Journal of High Energy Physics, 2010
We give a concise summary of the impressive recent development unifying a number of different fun... more 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 (βensemble), , 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. 1. Introduction. A renewed interest to SW theory [1]-[14] and to Nekrasov functions [15]-[19] is caused by the AGT conjecture [20]-[31], which connects the subject to the classical field of conformal field theory [32]-[38]. 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 [20]-[31] and older [39]-[51] papers. The main emphasize will be put on the description of Nekrasov functions in terms of SW theory, where the exact prepotential is expressed through the exact Bohr-Sommerfeld integrals, and the integrand is provided by the 1-point function of conformal matrix model (or quiver β-ensemble) in the Dijkgraaf-Vafa (DV) phase [52]-[55]. This conjecture, explicitly formulated in and further investigated , makes the picture complete, resolves the remaining uncertainties (about the shape of the second deformation) in and finalizes the program [3] to reformulate SW theory of 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. 2. Nekrasov functions from BS/SW periods. The central object of emerging unification is the exact SW-Nekrasov prepotential F ( a|ǫ 1 , ǫ 2 ) = ǫ 1 ǫ 2 log Z N ek , which now has a number of different interpretations: (A) Z N ek = sum over chains of Young diagrams = generalized hypergeometric series (B) Z N ek = sum over partitions with Plancherel like measure = discretized matrix model (C) Z N ek = B = conformal block, depending on a number of external and internal dimensions (D) Z N ek = partition function of conformal β -ensemble in DV phase = generalized Dotsenko -Fateev integrals (E) Z N ek = solution to consistent set of SW equations a I = AI dS ǫ1,ǫ2 , ∂F ∂a I = BI dS ǫ1,ǫ2 ,
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Papers by A. Mironov