2. Symplectic manifolds

By a symplectic manifold, we mean an even dimensional smooth manifold M<sup>2</sup><sup>n</sup> 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<sup>n</sup> never vanishes.

Examples: (1) The 2n-dimensional Euclidean space <sup>R</sup><sup>2</sup><sup>n</sup> with coordinates <sup>ð</sup>x1, …, xn, <sup>y</sup>1, …, yn<sup>Þ</sup> admits symplectic form <sup>ω</sup><sup>0</sup> <sup>¼</sup> ∑ n i¼1 dxi∧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 ¼ xi∘π,i ¼ 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

$$
\omega = \sum\_{i=1}^{n} dq\_i \wedge dq\_j. \tag{1}
$$

(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 ω<sup>0</sup> on R<sup>2</sup><sup>n</sup>.

A symplectomorphism of ðM, ωÞ is a diffeomorphism φ∈Dif fðMÞ which preserves the symplectic form φ� 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 <sup>ω</sup>ðX, �Þ∈Ω<sup>1</sup> ðMÞ. A vector field X∈ΓðTMÞ is called symplectic if ωðX, �Þ is closed.

Let M be closed, i.e., compact and without boundary. Let φ : R ! DiffðMÞ, t↦φ<sup>t</sup> be a smooth family of diffeomorphisms generated by a family of vector fields Xt∈ΓðTMÞ via,

Symplectic Manifolds: Gromov-Witten Invariants on Symplectic and Almost Contact Metric Manifolds http://dx.doi.org/10.5772/65663 91

$$\frac{d}{dt}\phi\_t = X\_t \circ \phi\_t,\\ \phi\_0 = \text{id}.\tag{2}$$

Then φ<sup>t</sup> ∈SympðMÞ if and only if Xt∈ΓðTM, ωÞ the space of symplectic vector fields on M. Moreover, if X,Y∈ΓðTM, ωÞ, then ½X,Y�∈ΓðTM, ωÞ and ωð½X, Y�, �Þ ¼ dH, where H ¼ ωðX, YÞ : M ! R. Let H : M ! R be a smooth function. Then the vector field XH on M determined by ωðXH, �Þ ¼ dH is called the Hamiltonian vector field associated with H. If M is closed, then XH generates a smooth 1-parameter group of diffeomorphisms φ<sup>t</sup> <sup>H</sup>∈DiffðMÞ such that

$$\frac{d}{dt}\phi\_H^t = X\_H \circ \phi\_H^t, \phi\_H^0 = \text{id.}\tag{3}$$

This {φ<sup>t</sup> <sup>H</sup>} is called the Hamiltonian flow associated with H. The flux homomorphism Flux is defined by

$$\text{Flux}\{\phi\_H^t\} = \int\_0 \omega(X\_t, \cdot) dt. \tag{4}$$

Theorem 2.2 ([6]). φ∈SymðMÞ is a Hamiltonian symplectomorphism if and only if there is a homotopy ½0, 1� ! SymðMÞ, t↦φ<sup>t</sup> such that φ<sup>0</sup> ¼ id, φ<sup>1</sup> ¼ φ, and Fluxðfφ<sup>t</sup> gÞ ¼ 0.

#### 2.1. Quantum cohomology

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,

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

By a symplectic manifold, we mean an even dimensional smooth manifold M<sup>2</sup><sup>n</sup> together with a global 2-form ω which is closed and nondegenerate, that is, the exterior derivative dω ¼ 0 and

Examples: (1) The 2n-dimensional Euclidean space <sup>R</sup><sup>2</sup><sup>n</sup> with coordinates <sup>ð</sup>x1, …, xn, <sup>y</sup>1, …, yn<sup>Þ</sup>

Then qi ¼ xi∘π,i ¼ 1, 2, …, n together with fiber coordinates p1, …, pn give local coordinates of

M is given by

<sup>ω</sup> <sup>¼</sup> ∑ n

i¼1 dqi ∧dqj

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

A symplectomorphism of ðM, ωÞ is a diffeomorphism φ∈Dif fðMÞ which preserves the

ω is nondegenerate, there is a bijection between the vector fields X∈ΓðTMÞ and 1-forms

Let M be closed, i.e., compact and without boundary. Let φ : R ! DiffðMÞ, t↦φ<sup>t</sup> be a smooth

ðMÞ. A vector field X∈ΓðTMÞ is called symplectic if ωðX, �Þ is closed.

family of diffeomorphisms generated by a family of vector fields Xt∈ΓðTMÞ via,

M ! M be the projection map and x1, …, xn are local coordinates of M.

ω ¼ ω. Denote by SymðMÞ the group of symplectomorphims of M. Since

M has a natural symplectic form

: (1)

and Floer type cohomology with coefficients in a Novikov ring [7, 9, 13].

closed fundamental 2-form.

90 Manifolds - Current Research Areas

2. Symplectic manifolds

admits symplectic form <sup>ω</sup><sup>0</sup> <sup>¼</sup> ∑

as follows. Let π : T�

form ω<sup>0</sup> on R<sup>2</sup><sup>n</sup>.

<sup>ω</sup>ðX, �Þ∈Ω<sup>1</sup>

symplectic form φ�

T�

the n-fold wedge product ω<sup>n</sup> never vanishes.

M. The natural symplectic form on T�

(3) Every Kähler manifold is symplectic.

n

i¼1

(2) Let M be a smooth manifold. Then its cotangent bundle T�

dxi∧dyi .

> Let ðM, ωÞ be a compact symplectic manifold. An almost complex structure is an automorphism of TM such that J <sup>2</sup> <sup>¼</sup> <sup>−</sup>I. The form <sup>ω</sup> is said to tame <sup>J</sup> if <sup>ω</sup>ðv, Jv<sup>Þ</sup> <sup>&</sup>gt; 0 for every <sup>v</sup>≠0. The set IτðM, ωÞ of almost complex structures tamed by ω is nonempty and contractible. Thus the Chern classes of TM are independent of the choice J∈IτðM, ωÞ. A smooth map φ : ðM1, J1Þ!ðM2, J2Þ from M<sup>1</sup> to M<sup>2</sup> is ðJ1, J2Þ-holomorphic if and only if

$$d\phi\_x \circ l\_1 = l\_2 \circ d\phi\_x \tag{5}$$

Hereafter, we denote by H2ðMÞ the image of Hurewicz homomorphism π2M ! H2ðM, ZÞ. A ði, JÞ-holomorphic map u : ðΣ, z1,…, zkÞ ! 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∈H2ðMÞ let Mg, <sup>k</sup>ðM, A; JÞ be the moduli space of ðj, JÞ-holomorphic stable maps which represent A.

Let <sup>B</sup> :<sup>¼</sup> <sup>C</sup><sup>∞</sup>ðΣ, <sup>M</sup>; <sup>A</sup><sup>Þ</sup> be the space of smooth maps

$$
u: \Sigma \to M\tag{6}$$

which represent A∈H2ðMÞ.

Let us consider infinite dimensional vector bundle E ! B whose fiber at u is the space Eu <sup>¼</sup> <sup>Ω</sup><sup>0</sup>,<sup>1</sup> ðΣ, u�TMÞ of smooth J-antilinear 1-forms on Σ with values in u�TM. The map ∂<sup>J</sup> : B ! E given by

$$\overline{\Theta}\_{\!\!\!\!/}(\mu) = \frac{1}{2} (d\mu + \mathsf{J} \circ d\mathsf{u} \bullet \mathsf{j}) \tag{7}$$

is a section of the bundle. The zero set of the section ∂<sup>I</sup> is the moduli space Mg, <sup>k</sup>ðM, A; JÞ.

For an element u∈Mg, <sup>k</sup>ðM, A; JÞ we denote by

$$D\_{\boldsymbol{u}}: \Omega^0(\Sigma, \boldsymbol{\mu}^\*TM) = T\_{\boldsymbol{u}}B \to \Omega^{0,1}(\Sigma, \boldsymbol{\mu}^\*TM) \tag{8}$$

the composition of the derivative

$$d(\overline{\partial}\_l)\_u: T\_u B \longrightarrow T\_{(u,0)} E \tag{9}$$

with the projection to fiber <sup>T</sup>ðu, <sup>0</sup>Þ<sup>E</sup> ! <sup>Ω</sup><sup>0</sup>,<sup>1</sup> ðΣ, u�TMÞ. Then the virtual dimension of Mg, <sup>k</sup>ðM, A; JÞ is

$$\begin{array}{c} \dim \mathcal{M}\_{\mathfrak{g},k}(M,A;I) = \text{index}D\_{\mathfrak{u}} : \Omega^0(\Sigma, \mathfrak{u}^\*TM) \to \Omega^{0,1}(\Sigma, \mathfrak{u}^\*TM) \\ = 2c\_1(TM)A + n(2 - 2\mathfrak{g}) + (6\mathfrak{g} - 6) + 2k. \end{array} \tag{10}$$

Theorem 2.1.1. For a generic almost complex structure J∈IτðM, ωÞ the moduli space Mg, <sup>k</sup>ðM, A; JÞ is a compact stratified manifold of virtual dimension,

$$\dim \mathcal{M}\_{\mathbb{S},k}(M, A; \mathbf{J}) = 2\mathbf{c}\_1(TM)A + n(\mathbf{2} - \mathbf{2g}) + (\mathbf{6g} - \mathbf{g}) + 2\mathbf{k}.\tag{11}$$

For some technical reasons, we assume that c1ðAÞ≥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

$$\text{ev}\,\mathcal{M}\_{\S,k}(M,A;I) \to M^k,\\ \text{ev}([\mu;z\_1,\ldots,z\_k]) = (\mu(z\_1),\ldots,\mu(z\_k)).\tag{12}$$

Then the image ImðevÞ is well defined, up to cobordism on J, as a dimMg, <sup>k</sup>ðM, A; JÞ : ≡mdimensional homology class in M<sup>k</sup> .

Definition. The Gromov-Witten invariant Φ<sup>M</sup>,<sup>A</sup> <sup>g</sup>, <sup>k</sup> is defined by

$$\Phi^{M,A}\_{\mathfrak{g},k} : H^m(M^k) \to Q, \Phi^{M,A}\_{\mathfrak{g},k}(a) = \int\_{\mathcal{M}\_{\mathfrak{g},k}(M,A;I)} \mathrm{ev}^\* \bullet \alpha \tag{13}$$

where <sup>α</sup> <sup>¼</sup> PDðaÞ∈H2nk−mðMk <sup>Þ</sup> and • is the intersection number of ev and <sup>α</sup> in Mk .

The minimal Chern number N of ðM, ωÞ is the integer N :¼ min }7Bλjc1ðAÞ ¼ λ≥0, A∈H2ðMÞ}. We define the quantum product <sup>a</sup> � <sup>b</sup> of <sup>a</sup>∈H<sup>k</sup> ðMÞ and b∈H ðMÞ as the formal sum

Symplectic Manifolds: Gromov-Witten Invariants on Symplectic and Almost Contact Metric Manifolds http://dx.doi.org/10.5772/65663 93

$$a \ast b = \sum\_{A \in H\_2(M)} (a \ast b)\_A q^{c\_1(A)/N} \tag{14}$$

where <sup>q</sup> is an auxiliary variable of degree 2<sup>N</sup> and <sup>ð</sup><sup>a</sup> � <sup>b</sup>ÞA∈H<sup>k</sup>þl−2c1ðA<sup>Þ</sup> ðMÞ is defined by

$$\int\_{\mathcal{C}} (a \ast b)\_A = \Phi\_{0,\lambda}^{M,A}(a \otimes b \otimes r) \tag{15}$$

for C∈Hkþl−2c1ð<sup>A</sup>ÞðMÞ, r ¼ PDðCÞ. Hereafter, we use the Gromov-Witten invariants of g ¼ 0 and k ¼ 3. Then the quantum product a � b is an element of

QH� :¼ H� ðMÞ⊗Q½q� (16)

where Q½q� is the ring of Laureut polynomials of the auxiliary variable q.

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

Let us consider infinite dimensional vector bundle E ! B whose fiber at u is the space

<sup>∂</sup>JðuÞ ¼ <sup>1</sup> 2

is a section of the bundle. The zero set of the section ∂<sup>I</sup> is the moduli space Mg, <sup>k</sup>ðM, A; JÞ.

TMÞ ¼ TuB ! <sup>Ω</sup><sup>0</sup>,<sup>1</sup>

Theorem 2.1.1. For a generic almost complex structure J∈IτðM, ωÞ the moduli space Mg, <sup>k</sup>ðM, A; JÞ is

For some technical reasons, we assume that c1ðAÞ≥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

Then the image ImðevÞ is well defined, up to cobordism on J, as a dimMg, <sup>k</sup>ðM, A; JÞ : ≡m-

The minimal Chern number N of ðM, ωÞ is the integer N :¼ min }7Bλjc1ðAÞ ¼ λ≥0, A∈H2ðMÞ}.

Þ ! <sup>Q</sup>, <sup>Φ</sup><sup>M</sup>,<sup>A</sup>

<sup>g</sup>, <sup>k</sup> is defined by

<sup>g</sup>, <sup>k</sup> <sup>ð</sup>aÞ ¼ ∫

<sup>Þ</sup> and • is the intersection number of ev and <sup>α</sup> in Mk

<sup>M</sup>g, <sup>k</sup> <sup>ð</sup><sup>M</sup>, <sup>A</sup>; <sup>J</sup><sup>Þ</sup>

ðΣ, u�

<sup>ð</sup>Σ, <sup>u</sup>�TMÞ ! <sup>Ω</sup><sup>0</sup>,<sup>1</sup>

dimMg, <sup>k</sup>ðM, A; JÞ ¼ 2c1ðTMÞA þ nð2−2gÞþð6g−gÞ þ 2k: (11)

<sup>¼</sup> <sup>2</sup>c1ðTMÞ<sup>A</sup> <sup>þ</sup> <sup>n</sup>ð2−2gÞþð6g−6Þ þ <sup>2</sup>k: (10)

dð∂JÞ<sup>u</sup> : TuB ! Tðu,0ÞE (9)

, evð½u; z1, …, zk�Þ ¼ ðuðz1Þ, …, uðzkÞÞ: (12)

ev�

ðMÞ and b∈H ðMÞ as the formal sum

•α (13)

.

ðΣ, u�TMÞ. Then the virtual dimension of

ðΣ, u�TMÞ

ðΣ, u�TMÞ of smooth J-antilinear 1-forms on Σ with values in u�TM. The map

ðdu þ J∘du∘jÞ (7)

TMÞ (8)

Eu <sup>¼</sup> <sup>Ω</sup><sup>0</sup>,<sup>1</sup>

∂<sup>J</sup> : B ! E given by

92 Manifolds - Current Research Areas

Mg, <sup>k</sup>ðM, A; JÞ is

the evaluation map by

dimensional homology class in M<sup>k</sup>

where <sup>α</sup> <sup>¼</sup> PDðaÞ∈H2nk−mðMk

Definition. The Gromov-Witten invariant Φ<sup>M</sup>,<sup>A</sup>

We define the quantum product <sup>a</sup> � <sup>b</sup> of <sup>a</sup>∈H<sup>k</sup>

Φ<sup>M</sup>,<sup>A</sup>

<sup>g</sup>, <sup>k</sup> : <sup>H</sup><sup>m</sup>ðMk

For an element u∈Mg, <sup>k</sup>ðM, A; JÞ we denote by

with the projection to fiber <sup>T</sup>ðu, <sup>0</sup>Þ<sup>E</sup> ! <sup>Ω</sup><sup>0</sup>,<sup>1</sup>

a compact stratified manifold of virtual dimension,

evM: <sup>g</sup>, <sup>k</sup>ðM, <sup>A</sup>; <sup>J</sup>Þ ! Mk

.

the composition of the derivative

Du : Ω<sup>0</sup>

ðΣ, u�

dimMg, <sup>k</sup>ðM, <sup>A</sup>; <sup>J</sup>Þ ¼ indexDu : <sup>Ω</sup><sup>0</sup>

$$\ast : \mathbb{Q}H^\*(M) \otimes \mathbb{Q}H^\*(M) \to \mathbb{Q}H^\*(M). \tag{17}$$

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

Theorem 2.1.2. Let ðM, ωÞ be a compact semipositive symplectic manifold. Then the quantum cohomology ðQH� ðMÞ, þ , �Þ is a ring.

Remark. For A ¼ 0∈H2ðMÞ, the all J-holomorphic maps in the class A are constant. Thus <sup>ð</sup><sup>a</sup> � <sup>b</sup>Þ<sup>0</sup> <sup>¼</sup> <sup>a</sup>∪ <sup>b</sup>. The constant term of <sup>a</sup> � <sup>b</sup> is the usual cup product <sup>a</sup>∪ <sup>b</sup>.

We defined the Novikov ring Λ<sup>ω</sup> by the set of functions λ : H2ðMÞ ! Q that satisfy the finiteness condition

$$\#\left\{A\in H\_2(M)|\lambda(A)\neq 0,\omega(A)<\mathcal{c}\right\}<\infty\tag{18}$$

for every c∈R. The grading is given by degðAÞ ¼ 2c1ðAÞ.

Examples ([5]). (1) Let p∈H<sup>2</sup> <sup>ð</sup>CP<sup>n</sup><sup>Þ</sup> and <sup>A</sup>∈H2ðCPn<sup>Þ</sup> be the standard generators. There is a unique complex line through two distinct points in <sup>C</sup>P<sup>n</sup> and so <sup>p</sup> � pn <sup>¼</sup> <sup>q</sup>. The quantum cohomology of CP<sup>n</sup> is

$$\mathbb{Q}H^\*(\mathbb{C}P^\eta; \mathbb{Q}[\eta]) = \frac{\mathbb{Q}[p, q]}{}. \tag{19}$$

(2) Let <sup>G</sup>ðk, <sup>n</sup><sup>Þ</sup> be the Grassmannian of complex <sup>k</sup>-planes in <sup>C</sup><sup>n</sup>. There are two natural complex vector bundles <sup>C</sup><sup>k</sup> ! <sup>E</sup> ! <sup>G</sup>ðk, <sup>n</sup><sup>Þ</sup> and <sup>C</sup><sup>n</sup>−<sup>k</sup> ! <sup>F</sup> ! <sup>G</sup>ðk, <sup>n</sup>Þ. Let xi <sup>¼</sup> ciðE� Þ and yi ¼ ciðF� Þ be Chern classes of the dual bundles E� and F� , respectively. Since E⊕F is trivial, ∑j <sup>i</sup>¼<sup>0</sup>xiyj<sup>−</sup><sup>i</sup> <sup>¼</sup> 0, <sup>j</sup> <sup>¼</sup> <sup>1</sup>, …, <sup>n</sup>. By computation xk � yn<sup>−</sup><sup>k</sup> ¼ ð−1<sup>Þ</sup> n−k q. The quantum cohomology of Gðk, nÞ is

$$QH^\*\left(G(k,n);\mathbb{Q}[q]\right) = \frac{\mathbb{Q}[x\_1,\ldots,x\_k,q]}{<\,y\_{n-k+1},\ldots,y\_{n-1},y\_n+(-1)^{n-k}q>}.\tag{20}$$

Let {e0, …,en} be an integral basis of H� <sup>ð</sup>M<sup>Þ</sup> such that <sup>e</sup><sup>0</sup> <sup>¼</sup> <sup>1</sup>∈H<sup>0</sup> ðMÞ and each ei has pure degree. We introduce n þ 1 formal variables t0, …, tn and the linear polynomial at in t0, …, tn with coefficients in H� ðMÞ by at ¼ t0e<sup>0</sup> þ ⋯ þ tnen. The Gromov-Witten potential of ðM, ωÞ is a formal power series in variables t0, …, tn with coefficients in the Novikov ring Λω

$$\begin{split} \boldsymbol{\Phi}^{\mathcal{M}}(t) &= \sum\_{k \ge 3} \sum\_{A} \frac{1}{k!} \boldsymbol{\Phi}^{\mathcal{M},A}\_{0,k}(a\_t, \dots, a\_t) q^{\frac{c\_1(A)}{N}} \\ &= \sum\_{k\_0 + \dots + k\_n \ge 3} \sum\_A \frac{\varepsilon(k\_0, \dots, k\_n)}{k\_0! \dots k\_n!} \boldsymbol{\Phi}^{\mathcal{M},A}\_{0,k}(\boldsymbol{\epsilon}^{k\_0}\_0 \otimes \dots \otimes \boldsymbol{\epsilon}^{k\_n}\_n) \cdot (t^0) k\_0 \dots (t^n)^{k\_n} q^{c\_1(A)/N} . \end{split} \tag{21}$$

Examples ([4]). (1) Φ<sup>C</sup>P<sup>1</sup> <sup>ð</sup>tÞ ¼ <sup>1</sup> 2 t0 <sup>2</sup>t<sup>1</sup> <sup>þ</sup> <sup>e</sup><sup>t</sup><sup>1</sup> <sup>−</sup>1−t1<sup>−</sup> <sup>t</sup><sup>1</sup> 2 2 � �:

$$(\text{2) } \Phi^{\mathbb{CP}^n}(t) = \frac{1}{6} \sum\_{i+j+k=n} t\_i t\_j t\_k + \sum\_{d>0k\_2} \sum\_{\dots k\_n} N\_d(k\_2 \dots k\_n) \cdot \frac{t\_2^{k\_2} \dots t\_n^{k\_n}}{k\_2! \dots k\_n!} e^{d t\_1} q^d,$$

where Ndðk2…knÞ ¼ <sup>Φ</sup><sup>C</sup>P<sup>n</sup>, , dA <sup>0</sup>, <sup>k</sup> <sup>ð</sup>p<sup>2</sup>…p<sup>2</sup>, …, <sup>p</sup><sup>n</sup>…pnÞ.

We define a nonsingular matrix <sup>ð</sup>gij<sup>Þ</sup> by gij <sup>¼</sup> ∫ M ei∪ ej and denote by <sup>ð</sup>gij<sup>Þ</sup> its inverse matrix.

Theorem 2.1.3 ([4, 5]). The Gromov-Witten potential <sup>Φ</sup><sup>M</sup> of <sup>ð</sup>M, <sup>ω</sup><sup>Þ</sup> satisfies the WDVV-equations:

$$\sum\_{\nu,\mu} \partial\_{l\_l} \partial\_{l\_j} \partial\_{l\_\nu} \Phi^M(t) g^{\nu \mu} \partial\_{l\_\mu} \partial\_{l\_l} \partial\_l \quad \Phi^M(t) = \varepsilon\_{\bar{\eta}k} \cdot \sum\_{\nu,\mu} \partial\_{l\_{\bar{l}}} \partial\_{l\_k} \partial\_{l\_{\bar{\nu}}} \Phi^M(t) g^{\nu \mu} \partial\_{l\_{\bar{\mu}}} \partial\_{l\_l} \partial\_l \quad \Phi^M(t), \tag{22}$$

where εijk ¼ ð−1Þ degðe1ÞðdegðejÞþdegðekÞÞ.

#### 2.2. Floer cohomology

Let a compact symplectic manifold ðM, ωÞ be semipositive. Let Htþ<sup>1</sup> : M ! R be a smooth 1 periodic family of Hamiltonian functions. The Hamiltonian vector field Xt is defined by ωðXt, �Þ ¼ dHt. The solutions of the Hamiltonian differential equation x\_ðtÞ ¼ XtðxðtÞÞ generate a family of Hamiltonian symplectomorphisms <sup>φ</sup><sup>t</sup> : <sup>M</sup> ! <sup>M</sup> satisfying <sup>d</sup> dt φ<sup>t</sup> ¼ Xt∘φ<sup>t</sup> and φ<sup>0</sup> ¼ id. For every contractible loop x : R=Z ! M, there is a smooth map u : D :¼ }7Bz∈Cjjzj≤1} ! M such that <sup>u</sup>ðe<sup>2</sup>πitÞ ¼ <sup>x</sup>ðtÞ. Two such maps <sup>u</sup><sup>1</sup> and <sup>u</sup><sup>2</sup> are called equivalent if their boundary sumðu1Þ#ð−u2Þ is homologus to zero in H2ðMÞ. Denote by ðx, ½u1�Þ ¼ ðx, ½u2�Þ for equivalent pairs, LM the space of contractible loops and LM~ the space of equivalence classes. Then LMg ! LM is a covering space whose covering transformation group is H2ðMÞ via, Aðx, ½u�Þ ¼ ðx, ½A#u�Þ for each A∈H2ðMÞ and ðx, ½u�Þ∈LMg.

Definition. The symplectic action functional aH is defined by

Symplectic Manifolds: Gromov-Witten Invariants on Symplectic and Almost Contact Metric Manifolds http://dx.doi.org/10.5772/65663 95

$$a\_H: \overleftarrow{L\!M} \to R, a\_H(\mathbf{x}, [\!u\!u\!]) = -\underset{D}{\!\!\! } \mu^\* \omega \cdot \underset{\!\!\! }{\text{\!\!\! }} H\_t\left(\mathbf{x}(t)\right) dt. \tag{23}$$

For each element ~x :¼ ðx, ½u�Þ∈LMg and ξ∈T~xLMg, we have

$$d a\_H(\mathbf{x}, [u])(\xi) = \int\_0 \omega \left( \dot{\mathbf{x}}(t) \mathbf{-} \mathbf{X}\_t \left( \mathbf{x}(t) \right), \xi \right) dt. \tag{24}$$

Thus the critical points of aH are one-to-one correspondence with the periodic solutions of x\_ðtÞ−Xt � xðtÞ � <sup>¼</sup> 0. Denote by PH<sup>~</sup> <sup>⊂</sup>LM<sup>~</sup> the critical points of aH and by PH⊂LM the set of periodic solutions.

The gradient flow lines of aH are the solutions <sup>u</sup> : <sup>R</sup><sup>2</sup> ! <sup>M</sup> of the partial differential equation ∂<sup>u</sup> þ JðuÞ � ∂tu−XtðuÞ � ¼ 0

with conditions uðs, t þ 1Þ ¼ uðs, tÞ,

$$\lim\_{s \to \pm \infty} \mu(s, t) = \mathfrak{x}^{\pm}(t) \tag{25}$$

for some x–∈PH.

QH� �

Let {e0, …,en} be an integral basis of H�

<sup>¼</sup> ∑ k0þ⋯þkn≥3

∑ A

<sup>ð</sup>tÞ ¼ <sup>1</sup> 2 t0

titjtk <sup>þ</sup> ∑ d>0 ∑ k2…kn

We define a nonsingular matrix <sup>ð</sup>gij<sup>Þ</sup> by gij <sup>¼</sup> ∫

degðe1ÞðdegðejÞþdegðekÞÞ.

with coefficients in H�

94 Manifolds - Current Research Areas

Examples ([4]). (1) Φ<sup>C</sup>P<sup>1</sup>

<sup>ð</sup>tÞ ¼ <sup>1</sup>

∑ <sup>υ</sup>, <sup>μ</sup> ∂ti ∂tj

2.2. Floer cohomology

where εijk ¼ ð−1Þ

LM

<sup>6</sup> ∑ iþjþk¼n

where Ndðk2…knÞ ¼ <sup>Φ</sup><sup>C</sup>P<sup>n</sup>, , dA

(2) Φ<sup>C</sup>P<sup>n</sup>

Gðk, nÞ; Q½q�

�

formal power series in variables t0, …, tn with coefficients in the Novikov ring Λω

k≥3 ∑ A 1 k! Φ<sup>M</sup>,<sup>A</sup>

<sup>2</sup>t<sup>1</sup> <sup>þ</sup> <sup>e</sup><sup>t</sup><sup>1</sup> <sup>−</sup>1−t1<sup>−</sup> <sup>t</sup><sup>1</sup>

Ndðk2…knÞ � <sup>t</sup>

<sup>0</sup>, <sup>k</sup> ðe k0 <sup>0</sup> ⊗⋯⊗e

� �

M

Theorem 2.1.3 ([4, 5]). The Gromov-Witten potential <sup>Φ</sup><sup>M</sup> of <sup>ð</sup>M, <sup>ω</sup><sup>Þ</sup> satisfies the WDVV-equations:

Let a compact symplectic manifold ðM, ωÞ be semipositive. Let Htþ<sup>1</sup> : M ! R be a smooth 1 periodic family of Hamiltonian functions. The Hamiltonian vector field Xt is defined by ωðXt, �Þ ¼ dHt. The solutions of the Hamiltonian differential equation x\_ðtÞ ¼ XtðxðtÞÞ generate

For every contractible loop x : R=Z ! M, there is a smooth map u : D :¼ }7Bz∈Cjjzj≤1} ! M such that <sup>u</sup>ðe<sup>2</sup>πitÞ ¼ <sup>x</sup>ðtÞ. Two such maps <sup>u</sup><sup>1</sup> and <sup>u</sup><sup>2</sup> are called equivalent if their boundary sumðu1Þ#ð−u2Þ is homologus to zero in H2ðMÞ. Denote by ðx, ½u1�Þ ¼ ðx, ½u2�Þ for equivalent pairs, LM the space of contractible loops and LM~ the space of equivalence classes. Then

g ! LM is a covering space whose covering transformation group is H2ðMÞ via,

g.

<sup>υ</sup>, <sup>μ</sup> ∂tj

2 2

k2 <sup>2</sup> …t kn n k2!…kn!

:

<sup>Φ</sup><sup>M</sup>ðtÞ ¼ ∑

<sup>0</sup>, <sup>k</sup> <sup>ð</sup>p<sup>2</sup>…p<sup>2</sup>, …, <sup>p</sup><sup>n</sup>…pnÞ.

<sup>∂</sup><sup>t</sup>υΦ<sup>M</sup>ðtÞgυμ∂<sup>t</sup><sup>μ</sup> <sup>∂</sup>tk∂<sup>t</sup> <sup>Φ</sup><sup>M</sup>ðtÞ ¼ <sup>ε</sup>ijk � ∑

a family of Hamiltonian symplectomorphisms <sup>φ</sup><sup>t</sup> : <sup>M</sup> ! <sup>M</sup> satisfying <sup>d</sup>

Aðx, ½u�Þ ¼ ðx, ½A#u�Þ for each A∈H2ðMÞ and ðx, ½u�Þ∈LM

Definition. The symplectic action functional aH is defined by

εðk0, …, knÞ <sup>k</sup>0!…kn! <sup>Φ</sup><sup>M</sup>,<sup>A</sup>

<sup>¼</sup> <sup>Q</sup>½x1, …, xk, <sup>q</sup>� <sup>&</sup>lt; yn<sup>−</sup>kþ<sup>1</sup>, …, yn<sup>−</sup>1, yn þ ð−1<sup>Þ</sup>

degree. We introduce n þ 1 formal variables t0, …, tn and the linear polynomial at in t0, …, tn

<sup>ð</sup>M<sup>Þ</sup> such that <sup>e</sup><sup>0</sup> <sup>¼</sup> <sup>1</sup>∈H<sup>0</sup>

<sup>0</sup>, <sup>k</sup> ðat, …, atÞq

kn <sup>n</sup> Þ�ðt 0 Þk0…ðt

e dt<sup>1</sup> q<sup>d</sup> ,

ðMÞ by at ¼ t0e<sup>0</sup> þ ⋯ þ tnen. The Gromov-Witten potential of ðM, ωÞ is a

c1ðAÞ N

n−k

<sup>n</sup>Þkn qc1ðAÞ=<sup>N</sup>:

ei∪ ej and denote by <sup>ð</sup>gij<sup>Þ</sup> its inverse matrix.

<sup>∂</sup>tk∂<sup>t</sup>υΦ<sup>M</sup>ðtÞgυμ∂<sup>t</sup><sup>μ</sup> <sup>∂</sup>ti

<sup>q</sup> <sup>&</sup>gt; : (20)

ðMÞ and each ei has pure

<sup>∂</sup><sup>t</sup> <sup>Φ</sup><sup>M</sup>ðtÞ, (22)

dt φ<sup>t</sup> ¼ Xt∘φ<sup>t</sup> and φ<sup>0</sup> ¼ id.

(21)

Let <sup>M</sup>ð~x<sup>−</sup> , <sup>~</sup>xþÞ be the space of such solutions <sup>u</sup> with <sup>~</sup>x<sup>þ</sup> <sup>¼</sup> <sup>~</sup>x<sup>−</sup> #u. This space is invariant under the shift uðs,tÞ↦uðs þ s0, tÞ for each s0∈R. For a generic Hamiltonian function, the space <sup>M</sup>ð~x<sup>−</sup> , ~xþÞ is a manifold of dimension

$$\dim \mathcal{M}(\tilde{\mathfrak{x}}^{-}, \tilde{\mathfrak{x}}^{+}) = \mu(\tilde{\mathfrak{x}}^{-}) \text{-} \mu(\tilde{\mathfrak{x}}^{+}) . \tag{26}$$

Here <sup>μ</sup> : PH<sup>~</sup> ! <sup>Z</sup> is a version of Maslov index defined by the path of symplectic matrices generated by the linearized Hamiltonian flow along xðtÞ.

Let μð~xÞ−μð~yÞ ¼ 1. Then Mð~x, ~yÞ is a one-dimensional manifold and the quotient by shift Mð~x, ~yÞ=R is finite. In this case, we denote by nð~x, ~yÞ ¼ # � <sup>M</sup>ð~x,~y<sup>Þ</sup> R � the number of connecting orbits from ~x to ~y counted with appropriate signs.

We define the Floer cochain group FC� <sup>ð</sup>M, <sup>H</sup><sup>Þ</sup> as the set of all functions <sup>ξ</sup> : PH<sup>~</sup> ! <sup>Q</sup> that satisfy the finiteness condition,

$$\#\{\tilde{\mathfrak{X}} \boxplus PH \, | \, \xi \left(\tilde{\mathfrak{x}}\right) \forall \mathbf{0}, a\_H(\tilde{\mathfrak{x}}) \leq c \; \big|\; \tag{27}$$

for every c∈R. The complex FC� ðM, HÞ is a Λω-module with action

$$(\lambda \* \xi)(\tilde{\mathfrak{x}}) := \sum\_{A} \lambda(A)\xi(A \mathfrak{X}). \tag{28}$$

The degree k part FC� ðM, HÞ consists of all ξ∈FC� ðM, HÞ that are nonzero only on elements ~x∈PHf with μð~xÞ ¼ k. Thus the dimension of FC� ðM, HÞ as a Λω-module is the number #ðPHÞ.

We define a coboundary operator δ : FC� <sup>ð</sup>M, <sup>H</sup>Þ ! FC<sup>k</sup>þ<sup>1</sup> ðM, HÞ by

$$\delta(\xi)(\tilde{\mathfrak{x}}) = \sum\_{\mu(\tilde{\mathfrak{y}})=k} \mathfrak{n}(\tilde{\mathfrak{x}}, \tilde{\mathfrak{y}}) \xi(\tilde{\mathfrak{y}}).\tag{29}$$

The coefficients of δðδðξÞð~xÞÞ are given by counting the numbers of pairs of connecting orbits from ~x to ~y where μð~xÞ−μð~yÞ ¼ 2 ¼ dimMð~x, ~yÞ. The quotient Mðx~, y~Þ=R is a one-dimensional oriented manifold that consists of pairs counted by δðδðξÞð~xÞÞ. Thus the numbers cancel out in pairs, so that δ δðξÞ ¼ 0.

Definition. The cochain complex ðFC� ðM, HÞ, δÞ induces its cohomology groups

$$FH^k(M,H) := \frac{\text{Ker}\delta : FC^k(M,H) \to FC^{k+1}(M,H)}{\text{Im}\delta : FC^{k-1}(M,H) \to FC^k(M,H)}\tag{30}$$

which are called the Floer cohomology groups of ðM, ω, H, JÞ.

Remark. By the usual cobordism argument, the Floer cohomology groups FH� ðM, HÞ are independent to the generic choices of H and J. Let f : M ! R be a Morse function such that the negative gradient flow of f with respect to the metric gð�, �Þ ¼ ωð�, J�Þ is Morse-Smale. Let H ¼ −εf : M ! R be the time-independent Hamiltonian. If ε is small, then the 1-periodic solutions of x\_ðtÞ−XH xðtÞ ¼ 0 are one-to-one correspondence with the critical points of f . Thus we have PH ¼ CritðfÞ and the Maslov type index can be normalized as

$$\mu(\mathbf{x}, [\mu]) = \text{ind}\_f(\mathbf{x}) \text{-} n \tag{31}$$

where u : D ! M is the constant map uðDÞ ¼ x.

We define a cochain complex MC� ðM; ΛωÞ as the graded Λω-module of all functions

$$\mathbb{X}: \text{Crit}(f)H\_2(M) \to \mathbb{Q} \tag{32}$$

that satisfy the finiteness condition

$$\#\{(\mathbf{x},A)|\xi(\mathbf{x},A)\neq 0,\omega(A)\geq \mathbf{c}\}<\infty\tag{33}$$

for every <sup>c</sup>∈R. The Λω-module structure is given by <sup>ð</sup><sup>λ</sup> � <sup>ξ</sup>Þðx, <sup>A</sup>Þ ¼ ∑ <sup>λ</sup>ðBÞξðx, <sup>A</sup> <sup>þ</sup> <sup>B</sup><sup>Þ</sup> and the grading degðx, AÞ ¼ indfðxÞ−2c1ðAÞ. The gradient flow lines u : R ! M of f are the solutions of u\_ðsÞ ¼ −∇fðuðsÞÞ. These solutions determine coboundary operator

$$\delta: \mathcal{M}\mathbb{C}^{k}(M; \Lambda\_{\omega}) \to \mathcal{M}\mathbb{C}^{k+1}(M; \Lambda\_{\omega}) \tag{34}$$

$$\delta(\xi)(\mathbf{x}, A) = \sum\_{y} \boldsymbol{\mu}\_{\mathcal{f}}(\mathbf{x}, y)\xi(y, A) \tag{35}$$

where nfðx, <sup>y</sup><sup>Þ</sup> is the number of connecting orbits <sup>u</sup> from <sup>x</sup> to <sup>y</sup> satisfying lim<sup>s</sup>!−<sup>∞</sup> uðsÞ ¼ x, lim<sup>s</sup>!þ<sup>∞</sup> uðsÞ ¼ y, counted with appropriate signs and indfðxÞ−indfðyÞ ¼ 1 .

Definition–Theorem 2.2.1. (1) The cochain complex ðMC� ðM; ΛωÞ, δÞ defines a cohomology group

$$\mathcal{M}H^\*(M;\Lambda\_\omega) := \frac{\text{Ker}\,\delta : \mathcal{M}\mathcal{C}^\*(M;\Lambda\_\omega) \to \mathcal{M}\mathcal{C}^{\*+1}(M;\Lambda\_\omega)}{\text{Im}\,\delta : \mathcal{M}\mathcal{C}^{\*-1}(M;\Lambda\_\omega) \to \mathcal{M}\mathcal{C}^\*(M;\Lambda\_\omega)}\tag{36}$$

which is called the Morse-Witten cohomology of M.

(2) MH� ðM; ΛωÞ is naturally isomorphic to the quantum cohomology QH� ðM; ΛωÞ.

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

$$\Phi: FH^\*(M, H) \to QH^\*(M; \Lambda\_\omega) \tag{37}$$

which is linear over the Novikov ring Λω.

We define a coboundary operator δ : FC�

pairs, so that δ

solutions of x\_ðtÞ−XH

lim<sup>s</sup>!þ<sup>∞</sup>

 δðξÞ ¼ 0.

96 Manifolds - Current Research Areas

Definition. The cochain complex ðFC�

 xðtÞ 

where u : D ! M is the constant map uðDÞ ¼ x.

We define a cochain complex MC�

that satisfy the finiteness condition

FH<sup>k</sup>

which are called the Floer cohomology groups of ðM, ω, H, JÞ.

<sup>ð</sup>M, <sup>H</sup>Þ ! FC<sup>k</sup>þ<sup>1</sup>

<sup>μ</sup>ð~yÞ¼<sup>k</sup>

The coefficients of δðδðξÞð~xÞÞ are given by counting the numbers of pairs of connecting orbits from ~x to ~y where μð~xÞ−μð~yÞ ¼ 2 ¼ dimMð~x, ~yÞ. The quotient Mðx~, y~Þ=R is a one-dimensional oriented manifold that consists of pairs counted by δðδðξÞð~xÞÞ. Thus the numbers cancel out in

<sup>δ</sup>ðξÞð~xÞ ¼ ∑

<sup>ð</sup>M, <sup>H</sup><sup>Þ</sup> :<sup>¼</sup> Ker<sup>δ</sup> : FCk

Imδ : FC<sup>k</sup>−<sup>1</sup>

independent to the generic choices of H and J. Let f : M ! R be a Morse function such that the negative gradient flow of f with respect to the metric gð�, �Þ ¼ ωð�, J�Þ is Morse-Smale. Let H ¼ −εf : M ! R be the time-independent Hamiltonian. If ε is small, then the 1-periodic

for every <sup>c</sup>∈R. The Λω-module structure is given by <sup>ð</sup><sup>λ</sup> � <sup>ξ</sup>Þðx, <sup>A</sup>Þ ¼ ∑ <sup>λ</sup>ðBÞξðx, <sup>A</sup> <sup>þ</sup> <sup>B</sup><sup>Þ</sup> and the grading degðx, AÞ ¼ indfðxÞ−2c1ðAÞ. The gradient flow lines u : R ! M of f are the solutions of

<sup>ð</sup>M; ΛωÞ ! MCkþ<sup>1</sup>

y

<sup>δ</sup>ðξÞðx, <sup>A</sup>Þ ¼ ∑

uðsÞ ¼ y, counted with appropriate signs and indfðxÞ−indfðyÞ ¼ 1 .

where nfðx, <sup>y</sup><sup>Þ</sup> is the number of connecting orbits <sup>u</sup> from <sup>x</sup> to <sup>y</sup> satisfying lim<sup>s</sup>!−<sup>∞</sup>

Remark. By the usual cobordism argument, the Floer cohomology groups FH�

Thus we have PH ¼ CritðfÞ and the Maslov type index can be normalized as

u\_ðsÞ ¼ −∇fðuðsÞÞ. These solutions determine coboundary operator δ : MC<sup>k</sup>

ðM, HÞ by

ðM, HÞ, δÞ induces its cohomology groups

¼ 0 are one-to-one correspondence with the critical points of f .

ðM; ΛωÞ as the graded Λω-module of all functions

μðx, ½u�Þ ¼ indfðxÞ−n (31)

ξ : CritðfÞH2ðMÞ ! Q (32)

#{ðx, AÞjξðx, AÞ≠0, ωðAÞ≥c} < ∞ (33)

ðM; ΛωÞ (34)

nfðx, yÞξðy, AÞ (35)

ðM, HÞ

<sup>ð</sup>M, <sup>H</sup><sup>Þ</sup> (30)

ðM, HÞ are

uðsÞ ¼ x,

<sup>ð</sup>M, <sup>H</sup>Þ ! FC<sup>k</sup>þ<sup>1</sup>

<sup>ð</sup>M, <sup>H</sup>Þ ! FC<sup>k</sup>

nð~x, ~yÞξð~yÞ: (29)

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

$$\mathbb{Z}FH^\*(M,H) \simeq QH^\*(M; \Lambda\_\omega) \simeq H^\*(M) \otimes \Lambda\_\omega,\tag{38}$$

The rank of FC� ðM, HÞ as a Λω-module must be at least equal to the dimension of H� ðMÞ. The rank is the number #ðPHÞ which is the number of the fixed points of φ.

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

$$\#(\text{Fix}(\phi)) \succeq \sum\_{j=0}^{2n} b\_j(M) \tag{39}$$

where bjðMÞ is the jth Betti number of M.
