The power of Nekrasov functions

The recent AGT suggestion (L. Alday et al. (2009) [1]) to use the set of Nekrasov functions (N. Nekrasov (2004) [2]) as a basis for a linear decomposition of generic conformal blocks works very well not only in the case of Virasoro symmetry, but also for conformal theories with extended chiral algebra. This is rather natural, because Nekrasov functions are introduced as expansion basis for generalized hypergeometric integrals, very similar to those which arise in expansion of Dotsenko-Fateev integrals in powers of alpha-parameters. Thus, the AGT conjecture is closely related to the old belief that conformal theory can be effectively described in the free field formalism, and it can actually be a key to clear formulating and proof this long-standing hypothesis. As an application of this kind of reasoning we use knowledge of the exact hypergeometric conformal block for complete proof of the AGT relation for a restricted class of external states. (C) 2009 Elsevier B.V. All rights reserved.