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Higher genus FJRW invariants of a Fermat cubic

Jun Li, Yefeng Shen and Jie Zhou

Geometry & Topology 27 (2023) 1845–1890

We reconstruct all-genus Fan–Jarvis–Ruan–Witten invariants of a Fermat cubic Landau–Ginzburg space (x13 + x23 + x33: [3μ3] ) 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.

FJRW invariants, Fermat cubic, quasimodular forms, LG/CY correspondence
Mathematical Subject Classification
Primary: 14N35
Received: 29 September 2020
Revised: 9 November 2021
Accepted: 11 December 2021
Published: 27 July 2023
Proposed: Jim Bryan
Seconded: Paul Seidel, Mark Gross
Jun Li
Shanghai Center for Mathematical Sciences
Fudan University
Yefeng Shen
Department of Mathematics
University of Oregon
Eugene, OR
United States
Jie Zhou
Yau Mathematical Sciences Center
Tsinghua University

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