Mirror symmetry for exceptional unimodular singularities

Si Li Tsinghua University Changzheng Li Sun Yat-Sen University Kyoji Saito The University of Tokyo Yefeng Shen Stanford University

Algebraic Geometry Mathematical Physics mathscidoc:1705.01003

J. Eur. Math. Soc. (JEMS) , 19, (4), 1189–1229, 2017
In this paper, we prove the mirror symmetry conjecture between the Saito–Givental theory of exceptional unimodular singularities on the Landau–Ginzburg B-side and the Fan–Jarvis– Ruan–Witten theory of their mirror partners on the Landau–Ginzburg A-side. On the B-side, we develop a perturbative method to compute the genus-0 correlation functions associated to the primitive forms. This is applied to the exceptional unimodular singularities, and we show that the numerical invariants match the orbifold-Grothendieck–Riemann–Roch and WDVV calculations in FJRW theory on the A-side. The coincidence of the full data at all genera is established by reconstruction techniques. Our result establishes the first examples of LG-LG mirror symmetry for weighted homogeneous polynomials of central charge greater than one (i.e. which contain negative degree deformation parameters).
Landau–Ginzburg model, mirror symmetry, singularity, FJRW theory, primitive form
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  title={Mirror symmetry for exceptional unimodular singularities},
  author={Si Li, Changzheng Li, Kyoji Saito, and Yefeng Shen},
  booktitle={J. Eur. Math. Soc. (JEMS) },
Si Li, Changzheng Li, Kyoji Saito, and Yefeng Shen. Mirror symmetry for exceptional unimodular singularities. 2017. Vol. 19. In J. Eur. Math. Soc. (JEMS) . pp.1189–1229. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170530154809808956769.
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