On non-conformal limit of the AGT relations

The Seiberg-Witten prepotentials for N = 2 SUSY gauge theories with N-f < 2N(c) fundamental multiplets are obtained from conformal N-f = 2N(c) theory by decoupling 2N(c) - N-f multiplets of heavy matter. This procedure can be lifted to the level of Nekrasov functions with arbitrary background parameters epsilon(1) and epsilon(2). The AGT relations imply that similar limit exists for conformal blocks (or, for generic N-c > 2, for the blocks in conformal theories with W-Nc chiral algebra). We consider the limit of the four-point function explicitly in the Virasoro case of N-c = 2, by bringing the dimensions or 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. (C) 2009 Elsevier B.V. All rights reserved.