abstract: 

Quantum Field Theory is a rich subject that forms the cornerstone of much of particle physics, quantum gravity, and condensed matter physics. Strongly coupled quantum field theories lie at the heart of many of the deepest problems in modern physics, ranging from quantum critical points, thermalization, and strange metals, to holographic duality, black holes, and the emergence of spacetime. 

Solvable models have had a long history of guiding our understanding of non-perturbative phenomena in quantum field theory. We begin by briefly reviewing some of the textbook solvable models, including integrable models such as the Sinh-Gordon model and large N models such as two-dimensional QCD with many colors. We then describe a remarkable recently discovered theory - the SYK model - which exhibits both novel field-theoretic features (a new large N limit) and novel physical features (quantum chaos). In this talk we will  present the solution to the  SYK model, giving formulas for all correlation functions. 

In the process of solving SYK, we  will need to develop new results and tools within conformal field theory. In particular, any correlation function in a conformal field theory can be expanded in a complete basis of conformal blocks, much in the same way as the familiar Fourier expansion of a function in a basis of plane waves. This framework of harmonic analysis for the conformal group  gives powerful control and insight into conformal field theory. We present several results in this direction, including explicit formulas for n-point conformal blocks and the 6j symbol for the conformal group, and show how they enter into the solution of SYK. More generally, we describe how these results may be applied to the conformal bootstrap - a program to find all conformal field theories through consistency relations of analyticity and crossing. We end with a discussion of directions for finding richer solvable models, and how they may shed light on our understanding of holographic duality.