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arXiv · 1312.7227

BCFG Drinfeld-Sokolov Hierarchies and FJRW-Theory

Si-Qi Liu·Yongbin Ruan·Youjin Zhang
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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.

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Si-Qi Liu, Yongbin Ruan, Youjin Zhang. 2013-12-27. BCFG Drinfeld-Sokolov Hierarchies and FJRW-Theory. https://arxiv.org/abs/1312.7227

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