BPS/CFT correspondence V: BPZ and KZ equations from qq-characters

TL;DR

This paper derives BPZ and KZ equations for surface defect partition functions in 4D ${ m extbf{N}=2}$ theories using $qq$-characters, linking them to Liouville and WZW models, and demonstrating new connections in gauge theory and 2D CFT.

Contribution

It introduces a novel method to derive BPZ and KZ equations from $qq$-characters for specific surface defects in ${ m extbf{N}=2}$ theories, establishing new links to 2D conformal field theories.

Findings

Partition functions satisfy BPZ equations in Liouville theory.

Orbifold defects solve KZ-like equations in WZW models.

Connections between 4D gauge theories and 2D CFTs are elucidated.

Abstract

We illustrate the use of the theory of $qq$-characters by deriving the BPZ and KZ-type equations for the partition functions of certain surface defects in quiver ${\mathcal N}=2$ theories. We generate a surface defect in the linear quiver theory by embedding it into a theory with additional node, with specific masses of the fundamental hypermultiplets. We prove that the supersymmetric partition function of this theory with $SU(2)^{r-3}$ gauge group verifies the celebrated Belavin-Polyakov-Zamolodchikov equation of two dimensional Liouville theory. We also study the $SU(N)$ theory with $2N$ fundamental hypermultiplets and the theory with adjoint hypermultiplet. We show that the regular orbifold defect in this theory solves the KZ-like equation of the WZW theory on a four punctured sphere and one-punctured torus, respectively. In the companion paper these equations will be mapped to the Knizhnik-Zamolodchikov equations