Construction of (higher-point) conformal blocks from generalised coherent states and their uses

For a Lagrangian quantum field theory, the equations determining its $n$-point correlation functions non-perturbatively form an infinite set of coupled, non-linear integral equations. This hierarchy simplifies drastically once conformal symmetry is imposed: it diagonalises when expressed through conformal partial waves. I will trace this connection following Mack's group-theoretic approach to conformal field theory (CFT). Building on this, I will first motivate the role of conformal blocks in solving for the dynamical content of a given CFT. I will then present a constructive approach to obtain them using generalized coherent states of the conformal group, or so-called oscillator representations. This framework makes the computation of higher-point and thermal conformal blocks in both two and four dimensions particularly transparent.