Paper detail

BCFG Drinfeld-Sokolov Hierarchies and FJRW-Theory

According to the ADE Witten conjecture, which is proved by Fan, Jarvis and Ruan, the total descendant potential of the FJRW invariants of an ADE singularity is a tau function of the corresponding mirror ADE Drinfeld-Sokolov hierarchy. In the present paper, we show that there is a finite group $Γ$ acting on a certain ADE singularity which induces an action on the corresponding FJRW-theory, and the $Γ$-invariant sector also satisfies the axioms of a cohomological field theory except the gluing loop axiom. On the other hand, we show that there is also a $Γ$-action on the mirror Drinfeld-Sokolov hierarchy, and the $Γ$-invariant flows yield the BCFG Drinfeld-Sokolov hierarchy. We prove that the total descendant potential of the $Γ$-invariant sector of a FJRW-theory is a tau function of the corresponding BCFG Drinfeld-Sokolov hierarchy.