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

TL;DR

This paper connects the Baxter Q-operator in N=2 gauge theory with degenerate fields in Toda and Liouville theories, deriving new difference-differential relations that generalize known equations in the Nekrasov-Shatashvili limit.

Contribution

It introduces a novel expression for the Q-operator in terms of gauge theory operators and extends the T-Q relation beyond the Nekrasov-Shatashvili limit.

Findings

Derived a mixed difference-differential relation for the Q-operator.

Expressed degenerate primary fields in Toda theory via the Q-operator.

Generalized the T-Q difference equation to the generic case.

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.