Matrix model version of AGT conjecture and generalized Selberg integrals

TL;DR

This paper proves a matrix model version of the AGT conjecture by connecting conformal blocks, Virasoro algebra coefficients, and generalized Selberg integrals, revealing deep relations in conformal field theory and matrix models.

Contribution

It demonstrates that conformal blocks can be represented through matrix models involving generalized Selberg integrals, confirming a matrix model version of the AGT conjecture.

Findings

Conformal blocks are expressed as multilinear combinations of generalized Selberg integrals.

The matrix model representation of conformal blocks aligns with the AGT conjecture.

Virasoro algebra coefficients relate to generalized Selberg integrals associated with Young diagrams.

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.