Analytic Functionals for Higher-Dimensional Conformal Bootstrap

The conformal bootstrap has been extremely successful for simple critical systems such as the 3D Ising model and vector models, where it has led to very precise determinations of critical exponents.

However, obtaining similarly reliable nonperturbative data for more complicated theories, especially conformal gauge theories, remains difficult. One practical obstruction is that the standard derivative basis converges slowly when the external operators have high conformal dimension. In this talk, I will describe an analytic functional approach to this problem. I will start from the one-dimensional story, where analytic functionals give bases dual to generalized free spectraand turn crossing into discrete sum rules. I will then explain how the construction can be lifted to higher dimensions using the factorization of two-dimensional global blocks together with dimensional reduction formulas for higher-dimensional conformal blocks. The result is a class of product analytic functionals acting directly on higher-dimensional crossing equations. I will discuss the convergence and positivity properties of these functionals and show that, unlike the standard derivative basis, they maintain fast convergence even when the external operators have high conformal dimension. I will then present numerical applications, including bounds and data extraction in the 3D Ising model, as well as extensions to bootstrap problems with global symmetry. I will end with possible applications to mixed-correlator bootstrap and strongly coupled conformal gauge theories.