Fixing a weakly unobstructed Lagrangian torus in a symplectic manifold X, we define a holomorphic function W known as the Floer potential. We construct a canonical A∞-functor from the Fukaya category of X to the category of matrix factorizations of W. It provides a unified way to construct matrix factorizations from Lagrangian Floer theory. The technique is applied to toric Fano manifolds to transform Lagrangian branes to matrix factorizations and prove homological mirror symmetry. Using the method, we also obtain an explicit expression of the matrix factorization mirror to the real locus of the complex projective space.
Bibliographical noteFunding Information:
The authors are grateful to Jonathan David Evans and Yanki Lekili for informing them about their very useful results on generation of Fukaya categories of Hamiltonian G-manifolds, and in particular toric Fano manifolds, which the authors crucially use to prove HMS of toric Fano manifolds. C.H. Cho and S.-C. Lau thank The Chinese University of Hong Kong for its hospitality, where part of the work was carried out. C.H. Cho thanks Yong-Geun Oh for helpful discussions. S.-C. Lau expresses his gratitude to Kwokwai Chan and Junwu Tu for useful explanations of their works. S.-C. Lau is grateful to Harvard University and Boston University . The work of the first author is supported by Samsung Science and Technology Foundation (Project Number SSTF-BA1402-05 ). The work of the second and third authors is supported by the Simons Foundation grant ( #385573 , Simons Collaboration on Homological Mirror Symmetry).
All Science Journal Classification (ASJC) codes
- Mathematical Physics
- Physics and Astronomy(all)
- Geometry and Topology