Matrix model conjecture for exact BS periods and Nekrasov functions

• Published in: The journal of high energy physics Vol. 2010; no. 2
• Main Authors: Mironov, A., Morozov, A., Shakirov, Sh
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01.02.2010; Springer Nature B.V
• Subjects: Classical and Quantum Gravitation; Elementary Particles; Hierarchies; High energy physics; Integrals; Physics; Physics and Astronomy; Quantum Field Theories; Quantum Field Theory; Quantum Physics; Relativity Theory; String Theory; Conformal and W Symmetry; Integrable Hierarchies; Matrix Models; Supersymmetric gauge theory
• ISSN: 1029-8479
• DOI: 10.1007/JHEP02(2010)030

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), , where ε and β are 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.