Lagrangian Floer theory and mirror symmetry on compact toric manifolds (Q2789580)

This monograph approaches Lagrangian Floer theory on toric manifolds, by means of mirror symmetry. It is a natural and necessary continuation of the authors' work on Lagrangian Floer theory: [Duke Math. J. 151, No. 1, 23--175 (2010; Zbl 1190.53078); Sel. Math., New Ser. 17, No. 3, 609--711 (2011; Zbl 1234.53023); Surveys in Differential Geometry 17, 229--298 (2012; Zbl 1382.53001)].NEWLINENEWLINENEWLINEThe main result is a version of mirror symmetry between the compact toric A-model (a Frobenius manifold defined by quantum cohomology) and the Landau-Ginzburg B-model (a Frobenius manifold defined by Saito's theory).NEWLINENEWLINENEWLINEThis monograph contains three chapters and an appendix.NEWLINENEWLINENEWLINEChapter 1 is an introduction and contains various notations which are needed.NEWLINENEWLINENEWLINEChapter 2, entitled {Ring isomorphism}, comprises several issues about: the completion of the Laurent polynomial ring over the Nivikov ring, the potential function which is parameterized by the group of \(T^{n}\)-invariant cycles, the bijectivity of Kodaira-Spencer map and many other interesting results.NEWLINENEWLINENEWLINEIn Chapter 3 (Coincidence of pairings), the proof of the main result is completed and the coincidence of the residue pairing and the inverse of the Hessian determinant modulo the higher-order term in the general case is proved. Also, it is shown that the residue pairing detects the second of the nontrivial elements of the cyclic cohomology.NEWLINENEWLINENEWLINEFinally, in the appendix the authors discuss the construction of the Kuranishi structure on the moduli space of pseudo-holomorphic disks which is invariant under the \(T^n\)-action and other symmetries.