Matrix model version of AGT conjecture and generalized Selberg integrals

Abstract

Operator product expansion (OPE) of two operators in two-dimensional conformal field theory includes a sum over Virasoro descendants of other operator with universal coefficients, dictated exclusively by properties of the Virasoro algebra and independent of choice of the particular conformal model. In the free field model, these coefficients arise only with a special "conservation" relation imposed on the three dimensions of the operators involved in OPE. We demonstrate that the coefficients for the three unconstrained dimensions arise in the free field formalism when additional Dotsenko-Fateev integrals are inserted between the positions of the two original operators in the product. If such coefficients are combined to form an $n$-point conformal block on Riemann sphere, one reproduces the earlier conjectured $\beta$-ensemble representation of conformal blocks, thus proving this (matrix model) version of the celebrated AGT relation. The statement can also be regarded as a relation between the $3j$-symbols of the Virasoro algebra and the slightly generalized Selberg integrals $I_Y$, associated with arbitrary Young diagrams. The conformal blocks are multilinear combinations of such integrals and the remaining part of the original AGT conjecture relates them to the Nekrasov functions which have exactly the same structure.

19 pages

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Cited by in corpus (13)

• On AGT Relations with Surface Operator Insertion and Stationary Limit of Beta-Ensembles
• Gauge theories on Omega-backgrounds from non commutative Seiberg-Witten curves
• A direct proof of AGT conjecture at beta = 1
• Brezin-Gross-Witten model as "pure gauge" limit of Selberg integrals
• Massive Scaling Limit of beta-Deformed Matrix Model of Selberg Type
• From Brezin-Hikami to Harer-Zagier formulas for Gaussian correlators
• Hitchin Equation, Irregular Singularity, and $N=2$ Asymptotical Free Theories
• Classical conformal blocks from TBA for the elliptic Calogero-Moser system
• Quantum Hitchin Systems via beta-deformed Matrix Models
• Generalized matrix models and AGT correspondence at all genera
• Seiberg-Witten curve via generalized matrix model
• Direct integration for general Omega backgrounds
• An(1) Affine Quiver Matrix Model