Mirror symmetry for exceptional unimodular singularities

  • Changzheng Li

    Sun Yat-Sen University, Guangzhou, China and Institute for Basic Science, Pohang, Korea
  • Si Li

    Tsinghua University, Beijing, China
  • Kyoji Saito

    The University of Tokyo, Japan
  • Yefeng Shen

    Stanford University, USA


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).

Cite this article

Changzheng Li, Si Li, Kyoji Saito, Yefeng Shen, Mirror symmetry for exceptional unimodular singularities. J. Eur. Math. Soc. 19 (2017), no. 4, pp. 1189–1229

DOI 10.4171/JEMS/691