On non-conformal limit of the AGT relations

The Seiberg-Witten prepotentials for N=2 SUSY gauge theories with Nf<2Nc fundamental multiplets are obtained from conformal Nf=2Nc theory by decoupling 2Nc−Nf multiplets of heavy matter. This procedure can be lifted to the level of Nekrasov functions with arbitrary background parameters ϵ1 and ϵ2. The AGT relations imply that similar limit exists for conformal blocks (or, for generic Nc>2, for the blocks in conformal theories with WNc chiral algebra). We consider the limit of the four-point function explicitly in the Virasoro case of Nc=2, by bringing the dimensions of external states to infinity. The calculation is performed entirely in terms of representation theory for the Virasoro algebra and reproduces the answers conjectured in [D. Gaiotto, arXiv:0908.0307] with the help of the brane-compactification analysis and computer simulations. In this limit, the conformal block involving four external primaries, corresponding to the theory with vanishing beta-function, turns either into a 2-point or 3-point function, with certain coherent rather than primary external states.