Quantum ${D_4}$ Drinfeld-Sokolov hierarchy and quantum singularity theory

In this paper we compute explicitly the double ramification hierarchy and its quantization for the $D_4$ Dubrovin-Saito cohomological field theory obtained applying the Givental-Teleman reconstruction theorem to the $D_4$ Coxeter group Frobenius manifold, or equivalently the $D_4$ Fan-Jarvis-Ruan-Witten cohomological field theory (with respect to the non-maximal diagonal symmetry group $\langle J\rangle = \mathbb{Z}_3$). We then prove its equivalence to the corresponding Dubrovin-Zhang hierarchy, which was known to coincide with the $D_4$ Drinfeld-Sokolov hierarchy. Our techniques provide hence an explicit quantization of the $D_4$ Drinfeld-Sokolov hierarchy. Moreover, since the DR hierarchy is well defined for partial CohFTs too, our approach immediately computes the DR hierarchies associated to the invariant sectors of the $D_4$ CohFT with respect to folding of the Dynkin diagram, the $B_3$ and $G_2$ Drinfeld-Sokolov hierarchies.