Open access peer-reviewed chapter

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

By Yong Seung Cho

Submitted: April 30th 2016Reviewed: September 7th 2016Published: January 18th 2017

DOI: 10.5772/65663

## 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 ϕand 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 M2ntogether with a global 2-form ωwhich is closed and nondegenerate, that is, the exterior derivative dω=0and the n-fold wedge product ωnnever vanishes.

Examples: (1) The 2n-dimensional Euclidean space R2nwith coordinates (x1,,xn,y1,,yn)admits symplectic form ω0=i=1ndxidyi.

(2) Let Mbe a smooth manifold. Then its cotangent bundle T*Mhas a natural symplectic form as follows. Let π:T*MMbe the projection map and x1,,xnare local coordinates of M. Then qi=xiπ,i=1,2,,ntogether with fiber coordinates p1,,pngive local coordinates of T*M. The natural symplectic form on T*Mis given by

ω=i=1ndqidqj.E1

(3) Every Kähler manifold is symplectic.

Darboux’s Theorem 2.1 ([6]). Every symplectic formωonMis locally diffeomorphic to the standard formω0onR2n.

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

ddtϕt=Xtϕt,ϕ0=id.E2

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:MRbe a smooth function. Then the vector field XHon Mdetermined by ω(XH,)=dHis called the Hamiltonian vector field associated with H. If Mis closed, then XHgenerates a smooth 1-parameter group of diffeomorphisms ϕHtDiff(M)such that

ddtϕHt=XHϕHt,ϕH0=id.E3

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

Flux{ϕHt}=01ω(Xt,)dt.E4

Theorem 2.2 ([6]). ϕSym(M)is a Hamiltonian symplectomorphism if and only if there is a homotopy[0,1]Sym(M), tϕtsuch thatϕ0=id, ϕ1=ϕ, andFlux({ϕt})=0.

### 2.1. Quantum cohomology

Let (M,ω)be a compact symplectic manifold. An almost complex structure is an automorphism of TMsuch that J2=I. The form ωis said to tame Jif ω(v,Jv)>0for every v0. The set Iτ(M,ω)of almost complex structures tamed by ωis nonempty and contractible. Thus the Chern classes of TMare independent of the choice JIτ(M,ω). A smooth map ϕ:(M1,J1)(M2,J2)from M1to M2is (J1,J2)-holomorphic if and only if

dϕxJ1=J2dϕxE5

Hereafter, we denote by H2(M)the image of Hurewicz homomorphism π2MH2(M,Z). A (i,J)-holomorphic map u:(Σ,z1,,zk)Mfrom a reduced Riemann surface (Σ,j)of genus gwith kmarked points to (M,J)is said to be stable if every component of Σof genus 0 (resp. 1), which is contracted by u, has at least 3 (resp. 1) marked or singular points on its component, and the kmarked 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:ΣME6

which represent AH2(M).

Let us consider infinite dimensional vector bundle EBwhose fiber at uis the space Eu=Ω0,1(Σ,u*TM)of smooth J-antilinear 1-forms on Σwith values in u*TM. The map ¯J:BEgiven by

¯J(u)=12(du+Jduj)E7

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

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

Du:Ω0(Σ,u*TM)=TuBΩ0,1(Σ,u*TM)E8

the composition of the derivative

d(¯J)u:TuBT(u,0)EE9

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(22g)+(6g6)+2k.E10

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

dimMg,k(M,A;J)=2c1(TM)A+n(22g)+(6gg)+2k.E11

For some technical reasons, we assume that c1(A)0if ω(A)>0and Ais represented by some J-holomorphic curves. In this case, we call the symplectic manifold Msemipositive. We define the evaluation map by

ev:Mg,k(M,A;J)Mk,ev([u;z1,,zk])=(u(z1),,u(zk)).E12

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,Ais defined by

Φg,kM,A:Hm(Mk)Q,Φg,kM,A(a)=Mg,k(M,A;J)ev*αE13

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

The minimal Chern number Nof (M,ω)is the integer N:=min {λ|c1(A)=λ0,AH2(M)}. We define the quantum product a*bof aHk(M)and bHl(M)as the formal sum

a*b=AH2(M)(a*b)Aqc1(A)/NE14

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

C(a*b)A=Φ0,3M,A(abr)E15

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

QH*:=H*(M)Q[q]E16

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).E17

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 Aare constant. Thus (a*b)0=ab. The constant term of a*bis the usual cup product ab.

We defined the Novikov ring Λωby the set of functions λ:H2(M)Qthat satisfy the finiteness condition

#{AH2(M)|λ(A)0,ω(A)<c}<E18

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 CPnand so p*pn=q. The quantum cohomology of CPnis

QH*(CPn;Q[q])=Q[p,q]<pn+1=q>.E19

(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 EFis 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]<ynk+1,,yn1,yn+(1)nkq>.E20

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

ΦM(t)=k3A1k!Φ0,kM,A(at,,at)qc1(A)N=k0++kn3Aε(k0,,kn)k0!kn!Φ0,kM,A(e0k0enkn)(t0)k0(tn)knqc1(A)/N.E21

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=Meiejand denote by (gij)its inverse matrix.

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

υ,μtitjtυΦM(t)gυμtμtktlΦM(t)=εijkυ,μtjtktυΦM(t)gυμtμtitlΦM(t),E22

whereεijk=(1)deg(e1)(deg(ej)+deg(ek)).

### 2.2. Floer cohomology

Let a compact symplectic manifold (M,ω)be semipositive. Let Ht+1:MRbe a smooth 1-periodic family of Hamiltonian functions. The Hamiltonian vector field Xtis defined by ω(Xt,)=dHt. The solutions of the Hamiltonian differential equation x˙(t)=Xt(x(t))generate a family of Hamiltonian symplectomorphisms ϕt:MMsatisfying ddtϕt=Xtϕtand ϕ0=id. For every contractible loop x:R/ZM, there is a smooth map u:D:={zC||z|1}Msuch that u(e2πit)=x(t). Two such maps u1and u2are called equivalent if their boundary sum(u1)#(u2)is homologus to zero in H2(M). Denote by (x,[u1])=(x,[u2])for equivalent pairs, LMthe space of contractible loops and LM˜the space of equivalence classes. Then LM˜LMis 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 aHis defined by

aH:LM˜R,aH(x,[u])=Du*ω01Ht(x(t))dt.E23

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

daH(x,[u])(ξ)=01ω(x˙(t)Xt(x(t)),ξ)dt.E24

Thus the critical points of aHare one-to-one correspondence with the periodic solutions of x˙(t)Xt(x(t))=0. Denote by PH˜LM˜the critical points of aHand by PHLMthe set of periodic solutions.

The gradient flow lines of aHare the solutions u:R2Mof the partial differential equation

u+J(u)(tuXt(u))=0E250

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

lims±u(s,t)=x±(t)E25

for some x±PH.

Let M(x˜,x˜+)be the space of such solutions uwith 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˜+).E26

Here μ:PH˜Zis 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˜)/Ris 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˜Qthat satisfy the finiteness condition,

#{x˜PH˜|ξ(x˜)0,aH(x˜)c}<E27

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

(λ*ξ)(x˜):=Aλ(A)ξ(A#x˜).E28

The degree kpart 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˜).E29

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˜)/Ris 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):=Kerδ:FCk(M,H)FCk+1(M,H)Imδ:FCk1(M,H)FCk(M,H)E30

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 Hand J. Let f:MRbe a Morse function such that the negative gradient flow of fwith respect to the metric g(,)=ω(,J)is Morse-Smale. Let H=εf:MRbe the time-independent Hamiltonian. If εis small, then the 1-periodic solutions of x˙(t)XH(x(t))=0are 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)nE31

where u:DMis 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)QE32

that satisfy the finiteness condition

#{(x,A)|ξ(x,A)0,ω(A)c}<E33

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:RMof fare the solutions of u˙(s)=f(u(s)). These solutions determine coboundary operator

δ:MCk(M;Λω)MCk+1(M;Λω)E34
δ(ξ)(x,A)=ynf(x,y)ξ(y,A)E35

where nf(x,y)is the number of connecting orbits ufrom xto ysatisfying 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):=Kerδ:MC*(M;Λω)MC*+1(M;Λω)Imδ:MC*1(M;Λω)MC*(M;Λω)E36

which is called the Morse-Witten cohomology ofM.

(2) MH*(M;Λω)is naturally isomorphic to the quantum cohomologyQH*(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)E37

which is linear over the Novikov ringΛω.

Let H:MRbe a generic Hamiltonian function and ϕ:MMthe Hamiltonian symplectomorphism of H. By Theorems 2.2.1 and 2.2.2

FH*(M,H)QH*(M;Λω)H*(M)ΛωE38

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 symplectomorphismϕ:MMhas only nondegenerate fixed points, then

#(Fix(ϕ))j=02nbj(M)E39

wherebj(M)is the jth Betti number ofM.

## 3. Almost contact metric manifolds

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

φx2=I+ηxξx,ηx(ξx)=1,E40

where I:TxMTxMis the identity map of the tangent space TxM.

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

g(X,Y)=g(φX,φY)+η(X)η(Y),X,YΓ(TM),E41

is called an almost contact metric manifold.

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

ϕ(X,Y)=g(X,φY)E42

for all X,YΓ(TM). The (2n+1)-form ϕnηdoes not vanish on M, and so Mis orientable. The Nijehuis tensor [8, 11] of the (1,1) type tensor φis the (1,2) type tensor field Nφdefined by

Nφ(X,Y)=[φX,φY][X,Y]φ[φX,Y]φ[X,φY]E43

for all X,YΓ(TM), where [X,Y]is the Lie bracket of Xand Y. An almost cocomplex structure (φ,ξ,η)on Mis 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=0and 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 Mis an integrable almost cosymplectic manifold,

5. Sasakian if Mis a normal almost Sasakian manifold,

6. a C-manifold if Mis 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×S1where Mis 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 Mis defined by

g((X1,Y1),(X2,Y2))=g1(X1,X2)+g2(Y1,Y2).E44

An almost complex structure on Mis defined by

J(X,Y)=(φ1(X)+η2(Y)ξ1,φ2(Y)η1(X)ξ2).E45

Then J2=Iand the fundamental 2-form ϕon Mis ϕ=ϕ1+ϕ2+η1η2. If ϕ1, ϕ2and η1and η2are closed, then ϕis 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. Ifϕiandηi, i= 1,2, are closed, thenϕis closed.

2. Jis an almost complex structure onM.

3. IfMi, i=1,2, are cosymplectic, thenMis 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=ϕ1on M1.

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

1. IfM2is contact, thenMis an almostC-manifold.

2. IfM2is aC-manifold, thenMis an almostC-manifold.

3. IfM2is almost cosymplectic, thenMis 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 HM. As the symplectic manifolds, a (1,1) type tensor field φ:HHwith φ2=Iis said to be tamed by ϕif ϕ(X,φX)>0for XH\{0}is said to be compatible if ϕ(φX,φY)=ϕ(X,Y).

Assume that the almost contact metric manifold Mhas a closed fundamental 2-form ϕ, i.e., dϕ=0. An almost contact metric manifold Mwith the ϕis 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 structureφon the distribution, CnHM, the moduli spaceM0,3(M;A,φ)is a compact stratified manifold with virtual dimension2c1(H)[A]+2n.

Consider the evaluation map given by

ev:M0,3(M;A,φ)M3,E46
ev(Σ;z1,z2,z3,u)=(u(z1),u(z2),u(z3)).E47

We have a Gromov-Witten type invariant given by

Φ0,3M,A,φ:H*(M3)QE48
Φ0,3M,A,φ(α)=M0,3(M;A,φ)ev*(α)=ev*[M0,3(M;A,φ)]PD(α)E49

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=AH2(M)(α*β)Aqc1(H)[A]/N,E50

where Nis called the minimal Chern number defined by

<c1(H),H2(M)>=NZE51

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(αβγ),γ=PD(C).E52

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

QH*(M):=H*(M)Q[q]E53

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

Theorem 3.1.2. The quantum type cohomologyQH*(M)of the manifoldMis 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 classA=A1+A2H2(M1)H2(2)H2(M).

Lemma 3.1.3. Let(M,g,φ,η,ξ,ϕ)be the productM=M1×M2constructed as above. For a generic almost cocomplex structureφonM

(1) the moduli spaceM0,3(M;A,φ)is homeomorphic to the product

M0,3(M1,A1,J1)×M0,3(M2,A2,φ),E54
dimM0,3(M,A,φ)=2[c1(TM1)(A1)+c1(H2)(A2)]+2(n1+n2).E55

Theorem 3.1.4.For the product(M,g,φ,η,ξ,ϕ)=(M1,g1,J1,ω1)×(M2,g2,φ2,η2,ξ2,ϕ2), ifA=A1+A2H2(M1)H2(M2)H2(M), then the Gromov-Witten type invariants satisfy the following equality

Φ0,3M,A,φ=Φ0,3M1,A1,J1Φ0,3M2,A2,φ2.E56

The complex (n1+n2)-dimensional vector bundle

TM1H2M=M1×M2E57

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

The minimal Chern numbers N1and N2are given by N1Z=<c1(TM1),H2(M1)>and

N2Z=<c1(H2),H2(M2)>.E58

For cohomology classes

α=α1α2Hk1(M1)Hk2(M2)Hk(M),E59
β=β1β2Hl1(M1)Hl2(M2)Hl(M),E60

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

α*β=A1H2(M1)A2H2(M2)(α1*β1)A1qc1(A1)/N1(α2*β2)A2qc1(A2)/N2E61

where qiis a degree 2Niauxiliary 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(αiβiγi)E62

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

The quantum type cohomology of Mis defined by the tensor product

QH*(M)=H*(M)Q[q1,q2],E63

where Q[q1,q2]is the ring of Laurent polynomials of variables q1and q2with 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].E64

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

QH*(M)=QH*(M1)QH*(M2).E65

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,,ek1of H*(M1)and f0,f1,,fk2of 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+1variables t0,t1,,tk1, with coefficients in H*(M1)

at:=t0e0+t1e1++tk1ek1,E66

and a linear polynomial of k2+1variables s0,s1,,sk2with coefficients in H*(M2)

as:=s0f0+s1f1++sk2fk2.E67

By choosing the coefficients in Q, the cohomology of Mis

H*(M)H*(M1)H*(M2).E68

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 Λωas follows:

Ψ0M(t,s)=Am1m!Φ0,mM,A,φ(atas,,atas)eAϕ=A1m11m1!Φ0,m1M1,A1,J1(at,,at)eA1ω1A2m21m2!Φ0,m2M2,A2,J2(as,,as)eA2ϕ2=Ψ0M1(t)Ψ0M2(s).E69

Theorem 3.1.6. The rational Gromov-Witten type potential of(M,φ)is the product of the rational Gromov-Witten potentials ofM1andM2, that is,

Ψ0M(t,s)=Ψ0M1(t)Ψ0M2(s).E70

### 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:MRbe a smooth 1-periodic family of Hamiltonian functions. Denoted by Xt:MTMthe Hamiltonian vector field of Ht.

The vector fields Xtgenerate a family of Hamiltonian contactomorphisms ψt:MMsatisfying ddtψt=Xtψtand ψ0=id.

Let a:R/ZMbe 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:DMare called equivalent if their boundary sum u1#(u2):S2Mis 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 LMof contractible loops in Mwhose group of deck transformation is H2(M).

The symplectic type action functional aH:LM˜Ris defined by

aH(a,[u])=Du*ϕ01Ht(a(t))dt,E71

then satisfies aH(A#a˜)=aH(a˜)ϕ(A).

Lemma 3.2.1. Let(M,ϕ)the manifold with a closed fundamental 2-formϕand fix a Hamiltonian functionHC(R/Z×M). Let(a,[u])LMandVTaLM=C(R/Z,a*TM). Then

(daH)(a,[u])(V)=01ϕ(a˙XHt(a),V)dt.E72

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

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

u:R2M,(s,t)u(s,t)E73

of the partial differential equation

s(u)+φ(u)(tuXt(u))=0E74

with periodicity condition

u(s,t+1)=u(s,t)E75

and limit condition

limsu(s,t)=a(t),lims+u(s,t)=b(t),E76

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˜.E77

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˜),E78

where the function μ:P(H)˜Zis 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 HtHis a C2-small Morse function, then a critical point (a,[u])of Htis a constant map u(D)=awith index indH(a).

If μ(a˜)μ(b˜)=1, then the space M(a˜,b˜)is a one-dimensional manifold with Raction by time shift and the quotient M(a˜,b˜)/Ris 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),E79

where the connection orbits are to be counted with signs determined by a system of coherent orientation sof 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)˜RE80

that satisfy the finiteness condition

#{x˜P(H)˜|ξ(x˜)0,aH(x˜)c}<E81

for all cR.

Now we define a coboundary operator

δk:FCk(M,H)FCk+1(M,H),E82
(δkξ)(a˜)=μ(a˜)=μ(b˜)+1η(a˜,b˜)ξ(b˜)E83

where ξFCk(M,H), μ(a˜)=k+1and 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 satisfyδk+1δk=0, for allk.

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

FH*(M,ϕ,H,φ):=KerδImδE84

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

(2) The Floer type cohomology groupFH*(M,ϕ,H,φ)is a module over Novikov ringΛϕand is independent of the generic choices ofHandφ.

## 4. Quantum and Floer type cohomologies

In this section, we assume that our manifold Mis a compact either almost cosymplectic or contact or C-manifold. In Section 3.1, we study quantum type cohomology of Mand 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-formϕ. 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).E85

Proof. Let h:MRbe a Morse function such that the negative gradient flow of hwith respect to the metric ϕ(,φ())+ηηis Morse-Smale and consider the time-independent Hamiltonian

Ht:=εh,tR.E86

If εis sufficiently small, then the 1-periodic solutions of

a˙(t)=Xt(a(t))E87

are precisely the critical point of h. The index is

μ(a,ua)=nindh(a)=indh(a)nE88

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

The downward gradient flow lines u:RMof hare solutions of the ordinary differential equation

u˙(s)=J(u)Xt(u).E89

These solutions determine a coboundary operator

δ:C*(M,h,Λϕ)C*(M,h,Λϕ).E90

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 Λϕmodule of all functions

ξ:Crit(h)×H2(M)RE91

that satisfy the finiteness condition

#{(a,A)|ξ(a,A)0,ϕ(A)c}<E92

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

(ν*ξ)(a,A)=Bν(B)ξ(a,A+B),E93

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

(δξ)(a,A)=bnh(a,b)ξ(b,A),(a,A)Crit(h)×H2(M),E94

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

{u˙(s)+u(s)=0,limsu(s)=a,lims+u(s)=b,E95

counted with appropriate signs.

Here we assume that the gradient flow of his Morse-Smale and so the number of connecting orbits is finite when indh(a)indh(b)=1. Then the coboundary operator δis a Λϕ-module homomorphism of degree one and satisfies δδ=0. Its cohomology is canonically isomorphic to the quantum type cohomology of Mwith coefficients in Λϕ.

For each element a˜P(H)˜we denote M(a˜,H,φ)by the space of perturbed φ-cohomomorphic maps u:CMsuch that u(re2πit)converges to a periodic solution a(t)of the Hamiltonian system Htas r. The space M(a˜,H,φ)has dimension nμ(a˜). Now fix a Morse function h:MRsuch that the downward gradient flow u:RMsatisfying (95) is Morse-Smale. For a critical point bCrit(h)the unstable manifold Wu(b,h)of bhas dimension indh(b)and codimension 2nindh(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.E96

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)E97
(Φξ)(b,A)Λindh(b)=μ(a˜)+nn(b,a˜)ξ(A#a˜)E98

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

Φ:FH*(M,Λϕ)H*(M,h,Λϕ)=KerδImδQH*(M,Λϕ).E99

Similarly, we can construct a chain map,

Ψ:C*(M,h,Λϕ)FC*(M,H)E100
(Ψξ)(a˜):=μ(a˜)+n=indh(b)2c1(A)n((A)#a˜,b)ξ(b,A).E101

Then ΦΨand ΨΦare 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 ϕusing 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 Nbe a quintic hypersurface in CP4which is called a Calabi-Yau threefold. Then Nis symply connected, c1(TN)=0and its Betti numbers b0=b6=1, b1=b5=0, b2=b4=1and b3=204.

Let Abe 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 Nis QH*(N)=H*(N)Λwhere Λis the universal Novikov ring [5].

Let (N,g1,ω1,J1)be the standard Kähler structure on Nand (S1,g2,φ2=0,η2=dθ,ξ2=ddθ,ϕ2=0)the standard contact structure on S1. Then the product M=N×S1has a canonical cosymplectic structure (M,g,φ,η,ξ,ϕ)as in Section 3. The quantum type cohomology of Mis

QH*(M)=QH*(N)QH*(S1)E102

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

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

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Yong Seung Cho (January 18th 2017). Symplectic Manifolds: Gromov-Witten Invariants on Symplectic and Almost Contact Metric Manifolds, Manifolds - Current Research Areas, Paul Bracken, IntechOpen, DOI: 10.5772/65663. Available from:

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