On the Drinfeld-Sokolov hierarchy of type $$E_6^{(1)}$$ and its topological solution

Abstract

In this paper we report some explicit evolutionary PDEs of the Drinfeld-Sokolov hierarchy of type $$E_6^{(1)}$$, and show how the unknown functions in these PDEs are related to the tau function. Moreover, for this hierarchy we compute its topological solution of formal series up to a certain degree, whose coefficients of monomials give the Fan-Jarvis-Ruan-Witten invariants for the E6 simple singularity. Based on such results we also derive several explicit evolutionary PDEs and some low-degree terms of the topological solution for the Drinfeld-Sokolov hierarchy of type $$F_4^{(1)}$$.