Conformal blocks in D>2. From Conformal Bootstrap to the new class of Calogero-Sutherland models.
I will start with a short overview of conformal bootstrap program in D>2. Conformal bootstrap is a nonperturbative symmetry-based approach to CFT. It starts with formal definition of CFT as a set of self consistent CFT data and then analyses all consequences of symmetries. Particularly combining unitarity with crossing one can produce bounds on anomalous dimensions and structure constants. One of the most famous result in this direction - the current most precise determination of critical exponents in the 3d Ising model (arXiv:1603.04436).
The only one analytical input in the numerical bootstrap program is explicit form of conformal blocks. In the main part of my talk I will present our universal approach to general conformal blocks through the harmonic analysis of certain bundles over a coset of the conformal group. The resulting Casimir equations are given by a matrix version of the Calogero-Sutherland Hamiltonian that describes the scattering of interacting spinning particles in a 1-dimensional external potential. The approach will be illustrated in several examples including fermionic seed blocks in 3D CFT where they take a very simple form.