Symplectic Manifolds: Gromov-Witten Invariants on Symplectic and Almost Contact Metric Manifolds Symplectic Manifolds: Gromov-Witten Invariants on Symplectic and Almost Contact Metric Manifolds

In this chapter, we introduce Gromov-Witten invariant, quantum cohomology, Gromov- Witten potential, and Floer cohomology on symplectic manifolds, and in connection with these, we describe Gromov-Witten type invariant, quantum type cohomology, Gromov-Witten type potential and Floer type cohomology on almost contact metric manifolds. On the product of a symplectic manifold and an almost contact metric manifold, we induce some relations between Gromov-Witten type invariant and quantum cohomology and quantum type invariant. We show that the quantum type coho- mology is isomorphic to the Floer type cohomology.


Introduction
The symplectic structures of symplectic manifolds ðM, ω, JÞ are, by Darboux's theorem 2.1, locally equivalent to the standard symplectic structure on Euclidean space.
In Section 3, we introduce almost contact metric manifolds ðM, g, ϕ, η, ξ, φÞ with a closed fundamental 2-form φ and their product [4,7,8]. In Section 3.1, we study ϕ-coholomorphic map, moduli space of ϕ-coholomorphic maps which represent a homology class of dimension two, Gromov-Witten type cohomology, quantum type product and quantum type cohomology, Gromov-Witten type potentials on the product of a symplectic manifold, and an almost contact metric manifold [5,6,13]. In Section 3.2, we investigate the symplectic type action functional on the universal covering space of the contractible loops, its gradient flow line, the moduli space of the connecting flow orbits between critical loops, Floer type cochain complex, and Floer type cohomology with coefficients in a Novikov ring [7,9,13].
In Section 4, as conclusions we show that the Floer type cohomology and the quantum type cohomology of an almost contact metric manifold with a closed fundamental 2-form are isomorphic [7,13], and present some examples of almost contact metric manifolds with a closed fundamental 2-form.

Symplectic manifolds
By a symplectic manifold, we mean an even dimensional smooth manifold M 2n together with a global 2-form ω which is closed and nondegenerate, that is, the exterior derivative dω ¼ 0 and the n-fold wedge product ω n never vanishes.
(2) Let M be a smooth manifold. Then its cotangent bundle T Ã M has a natural symplectic form as follows. Let π : T Ã M ! M be the projection map and x 1 , …, x n are local coordinates of M. Then q i ¼ x i ∘π,i ¼ 1, 2, …, n together with fiber coordinates p 1 , …, p n give local coordinates of T Ã M. The natural symplectic form on T Ã M is given by (3) Every Kähler manifold is symplectic.
A symplectomorphism of ðM, ωÞ is a diffeomorphism φ∈Dif f ðMÞ which preserves the symplectic form φ Ã ω ¼ ω. Denote by SymðMÞ the group of symplectomorphims of M. Since ω is nondegenerate, there is a bijection between the vector fields X∈ΓðTMÞ and 1-forms ωðX, ÁÞ∈Ω 1 ðMÞ. A vector field X∈ΓðTMÞ is called symplectic if ωðX, ÁÞ is closed.
Let M be closed, i.e., compact and without boundary. Let φ : R ! DiffðMÞ, t↦φ t be a smooth family of diffeomorphisms generated by a family of vector fields X t ∈ΓðTMÞ via, This {φ t H } is called the Hamiltonian flow associated with H. The flux homomorphism Flux is defined by Theorem 2.2 ( [6]). φ∈SymðMÞ is a Hamiltonian symplectomorphism if and only if there is a homotopy ½0, 1 ! SymðMÞ, t↦φ t such that φ 0 ¼ id, φ 1 ¼ φ, and Fluxðfφ t gÞ ¼ 0.

Quantum cohomology
Let ðM, ωÞ be a compact symplectic manifold. An almost complex structure is an automorphism of TM such that J 2 ¼ −I. The form ω is said to tame J if ωðv, JvÞ > 0 for every v≠0. The set I τ ðM, ωÞ of almost complex structures tamed by ω is nonempty and contractible. Thus the Chern classes of TM are independent of the choice J∈I τ ðM, ωÞ. A smooth map φ : Hereafter, we denote by H 2 ðMÞ the image of Hurewicz homomorphism π 2 M ! H 2 ðM, ZÞ. A ði, JÞ-holomorphic map u : ðΣ, z 1 , …, z k Þ ! M from a reduced Riemann surface ðΣ, jÞ of genus g with k marked points to ðM, JÞ is said to be stable if every component of Σ of genus 0 (resp. 1), which is contracted by u, has at least 3 (resp. 1) marked or singular points on its component, and the k marked points are distinct and nonsingular on Σ. For a two-dimensional homology class A∈H 2 ðMÞ let M g, k ðM, A; JÞ be the moduli space of ðj, JÞ-holomorphic stable maps which represent A.
Let B :¼ C ∞ ðΣ, M; AÞ be the space of smooth maps which represent A∈H 2 ðMÞ.
Let us consider infinite dimensional vector bundle E ! B whose fiber at u is the space E u ¼ Ω 0, 1 ðΣ, u Ã TMÞ of smooth J-antilinear 1-forms on Σ with values in u Ã TM. The map ∂ J : B ! E given by is a section of the bundle. The zero set of the section ∂ I is the moduli space M g, k ðM, A; JÞ.
For an element u∈M g, k ðM, A; JÞ we denote by the composition of the derivative with the projection to fiber T ðu, 0Þ E ! Ω 0, 1 ðΣ, u Ã TMÞ. Then the virtual dimension of M g, k ðM, A; JÞ is For some technical reasons, we assume that c 1 ðAÞ≥0 if ωðAÞ > 0 and A is represented by some J-holomorphic curves. In this case, we call the symplectic manifold M semipositive. We define the evaluation map by ev : M g, k ðM, A; JÞ ! M k , evð½u; z 1 , …, z k Þ ¼ ðuðz 1 Þ, …, uðz k ÞÞ: Then the image ImðevÞ is well defined, up to cobordism on J, as a dimM g, k ðM, A; JÞ : ≡mdimensional homology class in M k .
Definition. The Gromov-Witten invariant Φ M, A g, k is defined by where α ¼ PDðaÞ∈H 2nk−m ðM k Þ and • is the intersection number of ev and α in M k .
The minimal Chern number N of ðM, ωÞ is the integer N :¼ min }7Bλjc 1 ðAÞ ¼ λ≥0, A∈H 2 ðMÞ}. We define the quantum product a Ã b of a∈H k ðMÞ and b∈H ðMÞ as the formal sum where q is an auxiliary variable of degree 2N and ða Ã bÞ A ∈H kþl−2c1ðAÞ ðMÞ is defined by for C∈H kþl−2c1ðAÞ ðMÞ, r ¼ PDðCÞ. Hereafter, we use the Gromov-Witten invariants of g ¼ 0 and k ¼ 3. Then the quantum product a Ã b is an element of where Q½q is the ring of Laureut polynomials of the auxiliary variable q.
The constant term of a Ã b is the usual cup product a ∪ b.
We defined the Novikov ring Λ ω by the set of functions λ : H 2 ðMÞ ! Q that satisfy the finiteness condition for every c∈R. The grading is given by degðAÞ ¼ 2c 1 ðAÞ.
Examples ( [5]). (1) Let p∈H 2 ðCP n Þ and A∈H 2 ðCP n Þ be the standard generators. There is a unique complex line through two distinct points in CP n and so p Ã p n ¼ q. The quantum cohomology of CP n is (2) Let Gðk, nÞ be the Grassmannian of complex k-planes in C n . There are two natural complex vector bundles C k ! E ! Gðk, nÞ and C n−k ! F ! Gðk, nÞ. Let x i ¼ c i ðE Ã Þ and y i ¼ c i ðF Ã Þ be Chern classes of the dual bundles E Ã and F Ã , respectively. Since E⊕F is trivial, By computation x k Ã y n−k ¼ ð−1Þ n−k q. The quantum cohomology of Gðk, nÞ is Let {e 0 , …, e n } be an integral basis of H Ã ðMÞ such that e 0 ¼ 1∈H 0 ðMÞ and each e i has pure degree. We introduce n þ 1 formal variables t 0 , …, t n and the linear polynomial a t in t 0 , …, t n with coefficients in H Ã ðMÞ by a t ¼ t 0 e 0 þ ⋯ þ t n e n . The Gromov-Witten potential of ðM, ωÞ is a formal power series in variables t 0 , …, t n with coefficients in the Novikov ring Λ ω We define a nonsingular matrix ðg ij Þ by g ij ¼ ∫ M e i ∪ e j and denote by ðg ij Þ its inverse matrix.

Floer cohomology
Let a compact symplectic manifold ðM, ωÞ be semipositive. Let H tþ1 : M ! R be a smooth 1periodic family of Hamiltonian functions. The Hamiltonian vector field X t is defined by ωðX t , ÁÞ ¼ dH t . The solutions of the Hamiltonian differential equation _ xðtÞ ¼ X t ðxðtÞÞ generate a family of Hamiltonian symplectomorphisms φ t : Two such maps u 1 and u 2 are called equivalent if their boundary sumðu 1 Þ#ð−u 2 Þ is homologus to zero in H 2 ðMÞ. Denote by ðx, ½u 1 Þ ¼ ðx, ½u 2 Þ for equivalent pairs, LM the space of contractible loops andL M the space of equivalence classes. Then g LM ! LM is a covering space whose covering transformation group is H 2 ðMÞ via, Aðx, ½uÞ ¼ ðx, ½A#uÞ for each A∈H 2 ðMÞ and ðx, ½uÞ∈ g LM.
Definition. The symplectic action functional a H is defined by Thus the critical points of a H are one-to-one correspondence with the periodic solutions of The gradient flow lines of a H are the solutions u : R 2 ! M of the partial differential equation for some x -∈PH.
This space is invariant under the shift uðs, tÞ↦uðs þ s 0 , tÞ for each s 0 ∈R. For a generic Hamiltonian function, the space Here μ :PH ! Z is a version of Maslov index defined by the path of symplectic matrices generated by the linearized Hamiltonian flow along xðtÞ. The coefficients of δðδðξÞðxÞÞ are given by counting the numbers of pairs of connecting orbits fromx toỹ where μðxÞ−μðỹÞ ¼ 2 ¼ dimMðx,ỹÞ. The quotient Mðx,ỹÞ=R is a one-dimensional oriented manifold that consists of pairs counted by δðδðξÞðxÞÞ. Thus the numbers cancel out in pairs, so that δ δðξÞ ¼ 0.
Definition. The cochain complex ðFC Ã ðM, HÞ, δÞ induces its cohomology groups The rank of FC Ã ðM, HÞ as a Λ ω -module must be at least equal to the dimension of H Ã ðMÞ. The rank is the number #ðPHÞ which is the number of the fixed points of φ.

Almost contact metric manifolds
Let be a real ð2n þ 1Þ-dimensional smooth manifold. An almost cocomplex structure on M is defined by a smooth ð1, 1Þ type tensor ϕ, a smooth vector field ξ, and a smooth 1-form η on M such that for each point x∈M, where I : T x M ! T x M is the identity map of the tangent space T x M.
A Riemannian manifold M with a metric tensor g and with an almost co-complex structure ðϕ, ξ, ηÞ such that gðX, YÞ ¼ gðϕX, ϕYÞ þ ηðXÞηðYÞ, X, Y∈ΓðTMÞ, is called an almost contact metric manifold. The fundamental 2-form φ of an almost contact metric manifold ðM, g, ϕ, ξ, ηÞ is defined by for all X, Y∈ΓðTMÞ. The ð2n þ 1Þ-form φ n ∧η does not vanish on M, and so M is orientable. The Nijehuis tensor [8,11] of the (1,1) type tensor ϕ is the (1,2) type tensor field N ϕ defined by for all X, Y∈ΓðTMÞ, where ½X, Y is the Lie bracket of X and Y. An almost cocomplex structure ðϕ, ξ, ηÞ on M is said to be integrable if the tensor field N ϕ ¼ 0, and is normal if N ϕ þ 2dη⊗ξ ¼ 0.

Sasakian if M is a normal almost Sasakian manifold,
6. a C-manifold if M is a normal almost C-manifold.
An example of compact Sasakian manifolds is an odd-dimensional unit sphere S 2nþ1 , and the one of the co-Kähler (almost cosymplectic) manifolds is a product MS 1 where M is a compact Kähler (symplectic) manifold, respectively.

J is an almost complex structure on M.
3. If M i , i ¼ 1, 2, are cosymplectic, then M is Kähler.
1. If M 2 is contact, then M is an almost C-manifold.

2.
If M 2 is a C-manifold, then M is an almost C-manifold.
3. If M 2 is almost cosymplectic, then M is almost cosymplectic.

Quantum type cohomology
In [10,11] we have studied the quantum type cohomology on contact manifolds. In this section, we want to introduce the quantum type cohomologies on almost cosymplectic, contact, and C-manifolds.
Let A∈H 2 ðM; ZÞ be a two-dimensional integral homology class in M. Let M 0, 3 ðM; A, ϕÞ be the moduli space of stable rational ϕ-coholomorphic maps with three marked points, which represent class A.
We have a Gromov-Witten type invariant given by which is the number of these intersection points counted with signs according to their orientations.
We define a quantum type product Ã on H Ã ðMÞ, for α∈H k ðMÞ and β∈H ðMÞ, where N is called the minimal Chern number defined by The ðα Ã βÞ A ∈H kþl−2c1ðHÞ½A ðMÞ is defined for each C∈H kþl−2c1ðHÞ½A ðMÞ, We denote a quantum type cohomology [11,13] of M by where Q½q is the ring of Laurent polynomials in q of degree 2N with coefficients in the rational numbers Q. By linearly extending the product Ã on QH Ã ðMÞ, we have Let ðM 2n1 1 , g 1 , J 1 , ω 1 Þ be a symplectic manifold and ðM 2n2þ1 2 , g 2 , ϕ 2 , η 2 , ξ 2, , φ 2 Þ be an either almost cosymplectic or contact or C-manifold.
The quantum type cohomology of M is defined by the tensor product where Q½q 1 , q 2 is the ring of Laurent polynomials of variables q 1 and q 2 with coefficients in Q.
Extend the product Ã linearly on the quantum cohomology QH Ã ðMÞ; similarly, we define the quantum cohomology rings Theorem 3.1.5. There is a natural ring isomorphism between quantum type cohomology rings constructed as above, Let ðM, g, ϕ, φÞ be the product of a compact symplectic manifold ðM 2n1 1 , g 1 , J 1 , ω 1 Þ and an either almost cosymplectic or contact or C-manifold ðM 2n2þ1 bases, e 0 , e 1 , …, e k1 of H Ã ðM 1 Þ and f 0 , f 1 , …, f k2 of H Ã ðM 2 Þ such that e 0 ¼ 1∈H 0 ðM 1 Þ, f 0 ¼ 1∈H 0 ðM 2 Þ and each basis element has a pure degree. We introduce a linear polynomial of k 1 þ 1 variables t 0 , t 1 , …, t k1 , with coefficients in H Ã ðM 1 Þ and a linear polynomial of k 2 þ 1 variables s 0 , s 1 , ⋯, s k2 with coefficients in H Ã ðM 2 Þ a s : By choosing the coefficients in Q, the cohomology of M is Then, H Ã ðMÞ has an integral basis {e i ⊗f i ji ¼ 0, …, k 1 , j ¼ 0, …, k 2 }. The rational Gromov-Witten type potential of the product ðM, ωÞ is a formal power series in the variables {t i , s j ji ¼ 0, …, k 1 , j ¼ 0, …, k 2 } with coefficients in the Novikov ring Λ ω as follows: Theorem 3.1.6. The rational Gromov-Witten type potential of ðM, ϕÞ is the product of the rational Gromov-Witten potentials of M 1 and M 2 , that is,

Floer type cohomology
In this subsection, we assume that our manifold ðM 2nþ1 , g, ϕ, η, ξ, φÞ is either a almost cosymplectic, contact, or C-manifold. Letã :¼ ða, ½uÞ be an equivalence class and denoted by g LM the space of equivalence classes.
The space g LM is the universal covering space of the space LM of contractible loops in M whose group of deck transformation is H 2 ðMÞ.
The symplectic type action functional a H : g LM ! R is defined by where a, b∈PðHÞ. If H t ≡H is a C 2 -small Morse function, then a critical point ða, ½uÞ of H t is a constant map uðDÞ ¼ a with index ind H ðaÞ.
If μðãÞ−μðbÞ ¼ 1, then the space Mðã,bÞ is a one-dimensional manifold with R action by time shift and the quotient Mðã,bÞ=R is a finite set. In fact, μðãÞ∈π 1 ðUðnÞÞ≃Z.
If μðãÞ−μðbÞ ¼ 1,ã,b∈PðHÞ, then we denote where the connection orbits are to be counted with signs determined by a system of coherent where ξ∈FC k ðM, HÞ, μðãÞ ¼ k þ 1 and μðbÞ ¼ k. (2) The Floer type cohomology group FH Ã ðM, φ, H, ϕÞ is a module over Novikov ring Λ φ and is independent of the generic choices of H and ϕ.

Quantum and Floer type cohomologies
In this section, we assume that our manifold M is a compact either almost cosymplectic or contact or C-manifold. In Section 3.1, we study quantum type cohomology of M and in Section 3.2 Floer type cohomology of M. Consequently, we have: where u a : D ! M is the constant map u a ðzÞ ¼ a.
The downward gradient flow lines u : R ! M of h are solutions of the ordinary differential equation These solutions determine a coboundary operator This coboundary operator is defined on the same cochain complex as the Floer coboundary δ, and the cochain complex has the same grading for both complex C Ã ðM, h, Λ φ Þ, which can be identified with the graded Λ φ module of all functions for all c∈R. The Λ φ -module structure is given by the grading is degða, AÞ ¼ ind h ðaÞ−2c 1 ðAÞ, and the coboundary operator δ is defined by counted with appropriate signs.
Here we assume that the gradient flow of h is Morse-Smale and so the number of connecting orbits is finite when ind h ðaÞ−ind h ðbÞ ¼ 1. Then the coboundary operator δ is a Λ φ -module homomorphism of degree one and satisfies δ∘δ ¼ 0. Its cohomology is canonically isomorphic to the quantum type cohomology of M with coefficients in Λ φ .
which is a Λ φ -module homomorphism and raises the degree by n. The chain map Φ induces a homomorphism on cohomology Then Φ∘Ψ and Ψ∘Φ are chain homotopic to the identity. Thus we have an isomorphism Φ.
We have studied the Gromov-Witten invariants on symplectic manifolds ðM, ω, JÞ using the theory of J-holomorphic curves, and the Gromov-Witten type invariants on almost contact metric manifolds ðN, g, ϕ, η, ξ, φÞ with a closed fundamental 2-form φ using the theory of ϕcoholomorphic curves. We also have some relations between them. We can apply the theories to many cases.