Argyres-Douglas theories, S-duality and AGT correspondence

We propose a Nekrasov-type formula for the instanton partition functions of four-dimensional N=2 U(2) gauge theories coupled to (A_1,D_{2n}) Argyres-Douglas theories. This is carried out by extending generalized AGT correspondence case group, which requires us define irregular states direct sum Virasoro and Heisenberg algebras. Using our formula, one can evaluate contribution theory at each fixed point on moduli space. As an application, we function (A_3,A_3) find it in peculiar relation that SU(2) with four fundamental flavors. From this relation, read off how S-duality group acts UV coupling theory.