Solving Conformal Field Theories with the Functional Bootstrap

Conformal Field Theories (CFTs) have a wide range of experimental and theoretical applications: describing classical andquantum critical phenomena, where they determine critical exponents; as low (or high) energy limits of ordinary quantumfield theories; and as theories of quantum gravity in disguise via the AdS/CFT correspondence.Unfortunately, most interesting CFTs are strongly interacting and difficult to analyse. On the one hand, perturbative andrenormalization group methods usually involve approximations that are hard to control and which require difficultresummations. On the other hand, numerical simulations of the underlying systems are difficult near the critical point and canaccess only a limited set of observables.The conformal bootstrap program is a new approach. It exploits basic consistency conditions which are encoded into aformidable set of bootstrap equations, to map out and determine the space of CFTs. A longstanding conjecture states thatthese equations actually provide a fully non-perturbative definition of CFTs. In this project we will develop a groundbreakingset of tools ? analytic extremal functionals ? to extract information from the bootstrap equations. This Functional Bootstraphas the potential to greatly deepen our understanding of CFTs as well as to determine incredibly precise bounds on thespace of theories. Our main goals are A) to fully develop the functional bootstrap for the simpler and mostly unexplored one-dimensional setting, relevant for critical systems such as spin models with long-range interactions and line defects inconformal gauge theories, leading to analytic insights and effective numerical solutions of these systems; and B) to establishfunctionals as the default technique for higher dimensional applications by developing the formalism, obtaining generalanalytic bounds and integrating into existing numerical workflows to obtain highly accurate determinations of criticalexponents