Conformal blocks as Dotsenko-Fateev Integral Discriminants

TL;DR

This paper provides further evidence that Dotsenko-Fateev integral discriminants precisely match conformal blocks, supporting the AGT conjecture and extending the correspondence to multi-point functions and torus cases.

Contribution

It demonstrates the equivalence for a broad family of 4-point spherical conformal blocks up to level 3 and extends the analysis to multi-point and torus conformal functions.

Findings

Confirmed the identity for 4-point blocks up to level 3.

Extended the correspondence to multi-point spherical functions.

Commented on similar descriptions for 1-point functions on a torus.

Abstract

As anticipated in [1], elaborated in [2-4], and explicitly formulated in [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 DV 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 multi-point spherical functions and comment on a similar description of the 1-point function on a torus.