From global to heavy-light: 5-point conformal blocks

We consider virasoro conformal blocks in the large central charge limit. there are different regimes depending on the behavior of the conformal dimensions. the most simple regime is reduced to the global $$ slleft(2,mathrm{mathbb{c}}right) $$ conformal blocks while the most complicated one is known as the classical conformal blocks. recently, fitzpatrick, kaplan, and walters showed that the two regimes are related through the intermediate stage of the so-called heavy-light semiclassical limit. we study this idea in the particular case of the 5-point conformal block. to find the 5-point global block we use the projector technique and the casimir operator approach. furthermore, we discuss the relation between the global and the heavy-light limits and construct the heavy-light block from the global block. in this way we reproduce our previous results for the 5-point perturbative classical block obtained by means of the monodromy method.