delta_0^+,\Delta_0^-)=(4,2))$ and GFVF $((\Delta_0^+,\Delta_0^-)=(4,5))$. The red vertical line corresponds to the approximate dimension of the lightest singlet operator in the interacting $O(2)$ model: $\Delta_0^+=1.5117$. The blue shading shows the region excluded by bootstrapping the four point function of identical parity odd scalars with the dimension $\Delta_0^-$. See text for more details. The best bound has been computed at $\Lambda=23$ while gray lines correspond to $\Lambda=11,15,19$.