BCFG Drinfeld-Sokolov Hierarchies and FJRW-Theory

TL;DR

This paper establishes a connection between FJRW-theory invariants of ADE singularities and BCFG Drinfeld-Sokolov hierarchies via group actions, extending the ADE correspondence to BCFG types.

Contribution

It introduces a group action framework linking FJRW-theory invariants to BCFG hierarchies, generalizing the ADE case and demonstrating the tau function property for invariant sectors.

Findings

The $ ext{Gamma}$-invariant sector of FJRW-theory forms a cohomological field theory.

The $ ext{Gamma}$-invariant flows produce the BCFG Drinfeld-Sokolov hierarchy.

The total descendant potential of the invariant sector is a tau function of the BCFG hierarchy.

Abstract

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 $\Gamma$ acting on a certain ADE singularity which induces an action on the corresponding FJRW-theory, and the $\Gamma$-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 $\Gamma$-action on the mirror Drinfeld-Sokolov hierarchy, and the $\Gamma$-invariant flows yield the BCFG Drinfeld-Sokolov hierarchy. We prove that the total descendant potential of the $\Gamma$-invariant sector of a FJRW-theory is a tau function of the corresponding BCFG Drinfeld-Sokolov hierarchy.