As anticipated in Ref. 1,, 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.

We show that partition functions of various matrix models can be obtained by acting on elementary functions with exponents ofŴ operators. A number of illustrations is given, including the Gaussian Hermitian matrix model, Hermitian model in external field and the Hurwitz-Kontsevitch model, for which we suggest an elegant matrix-model representation. In all these examples, the relevantŴ operators belong to theŴ (3) algebra.

We extend the proof from [25],which interprets the AGT relation as the Hubbard-Stratonovich duality relation to the case of 5d gauge theories. This involves an additional q-deformation. Not surprisingly, the extension turns out to be trivial: it is enough to substitute all relevant numbers by q-numbers in all the formulas, Dotsenko-Fateev integrals by the Jackson sums and the Jack polynomials by the MacDonald ones. The problem with extra poles in individual Nekrasov functions continues to exist, therefore, such a proof works only for β = 1, i.e. for q = t in MacDonald's notation. For β = 1 the conformal blocks are related in this way to a non-Nekrasov decomposition of the LMNS partition function into a double sum over Young diagrams. * Lebedev Physics Institute and ITEP

We give a concise summary of the impressive recent development unifying a number of different fundamental subjects. The quiver Nekrasov functions (generalized hypergeometric series) form a full basis for all conformal blocks of the Virasoro algebra and are sufficient to provide the same for some (special) conformal blocks of W-algebras. They can be described in terms of Seiberg-Witten theory, with the SW differential given by the 1-point resolvent in the DV phase of the quiver (discrete or conformal) matrix model (β-, where ε and β are related to the LNS parameters ǫ1 and ǫ2. This provides explicit formulas for conformal blocks in terms of analytically continued contour integrals and resolves the old puzzle of the free-field description of generic conformal blocks through the Dotsenko-Fateev integrals. Most important, this completes the GKMMM description of SW theory in terms of integrability theory with the help of exact BS integrals, and provides an extended manifestation of the basic principle which states that the effective actions are the tau-functions of integrable hierarchies. Today it is clear that a new unification of the fundamental importance emerges, bringing together at a principally new level the CFT, the theory of loop algebras, SW theory, quantization theory, Baxter equations, DV phase of matrix models, loop equations, the theory of hypergeometric functions, symmetric groups, Hurwitz theory, Kontsevich models and modern combinatorics. This unification is capable to resolve a number of long-standing problems in each of the fields. The goal of this paper is to briefly summarize our knowledge about this emerging pattern, which is scattered and expressed in length in a number of fresh [29,30], makes the picture complete, resolves the remaining uncertainties (about the shape of the second deformation) in [56,57] and finalizes the program [3] to reformulate SW theory of [1,2] in terms of the BS integrals and underlying integrable systems. We do not discuss long formulas, checks and even evidence in favor of all these conjectures: all calculations in these fields remain long and tedious, and most statements still need to be checked and proved, however, the entire picture is starting to get relatively clear.

The AGT conjecture claims an equivalence of conformal blocks in 2d CFT and sums of Nekrasov functions (instantonic sums in 4d SUSY gauge theory). The conformal blocks can be presented as Dotsenko-Fateev β-ensembles, hence, the AGT conjecture implies the equality between Dotsenko-Fateev β-ensembles and the Nekrasov functions. In this paper, we prove it in a particular case of β = 1 (which corresponds to c = 1 at the conformal side and to ǫ 1 + ǫ 2 = 0 at the gauge theory side) in a very direct way. The central role is played by representation of the Nekrasov functions through correlators of characters (Schur polynomials) in the Selberg matrix models. We mostly concentrate on the case of SU (2) with 4 fundamentals, the extension to other cases being straightforward. The most obscure part is extending to an arbitrary β: for β = 1, the Selberg integrals that we use do not reproduce single Nekrasov functions, but only sums of them. * Lebedev Physics Institute and ITEP, Moscow, Russia;