Carving Out the Space of Conformal Field Theories

We study the constraints of conformal symmetry and unitarity in Conformal Field Theories (CFTs). Crossing symmetry of four-point functions implies universal bounds on operator dimensions and three-point function coefficients. These bounds can be extracted by solving a class of infinite-dimensional convex optimization problems, giving quantitative, nonperturbative results about potentially strongly coupled theories. Our results include general bounds on operator dimensions in 4d CFTs with concrete phenomenological implications, and novel determinations of critical exponents in the 3d Ising Model. We also introduce new techniques for computing conformal blocks of higher spin operators, paving the way for further studies.