Lagrangian Floer theory on compact toric manifolds: survey

Abstract

This article is a survey of a series of papers [FOOO3,FOOO4,FOOO5] in which we developed the method of calculation of Floer cohomology of Lagrangian torus orbits in compact toric manifolds, and its applications to symplectic topology and to mirror symmetry. In this article we summarize the main ingredients of calculation and illustrate them by examples. The second half of the survey is devoted to discussion of the most recent result from [FOOO5] (arXiv:1009.1648) where the mirror symmetry between the two Frobenius manifolds arising from the big quantum cohomology and from the K. Saito theory of singularities was established.

60 pages, 5 figures

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Cited by in corpus (7)

• Homological Mirror Symmetry for Calabi-Yau hypersurfaces in projective space
• Quasi-states, quasi-morphisms, and the moment map
• Holomorphic orbidiscs and Lagrangian Floer cohomology of symplectic toric orbifolds
• Low-area Floer theory and non-displaceability
• Homological mirror symmetry for punctured spheres
• Remarks on Lagrangian intersections in toric manifolds
• SYZ for index 1 Fano hypersurfaces in projective space