Higher genus FJRW invariants of a Fermat cubic

Li, Jun¹ ; Shen, Yefeng² ; Zhou, Jie³

¹ Shanghai Center for Mathematical Sciences, Fudan University, Shanghai, China
² Department of Mathematics, University of Oregon, Eugene, OR, United States
³ Yau Mathematical Sciences Center, Tsinghua University, Beijing, China

Geometry & topology, Tome 27 (2023) no. 5, pp. 1845-1890.

Résumé

We reconstruct all-genus Fan–Jarvis–Ruan–Witten invariants of a Fermat cubic Landau–Ginzburg space $(x_1^3+x_2^3+x_3^3:[{ℂ}^3∕{\mu }_3]\to ℂ)$ 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.

Détail

DOI : 10.2140/gt.2023.27.1845

Keywords: FJRW invariants, Fermat cubic, quasimodular forms, LG/CY correspondence

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