Solving Conformal Field Theories with the Functional Bootstrap

Conformal Field Theories (CFTs) have a wide range of experimental and theoretical applications: describing classical and

quantum critical phenomena, where they determine critical exponents; as low (or high) energy limits of ordinary quantum

field 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 and

renormalization group methods usually involve approximations that are hard to control and which require difficult

resummations. On the other hand, numerical simulations of the underlying systems are difficult near the critical point and can

access only a limited set of observables.

The conformal bootstrap program is a new approach. It exploits basic consistency conditions which are encoded into a

formidable set of bootstrap equations, to map out and determine the space of CFTs. A longstanding conjecture states that

these equations actually provide a fully non-perturbative definition of CFTs. In this project we will develop a groundbreaking

set of tools ? analytic extremal functionals ? to extract information from the bootstrap equations. This Functional Bootstrap

has the potential to greatly deepen our understanding of CFTs as well as to determine incredibly precise bounds on the

space 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 in

conformal gauge theories, leading to analytic insights and effective numerical solutions of these systems; and B) to establish

functionals as the default technique for higher dimensional applications by developing the formalism, obtaining general

analytic bounds and integrating into existing numerical workflows to obtain highly accurate determinations of critical

exponents