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.
Highlights
JHEP11(2016)058 of a degenerate primary field in the conformal block [36]
Restricting to the case of Liouville theory, starting from the second order differential equation satisfied by the multi-points conformal blocks including a degenerate field V−b/2 [32] we derive the analogue equation satisfied by the gauge theory partition function with Q operator insertion. We show that this equation leads to a mixed linear difference-differential equation for Q operators which is a direct generalization of the T − Q equation from NS limit to the case of generic Ω-Background
Note that the parameters of the first gauge factor are chosen to be a0,u = a0,u − 1δ1,u, where a0,u are the parameters of the “frozen node” corresponding to the n antifundamental hypermultiplets. It has been shown in [36] that under such choice of parameters all n-tuples of Young diagrams Y0,u corresponding to the special node0 give no contribution to the partition function unless the first diagram Y0,1 consists of a single column while the remaining n − 1 diagrams are empty
Summary
JHEP11(2016)058 of a degenerate primary field in the conformal block [36]. In section 3, restricting to the case of Liouville theory, starting from the second order differential equation satisfied by the multi-points conformal blocks including a degenerate field V−b/2 [32] we derive the analogue equation satisfied by the gauge theory partition function with Q operator insertion. 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 Qoperator.
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