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Matrix Model Conjecture for Exact BS Periods and Nekrasov Functions

A. Mironov, A. Morozov, Sh. Shakirov

Abstract

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-ensemble), dS = ydz + O(ε^2) = \sum_p ε^{2p} ρ_β^{(p|1)}(z), where εand βare related to the LNS parameters ε_1 and ε_2. 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.

Matrix Model Conjecture for Exact BS Periods and Nekrasov Functions

Abstract

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-ensemble), dS = ydz + O(ε^2) = \sum_p ε^{2p} ρ_β^{(p|1)}(z), where εand βare related to the LNS parameters ε_1 and ε_2. 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.

Paper Structure

This paper contains 64 equations, 2 figures.

Figures (2)

  • Figure 1: Potential $-W(z)$ for four masses ${\vec{\alpha}} = (3,2,1,3)$ situated at positions ${\vec{q}} = (0.5, 1.5, 2.5, 3)$. The extrema of $W(z)$ are filled with the eigenvalues with filling fractions $N_1, N_2, N_3$.
  • Figure 2: Diagram for the five-point conformal block, with the dimensions of external states corresponding to the parameters $\alpha$ and the dimensions of intermediate states corresponding to the parameters $a_1 = N_1 - N_2$, $a_2 = N_2 - N_3$.