Floer Theory Of Landau-Ginzburg Model

Lagrangian intersection Floer theory plays an important role in symplectic geom-etry and mathematical physics.It is originated from A.Floer's attempt[19],[20],[21]to prove Arnold conjecture[22].For a Hamiltonian diffeomorphism on a closed sym-plectic manifold M,Arnold conjectured that the number of its fixed points is bounded below by the minimum number of critical points of a Morse function on M.The weak version of Arnold conjecture says that the number of its fixed points is bounded below by the sum of Betti numbers of M.Floer regarded these fixed points as intersection points of two Lagrangian submanifolds in the product manifold M x M.For a given symplectic manifold P with two Lagrangian submanifolds L0,L1,Floer consider such J-holomorphic maps from R ×[0,1]to P,which have finite energy,and map R × {0} and R × {1} to L0 and L1 respectively.By studying the moduli spaces of such solutions,Floer tried to define a cohomology group.In the original study of Floer,?2(P,L0)= 0,and L1 is the image of L0 by some Hamiltonian diffeomorphism.In this case,Floer cohomology is well defined.Floer then solved Arnold conjecture in this special case.Oh then generalized Floer's construction to the case of monotone Lagrangian sub-manifolds(see[23],[24])and generalized Floer's inequality(see[25]).He also studied Floer theory on the cotangent bundle(see[26]).In order to overcome the difficulties to define the Floer cohomology in general case,Fukaya,Oh,Ohta,and Ono[27]sys-tematically developed the theory of anomaly and obstruction.One of the important application of this theory is Lagrangian Floer theory on compact toric manifolds(see[28],[29]).The theory of Fukaya category is closely related to Floer theory.In general,the objects of Fukaya category of a symplectic manifold P are suitable Lagrangian sub-manifolds.Given two Lagrangian submanifolds among them,the morphism spaces is generated by their intersections.There are composition maps on the products of mor-phism spaces,satisfying certain relations.The first order map is defined using the Floer differential.In 1994,Kontsevich proposed a famous conjecture known as homological mirror symmetry(see[6]),which related the Fukaya category of a symplectic manifold whose first Chern class is zero to the derived category of coherent sheaves on its dual com-plex algebraic variety.Thus the study of Floer theory and Fukaya category plays an important role in the theory of Mirror symmetry.In order to generalize the homological mirror symmetry to some other objects,e.g.,Fano manifolds,the concept of Landau-Ginzburg model(LG model)is introduced.The idea is to introduce a super-potential to deform the derived category of coherent sheaves.Landau-Ginzburg B model was constructed by Orlov[9],[10].He introduced the method of matrix factorization to study the non-affine case.One candidate of Landau-Ginzburg A model is Seidel-Fukaya category,which has been established by Seidel(see[11],[12],[13]).Given an exact Morse fibration,we can suitably choose some vanishing cycles to be the objects of the category.We can study the algebraic properties of this A? category by studying J-holomorphic sections of the fibration.He also studied a related theory called Fukaya category of Lefschetz fibrations(see[14]).By studying(2,2)supersymmetry,a new type of A model is studied by Hori,Iqbal,and Vafa[15]in their physics literature.They consider a perturbed Chauchy-Riemann equation on a non-compact Kahler manifold.The perturbation is decided by a holomorphic function on it.It is also required that the two pieces of boundaries of R ×[0,1]are mapped to the wave-front trajectory emanating from two different critical points of the holomorphic function by the solution.Fan,Jarvis,and Ruan[16],[17],[18]have recently developed a new theory known as FJRW theory,by studying the Witten equation on orbifold line bundles.This theory can be regarded as a closed string version of LG A model.Our study of Floer theory of Landau-Ginzburg model is inspired by the article of Hori,Iqbal,and Vafa as well as FJRW theory,which are mentioned above.We state main results in the paper as follows:the first part is about the study of general setting of Landau-Ginzburg model on noncompact Kahler manifolds;the second part is the construction of LG-Floer cohomology of Cn,as well as an example of C*case.We first introduce the notion of tame condition of LG model(M,h,W).We then give two examples of models which satisfies the tame condition:one is the non-degenerate,quasi-homogeneous polynomial on Cn,the other is the convenient and non-degenerate Laurent polynomial on(C*)n.We then define the LG-Floer equation for Landau-Ginzburg model satisfying the tame condition.We can associate to each critical point of W a Lagrangian submanifold called the Lefschetz thimble attached to it.The two pieces of boundaries of R ×[0,1]should be mapped to two Lefschetz thimbles by the solutions.We introduce the notion of nice perturbation of a non-degenerate,quasi-homogeneous polynomial W1 on Cn.A nice perturbation of W1 is a holomorphic Morse function on Cn,with certain asymptotic behaviors the same to W1.Fix a nice perturbation W,we can deform the Kahler form in a fixed Kahler class to make sure that the two Lefschetz thimbles intersect transversally(at time 1).The metric h chosen should be equal to the standard metric outside a compact set,as well as on the two Lefschetz thimbles.Such a perturbation exists as a result of Sard-Smale theorem.By the help of estimate of non-degenerate,quasi-homogeneous polynomials in[16],as well as the property of Lefschetz thimbles,we can give a C0 estimate of the solutions.Then we study the compactification of moduli spaces.After that we study the Freholm theory.Then it is possible to construct the LG-Floer cohomology of the model(Cn,h,W).