Mirror symmetry for exceptional unimodular singularities

Abstract

In this paper, we prove the mirror symmetry conjecture between the Saito-Givental theory of exceptional unimodular singularities on Landau-Ginzburg B-side and the Fan-Jarvis-Ruan-Witten theory of their mirror partners on Landau-Ginzburg A-side. On the B-side, we compute the genus-zero generating function from a perturbative formula of primitive forms introduced by the first three authors recently. This computation matches 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 of all genera for weighted homogeneous polynomials of central charge greater than one (i.e. which contain negative degree deformation parameters).

A proof of Proposition 2.7 is added. Some minor mistakes corrected. To appear in J. Eur. Math. Soc

References in corpus (5)

• A Generalized Construction of Mirror Manifolds
• FJRW rings and Landau-Ginzburg Mirror Symmetry
• Landau-Ginzburg/Calabi-Yau correspondence for quintic three-folds via symplectic transformations
• A Landau--Ginzburg mirror theorem without concavity
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Cited by in corpus (7)

• Primitive forms via polyvector fields
• A Mirror Theorem between Landau-Ginzburg models
• Flat structures on Frobenius Manifolds in the case of irrelevant deformations
• Flat coordinates for Saito Frobenius manifolds and String theory
• Mirror symmetry and K3 surfaces
• From primitive form to mirror symmetry
• On Frobenius Manifolds from Gromov--Witten Theory of Orbifold Projective Lines with $r$ orbifold points