Higher Genus FJRW Invariants of a Fermat Cubic

Abstract

We reconstruct the all-genus Fan-Jarvis-Ruan-Witten invariants of a Fermat cubic Landau-Ginzburg space $(x_1^3+x_2^3+x_3^3: [\mathbb{C}^3/ \mathbold{\mu}_3]\to \mathbb{C})$ from genus-one primary invariants, using tautological relations and axioms of Cohomological Field Theories. These genus-one 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 quasi-modular forms. This transformation relates two non-semisimple 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.