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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)
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