BPZ equations for higher degenerate fields and non-perturbative Dyson-Schwinger equations

Abstract

A correlation function in the two-dimensional Liouville field theory satisfies a linear partial differential equation when it involves a degenerate field, due to the decoupling of the null descendant field. According to the Alday-Gaiotto-Tachikawa (AGT) correspondence, such a correlation function is related to the partition function of certain four-dimensional \(\mathcal{N}=2\) superconformal quiver gauge theories in the \(\Omega\)-background at a special point in the parameter space. Using the non-perturbative Dyson-Schwinger equations, we show that the instanton partition function satisfies a differential equation resulting from the constraints of the gauge field configurations. We demonstrate that the third order differential equations satisfied by the correlation function in the Liouville field theory with the next-to-simplest degenerate field and by the gauge theory partition function with the corresponding constraint can be identified, thereby confirming the main assertion of the BPS/CFT correspondence.