Aah-Boots · Analytic Advances for the Hybrid Bootstrap

The conformal bootstrap program has been spectacularly successful in producing rigorous numerical bounds on the dynamical data of conformal field theories (CFTs)—systems appearing ubiquitously in many areas of physics and experimental setups. The standard numerical bootstrap uses basic consistency principles to exclude huge regions of parameter space, often leaving small allowed ""islands"" that essentially determine low-lying CFT data, e.g. for the 3d Ising model.Despite its success, the bootstrap is still held back by its reliance on heavy numerical algorithms. The state-of-the-art bounds for 3d Ising required the order of 10 million CPU-hours. Not only is this slow and expensive, but numerical convergence is a scientific bottleneck. Particularly in 4d CFTs, slow convergence has widely been found to be an obstacle to obtaining precision islands.In light of these numerical limitations, it is natural to ask if the shape and position of bootstrap islands can be understood analytically. We will develop such an analytic understanding by rigorously proving the duality between optimal bounds and the extremal solutions that saturate them. A consequence is that bootstrap islands for a given system of correlators cannot shrink to zero size, answering an enduring foundational question.As well as significantly expanding theoretical understanding of the bootstrap, we will harness this analytic information to develop a hybrid (analytic/numerical) bootstrap method. The idea is to solve directly for extremal solutions and extract optimal bounds from them. This stands to be much faster than the usual semi-definite programming numerics, opening a new avenue for precision 4d CFT bootstrap with multi-correlator systems that would otherwise be intractable. We will apply this hybrid method to bootstrap the conformal window of QCD, bounding low-lying CFT data and the conformal window in the number of flavors Nf. We will further apply it to bootstrap interesting 4d N=1 and N=2 SCFTs.""