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

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

39 pages, paper 5 of 5; some typos fixed, refs added

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• Deforming SW curve
• Affine sl(N) conformal blocks from N=2 SU(N) gauge theories
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Cited by in corpus (4)

• On $W$-representations of $\beta$- and $q,t$-deformed matrix models
• Super instanton counting and localization
• Twisted reduction of quiver W-algebras
• VEV of $Q$-operator in $U(1)$ linear quiver 5d gauge theories