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ABSTRACT. In this paper, we prove the mirror symmetry conjecture between the SaitoGivental 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 develop a perturbative method to compute the genus-zero correlation functions associated to the primitive forms. This is applied to the exceptional unimodular singularities, and we show that the numerical invariants match the orbifoldGrothendieck-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).

We prove the Landau-Ginzburg mirror symmetry conjecture between invertible quasihomogeneous polynomial singularities at all genera. That is, we show that the FJRW theory (LG A-model) of such a polynomial is equivalent to the Saito-Givental theory (LG B-model) of the mirror polynomial.

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We prove the Landau-Ginzburg/Calabi-Yau correspondence between the Gromov-Witten theory of each elliptic orbifold curve and its Fan-Jarvis-Ruan-Witten theory counterpart via modularity. We show that the correlation functions in these two enumerative theories are different representations of the same set of quasi-modular forms, expanded around different points on the upper-half plane. We relate these two representations by the Cayley transform. Appendix A WDVV equations in FJRW theories 32Appendix B Analytic continuation of periods and connection matrices 34

We prove that the ancestor Gromov-Witten correlation functions of one-dimensional compact Calabi-Yau orbifolds are quasi-modular forms. This includes the pillowcase orbifold which can not yet be handled by using Milanov-Ruan's B-model technique. We first show that genus zero modularity is obtained from the phenomenon that the system of WDVV equations is essentially equivalent to the set of Ramanujan identities satisfied by the generators of the ring of quasi-modular forms for a certain modular group associated to the orbifold curve. Higher genus modularity then follows by using tautological relations.