Positive geometry in the diagonal limit of the conformal bootstrap

TL;DR

The paper develops a positive-geometry perspective on the diagonal conformal bootstrap in arbitrary dimensions, showing that spin induces a weighted Minkowski sum of cyclic polytopes, which becomes a cyclic polytope in the large $\Delta$ limit. It provides analytic forms for the diagonal conformal blocks, including exact and Watson-type asymptotics, and demonstrates that the asymptotic blocks closely approximate the exact ones even at moderate $\Delta$. By embedding the crossing equations into a projective polytope framework, it connects the standard linear-programming bootstrap with the cyclic-polytope picture, offering analytic and numerical pathways to study bounds and kinks. The work lays out explicit positivity criteria and crossing-compatibility conditions, enabling a promising synthesis of bootstrap numerics with positive-geometry methods for higher-dimensional CFTs.

Abstract

We consider the diagonal limit of the conformal bootstrap in arbitrary dimensions and investigate the question if physical theories are given in terms of cyclic polytopes. Recently, it has been pointed out that in $d=1$, the geometric understanding of the bootstrap equations for unitary theories leads to cyclic polytopes for which the faces can all be written down and, in principle, the intersection between the unitarity polytope and the crossing plane can be systematically explored. We find that in higher dimensions, the natural structure that emerges, due to the inclusion of spin, is the weighted Minkowski sum of cyclic polytopes. While it can be explicitly shown that for physical theories, the weighted Minkowski sum of cyclic polytopes is not a cyclic polytope, it also turns out that in the large conformal dimension limit it is indeed a cyclic polytope. We write down several analytic formulae in this limit and show that remarkably, in many cases, this works out to be very good approximation even for $O(1)$ conformal dimensions. Furthermore, we initiate a comparison between usual numerics obtained using linear programming and what arises from positive geometry considerations.

Figures (3)

• Figure 1: Plot of bounds on dimension of leading scalar in the OPE of $\phi\times \phi$ for $d=2$ using linear programming, keeping only scalars in the spectrum in the diagonal limit. The blue line is for 10 derivatives in the functional while the black is for 20 derivatives. In each case truncating the spectrum does not change the plot much. The green line is obtained in the non-diagonal limit rychkovvichi, including spins, while the red cross is the location of the 2d Ising model. The red line indicates what arises from considering the intersection of the $N=2$ cyclic polytope obtained using scalars in $d=2$ and the crossing plane.
• Figure 2: Plot of $\partial_z^2 \mathcal{F}_{d,\Delta,\ell}(z)$ as a function of $z$ for $\Delta_{\phi}=0.75$ in $d=2$. For $\ell\geq 4$ for unitary $\Delta\geq \ell$, the second derivative is positive in the region shown above.
• Figure 3: Ratio of \ref{['block']} and \ref{['blockapprox']} vs $z$ for various $\Delta$ and $\ell$. As is clear eq.(\ref{['blockapprox']}) gives an excellent approximation to the exact blocks for physical values of $\Delta,\ell$ to within a few percent.