Quantum Langlands correspondence for Painlevé II

We construct an irregular KZ-BPZ correspondence for quantum Painlevé II from a rank-three Wakimoto-Whittaker module. The irregular KZ equation and the quantum isomonodromy equation form a joint affine system whose Drinfeld-Sokolov reduction yields scalar BPZ equations. We obtain matrix-integral solutions of this system for one fundamental insertion. An exact quantum Langlands duality exchanges the fundamental insertion with the zero of the dual Whittaker differential and allows affine solutions to be reconstructed from the BPZ equations. This duality interchanges the classical and Nekrasov-Shatashvili limits, which yield the Painlevé II Lax connection and an irregular Gaudin oper, respectively.