Proving the AGT relation for N_f = 0,1,2 antifundamentals

TL;DR

Proves the AGT correspondence for SU(2) quivers with $N_f=0,1,2$ antifundamentals by equating recursion relations for irregular blocks, derived from decoupling limits of the Zamolodchikov elliptic 4-point block, with the Nekrasov partition functions. The method extends Fateev–Litvinov's torus one-point block proof to multiple antifundamental flavors via explicit recursions and large-$p$ asymptotics. The main contributions are (i) recursive constructions for irregular blocks with multiple $\mu$ parameters and their fusion-polynomial structure, (ii) matching recursions for $N_f=0,1,2$ with those of the Nekrasov functions, and (iii) explicit identifications $Z^{0} = \langle \Delta,\Lambda^2|\Delta,\Lambda^2\rangle$, $Z^{1} = \langle \Delta,\mu,\tfrac12 \Lambda|\Delta,\Lambda^2\rangle$, $Z^{2} = e^{-\Lambda^2/2}\langle \Delta,\mu_1,\tfrac12 \Lambda|\Delta,\mu_2,\tfrac12 \Lambda\rangle$, thereby extending the AGT framework to this nontrivial flavor regime.

Abstract

Using recursive relations satisfied by Nekrasov partition functions and by irregular conformal blocks we prove the AGT correspondence in the case of N=2 superconformal SU(2) quiver gauge theories with N_f = 0,1,2 antifundamental hypermultiplets