VEV of Baxter’s Q-operator in N = 2 gauge theory and the BPZ differential equation

In this short note using AGT correspondence we express the simplest fully degenerate primary fields of Toda field theory in terms of an analogue of Baxter’s Q-operator naturally emerging on the N = 2 gauge theory side. This quantity can be considered as a generating function of certain chiral operators constructed from the scalars of the N = 2 vector multiplets. In the special case of Liouville theory, exploring the second order differential equation satisfied by conformal blocks including a primary field which is degenerate at the second level (BPZ equation) we derive a mixed difference-differential relation for Q-operator. Thus we generalize the T -Q difference equation known in Nekrasov-Shatashvili limit of the Ω-background to the generic case.