Irregular KZ equations and Kac–Moody representations

In this paper we construct irregular representations of the affine Kac–Moody algebra sl (2, ℂ). We show how such irregular representations correspond to irregular Gaiotto–Teschner representations of the Virasoro algebra. The intertwiners for such representations satisfy a version of Knizhnik–Zamolodchikov (KZ) equations which we call irregular KZ equations. By connecting to 2d Liouville theory, we show how the conformal blocks governed by our irregular KZ equation correspond to 4d Argyres–Douglas theories with surface operator insertions. The corresponding flat connections describe braiding between such operators on the Gaiotto curve.

It is our pleasure to thank Boris Feigin, Xia Gu, Pavel Putrov and Xiaomeng Xu for helpful discussions and suggestions. The work of S G is supported by a Simons Collaboration Grant on New Structures in Low-Dimensional Topology, by the NSF Grant DMS-2245099, and by the U.S. Department of Energy, Office of Science, Office of High Energy Physics, under Award No. DE-SC0011632. The work of B H and Y L is supported by NSFC Grant 12250610187. The work of N R was supported by the Simons Collaboration ‘Categorical symmetries’, by the grant BMSTC and ACZSP (Grant No. Z221100002722017), by the Changjiang fund, and by the Project No 075-15-2024-631 funded by the ministery of Science and Higher Education of the Russian Federation.

All data that support the findings of this study are included within the article (and any supplementary files).