Orbifold Milnor lattice and orbifold intersection form (Q1702668)

Let \(f: (\mathbb C^n,0) \rightarrow (\mathbb C,0)\) be a germ of a holomorphic function with an isolated critical point at the origin and let \(G\) be a finite abelian group acting faithfully on \((\mathbb C^n,0)\) and preserving \(f\). The authors define the quantum cohomology group \(\mathcal H_{f,G}\) of the pair \((f,G)\) similarly to the paper [\textit{H. Fan} et al., Ann. Math. (2) 178, No. 1, 1--106 (2013; Zbl 1310.32032)], where this notion was introduced in the framework of quantum singularity theory (FJRW-theory) for quasihomogeneous function germs. The paper under review aims to transfer or adapt the basic notions and results of the usual singularity theory to FJRW-theory. Using the theory of group rings, the authors define consistently orbifold versions of the monodromy operator on \(\mathcal H_{f,G}\), the Milnor fibre and vanishing cohomology group (the Milnor lattice in \(\mathcal H_{f,G}\)), the Seifert form and the intersection form, and so on. In conclusion, they discuss some examples with invertible polynomials related to the orbifold Landau-Ginzburg models and the Berglund-Hübsh-Henningson duality (see, e.g., [\textit{P. Berglund} and \textit{M. Henningson}, Nucl. Phys., B 433, No. 2, 311--332 (1995; Zbl 0899.58068)]), the behavior of the Milnor lattice under the corresponding mirror symmetry, etc.