Seidel’s morphism of toric $4$-manifolds

Abstract

Following McDuff and Tolman's work on toric manifolds [McDT06], we focus on 4-dimensional NEF toric manifolds and we show that even though Seidel's elements consist of infinitely many contributions, they can be expressed by closed formulas. From these formulas, we then deduce the expression of the quantum homology ring of these manifolds as well as their Landau-Ginzburg superpotential. We also give explicit formulas for the Seidel elements in some non-NEF cases. These results are closely related to recent work by Fukaya, Oh, Ohta, and Ono [FOOO11], Gonz\'alez and Iritani [GI11], and Chan, Lau, Leung, and Tseng [CLLT12]. The main difference is that in the 4-dimensional case the methods we use are more elementary: they do not rely on open Gromov-Witten invariants nor mirror maps. We only use the definition of Seidel's elements and specific closed Gromov-Witten invariants which we compute via localization. So, unlike Alice, the computations contained in this paper are not particularly pretty but they do stay on their side of the mirror. This makes the resulting formulas directly readable from the moment polytope.