Conformal Blocks as Dotsenko–fateev Integral Discriminants

As anticipated in Ref. 1, elaborated in Refs. 2–4, and explicitly formulated in Ref. 5, the Dotsenko–Fateev integral discriminant coincides with conformal blocks, thus providing an elegant approach to the AGT conjecture, without any reference to an auxiliary subject of Nekrasov functions. Internal dimensions of conformal blocks in this identification are associated with the choice of contours: parameters of the Dijkgraaf–Vafa phase of the corresponding matrix models. In this paper, we provide further evidence in support of this identity for the 6-parametric family of the 4-point spherical conformal blocks, up to level 3 and for arbitrary values of external dimensions and central charges. We also extend this result to multipoint spherical functions and comment on a similar description of the 1-point function on a torus.