Matrix Model Conjecture for Exact BS Periods and Nekrasov Functions

TL;DR

This paper unifies various subjects by linking quiver Nekrasov functions, conformal blocks, and Seiberg-Witten theory through matrix models, providing explicit formulas and resolving longstanding puzzles in conformal field theory and integrability.

Contribution

It introduces a matrix model approach to express conformal blocks and Seiberg-Witten theory, completing the integrability description of SW theory via exact Bethe-Salpeter integrals.

Findings

Explicit formulas for conformal blocks via contour integrals

Resolution of the free-field description puzzle for conformal blocks

Extension of the SW theory description through integrability and tau-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 (\beta-ensemble), dS = ydz + O(\epsilon^2) = \sum_p \epsilon^{2p} \rho_\beta^{(p|1)}(z), where \epsilon and \beta are related to the LNS parameters \epsilon_1 and \epsilon_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.