The Polyakov Bootstrap: Critical exponents without Feynman diagrams

In 1974, Polyakov had proposed a program to bootstrap conformal field theories. For a very long time this lay essentially ignored. It turns out that in a very precise way he had anticipated Witten diagrams, which are the Feynman diagrams in AdS space, as the building blocks for the “crossing-symmetric” bootstrap program. I will show how insights from AdS/CFT allow us to write down a basis in Mellin space and how to use this basis to extract multiloop Feynman diagram results for critical exponents and OPE coefficients for the Wilson-Fisher fixed point completely algebraically without encountering any infinities (and without calculating any Feynman diagrams!).