Quantum [formula omitted] Drinfeld–Sokolov hierarchy and quantum singularity theory

Abstract

In this paper we compute explicitly the double ramification hierarchy and its quantization for the D4 Dubrovin–Saito cohomological field theory obtained applying the Givental–Teleman reconstruction theorem to the D4 Coxeter group Frobenius manifold, or equivalently the D4 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 D4 Drinfeld–Sokolov hierarchy. Our techniques provide hence an explicit quantization of the D4 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 D4 CohFT with respect to folding of the Dynkin diagram, the B3 and G2 Drinfeld–Sokolov hierarchies.