Carving Out the Space of 4D CFTs

TL;DR

This work develops a robust four-dimensional conformal bootstrap framework that leverages semidefinite programming to bound operator dimensions, OPE coefficients, central charges, and current two-point functions in general CFTs and N=1 SCFTs. By avoiding discretization over operator dimensions and exploiting polynomial approximations of conformal blocks, the approach yields significantly stronger bounds, especially in theories with SO(N) and SU(N) global symmetries, and yields actionable constraints on phenomenologically relevant scenarios like conformal technicolor. The results sharpen our understanding of how symmetry and unitarity constrain the spectrum, reveal near-universal behaviors in certain limits (e.g., d→1), and connect to known theories such as SQCD in the conformal window, with implications for AdS/CFT and beyond-the-Standard-Model model-building. The methodology also opens pathways to stronger, broader bounds by incorporating higher-spin blocks and exploring 3D/6D extensions, potentially illuminating the landscape of consistent quantum field theories.

Abstract

We introduce a new numerical algorithm based on semidefinite programming to efficiently compute bounds on operator dimensions, central charges, and OPE coefficients in 4D conformal and N=1 superconformal field theories. Using our algorithm, we dramatically improve previous bounds on a number of CFT quantities, particularly for theories with global symmetries. In the case of SO(4) or SU(2) symmetry, our bounds severely constrain models of conformal technicolor. In N=1 superconformal theories, we place strong bounds on dim(Phi*Phi), where Phi is a chiral operator. These bounds asymptote to the line dim(Phi*Phi) <= 2 dim(Phi) near dim(Phi) ~ 1, forbidding positive anomalous dimensions in this region. We also place novel upper and lower bounds on OPE coefficients of protected operators in the Phi x Phi OPE. Finally, we find examples of lower bounds on central charges and flavor current two-point functions that scale with the size of global symmetry representations. In the case of N=1 theories with an SU(N) flavor symmetry, our bounds on current two-point functions lie within an O(1) factor of the values realized in supersymmetric QCD in the conformal window.

Figures (22)

• Figure 1: The 'search space' $\mathcal{S}$ (shown in blue) is the intersection of the hyperplane $\alpha(F_{\Delta_0,\ell_0})=1$ with the convex cone of linear functionals $\alpha$ satisfying $\alpha(F_{\Delta,\ell})\geq 0$ for all $\{\Delta,\ell\}$ in the spectrum. Previous methods discretized $\Delta$ to some finite set $\{\Delta_i\}$, thus approximating $\mathcal{S}$ as an intersection of a finite number of hyperplanes and half-spaces (left). Our approach is to approximate $\mathcal{S}$ as the intersection of a smaller number of curved spaces --- specifically cones of semidefinite matrices (right). Such intersections are sometimes called spectrahedra.
• Figure 2: An upper bound on the dimension of $\phi^2$, the lowest dimension scalar appearing in $\phi\times\phi$. Curves for $k=2,\dots,11$ are shown, with the $k=11$ bound being the strongest.
• Figure 3: An upper bound on the dimension of $\phi^\dag\phi$, the lowest dimension singlet scalar appearing in $\phi^\dag\times\phi$, where $\phi$ transforms in the fundamental representation of an $\mathrm{SO}(4)$ or an $\mathrm{SU}(2)$ global symmetry. Curves are shown for $k=2,\dots,11$. The bounds for $\mathrm{SO}(4)$ and $\mathrm{SU}(2)$ are identical in each case. The strongest bound crosses $\Delta_0=4$ around $d=1.52$.
• Figure 4: An upper bound on the dimension of $|\phi|^2$, the lowest dimension singlet scalar appearing in $\phi^T\times\phi$ (or $\phi^\dag\times\phi)$, where $\phi$ transforms in the fundamental representation of an $\mathrm{SO}(N)$ global symmetry or an $\mathrm{SU}(N/2)$ global symmetry (when $N\geq 4$ is even). Curves are shown for $N=2,\dots,14$, with $N=2$ being the strongest bound.
• Figure 5: Viable regions for conformal technicolor models in the flavor-generic (red) and flavor-optimistic (cross-hatched green) cases are shown superimposed with our bound (blue, excluding the gray-shaded region). Regions for $c(\Lambda_\mathrm{UV})=1,\ 0.1,$ and $0.01$ are shown in successively lighter shades of each color, with the largest region corresponding to $c(\Lambda_\mathrm{UV})=0.01$ in each case. Flavor-generic models are ruled out.