## The Witten equation, mirror symmetry, and quantum singularity theory.(English)Zbl 1310.32032

The first result of the manuscript, as it is announced by the authors, is the proof of the so-called Big Witten’s conjecture. This is the generalization of two other conjectures of Witten proved by M. Kontsevich in [Commun. Math. Phys. 147, No. 1, 1–23 (1992; Zbl 0756.35081)] and C. Faber et al. [Ann. Sci. Éc. Norm. Supér. (4) 43, No. 4, 621–658 (2010; Zbl 1203.53090)], respectively. As for the latter two conjectures the first part of the Big Witten’s conjecture was the existence of a certain moduli space. The authors construct such an appropriate moduli space $$\mathcal{W}_{g,k}$$, what allows one to formulate rigorously the Big Witten’s conjecture.
The initial data for the moduli space $$\mathcal{W}_{g,k}$$ is taken to be a quasi-homogeneous polynomial $$W: \mathbb{C}^N \to \mathbb{C}$$ defining an isolated singularity, and a symmetry group $$G$$ of $$W$$. As the space $$\mathcal{W}_{g,k}$$ consists of the genus $$g$$ curves with an additional structure at the $$k$$ marked points defined by $$(W,G)$$. A subtle question, resolved by the authors in an other paper, is the existence of a fundamental cycle of the constructed moduli space. The authors construct axiomatically a cohomological field theory on $$\mathcal{W}_{g,k}$$ and show that for $$W$$ defining an ADE-singularity the partition function of this cohomological field theory is a $$\tau$$-function of a Kac-Wakimoto hierarchy. This finalizes the Big Witten’s conjecture.
However equally important is the construction itself of the cohomological field theory on $$\mathcal{W}_{g,k}$$. After the first appearance online as a preprint in 2007 such cohomological field theories are now known as Fan-Jarvis-Ruan-Witten theories. These theories were heavily investigated since that time and appeared to be an important part of mirror symmetry, known now as the Landau-Ginzburg mirror symmetry.

### MSC:

 32S25 Complex surface and hypersurface singularities 32S30 Deformations of complex singularities; vanishing cycles 14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)

### Citations:

Zbl 0756.35081; Zbl 1203.53090

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### References:

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