Orbifold Landau-Ginzburg镜像对称

Landau-Ginzburg mirror symmetry implies amazing correspondence between totally different mathematical theory on singularities. More explicitly, fix a specific singularity, its Fan-Jarvis-Ruan-Witten or Polishchuk-Vaintrob theory is supposed to be equivalent to Saito-Givental theory of its mirror singularity. Since the birth of the first mathematical construction of LG A model-the FJRW theory introduced by Fan, Jarvis and Ruan, LG mirror symmetry conjecture becomes a popular topic in mathematical physics. Nowadays, the proof of this conjecture is almost completed. The next task is the orbifold generalization of this conjecture. Based on the proposal of Kontsevich-Katzarkov-Pantev, mathematician generalize Saito theory(LG B model) to some orbifold cases, via the technique from noncommutative geometry. In this project, we will study the orbifold version of Landau-Ginzburg mirror symmetry, including the isomorphism of Frobenius algebra between A and B model, and the orbifold Landau-Ginzburg mirror symmetry conjecture in these specific cases.

Landau-Ginzburg镜像对称揭示了奇点的FJRW或PV理论与其镜像奇点的Saito-Givental理论的对应。自范辉军-阮勇斌-Jarvis引进FJRW理论，完成Landau-Ginzburg A模型数学构造之日起，Landau-Ginzburg镜像对称的研究一直是数学物理领域的热点之一。时至今日，此猜想的光滑情形的证明已基本完成，一个自然问题是如何在orbifold情形构造该LG镜像对称。此情形的困难在于B模型Saito理论的orbifold版本的数学构造。以Kontsevich-Katzarkov-Pantev为首的数学家引进了非交换几何技术，在这个问题上取得重大进展。本项目希望在此基础上研究在这些情形下orbifold Landau-Ginzburg镜像对称， 包括Frobenius代数同构问题及一些非平凡情形Frobenius流形的同构问题，为后续研究打下基础。

本项目研究目标是orbifold Landau-Ginzburg镜像对称猜想： 给定一个可逆拟齐次多项式W以及一个W的admissible阿贝尔对称群G，还有其镜像多项式WT及镜像群GT，研究（W，G）上的A模型量子理论（Fan-Jarvis-Ruan-Witten理论，Polishchuk-Vaintrob理论或Kiem-Li理论）与（WT，GT）上的B模型量子理论（由矩阵分解范畴构造）之间的等价关系。本人在项目期间与合作者证明了Landau-Ginzburg A模(Polishchuk-Vaintrob理论)与B模在可逆多项式任意对称群情形的Frobenius代数的同构。