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arXiv · 1711.11582

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

Nikita Nekrasov
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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

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Nikita Nekrasov. 2017-12-04. BPS/CFT correspondence V: BPZ and KZ equations from qq-characters. https://arxiv.org/abs/1711.11582

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