On AGT description of N=2 SCFT with N_f=4

TL;DR

The paper demonstrates that the Weyl symmetry of the ${ m SO}(8)$ flavor group in the ${ m SU}(2)$ ${ m N}=2$ theory with $N_f=4$ is encoded in Liouville theory through nontrivial functional relations among four-point functions, providing a consistency check of the AGT correspondence. It elucidates the analytic extension of Liouville integral representations, culminating in a symmetry identity (the Postalin equation) that aligns Liouville four-point functions with the gauge theory’s Weyl actions. Additionally, it links surface operators to WZW theory via Hamiltonian reduction, offering a second route to the symmetry through five-point Liouville functions and KZ equations, and discusses broader interconnections between Liouville, WZW, and potential Langlands-type structures. These results deepen our understanding of how 4D ${ m N}=2$ gauge-theory dualities mirror 2D CFT structures and suggest new avenues for interpreting observables in terms of WZW correlators and degenerate insertions.

Abstract

We consider Alday-Gaiotto-Tachikawa (AGT) realization of the Nekrasov partition function of N=2 SCFT. We focus our attention on the SU(2) theory with N_f=4 flavor symmetry, whose partition function, according to AGT, is given by the Liouville four-point function on the sphere. The gauge theory with N_f=4 is known to exhibit SO(8) symmetry. We explain how the Weyl symmetry transformations of SO(8) flavor symmetry are realized in the Liouville theory picture. This is associated to functional properties of the Liouville four-point function that are a priori unexpected. In turn, this can be thought of as a non-trivial consistency check of AGT conjecture. We also make some comments on elementary surface operators and WZW theory.