We develop new tools for isolating CFTs using the numerical bootstrap. A “cutting surface” algorithm for scanning OPE coefficients makes it possible to find islands in high-dimensional spaces. Together with recent progress in large-scale semidefinite programming, this enables bootstrap studies of much larger systems of correlation functions than was previously practical. We apply these methods to correlation functions of charge-0, 1, and 2 scalars in the 3d O(2) model, computing new precise values for scaling dimensions and OPE coefficients in this theory. Our new determinations of scaling dimensions are consistent with and improve upon existing Monte Carlo simulations, sharpening the existing decades-old 8σ discrepancy between theory and experiment.

We study the critical $O(3)$ model using the numerical conformal bootstrap. In particular, we use a recently developed cutting-surface algorithm to efficiently map out the allowed space of conformal field theory data from correlators involving the leading $O(3)$ singlet $s$, vector $\varphi$, and rank-2 symmetric tensor $t$. We determine their scaling dimensions to be $({\mathrm{\Delta }}_{\varphi },{\mathrm{\Delta }}_s,{\mathrm{\Delta }}_t)=(0.518942(51),1.59489(59),1.20954(23))$, and also bound various operator product expansion coefficients. We additionally introduce a new “tip-finding” algorithm to compute an upper bound on the leading rank-4 symmetric tensor $t_4$, which we find to be relevant with ${\mathrm{\Delta }}_{t_4}<2.99056$. The conformal bootstrap thus provides a numerical proof that systems described by the critical $O(3)$ model, such as classical Heisenberg ferromagnets at the Curie transition, are unstable to cubic anisotropy.

We bound EFT coefficients appearing in 2 → 2 photon scattering amplitudes in four dimensions. After reviewing unitarity and positivity conditions in this context, we use dispersion relations and crossing symmetry to compute sum rules and null constraints. This allows us to derive new rigorous bounds on operators with four, six, and eight derivatives, including two-sided bounds on their ratios. Comparing with a number of partial UV completions, we find that some of our bounds are saturated by the amplitudes that arise from integrating out a massive scalar or axion, while others suggest the existence of unknown amplitudes.

The black hole weak gravity conjecture (WGC) is a set of linear inequalities on the four-derivative corrections to Einstein-Maxwell theory. Remarkably, in four dimensions, these combinations appear in the 2 → 2 photon amplitudes, leading to the hope that the conjecture might be supported using dispersion relations. However, the presence of a pole arising in the forward limit due to graviton exchange greatly complicates the use of such arguments. In this paper, we apply recently developed numerical techniques to handle the graviton pole, and we find that standard dispersive arguments are not strong enough to imply the black hole WGC. Specifically, under a fairly typical set of assumptions, including weak coupling of the EFT and Regge boundedness, a small violation of the black hole WGC is consistent with unitarity and causality. We quantify the size of this violation, which vanishes in the limit where gravity decouples and also depends logarithmically on an infrared cutoff. We discuss the meaning of these bounds in various scenarios. We also implement a method for bounding amplitudes without manifestly positive spectral densities, which could be applied to any system of non-identical states, and we use it to improve bounds on the EFT of pure photons in absence of gravity.