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Mathematics » "Manifolds - Current Research Areas", book edited by Paul Bracken, ISBN 978-953-51-2872-4, Print ISBN 978-953-51-2871-7, Published: January 18, 2017 under CC BY 3.0 license. © The Author(s).

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

By Yong Seung Cho
DOI: 10.5772/65663

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

Yong Seung Cho
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Abstract

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 cohomology is isomorphic to the Floer type cohomology.

Keywords: symplectic manifold, Gromov-Witten invariant, quantum cohomology, Gromov-Witten potential, Floer cohomology, almost contact metric manifold, Gromov-Witten type invariant, quantum type cohomology, Gromov-Witten type potential, Floer type cohomology

1. Introduction

The symplectic structures of symplectic manifolds (M, ω, J) are, by Darboux’s theorem 2.1, locally equivalent to the standard symplectic structure on Euclidean space.

In Section 2, we introduce basic definitions on symplectic manifolds [15, 1013] and flux homomorphism. In Section 2.1, we recall J-holomorphic curve, moduli space of J-holomorphic curves, Gromov-Witten invariant and Gromov-Witten potential, quantum product and quantum cohomology, and in Section 2.2, symplectic action functional and its gradient flow line, Maslov type index of critical loop, Floer cochain complex and Floer cohomology, and theorem of Arnold conjecture.

In Section 3, we introduce almost contact metric manifolds (M, g, φ, η, ξ, ϕ) 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.

2. Symplectic manifolds

By a symplectic manifold, we mean an even dimensional smooth manifold M2n together with a global 2-form ω which is closed and nondegenerate, that is, the exterior derivative dω=0 and the n-fold wedge product ωn never vanishes.

Examples: (1) The 2n -dimensional Euclidean space R2n with coordinates (x1, , xn, y1,, yn) admits symplectic form ω0= i=1ndxi dyi .

(2) Let M be a smooth manifold. Then its cotangent bundle T*M has a natural symplectic form as follows. Let π : T*M M be the projection map and x1, , xn are local coordinates of M . Then qi=xiπ , i=1, 2, , n together with fiber coordinates p1, , pn give local coordinates of T*M . The natural symplectic form on T*M is given by

 ω= ∑i=1ndqi ∧dqj. (1)

(3) Every Kähler manifold is symplectic.

Darboux’s Theorem 2.1 ([6]). Every symplectic form ω on M is locally diffeomorphic to the standard form ω0 on R2n .

A symplectomorphism of (M, ω) is a diffeomorphism ϕDiff (M) which preserves the symplectic form ϕ*ω=ω . Denote by Sym(M) the group of symplectomorphims of M . Since ω is nondegenerate, there is a bijection between the vector fields XΓ(TM) and 1-forms ω(X, )Ω1(M) . A vector field XΓ(TM) is called symplectic if ω(X, ) is closed.

Let M be closed, i.e., compact and without boundary. Let ϕ: RDiff (M) , tϕt be a smooth family of diffeomorphisms generated by a family of vector fields XtΓ(TM) via,

 ddtϕt=Xt∘ϕt,ϕ0=id. (2)

Then ϕtSymp(M) if and only if XtΓ(TM, ω) the space of symplectic vector fields on M . Moreover, if X , YΓ(TM, ω) , then [X, Y]Γ(TM, ω) and ω([X,Y], )=dH , where H=ω(X,Y) :MR . Let H :MR be a smooth function. Then the vector field XH on M determined by ω(XH, )=dH is called the Hamiltonian vector field associated with H . If M is closed, then XH generates a smooth 1-parameter group of diffeomorphisms ϕHtDiff(M) such that

 ddtϕHt=XH∘ϕHt, ϕH0=id. (3)

This {ϕHt} is called the Hamiltonian flow associated with H . The flux homomorphism Flux is defined by

 Flux {ϕHt}=∫01ω(Xt, ⋅ ) dt. (4)

Theorem 2.2 ([6]). ϕSym(M) is a Hamiltonian symplectomorphism if and only if there is a homotopy [0,1]Sym(M) , tϕt such that ϕ0=id , ϕ1=ϕ , and Flux ({ϕt})=0 .

2.1. Quantum cohomology

Let (M, ω) be a compact symplectic manifold. An almost complex structure is an automorphism of TM such that J2=I . The form ω is said to tame J if ω(v, Jv)>0 for every v0 . The set Iτ(M,ω) of almost complex structures tamed by ω is nonempty and contractible. Thus the Chern classes of TM are independent of the choice JIτ(M, ω) . A smooth map ϕ :(M1, J1)(M2, J2) from M1 to M2 is (J1, J2) -holomorphic if and only if

 dϕx∘J1=J2∘dϕx (5)

Hereafter, we denote by H2(M) the image of Hurewicz homomorphism π2MH2(M,Z) . A (i, J) -holomorphic map u :(Σ, z1, , zk)M from a reduced Riemann surface (Σ, j) of genus g with k marked points to (M, J) 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 AH2(M) let Mg,k(M, A;J) be the moduli space of (j, J) -holomorphic stable maps which represent A .

Let B:=C(Σ, M;A) be the space of smooth maps

 u :Σ →M (6)

which represent A H2(M) .

Let us consider infinite dimensional vector bundle EB whose fiber at u is the space Eu=Ω0,1(Σ, u*TM) of smooth J -antilinear 1-forms on Σ with values in u*TM . The map ¯J :BE given by

 ∂¯J(u)= 12(du+J∘du∘j) (7)

is a section of the bundle. The zero set of the section ¯I is the moduli space Mg,k(M,A;J) .

For an element uMg,k(M,A;J) we denote by

 Du : Ω0(Σ, u*TM)=TuB→Ω0,1(Σ,u*TM) (8)

the composition of the derivative

 d(∂¯J)u : TuB→ T(u,0)E (9)

with the projection to fiber T(u,0)EΩ0,1(Σ, u*TM) . Then the virtual dimension of Mg,k(M,A;J) is

 dimMg,k(M,A;J)=indexDu : Ω0(Σ, u*TM)→Ω0,1(Σ, u*TM)=2c1(TM)A+n(2−2g)+(6g−6)+2k. (10)

Theorem 2.1.1. For a generic almost complex structure J Iτ(M,ω) the moduli space Mg,k(M,A;J) is a compact stratified manifold of virtual dimension,

 dimMg,k(M,A;J)=2c1(TM)A+n(2−2g)+(6g−g)+2k. (11)

For some technical reasons, we assume that c1(A)0 if ω(A)>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 : Mg,k(M,A;J)→Mk,ev([u; z1, …, zk])=(u(z1), …, u(zk)). (12)

Then the image Im(ev) is well defined, up to cobordism on J , as a dimMg,k(M,A;J):m -dimensional homology class in Mk .

Definition. The Gromov-Witten invariant Φg,kM,A is defined by

 Φg,kM,A : Hm(Mk)→Q, Φg,kM,A(a)=∫Mg,k(M,A;J)ev*•α (13)

where α=PD(a)H2nkm(Mk) and • is the intersection number of ev and α in Mk .

The minimal Chern number N of (M, ω) is the integer N:=min {λ |c1(A)=λ0, A H2(M)} . We define the quantum product a*b of aHk(M) and bHl(M) as the formal sum

 a*b=∑A∈H2(M)(a*b)A qc1(A)/N (14)

where q is an auxiliary variable of degree 2N and (a*b)AHk+l2c1(A)(M) is defined by

 ∫C(a*b)A=Φ0,3M, A(a⊗b⊗r) (15)

for CHk+l2c1(A)(M) , r=PD(C) . Hereafter, we use the Gromov-Witten invariants of g=0 and k=3 . Then the quantum product a*b is an element of

 QH*:=H*(M)⊗Q[q] (16)

where Q[q] is the ring of Laureut polynomials of the auxiliary variable q .

Extending * by linearity, we get a product called quantum product

 * :QH*(M)⊗QH*(M)→QH*(M). (17)

It turns out that * is distributive over addition, skew-commutative, and associative.

Theorem 2.1.2. Let (M, ω) be a compact semipositive symplectic manifold. Then the quantum cohomology (QH*(M),+,*) is a ring.

Remark. For A=0H2(M) , the all J -holomorphic maps in the class A are constant. Thus (a*b)0=ab . The constant term of a*b is the usual cup product ab .

We defined the Novikov ring Λω by the set of functions λ :H2(M)Q that satisfy the finiteness condition

 #{A∈H2(M)| λ(A)≠0, ω(A)

for every cR . The grading is given by deg(A)=2c1(A) .

Examples ([5]). (1) Let pH2(CPn) and AH2(CPn) be the standard generators. There is a unique complex line through two distinct points in CPn and so p*pn=q . The quantum cohomology of CPn is

 QH*( CPn; Q[q])=Q[p,q]. (19)

(2) Let G(k,n) be the Grassmannian of complex k -planes in Cn . There are two natural complex vector bundles CkEG(k,n) and CnkFG(k,n) . Let xi=ci(E*) and yi=ci(F*) be Chern classes of the dual bundles E* and F* , respectively. Since EF is trivial, i=0jxiyji=0 , j=1, , n . By computation xk*ynk=(1)nkq . The quantum cohomology of G(k,n) is

 QH*(G(k,n); Q[q])=Q[x1, …, xk, q]. (20)

Let {e0,,en} be an integral basis of H*(M) such that e0=1H0(M) and each ei has pure degree. We introduce n+1 formal variables t0,, tn and the linear polynomial at in t0,, tn with coefficients in H*(M) by at=t0e0++tnen . The Gromov-Witten potential of (M, ω) is a formal power series in variables t0,, tn with coefficients in the Novikov ring Λω

 ΦM(t)=∑k≥3∑A1k!Φ0,kM,A(at,…,at)qc1(A)N= ∑k0+⋯+kn≥3∑Aε(k0,…,kn)k0!…kn!Φ0,kM,A(e0k0⊗⋯⊗enkn)⋅(t0)k0…(tn)knqc1(A)/N. (21)

Examples ([4]). (1) ΦCP1(t)=12t02t1+(et11t1t122).

(2) ΦCPn(t)=16i+j+k=ntitjtk+d>0k2knNd(k2kn)t2k2tnknk2!kn!edt1qd,

where Nd(k2kn)=Φ0,kCPn,, dA (p2p2, , pnpn) .

We define a nonsingular matrix (gij) by gij=Meiej and denote by (gij) its inverse matrix.

Theorem 2.1.3 ([4, 5]). The Gromov-Witten potential ΦM of (M, ω) satisfies the WDVV-equations:

 ∑υ, μ∂ti∂tj∂tυΦM(t)gυμ∂tμ∂tk∂tlΦM(t) = εijk⋅∑υ, μ∂tj∂tk∂tυΦM(t)gυμ∂tμ∂ti∂tlΦM(t), (22)

where εijk=(1)deg(e1)(deg(ej)+deg(ek)) .

2.2. Floer cohomology

Let a compact symplectic manifold (M, ω) be semipositive. Let Ht+1 : MR be a smooth 1-periodic family of Hamiltonian functions. The Hamiltonian vector field Xt is defined by ω(Xt, )=dHt . The solutions of the Hamiltonian differential equation x˙(t)=Xt(x(t)) generate a family of Hamiltonian symplectomorphisms ϕt : MM satisfying ddtϕt=Xtϕt and ϕ0=id . For every contractible loop x : R/Z M , there is a smooth map u :D:={z C ||z|1}M such that u(e2πit)=x(t) . Two such maps u1 and u2 are called equivalent if their boundary sum (u1)#(u2) is homologus to zero in H2(M) . Denote by (x, [u1])=(x, [u2]) for equivalent pairs, LM the space of contractible loops and LM˜ the space of equivalence classes. Then LM˜LM is a covering space whose covering transformation group is H2(M) via, A(x, [u])=(x, [A#u]) for each AH2(M) and (x, [u])LM˜ .

Definition. The symplectic action functional aH is defined by

 aH : LM˜→R,aH(x, [u])=−∫Du*ω− ∫01Ht(x(t))dt. (23)

For each element x˜:=(x,[u])LM˜ and ξTx˜LM˜ , we have

 daH(x,[u])(ξ)=∫01ω(x˙(t)−Xt(x(t)), ξ)dt. (24)

Thus the critical points of aH are one-to-one correspondence with the periodic solutions of x˙(t)Xt(x(t))=0 . Denote by PH˜LM˜ the critical points of aH and by PHLM the set of periodic solutions.

The gradient flow lines of aH are the solutions u :R2M of the partial differential equation

u+J(u)(tuXt(u))=0

with conditions u(s, t+1)=u(s,t) ,

 lims→±∞u(s,t)=x±(t) (25)

for some x±PH .

Let M(x˜, x˜+) be the space of such solutions u with x˜+=x˜#u . This space is invariant under the shift u(s,t)u(s+s0,t) for each s0R . For a generic Hamiltonian function, the space M(x˜, x˜+) is a manifold of dimension

 dimM(x˜−, x˜+)=μ(x˜−)−μ(x˜+). (26)

Here μ :PH˜Z is a version of Maslov index defined by the path of symplectic matrices generated by the linearized Hamiltonian flow along x(t) .

Let μ(x˜)μ(y˜)=1 . Then M(x˜, y˜) is a one-dimensional manifold and the quotient by shift M(x˜, y˜)/R is finite. In this case, we denote by n(x˜, y˜)=#(M(x˜, y˜)R) the number of connecting orbits from x˜ to y˜ counted with appropriate signs.

We define the Floer cochain group FC*(M,H) as the set of all functions ξ : PH˜Q that satisfy the finiteness condition,

 #{x˜∈PH˜| ξ(x˜)≠0, aH(x˜)≤c}<∞ (27)

for every cR . The complex FC*(M,H) is a Λω -module with action

 (λ*ξ)(x˜):=∑Aλ(A)ξ(A#x˜). (28)

The degree k part FC*(M,H) consists of all ξFC*(M,H) that are nonzero only on elements x˜PH˜ with μ(x˜)=k . Thus the dimension of FC*(M,H) as a Λω -module is the number #(PH) .

We define a coboundary operator δ : FC*(M,H)FCk+1(M,H) by

 δ(ξ)(x˜)=∑μ(y˜)=kn(x˜, y˜)ξ(y˜). (29)

The coefficients of δ(δ(ξ)(x˜)) are given by counting the numbers of pairs of connecting orbits from x˜ to y˜ where μ(x˜)μ(y˜)=2=dimM(x˜, y˜) . The quotient M(x˜,y˜)/R is a one-dimensional oriented manifold that consists of pairs counted by δ(δ(ξ)(x˜)) . Thus the numbers cancel out in pairs, so that δ(δ(ξ))=0 .

Definition. The cochain complex (FC*(M,H), δ) induces its cohomology groups

 FHk(M,H):=Ker δ :FCk(M, H)→FCk+1(M,H)Im δ :FCk−1(M, H)→FCk(M,H) (30)

which are called the Floer cohomology groups of (M, ω, H, J) .

Remark. By the usual cobordism argument, the Floer cohomology groups FH*(M,H) are independent to the generic choices of H and J . Let f :MR be a Morse function such that the negative gradient flow of f with respect to the metric g( , )=ω(, J) is Morse-Smale. Let H=εf :M R be the time-independent Hamiltonian. If ε is small, then the 1-periodic solutions of x˙(t)XH(x(t))=0 are one-to-one correspondence with the critical points of f . Thus we have PH=Crit(f) and the Maslov type index can be normalized as

 μ(x,[u])=indf(x)−n (31)

where u :DM is the constant map u(D)=x .

We define a cochain complex MC*(M;Λω) as the graded Λω -module of all functions

 ξ :Crit(f)×H2(M)→Q (32)

that satisfy the finiteness condition

 #{(x,A)| ξ(x, A)≠0, ω(A)≥c}<∞ (33)

for every cR . The Λω -module structure is given by (λ*ξ)(x,A)=λ(B)ξ(x, A+B) and the grading deg(x,A)=indf(x)2c1(A) . The gradient flow lines u :RM of f are the solutions of u˙(s)=f(u(s)) . These solutions determine coboundary operator

 δ :MCk(M; Λω)→MCk+1(M; Λω) (34)
 δ(ξ)(x, A)=∑ynf(x,y)ξ(y,A) (35)

where nf(x,y) is the number of connecting orbits u from x to y satisfying limsu(s)=x , lims+u(s)=y , counted with appropriate signs and indf(x)indf(y)=1 .

Definition–Theorem 2.2.1. (1) The cochain complex (MC*(M; Λω), δ) defines a cohomology group

 MH*(M; Λω):=Ker δ :MC*(M; Λω)→MC*+1(M;Λω)Im δ :MC*−1(M;Λω)→MC*(M;Λω) (36)

which is called the Morse-Witten cohomology of M .

(2) MH*(M; Λω) is naturally isomorphic to the quantum cohomology QH*(M; Λω) .

Theorem 2.2.2 ([5]). Let a compact symplectic manifold (M, ω) be semipositive. There is an isomorphism

 Φ :FH*(M,H)→QH*(M; Λω) (37)

which is linear over the Novikov ring Λω .

Let H :M R be a generic Hamiltonian function and ϕ :M M the Hamiltonian symplectomorphism of H . By Theorems 2.2.1 and 2.2.2

 FH*(M,H)≃ QH*(M;Λω)≃ H*(M)⊗Λω (38)

The rank of FC*(M,H) as a Λω -module must be at least equal to the dimension of H*(M) . The rank is the number #(PH) which is the number of the fixed points of ϕ .

Theorem 2.2.3 (Arnold conjecture). Let a compact symplectic manifold (M, ω) be semipositive. If a Hamiltonian symplectomorphism ϕ :MM has only nondegenerate fixed points, then

 # (Fix(ϕ))≥∑j=02nbj(M) (39)

where bj(M) is the jth Betti number of M .

3. Almost contact metric manifolds

Let be a real (2n+1) -dimensional smooth manifold. An almost cocomplex structure on M is defined by a smooth (1,1) type tensor φ , a smooth vector field ξ, and a smooth 1-form η on M such that for each point xM ,

 φx2=−I+ηx⊗ξx, ηx(ξx)=1, (40)

where I :TxMTxM is the identity map of the tangent space TxM .

A Riemannian manifold M with a metric tensor g and with an almost co-complex structure (φ, ξ, η) such that

 g(X, Y)=g(φX,φY)+η(X)η(Y), X,Y ∈ Γ(TM), (41)

is called an almost contact metric manifold.

The fundamental 2-form ϕ of an almost contact metric manifold (M,g,φ,ξ,η) is defined by

 ϕ(X,Y)=g(X,φY) (42)

for all X,YΓ(TM) . The (2n+1) -form ϕnη 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 Nφ defined by

 Nφ(X,Y)=[φX,φY]−[X,Y]−φ[φX,Y]−φ[X,φY] (43)

for all X,YΓ(TM) , where [X,Y] is the Lie bracket of X and Y . An almost cocomplex structure (φ, ξ,η) on M is said to be integrable if the tensor field Nφ=0 , and is normal if Nφ+2dηξ=0 .

Definition. An almost contact metric manifold (M,g,φ,η,ξ,ϕ) is said to be

1. almost cosymplectic (or almost co-Kähler) if dϕ=0 and dη=0 ,

2. contact (or almost Sasakian) if ϕ=dη ,

3. an almost C -manifold if dϕ=0 , dη0 , and dηϕ ,

4. cosymplectic (co-Kähler) if M is an integrable almost cosymplectic manifold,

5. 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 S2n+1 , and the one of the co-Kähler (almost cosymplectic) manifolds is a product M×S1 where M is a compact Kähler (symplectic) manifold, respectively.

Let (M12n1+1, g1, φ1, η1, ξ1) and (M22n2+1, g2, φ2, η2, ξ2) be almost contact metric manifolds. For the product M:=M1×M2 , Riemannian metric on M is defined by

 g((X1,Y1),(X2,Y2))=g1(X1,X2)+g2(Y1,Y2). (44)

An almost complex structure on M is defined by

 J(X,Y)=(φ1(X)+η2(Y)ξ1, φ2(Y)−η1(X)ξ2). (45)

Then J2=I and the fundamental 2-form ϕ on M is ϕ=ϕ1+ϕ2+η1η2 . If ϕ1 , ϕ2 and η1 and η2 are closed, then ϕ is closed. Thus we have

Theorem 3.1. Let (M12n1+1, g1, φ1, η1, ξ1) be almost contact metric manifolds, j=1,2 , and (M,g, ϕ, J) be the product constructed as above.

1. If ϕi and ηi , i = 1,2, are closed, then ϕ is closed.

2. J is an almost complex structure on M .

3. If Mi , i=1,2 , are cosymplectic, then M is Kähler.

Let (M12n1, g1, J1) be a symplectic manifold, and (M22n2+1, g2, φ2, η2, ξ2) be an almost contact metric manifold. Then ξ1=η1=0 , and ω1=ϕ1 on M1 .

Theorem 3.2. Let (M, g, φ, η, ξ) be the product constructed as above.

1. If M2 is contact, then M is an almost C -manifold.

2. If M2 is a C -manifold, then M is an almost C -manifold.

3. If M2 is almost cosymplectic, then M is almost cosymplectic.

3.1. 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 (M2n+1, g, φ,η,ξ) be an almost contact metric manifold. Then the distribution H={XTM| η(X)=0} is an n -dimensional complex vector bundle on M .

Now fix the vector bundle H M . As the symplectic manifolds, a (1,1) type tensor field φ:HH with φ2=I is said to be tamed by ϕ if ϕ(X,φX)>0 for XH \{0} is said to be compatible if ϕ(φX, φY)=ϕ(X,Y) .

Assume that the almost contact metric manifold M has a closed fundamental 2-form ϕ , i.e., dϕ=0 . An almost contact metric manifold M with the ϕ is called semipositive if for every Aπ2(M) , ϕ(A)>0 , c1(H)(A)3n , then c1(H)(A)>0 [13]. A smooth map u :(Σ, j) (M, φ) from a Riemann surface (Σ, j) into (M, φ) is said to be φ -coholomorphic if duj=φdu .

Let AH2(M;Z) be a two-dimensional integral homology class in M . Let M0,3(M;A, φ) be the moduli space of stable rational φ -coholomorphic maps with three marked points, which represent class A .

Lemma 3.1.1. For a generic almost complex structure φ on the distribution, CnHM , the moduli space M0,3(M;A, φ) is a compact stratified manifold with virtual dimension 2 c1(H)[A]+2n .

Consider the evaluation map given by

 ev :M0,3(M;A,φ)→ M3, (46)
 ev(Σ; z1, z2, z3, u)=(u(z1), u(z2), u(z3)). (47)

We have a Gromov-Witten type invariant given by

 Φ0,3M,A,φ : H*(M3) → Q (48)
 Φ0,3M,A,φ(α)=∫M0,3(M;A,φ)ev*(α)=ev*[M0,3(M;A,φ)]⋅PD(α) (49)

which is the number of these intersection points counted with signs according to their orientations.

We define a quantum type product * on H*(M) , for αHk(M) and βHl(M) ,

 α*β=∑A∈H2(M)(α*β)Aqc1(H)[A]/N, (50)

where N is called the minimal Chern number defined by

 =NZ (51)

The (α*β)AHk+l2c1(H)[A](M) is defined for each CHk+l2c1(H)[A](M) ,

 ∫C(α*β)A=Φ0,3M,A,φ(α⊗ β⊗ γ), γ=PD(C). (52)

We denote a quantum type cohomology [11, 13] of M by

 QH*(M):=H*(M)⊗Q[q] (53)

where Q[q] is the ring of Laurent polynomials in q of degree 2N with coefficients in the rational numbers Q . By linearly extending the product * on QH*(M) , we have

Theorem 3.1.2. The quantum type cohomology QH*(M) of the manifold M is an associative ring under the product * .

Let (M12n1, g1, J1, ω1) be a symplectic manifold and (M22n2+1, g2, φ2, η2, ξ2,, ϕ2) be an either almost cosymplectic or contact or C -manifold.

Let the product (M2n+1, g, φ,η,ξ, ϕ) be construct as Theorem 3.2 where n=n1+n2 . Now we will only consider the free parts of the cohomologies. By the Künneth formula, H*(M)H*(M1)H*(M2) in particular, H2(M)H2(M1(H1(M1)H1(M2))H2(M2)) .

Assume that a two-dimensional class A=A1+A2H2(M1)H2(2)H2(M) .

Lemma 3.1.3. Let (M, g, φ, η, ξ, ϕ) be the product M=M1×M2 constructed as above. For a generic almost cocomplex structure φ on M

(1) the moduli space M0,3(M;A,φ) is homeomorphic to the product

 M0,3(M1, A1, J1)×M0,3(M2, A2,φ), (54)
 dimM0,3(M, A,φ)=2[c1(TM1)(A1)+c1(H2)(A2)]+2(n1+n2). (55)

Theorem 3.1.4. For the product (M, g, φ, η, ξ, ϕ)=(M1, g1, J1, ω1)×(M2, g2, φ2, η2, ξ2, ϕ2) , if A=A1+A2 H2(M1)H2(M2)H2(M) , then the Gromov-Witten type invariants satisfy the following equality

 Φ0,3M,A,φ=Φ0,3M1, A1, J1⋅Φ0,3M2, A2, φ2. (56)

The complex (n1+n2) -dimensional vector bundle

 TM1⊕H2→M=M1×M2 (57)

has the first Chern class c1(TM1H2)=c1(TM1)+c1(H2) .

The minimal Chern numbers N1 and N2 are given by N1Z=<c1(TM1), H2(M1)> and

 N2Z=. (58)

For cohomology classes

 α=α1⊗α2 ∈ Hk1(M1)⊗Hk2(M2)⊂Hk(M), (59)
 β=β1⊗β2 ∈ Hl1(M1)⊗Hl2(M2)⊂Hl(M), (60)

k1+k2=k , the quantum type product α*β is defined by

 α*β=∑A1∈H2(M1)A2∈H2(M2)(α1*β1)A1qc1(A1)/N1⊗(α2*β2)A2qc1(A2)/N2 (61)

where qi is a degree 2Ni auxiliary variable, i=1,2 , and the cohomology class (αi * βi)AiHki+li2c1(Ai)(Mi) is defined by the Gromov-Witten type invariants as follows:

 ∫Ci(αi*βi)Ai=Φ0,3Mi, Ai, φi(αi⊗βi⊗γi) (62)

where CiHki+li2c1(Ai)(Mi) , γi=PD(Ci) and φ1:=J1 , i=1,2 , respectively.

The quantum type cohomology of M is defined by the tensor product

 QH*(M)=H*(M)⊗Q[q1, q2], (63)

where Q[q1, q2] is the ring of Laurent polynomials of variables q1 and q2 with coefficients in Q . Extend the product * linearly on the quantum cohomology QH*(M) ; similarly, we define the quantum cohomology rings

 {QH*(M1)=H*(M1)⊗Q[q1],QH*(M2)=H*(M2)⊗Q[q2]. (64)

Theorem 3.1.5. There is a natural ring isomorphism between quantum type cohomology rings constructed as above,

 QH*(M)=QH*(M1)⊗QH*(M2). (65)

Let (M, g, φ, ϕ) be the product of a compact symplectic manifold (M12n1, g1, J1, ω1) and an either almost cosymplectic or contact or C -manifold (M22n2+1, g2, φ2, η2, ξ2, ϕ2) . We choose integral bases, e0, e1, , ek1 of H*(M1) and f0, f1, ,fk2 of H*(M2) such that e0=1H0(M1) , f0=1H0(M2) and each basis element has a pure degree. We introduce a linear polynomial of k1+1 variables t0, t1, , tk1 , with coefficients in H*(M1)

 at :=t0e0+t1e1+⋯+tk1ek1, (66)

and a linear polynomial of k2+1 variables s0, s1, , sk2 with coefficients in H*(M2)

 as := s0f0+s1f1+⋯+sk2fk2. (67)

By choosing the coefficients in Q , the cohomology of M is

 H*(M)≅H*(M1)⊗H*(M2). (68)

Then, H*(M) has an integral basis {eifi| i=0,, k1, j=0,, k2} . The rational Gromov-Witten type potential of the product (M, ω) is a formal power series in the variables {ti, sj| i=0,, k1, j=0,, k2} with coefficients in the Novikov ring Λω as follows:

 Ψ0M(t,s)=∑A∑m1m!Φ0,mM,A, φ(at⊗as, …, at⊗as)e−∫Aϕ=∑A1∑m11m1!Φ0,m1M1,A1,J1(at, …, at)e−∫A1ω1⋅∑A2∑m21m2!Φ0,m2M2,A2,J2(as, …, as)e−∫A2ϕ2= Ψ0M1(t)⋅Ψ0M2(s). (69)

Theorem 3.1.6. The rational Gromov-Witten type potential of (M, φ) is the product of the rational Gromov-Witten potentials of M1 and M2 , that is,

 Ψ0M(t,s)= Ψ0M1(t)⋅Ψ0M2(s). (70)

3.2. Floer type cohomology

In this subsection, we assume that our manifold (M2n+1, g, φ, η, ξ,ϕ) is either a almost cosymplectic, contact, or C -manifold.

Let Ht=Ht+1 :MR be a smooth 1-periodic family of Hamiltonian functions. Denoted by Xt :MTM the Hamiltonian vector field of Ht .

The vector fields Xt generate a family of Hamiltonian contactomorphisms ψt :MM satisfying ddtψt=Xtψt and ψ0=id .

Let a :R/ZM be a contractible loop, then there is a smooth map u :DM , defined on the unit disk D={zC | |z|1} , which satisfies u(e2πit)=a(t) . Two such maps u1, u2 :DM are called equivalent if their boundary sum u1#(u2) : S2M is homologus to zero in H2(M) .

Let a˜:=(a, [u]) be an equivalence class and denoted by LM˜ the space of equivalence classes. The space LM˜ is the universal covering space of the space LM of contractible loops in M whose group of deck transformation is H2(M) .

The symplectic type action functional aH :LM˜R is defined by

 aH(a, [u])=−∫Du*ϕ− ∫01Ht(a(t))dt, (71)

then satisfies aH(A#a˜)=aH(a˜)ϕ(A).

Lemma 3.2.1. Let (M, ϕ) the manifold with a closed fundamental 2-form ϕ and fix a Hamiltonian function HC(R/Z ×M) . Let (a, [u])LM and VTaLM=C(R/Z, a*TM) . Then

 (daH)(a, [u])(V)=∫01ϕ (a˙−XHt(a), V)dt. (72)

We denote by P(H)˜ LM˜ the set of critical points of aH and by P(H)LM the corresponding set of periodic solutions.

Consider the downward gradient flow lines of aH with respect to an L2 -norm on LM . The solutions are

 u : R2→M, (s,t)↦u(s,t) (73)

of the partial differential equation

 ∂s(u)+φ(u)(∂tu−Xt(u))=0 (74)

with periodicity condition

 u(s,t+1)=u(s,t) (75)

and limit condition

 lims→−∞u(s,t)=a(t), lims→+∞u(s,t)=b(t), (76)

where a, bP(H) .

Let M(a˜, b˜):=M(a˜, b˜, H, φ) be the space of all solutions u(s,t) satisfying (74)–(76) with

 a˜#u=b˜. (77)

The solutions are invariant under the action u(s,t)u(s+r,t) of the time shift rR . Equivalent classes of solutions are called Floer connecting orbits.

For a generic Hamiltonian function H , the space M(a˜, b˜) is a finite dimensional manifold of dimension

 dimM(a˜, b˜)=μ(a˜)−μ(b˜), (78)

where the function μ :P(H)˜ Z is a version of the Maslov index defined by the path of unitary matrices generated by the linealized Hamiltonian flow along a(t) on D .

If HtH is a C2 -small Morse function, then a critical point (a, [u]) of Ht is a constant map u(D)=a with index indH(a) .

If μ(a˜)μ(b˜)=1 , then the space M(a˜, b˜) is a one-dimensional manifold with R action by time shift and the quotient M(a˜, b˜)/R is a finite set. In fact, μ(a˜)π1(U(n))Z .

If μ(a˜)μ(b˜)=1 , a˜, b˜ P(H)˜ , then we denote

 η(a˜, b˜):=#(M(a˜, b˜)R), (79)

where the connection orbits are to be counted with signs determined by a system of coherent orientation s of the moduli space M(a˜, b˜) . These numbers give us a Floer type cochain complex.

Let FC*(M,H) be the set of functions

 ξ :P(H)˜→R (80)

that satisfy the finiteness condition

 #{x˜∈P(H)˜| ξ(x˜)≠0, aH(x˜)≤c}<∞ (81)

for all cR .

Now we define a coboundary operator

 δk :FCk(M,H)→ FCk+1(M,H), (82)
 (δkξ)(a˜)=∑μ(a˜)=μ(b˜)+1η(a˜, b˜)ξ(b˜) (83)

where ξFCk(M,H) , μ(a˜)=k+1 and μ(b˜)=k .

Lemma 3.2.2. Let (M, φ) be a semipositive almost contact metric manifold with a closed functional 2-forms. The coboundary operators satisfy δk+1δk=0 , for all k .

Definition - Theorem 3.2.3. (1) For a generic pair (H, φ) on M , the cochain complex (FC*, δ) defines cohomology groups

 FH*(M, ϕ, H, φ):=Ker δIm δ (84)

which are called the Floer type cohomology groups of the (M, ϕ, H, φ) .

(2) The Floer type cohomology group FH*(M, ϕ, H, φ) is a module over Novikov ring Λϕ and is independent of the generic choices of H and φ .

4. 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:

Theorem 4.1. Let (M, g, φ,η,ξ,ϕ) be a compact semipositive almost contact metric manifold with a closed fundamental 2-form ϕ . Then, for every regular pair (H, φ) , there is an isomorphism between Floer type cohomology and quantum type cohomology

 Φ :FH*(M, ϕ, H, φ)→ ∼ QH*(M, Λϕ). (85)

Proof. Let h :MR be a Morse function such that the negative gradient flow of h with respect to the metric ϕ ( , φ( ))+η η is Morse-Smale and consider the time-independent Hamiltonian

 Ht := −εh, t∈R. (86)

If ε is sufficiently small, then the 1-periodic solutions of

 a˙(t)=Xt(a(t)) (87)

are precisely the critical point of h . The index is

 μ(a, ua)=n−indh(a)= ind−h(a)−n (88)

where ua :DM is the constant map ua(z)=a .

The downward gradient flow lines u : RM of h are solutions of the ordinary differential equation

 u˙(s)=J(u)Xt(u). (89)

These solutions determine a coboundary operator

 δ : C*(M, h, Λϕ)→C*(M, h, Λϕ). (90)

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*(M, h, Λϕ) , which can be identified with the graded Λϕ module of all functions

 ξ :Crit(h)×H2(M)→R (91)

that satisfy the finiteness condition

 # {(a, A)| ξ(a, A)≠0, ϕ(A)≥c} < ∞ (92)

for all cR . The Λϕ -module structure is given by

 (ν* ξ )(a, A)= ∑Bν(B)ξ(a, A+B), (93)

the grading is deg(a, A)=indh(a)2c1(A) , and the coboundary operator δ is defined by

 (δξ)(a, A)= ∑bnh(a,b)ξ(b, A), (a, A)∈Crit(h)×H2(M), (94)

where nh(a, b) is the number of connecting orbits from a to b of shift equivalence classes of solutions of

 {u˙(s)+∇u(s)=0,lims→−∞u(s)=a, lims→+∞u(s)=b, (95)

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 indh(a)indh(b)=1 . Then the coboundary operator δ is a Λϕ -module homomorphism of degree one and satisfies δδ=0 . Its cohomology is canonically isomorphic to the quantum type cohomology of M with coefficients in Λϕ .

For each element a˜P(H)˜ we denote M(a˜, H, φ) by the space of perturbed φ -cohomomorphic maps u : CM such that u(re2πit) converges to a periodic solution a(t) of the Hamiltonian system Ht as r . The space M(a˜, H, φ) has dimension nμ(a˜) . Now fix a Morse function h :MR such that the downward gradient flow u :RM satisfying (95) is Morse-Smale. For a critical point b Crit (h) the unstable manifold Wu(b,h) of b has dimension indh(b) and codimension 2n indh(b) in the distribution D .

The submanifold M(b, a˜) of all uM(a˜, H, φ) with u(0)Wu(b) has dimension

 dimM(b, a˜)=indh(b)− μ(a˜)−n. (96)

If indh(b)=μ(a˜)+n , then M(b, a˜) is 0zero -dimensional and hence the numbers n(b, a˜) of its elements can be used to construct the chain map defined by

 Φ :FC*(M, H)→C*(M, h, Λϕ) (97)
 (Φξ)(b, A)Λ ∑indh(b)=μ(a˜)+nn(b, a˜) ξ(A#a˜) (98)

which is a Λϕ -module homomorphism and raises the degree by n . The chain map Φ induces a homomorphism on cohomology

 Φ :FH*(M, Λϕ)→H*(M,h, Λϕ)=Ker δIm δ ≃QH*(M, Λϕ). (99)

Similarly, we can construct a chain map,

 Ψ :C*(M, h, Λϕ)→FC*(M, H) (100)
 (Ψξ)(a˜):= ∑μ(a˜)+n=indh(b)−2c1(A)n((−A)#a˜, b) ξ(b, A). (101)

Then ΦΨ and ΨΦ are chain homotopic to the identity. Thus we have an isomorphism Φ .

We have studied the Gromov-Witten invariants on symplectic manifolds (M, ω, J) using the theory of J -holomorphic curves, and the Gromov-Witten type invariants on almost contact metric manifolds (N, g, φ, η, ξ, ϕ) with a closed fundamental 2-form ϕ using the theory of φ -coholomorphic curves. We also have some relations between them. We can apply the theories to many cases.

Examples 4.2.

1. The product of a symplectic manifold and a unit circle.

2. The circle bundles over symplectic manifolds.

3. The almost cosymplectic fibrations over symplectic manifolds.

4. The preimage of a regular value of a Morse function on a Kähler manifold.

5. The product of two cosymplectic manifolds is Kähler.

6. The symplectic fibrations over almost cosymplectic manifolds.

7. The number of a contactomorphism is greater than or equal to the sum of the Betti numbers of an almost contact metric manifold with a closed fundamental 2-form.

Examples 4.3. Let N be a quintic hypersurface in CP4 which is called a Calabi-Yau threefold. Then N is symply connected, c1(TN)=0 and its Betti numbers b0=b6=1 , b1=b5=0 , b2=b4=1 and b3=204 .

Let A be the standard generator in H2(N) and hH2(N) such that h(A)=1 . The moduli space M0,3(N,A) has the dimension zero. The Gromov-Witten invariant Φ0,3N, A(a1, a2, a3) is nonzero only when deg(ai)=2 , i=1,2,3 . In fact, Φ0,3N, A(h, h, h)=5 [4, 5]. The quantum cohomology of N is QH*(N)=H*(N)Λ where Λ is the universal Novikov ring [5].

Let (N, g1, ω1, J1) be the standard Kähler structure on N and (S1, g2, φ2=0, η2=dθ, ξ2=ddθ, ϕ2=0) the standard contact structure on S1 . Then the product M=N×S1 has a canonical cosymplectic structure (M, g, φ,η,ξ,ϕ) as in Section 3. The quantum type cohomology of M is

 QH*(M)=QH*(N)⊗QH*(S1) (102)

Let ψ1 :NN be a Hamiltonian symplectomorphism with nondegenerate critical points. Then #Fix(ψ1)i=06bi(N)=208 .

Let ψ2 :MM be a Hamiltonian contactomorphism with nondegenerate critical points. Then #Fix(ψ2)i=07bi(M)=416 .

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