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FJRW-rings and mirror symmetry. (English) Zbl 1250.81087
Landau-Ginzburg mirror symmetry is a mirror duality between a pair of Landau-Ginzburg models conjecturally at the Gepner point. The theory of the Landau-Ginzburg A-model is the Fan-Jarvis-Ruan-Witten theory [H. Fan, T. J. Jarvis and Y. Ruan, “The Witten equation, mirror symmetry and quantum singularity theory”, arXiv:0712.4021]. The B-model theory is the Saito-Givental’s theory. The Landau-Ginzburg model is given by a quasi-homogeneous polynomial \(W:(\mathbb C^*)^N \to \mathbb C\). The construction of the FJRW theory of the A-model involves a choice of an admissible group \(G\) acting on \((\mathbb C^*)^N\) which preserves \(W\). The mirror symmetry says when \(G\) is the maximal diagonal symmetry group, there is a dual polynomial \(W^T\), and the FJRW LG A-model of \(W/G\) is equivalent to the Saito-Givental LG B-model \(W^T\).

The authors prove the Landau-Ginzburg mirror symmetry for Arnol’d’s list of unimodal and bimodal quasi-homogeneous singularities, on the level of Frobenius structures, which involves \(3\)-point genus \(0\) correlators. The A-model structure is the FJRW ring \(\mathcal H_{W,G}\), while the B-model is the Milnor ring of \(W^T\). The paper under review proves they are isomorphic. The computation techniques in the proof rely on the axioms of the FJRW theories.

MSC:
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
14J33 Mirror symmetry (algebro-geometric aspects)
35Q55 NLS equations (nonlinear Schrödinger equations)
16L60 Quasi-Frobenius rings
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[1] Arnol’d, V., Gusein-Zade, S., Varchenko, A.: Singularities of Differentiable Maps Vols I, II. Basel-Boston: Birkhauser, 1985
[2] Berglund P., Hubsch T.: A generalized construction of mirror manifolds. Nucl. Phys. B 393, 377 (1993) · Zbl 1245.14039 · doi:10.1016/0550-3213(93)90250-S
[3] Fan, H., Jarvis, T.J., Ruan, Y.: The witten equation, mirror symmetry and quantum singularity theory. http://arxiv.org/abs/:0712.4021v3[math.AG] , 2009
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