Summary: We reconstruct all-genus Fan-Jarvis-Ruan-Witten invariants of a Fermat cubic Landau-Ginzburg space \((x_1^3+x_2^3+x_3^3: [\mathbb{C}^3/ \mu_3] \to\mathbb{C})\) from genus-one primary invariants, using tautological relations and axioms of cohomological field theories. The genus-one primary invariants satisfy a Chazy equation by the Belorousski-Pandharipande relation. They are completely determined by a single genus-one invariant, which can be obtained from cosection localization and intersection theory on moduli of three-spin curves.
We solve an all-genus Landau-Ginzburg/Calabi-Yau correspondence conjecture for the Fermat cubic Landau-Ginzburg space using Cayley transformation on quasimodular forms. This transformation relates two nonsemisimple CohFT theories: the Fan-Jarvis-Ruan-Witten theory of the Fermat cubic polynomial and the Gromov-Witten theory of the Fermat cubic curve. As a consequence, Fan-Jarvis-Ruan-Witten invariants at any genus can be computed using Gromov-Witten invariants of the elliptic curve. They also satisfy nice structures, including holomorphic anomaly equations and Virasoro constraints.