On AGT description of $ \mathcal{N} = 2 $ SCFT with N f = 4

Abstract

We consider Alday-Gaiotto-Tachikawa (AGT) realization of the Nekrasov partition function of $ \mathcal{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.