On non-conformal limit of the AGT relations

TL;DR

This paper explores the non-conformal limit of AGT relations by analyzing the behavior of conformal blocks in the Virasoro algebra as external dimensions tend to infinity, connecting gauge theory limits to conformal field theory structures.

Contribution

It explicitly computes the non-conformal limit of Virasoro conformal blocks, confirming conjectured behaviors using representation theory and extending the AGT correspondence beyond conformal cases.

Findings

Conformal blocks reduce to 2- or 3-point functions in the limit.

The analysis confirms conjectured limits from previous studies.

Representation theory provides a rigorous framework for these limits.

Abstract

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_{N_c} chiral algebra). We consider the limit of the four-point function explicitly in the Virasoro case of N_c=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 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.