Conformal blocks as Dotsenko-Fateev Integral Discriminants

TL;DR

This paper strengthens the case that conformal blocks can be represented as Dotsenko-Fateev integral discriminants in Dijkgraaf-Vafa phases, offering a self-contained route to a modified AGT correspondence without Nekrasov functions. It derives explicit mappings between DF parameters $(oldsymbol{eta},oldsymbol{ abla})$ and the Virasoro data $( riangle_i, riangle,c)$ for the 4-point sphere and extends the construction to multipoint blocks on the sphere, including a torus subsection. Systematic level-by-level checks (especially at $eta=1$) show that the DF integrals reproduce conformal blocks up to level 3 and, for arbitrary $eta$, the level-1 block with consistent central charge $c(eta) = 1 - 6igl( orac{}{}igr)^2$, supporting the proposed dictionary. The work also discusses obstacles related to contour choices and torus/higher-genus generalizations, outlining the required developments for a complete proof and broader applicability to $W$-algebras.

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.

Figures (8)

• Figure 1: Feynman-like diagram for a 4-point conformal block.
• Figure 2: Feynman-like diagram for a 4-point conformal block. Here $\Delta = \tilde{a} \left( \tilde{a} + \frac{1}{b}-b\right)$ and $\Delta_j = \tilde{\alpha}_j\left(\tilde{\alpha}_j + \frac{1}{b}-b\right)$ are the single internal and four external dimensions, respectively. The role of the screenings is to modify the free field selection rule at the vertices: instead of $\tilde{a} = \tilde{\alpha}_1 + \tilde{\alpha}_2$ and $\tilde{\alpha}_4 \cong \tilde{a} + \alpha_3$ one has $\tilde{a} = \tilde{\alpha}_1 + \tilde{\alpha}_2 + b N_1$ and $\tilde{\alpha}_4 = b-1/b-\tilde{\alpha}_1-\tilde{\alpha}_2-\tilde{\alpha}_3-b(N_1+N_2) \cong \tilde{a} + \tilde{\alpha}_3 + b N_2$.
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