Non-conformal limit of AGT relation from the 1-point torus conformal block

TL;DR

This paper investigates the AGT relation for a specific 4d N=2 supersymmetric gauge theory, analyzing the large mass limit of the associated conformal block to provide a non-trivial consistency check of the conjecture.

Contribution

It extends the analysis of the AGT relation to the large mass limit of the conformal block for the N=2 theory with an adjoint multiplet, confirming the conjecture's validity in this regime.

Findings

The large mass limit of the conformal block matches expectations from the AGT relation.

The analysis confirms the AGT conjecture for asymptotically free theories with an adjoint multiplet.

The limit behavior is consistent with previous cases involving fundamental multiplets.

Abstract

Given a 4d N=2 SUSY gauge theory, one can construct the Seiberg-Witten prepotentional, which involves a sum over instantons. Integrals over instanton moduli spaces require regularisation. For UV-finite theories the AGT conjecture favours particular, Nekrasov's way of regularization. It implies that Nekrasov's partition function equals conformal blocks in 2d theories with W_{N_c} chiral algebra. For $N_c=2$ and one adjoint multiplet it coincides with a torus 1-point Virasoro conformal block. We check the AGT relation between conformal dimension and adjoint multiplet's mass in this case and investigate the limit of the conformal block, which corresponds to the large mass limit of the 4d theory e.i. the asymptotically free 4d N=2 supersymmetric Yang-Mills theory. Though technically more involved, the limit is the same as in the case of fundamental multiplets, and this provides one more non-trivial check of AGT conjecture.