Torus conformal blocks and Casimir equations

The report discusses multipoint torus conformal blocks in an arbitrary channel. Each channel is characterized by a diagram containing a closed loop with legs (necklace) and trivalent vertices forming trees attached to the necklace. We show that blocks in an arbitrary channel can be constructed from a block in the "necklace" channel. For multipoint blocks in the "necklace" channel associated with the sl(2) algebra, the Casimir equations will be obtained and the plane limit will be analyzed when the modular parameter of the torus tends to zero. In this limit, it is shown that the Casimir equations are reduced to equations defining an (n+2)-point global block in a "comb" channel on a sphere. The report is based on arXiv:2205.05038.