Conformal blocks and generalized Selberg integrals

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 β-ensemble representation of conformal blocks. The statement can also be regarded as a relation between the 3j  -symbols of the Virasoro algebra and the slightly generalized Selberg integrals IY$I_Y$, associated with arbitrary Young diagrams. The conformal blocks are multilinear combinations of such integrals and the AGT conjecture relates them to the Nekrasov functions which have exactly the same structure.