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Integral structure for simple singularities. (English) Zbl 1471.14023

Let \(f\in \mathbb{C}[x_{1},x_{2},x_{3}]\) an \(ADE\) simple singularities and \(f^{T}\in \mathbb{C}[x_{1},x_{2},x_{3}]\) the corresponding Berglund-Hübsch dual. To construct a matrix model for the Fan-Jarvis-Ruan-Witten (FJRW) invariant of \(f^{T}\), similar to Kontesevich’s model as in [M. Kontsevich, Commun. Math. Phys. 147, No. 1, 1–23 (1992; Zbl 0756.35081)], one needs to find explicit identification of the generating function of FJRW invariants of \(f^{T}\) with a tau-function of a specific Kac-Wakimoto hierarchy. The identification involves rescaling parameters of the Kac-Wakimoto hierarchy and the precise value for the rescaling constants. Such a computation seems to be straightforward and it is not done in the paper. Instead, the authors explain in the introduction how the technical details leads to a problem in singularity theory.
On the other hand, in [H. Fan et al., Ann. Math. (2) 178, No. 1, 1–106 (2013; Zbl 1310.32032)] the authors showed that generating functions of \(FJRW\) invariants of \(f^{T}\) coincides with the total descendant potentials of \(f\) (or, t.d.p. of \(f\), for short). In [E. Frenkel et al., Funct. Anal. Other Math. 3, No. 1, 47–63 (2010; Zbl 1203.37108)] and [A. B. Givental and T. E. Milanov, Prog. Math. 232, 173–235 (2005; Zbl 1075.37025)] have proved that if \(f\) is a \(ADE\) singularity then t.d.p of \(f\) is a tau-function of the principal Kac-Wakimoto hierarchy of the same type \(ADE\). However, the problem is not solved yet, because the identification with the Milnor ring of the singularity and the Cartan subalgebra of the simple Lie algebra is not clearly explicit yet.
The contribution of the paper under review comes in showing that. The proofs are very well developed and they were done by approaching the ADE-singularity case by case. The paper offers a very pleasant reading.

MSC:

14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
32S30 Deformations of complex singularities; vanishing cycles
19L47 Equivariant \(K\)-theory
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References:

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