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 J=Z∕3Z). 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.