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

Abstract

In this short notes using AGT correspondence we express simplest fully degenerate primary fields of Toda field theory in terms an analogue of Baxter's $Q$-operator naturally emerging in ${\cal N}=2$ gauge theory side. This quantity can be considered as a generating function of simple trace chiral operators constructed from the scalars of the ${\cal N}=2$ vector multiplets. In the special case of Liouville theory, exploring the second order differential equation satisfied by conformal blocks including a degenerate at the second level primary field (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 $\Omega$-background to the generic case.