2.1.1. Kodaira’s Projective Embedding Theorem [4]

A compact Kähler manifold ( M , h , ω , J ) is Hodge if and only if M is holomorphically embedded in a projective space ℙ N for some N. □

Let M be a Hodge n -fold. Then the Hodge decompositions are for 0 ≤ k ≤ 2 n , 0 ≤ p , q ≤ n ,

where H p , q ( M ) is the complex subspace of H k ( M , ℂ ) of type ( p , q ) classes. In particular, H n ( M , ℂ ) = H n , 0 ( M ) ⊕ H n − 1 , 1 ( M ) ⊕ ⋯ ⊕ H 0 , n ( M ) .

Let Hdg 2 p ( M ) : = H p , p ( M ) ∩ H 2 p ( M , ℚ ) be the space of Hodge classes of M of degree 2 p , and Hdg * ( M ) : = ⊕ p = 0 n Hdg 2 p ( M ) which is called the spaces of all Hodge classes of M. Let H 2 p ( M , ℚ ) alg be the space of algebraic subvarieties of codimension p in M with rational coefficients. Then we had the following 2.1.2 Theorem, for the proof see section 4.9, Theorem 1 in [1].

2.1.2. Theorem [1]

Let M be a Hodge manifold of dimension n . Then for each 0 ≤ p ≤ n , there is an isomorphism

and consequently

□

Let ( M , ω , J ) be a Hodge manifold of dimension n . A Lagrangian subvariety of M is a subvariety L of M which is a maximal isotropic subvariety, hence L is a subvariety on which the Hodge 2-form ω vanishes which is maximal with this property. Thus, if a subvariety L ⊂ M is Lagrangian, then dim M = 2 dim L and the restriction ω | T L = 0 .

Examples

1) Every real curve in a Riemann surface is Lagrangian.

2) Every smooth manifold is Lagrangian in its cotangent bundle with the canonical symplectic form.

3) The product of Lagrangian subspaces is Lagrangian in the product of the spaces containing the Lagrangian subspaces.

Let ( M , ω , J ) be a Hodge manifold of dimension n . The Hodge decomposition of the middle dimensional cohomology of M is

and the Poincaré duality PD = ∩ [ M ] : H n ( M , ℂ ) → ˜ H n ( M , ℂ ) is an isomorphism.

Notations

Define PD ( H n − p , p   ( M ) ) : = H p , n − p   ( M ) , Hlg n − p , p   ( M ) : =   H n − p , p   ( M ) ∩ H n   ( M , ℚ ) , Hlg p , n − p ( M ) : = H p , n − p ( M ) ∩ H n ( M , ℚ ) , and

which are called the Lagrangian cohomology and homology groups of Lagrangian classes of M, respectively.

A Calabi-Yau n -fold is a Hodge manifold ( M , ω ) of dimension n satisfying one of following conditions, in this paper we use a Hodge manifold instead of a compact Kähler manifold:

1) The canonical line bundle of M is trivial, K M ≃ O M .

2) M has a holomorphic n -form Ω that vanishes nowhere.

3) The structure group of the tangent bundle TM is S U ( n ) .

4) M has a Hodge metric with global holonomy contained in S U ( n ) .

Remark

Let we have ( M , ω ) be a Calabi-Yau n -fold. Then we have the Hodge class [ ω ] ∈ H 2 ( M , ℤ ) , c 1 ( T M ) = 0 , and Hodge numbers h 1 , 0 = ⋯ = h n − 1 , 0 = 0 , and nonzero holomorphic n -form class [ Ω ] ∈ H n , 0 ( M ) = ℂ ( [ Ω ] ) ≃ ℂ .

The holomorphic n -form Ω is locally decomposable

Ω 1 ∧ ω = Ω 2 ∧ ω = 0 , and the 2 n -form

In physics the space time should be a 10-dimensional space [3][5][6]. The space is locally a product S 4 × M 6 , where S is the usual space-time and M is a compact 6-dimensional Riemannian manifold which is too small to be seen at macroscopic scale but is crucial for quantization. For the supersymmetry and consistency requirements we impose a condition on the metric of M that the holonomy group of the metric is S U ( 3 ) .

Then the M has the following properties:

1) The metric on M is Ricci flat and b 1 ( M ) = 0 ,

2) There is a complex structure on M such that c 1 ( T M ) = 0 and the metric is Kähler,

3) M is a Calabi-Yau 3-fold,

4) M has the good properties, such as supersymmetry, conformality and quantization called the superconformal field theory (SCFT).

Let ( M , ω ) be a Calabi-Yau n -fold. Then on M the associated superconformal algebra contains two U ( 1 ) subalgebras, one comes from infinitesimal rotation of the two supersymmetries in the left and the other in the right moving sector.

3.3.1. Representations of U ( 1 ) × U ( 1 )

The subalgebra U ( 1 ) × U ( 1 ) has a generator ( P , Q ) . However, there is no canonical choice of generator for each U ( 1 ) . If we change the order of the supersymmetries, then the sign of the generator is changed. The generator ( P , Q ) of U ( 1 ) × U ( 1 ) can be regarded as operators on the various Hilbert spaces of states. Then the Hilbert space decomposes into eigenspaces under the action of ( P , Q ) .

The U ( 1 ) × U ( 1 ) representation on the Hilbert spaces are:

The most interesting eigenspaces are the ( 1 , 1 ) -eigenspace H 1 ( M , T M ) and the ( − 1 , 1 ) -eigenspace H 1 ( M , Ω 1 M ) . Because H 1 ( M , T M ) is a tangent space of the moduli space C M ( M ) of the complex structures of M, and H 1 ( M , Ω 1 M ) is the tangent space of the moduli space K M ( M ) of the complexified Kähler structures of M.

3.3.2. Mirror Pairs

Let the generator P change to − P and Q leave to Q . Then the ( p , q ) -eigenspace H q ( M , ∧ p T M ) is changed to the ( − p , q ) -eigenspace H q ( M , Ω p M ) . However, in general they are not isomorphic. If there is a Calabi-Yau variety ( M 0 , ω 0 ) such that H q ( M , Ω p M ) ≃ H q ( M 0 , ∧ p T M 0 ) and H q ( M , ∧ p T M ) ≃ H q ( M 0 , Ω p M 0 ) for all p , q , then the Hodge diamonds of ( M , ω ) and ( M 0 , ω 0 ) are symmetric and they are called a mirror pair.

Let ( M , ω ) and ( M 0 , ω 0 ) be a mirror pair of Calabi-Yau n -folds. Then for all p and q ,

Similarly, we have H n − p , p ( M ) ≃ H p , p ( M 0 ) and H n ( M , ℂ ) ≃ H mid ( M 0 , ℂ ) .

Thus we have:

Theorem

Let ( M , ω ) and ( M 0 , ω 0 ) be a mirror pair of Calabi-Yau n -folds. Then we have the following isomorphisms;

□

Let ( M , ω ) be a Hodge manifold of dimension n . A Hodge class [ α ] ∈ Hdg 2 p ( M ) can be locally written by

Let the Poincaré dual

have the image PD ( Hdg 2 p   ( M ) ) = H n − p , n − p   ( M ) ∩ H 2 n − 2 p   ( M , ℚ ) : = Hdg 2 n − 2 p   ( M ) , and A = PD ( [ α ] ) = [ α ] ∩ [ M ] . Then the A can be locally written by

where the α ′ I are determined by [ α ] ∩ [ M ] . By the gluing argument of local pieces, we have the Z J , which is the subvariety locally spanned by { z j 1 , ⋯ , z j n − p } and z i 1 , ⋯ , z i p constants. Thus as a homology A ∈ H 2 n − 2 p ( M , ℚ ) alg and as a cohomology [ A ] = [ α ] ∈ H 2 p ( M , ℚ ) alg is an algebraic cycle in M .

Theorem

Let M be a Hodge manifold of dimension n . If a Hodge class [ α ] ∈ Hdg 2 p ( M ) has a local form α = ∑ | I | = p α I d z I ∧ d z ¯ I , then we have

and as a cohomology class

□

Note. Theorem 4.1.1 is another version in Theorem 2.1.2.

Let ( M , ω , J ) be a Hodge manifold of dimension n . A Lagrangian class [ β ] ∈ Hlg n − p , p ( M ) = H n − p , p ( M ) ∩ H n ( M , ℚ ) can be locally written by a form:

Let PD : Hlg n − p , p ( M ) → Hlg p , n − p ( M ) be the restriction of Poincaré duality. For a Lagrangian class [ β ] ∈ Hlg n − p , p ( M ) , the image PD ( [ β ] ) = [ β ] ∩ [ M ] : = L β can be locally written by:

which is the space generated by the elements with real coefficients in the space generated by 〈 z ¯ i 1 , ⋯ , z ¯ i p , z j 1 , ⋯ , z j n − p 〉 with complex coefficients. Then the space

L β has a real n -dimension. Since ω = i 2 ∑ j d z j ∧ d z ¯ j and the tangent space T L β is generated by:

where j = i 1 , ⋯ , i p and k = j 1 , ⋯ , j n − p , the ω = 0 on T L β and T L β ⊕ J ( T L β ) = T M | L β . By the analytic continuation we have that the L β is a Lagrangian subspace in M.

The submanifold L β corresponding to a Lagrangian class [ β ] ∈ Hlg n − p , p   ( M ) is a ( n − p , p ) -type Lagrangian subvariety of M. Then we have:

Theorem

Let ( M , ω , J ) be a compact Kähler manifold of dimension n . Then for each Lagrangian class [ β ] ∈ Hlg n − p , p ( M ) , the image PD ( [ β ] ) : = L β ⊂ M is represented as a Lagrangian subspace of M and as a cohomology cycle [ L β ] = PD − 1 PD ( [ β ] ) = [ β ] ∈ H n ( M , ℚ ) lag , that is,

and clearly H n ( M , ℚ ) lag ⊂ Hlg n ( M ) . Thus, we have Hlg n ( M ) = H n ( M , ℚ ) lag .

□

If Calabi-Yau manifolds ( M , ω , J ) and ( M 0 , ω 0 , J 0 ) are a mirror pair, then they have isomorphic superconformal field theories, especially the Kähler moduli spce of M and the complex moduli space of M 0 are isomorphic and vice versa, also they have the isomorphism of 4.3.1. The isomorphisms are collectively called the mirror map of mirror pair.

Let ( M , ω , J ) and ( M 0 , ω 0 , J 0 ) be a mirror pair of Calabi-Yau n -folds.

4.3.1. Mirror Maps

The mirror maps, for 0 ≤ p ≤ n ,

and

are isomorphic. The mirror maps induce the following commutative diagram of isomorphisms: for 0 ≤ p ≤ n ,

(*)

where ∩ [ M ] and ∩ [ M 0 ] are the Poincaré duals.

4.3.2. Local Descriptions

Let { z 1 , ⋯ , z n } and { w 1 , ⋯ , w n } be local holomorphic coordinates of M and M 0 , respectively, and z j = x j + i y j , w j = s j + i t j for j = 1 , ⋯ , n . Let [ α ] ∈ Hdg 2 p ( M ) be a Hodge class with local form

Then the image m p ( [ α ] ) : = [ β ] has a local form:

MirrorMaps

The mirror map is defined by the composition:

where Ω 0 = d w 1 ∧ ⋯ ∧ d w n is a nowhere vanishing holomorphic n -form on M 0 and M 0 is Calabi-Yau, and J = { 1 , ⋯ , n } − I .

While the Poincaré dual A : = [ α ] ∩ [ M ] ∈ H 2 n − 2 p ( M ) is represented by a codimension p subvariety of M with local coordinates { z j 1 , ⋯ , z j n − p } where { j 1 , ⋯ , j n − p } = { 1 , ⋯ , n } − I .

As a cohomology class [ A ] = [ α ] ∈ H 2 p ( M , ℚ ) .

Lagrangians

Let m p : H 2 p ( M , ℚ ) alg → H n − p , p ( M 0 , ℚ ) lag be the mirror map and for [ α ] ∈ H 2 p ( M , ℚ ) alg , [ β ] : = m p ( [ α ] ) . Then [ β ] is locally represented by:

The Poincaré dual B : = [ β ] ∩ [ M 0 ] is a subvariety of M 0 which is locally represented by ( s i 1 − i t i 1 , ⋯ , s i p − i t i p , s j 1 + i t j 1 , ⋯ , s j n − p + i t j n − p ) with real coefficients.

The tangent space of B is the space spanned by:

Since the fundamental 2-form on M 0 is

clearly we have ω 0 ≡ 0 on B. Thus B is Lagrangian in M 0 and [ B ] ∈ H p , n − p ( M 0 ) lag .

Theorem

Let ( M , ω , J ) and ( M 0 , ω 0 , J 0 ) be a mirror pair of Calabi-Yau n -folds. Then the mirror maps induce the following commutative diagram of isomorphisms: for 0 ≤ p ≤ n ,

(**)

where ∩ [ M ] and ∩ [ M 0 ] are the Poincaré duals. Similarly, there is a one-to-one correspondence between the Lagrangian subvarieties of M and the algebraic subvarieties of M 0 . □

In theorem 4.3.2, H 2 p ( M , ℚ ) alg = Hdg 2 p ( M ) = H p , q ( M , ℚ ) , the m p comes from mirror symmetry, and we define the map m p by the composition

of isomorphisms.

4.4.1. Hodge and Algebraic Classes

On the Hodge manifolds the equality of Hodge classes and algebraic cycles comes from the Lefschetz decomposition, and the one of Lagrangian classes and Lagrangian varieties comes from the Hodge decomposition. The one-to-one correspondence comes from the mirror pair of Calabi-Yau n -folds.

4.4.2. Hodge and Algebraic Cycles

The diagram (*) in (4.3.1) is algebraic, and (**) of (4.3.2.3) is a geometric correspondence.

4.4.3. Homological Mirror Symmetry

In 1994, Kontsevich proposed a homological version of mirror symmetry: A pair ( M , ω ) and ( M 0 , ω 0 ) is homological mirror symmetric if and only if the derived category of coherent sheaves on ( M 0 , ω 0 ) is equivalent to the derived category of Lagrangian submanifolds of ( M , ω ) [10]. 4.3.2 Theorem implies the homological mirror symmetry.

Local descriptions of H i and L i are as follows: Let { z 1 , z 2 , z 3 } be local holomorphic coordinates on M and { w 1 , w 2 , w 3 } be local holomorphic coordinates on M 0 , w j = s j + i t j , j = 1 , 2 , 3 . Then the algebraic subvarieties in M are

where H 1 = H , H 2 = H ∪ H , H 3 = H ∪ H ∪ H in the cohomologies. These correspond to complex submanifolds of increasing codimension in M.

The Lagrangian subvarieties in M 0 are

where the generators are taken with real coefficients, so that each L j is a real three-dimensional submanifold of M 0 . Since ω 0 = i 2 ∑ j = 1 n d w j ∧ d w ¯ j , the restriction ω 0 | L j = 0 and L j is Lagrangian.