IMPERIAL COLLEGE LONDON

DEPARTMENT OF

Introduction to Mirror in Aspects of Topological

Shu Chen

September, 2020

Supervised by Professor Daniel Waldram

Submitted in partial fulfillment of the requirements for the degree of Master of Science of

Imperial College London

1 Acknowledgements

I would like to thank professor Danial Waldram for accepting my request to be my dissertation supervisor and giving me this topic. During this special time in Covid-19, Prof. Waldram patiently answered my question face-to-face online every time when I asked him questions. I would also like to thank my parents and my family for being 100% supportive of me.

2 Abstract

Under the compactification by the Calabi-Yau threefold, the shows there is duality called mirror symmetry, which implies there is an isomorphism between two string under the compactifications of two topologically different internal manifolds. By twisting the topological string theory in two methods, the twisted theories named 퐴-model and 퐵-model have an isomorphism to each other under the mirror symmetry.

3 Contents

1 Introduction ...... 6

2 Calabi-Yau Compactification ...... 8

2.1 Complex Manifolds ...... 9

2.2 Complex Differential Forms ...... 12

2.3 Kählar Manifolds ...... 15

2.4 of Kählar Manifolds ...... 20

2.5 휕̅-cohomology Groups and Hodge Numbers ...... 21

2.6 Calabi-Yau Manifolds ...... 24

2.7 Spaces of Calabi-Yau Manifolds ...... 26

3 Topological String Theory ...... 29

3.1 ...... 29

3.2 Supersymmetric Nonlinear ...... 31

3.3 R- ...... 33

3.4 Twisting Supersymmetric Sigma Model ...... 34

3.5 Cohomological Theory ...... 36

3.6 퐴-model ...... 40

3.7 퐵-model ...... 43

3.8 Coupling to ...... 46

4

4 Mirror Symmetry ...... 48

4.1 Brief Introduction to Mirror Symmetry ...... 48

4.2 Mirror Symmetry in Aspects of T-duality ...... 50

4.3 Yukawa Couplings in Mirror Symmetry ...... 52

5 Conclusion ...... 55

References ...... 56

5 1 Introduction

Nowadays, there are two pillars of modern physics, which are Einstein’s general theory of relativity and the with the . However these two theories cannot be unified together properly to become “the ”. There are several attempts of such great grand unification, and is one of them and it seems to be the most successful one [6][7]. The superstring theory is a theory in which the elementary bosonic are considered to be the modes of the 1-dimensional string moving in the , and the fundamental are the super partners of these under the . The superstring theory predicts the existence of gravity and is also able to fit into the quantum mechanical theory we have already obtained. Apart for these, there are also some particular predictions it made, one of the most famous prediction is that our should be in a spacetime with 10 or 11 (we will focus on the 10-dimensional case, and the

11-dimensional theory is called M-theory [15]). It shows that the 6 extra spatial dimensions should be conpactified in a very small scale, and the corresponding compact space is called a Calabi-Yau manifold. Calabi in 1957 first conjectured the existence of such kind of manifolds [22] and then it was proved by Yau in 1977 [1][21].

There is a symmetry relation among the Calabi-Yau manifolds, which is called the mirror symmetry [11]. It implies a duality between the string theories with two topologically different

Calabi-Yau manifolds. The mirror symmetry first came to people’s sight in 1989 with the work of

Greene and Plesser [2] and Candelas, Lyker and Schimmrigk [3]. Mirror symmetry also gives an isomorphism between two superstring theories, type-IIA and type-IIB, under different internal

6 space [12], and it can be understand in aspect of T-duality [4]. Apart from that the mirror symmetry would also play important role in the topological string theory. In topological string theory, we can twist the theory in two different ways to obtain two theories, 퐴-model and

퐵-model. In 1991, Witten found that the mirror symmetry can also make the duality between two twisted models [5], and it links the complex structure on 퐵-model to the Kählar structure on

퐴-model.

In this dissertation, we will discuss what the Calabi-Yau compactification is and some corresponding properties of such compact manifold. Then the topological string theory will be introduced, and also the two models obtained by twisting the topological theory and their related knowledge will be given. Finally, the mirror symmetry will be explained and we will look at this duality in a few different aspects.

7 2 Calabi-Yau Compactification

In the superstring theory, the conformal invariance of the SCFT coupled to a theory requires a 10-dimensional spacetime rather than 4-dimensional one in which we feel in the ordinary life, so a way of compactification on the spacetime manifold is needed. Therefore the

whole spacetime should be a manifold which can be expressed in the form of 푀1,3 × 푀6, where

푀1,3 is the 4-dimensional Minkowski spacetime manifold and 푀6 is the 6-dimensionl compact

manifold for the spatial compactification. There are some requirements on 푀6. Firstly, the compact manifold should be a solution of the Einstein’s field equation, which also means

the manifold should be Ricci-flat. On the other hand, 푀6 is also required to preserve some rather than break them all [18][19], so it implies that the manifold needs to be a

Kählar manifold. All the requirements that manifold 푀6 should be a compact Ricci-flat

6-dimensional Kählar manifold, that is to say, we need a Calabi-Yau three-fold for the extra spatial dimensions.

In this chapter, we will introduce the complex manifolds and the Kählar manifolds, which are the keys to give the definition of the Calabi-Yau manifolds, and the corresponding knowledge of the differential manifold for each case will be introduced as well (such as complex differential forms, cohomology group, homology, Hodge diamond and etc.)[8][13][29][30][31]. After all the necessary knowledge is given, the definition of the Calabi-Yau manifolds will be illustrated, and some of its properties (such as topological invariant, moduli spaces [9][10] and etc.) will also be introduced.

8 2.1 Complex Manifolds

Before we define a complex manifold, we need to first define a holomorphic map on ℂ푚. A

푚 complex function : ℂ → ℂ is holomorphic, if 푓 = 푓1 + 푖푓2 satisfies the Cauchy-Riemann relations for each 푧휇 = 푥휇 + 푖푦휇,

휕푓 휕푓 휕푓 휕푓 1 = 2 , 2 = − 1 (2.1.1) 휕푥휇 휕푦휇 휕푥휇 휕푦휇

Then an m-dimensional complex manifold M is defined as the following axioms:

(i) is a .

(ii) is provided with a family of pairs {(푈푖, 휑푖)}.

(iii) {푈푖} is a family of open sets which covers . The map 휑푖 is a homeomorphism

푚 from 푈푖 to an open subset 푈 of ℂ . [Hence, is even dimensional.]

−1 (iv) Given 푈푖 and 푈푗 such that 푈푖 ∩ 푈푗 ≠ ∅, the map 휓푗푖 = 휑푗 ∘ 휑푖 from 휑푖(푈푖 ∩ 푈푗)

to 휑푗(푈푖 ∩ 푈푗) is holomorphic.

The number 푚 is the complex of 푀, and is denoted as dimℂ 푀 = 푚, and its real dimension is 2푚. The Axioms ensure that calculus on the complex manifold can exist without the dependence of the any chosen coordinates, and the manifold is differentiable. Complex manifold can also preserve its orientation, so they are also orientable. We can locate a point 푝 on 푀 by using the coordinate 푧휇 = 휑(푝) = 푥휇 + 푖푦휇 in a chart (푈, 휑), where 1 ≤ 휇 ≤ 푚. Then the

tangent space 푇푝푀 of complex manifold 푀 is naturally spanned by 2푚 vectors

휕 휕 휕 휕 { , … , , , … , } (2.1.2) 휕푥1 휕푥푚 휕푦1 휕푦푚

∗ and its co-tangent space 푇푝 푀 is spanned by

9 {푑푥1, … , 푑푥푚, 푑푦1, … , 푑푦푚 }. (2.1.3)

Let us define another 2푚 basic vecoters:

휕 1 휕 휕 ≡ ( − 푖 ) (2.1.4.a) 휕푧휇 2 휕푥휇 휕푦휇

휕 1 휕 휕 ≡ ( + 푖 ), (2.1.4.b) 휕푧̅휇 2 휕푥휇 휕푦휇 and corresponding 2푚 one-forms are defined as:

푑푧휇 ≡ 푑푥휇 + 푖푑푦휇 (2.1.5.a)

푑푧̅휇 ≡ 푑푥휇 − 푖푑푦휇 (2.1.5.b)

These are called the holomorphic bases (ones without bar) and anti-holomorphic bases (ones with bar). By using these vector and co-vector bases, we can then define a so-called almost complex structure which will play an essential role in the following sections.

We define a real field of type (1,1) on point 푝 of a complex manifold 푀, 퐽푝 ∶ 푇푝푀 →

푇푝푀 such that

휕 휕 휕 휕 퐽 ( ) = , 퐽 ( ) = − . (2.1.6) 푝 휕푥휇 휕푦휇 푝 휕푦휇 휕푥휇

Note that

2 퐽푝 = −푖 푑푇푝푀 (2.1.7)

퐽푝 is the almost complex structure of 푀 at point 푝,and it corresponds to the multiplication of ±푖

[17]. This structure is also independent of the charts chosen, which can be proven by finding its action of the overlapping parts of any two charts in a complex manifold [8]. The almost complex

10 structure can be only defined globally on a complex manifold.

By acting the almost complex structure to the bases that we defined in (2.1.4), we can easily find

휇 휇 that the bases 휕/휕푧 and 휕/휕푧̅ are both the unit eigenvectors of 퐽푝 with eigenvalues +푖 and

– 푖 respectively.

휕 휕 휕 휕 퐽 ( ) = 푖 퐽 ( ) = −푖 (2.1.8) 푝 휕푧 휇 휕푧 휇 푝 휕푧̅ 휇 휕푧̅ 휇

Therefore we can diagonalise 퐽푝 as a in the (anti-)holomorphic bases:

휕 휕 푖퐼 0 휇 휇 푚 (2.1.9) 퐽푝 = 푖푑푧 ⊗ 휇 − 푖푑푧̅ ⊗ 휇 = ( ), 휕푧 휕푧̅ 0 −푖퐼푚

where 퐼푚 is the 푚 × 푚 identity matrix.

ℂ For a tangent space 푇푝푀 of a complex manifold 푀, we can decompose it into two disjoint tangent spaces

ℂ + − 푇푝푀 = 푇푝푀 ⊕ 푇푝푀 (2.1.10)

+ − where 푇푝푀 is the tangent space with holomorphic bases and 푇푝푀 is the one with

anti-holomorphic bases. The vectors in each tangent space are the eigenvectors of 퐽푝 with the eigenvalues ±푖 respectively, so we could give the definition:

± ℂ 푇푝푀 = {푍 ∈ 푇푝푀 |퐽푝푍 = ±푖푍}. (2.1.11)

ℂ + − Therefore a complex tangent vector 푍 ∈ 푇푝푀 can be uniquely decomposed as 푍 = 푍 + 푍 ,

± ± + − where 푍 ∈ 푇푝푀 , and 푍 is defined as a holomorphic vector and 푍 is defined as an

± anti-holomorphic vector. We need to note that 푇푝푀 are in the same dimensions, and their

11 ℂ dimensions are both half of dimℂ 푇푝푀 = dimℂ 푀.

Until here, we have defined an almost complex structure in a complex manifold. It is obvious that

every complex manifold has an almost complex structure, as 퐽푝 can be directly related to the multiplication by ±푖, but an almost complex manifold, which is defined as a differential manifold with an almost complex structure, is not always a complex manifold [8]. It needs some restrictions.

We then define a tensor field of rank (1, 2) on an almost complex manifold, 푁 ∶ 푇푝푀 × 푇푝푀 →

푇푝푀:

푁(푋, 푌) ≡ [푋, 푌] + 퐽[퐽푋, 푌] + 퐽[푋, 퐽푌] − [퐽푋, 퐽푌] (2.1.12) which is called Nijenhuis tensor, and [푋, 푌] is the Lie bracket of two vectors. If we have a vanishing Nijenhuis tensor on a manifold 푀, the almost complex structure 퐽 is integrable. Then there is another theorem, which can be proved with complicated mathematical concepts [14], showing that: (Newlander and Nirdnberg 1957) Let (푀, 퐽) be a 2푚-dimensinal almost complex manifold. If 퐽 is integrable, the manifold 푀 is a complex manifold with the almost complex structure 퐽. Hence we can say a vanishing Nijenhuis tensor an almost complex manifold to be a complex manifold.

2.2 Complex Differential Forms

Same as real manifolds, complex manifolds also have differential forms. If we have a complex

12 푞 ℂ manifold 푀 with dimℂ 푀 = 푚, a complex 푞-form 휔 ∈ Ω푝(푀) at point 푝 can be defined as

푞 ℝ 휔 = 휂 + 푖휁, where 휂 and 휁 are both real 푞-forms 휂, 휁 ∈ Ω푝(푀) at 푝, and its complex conjugate is 휔̅ = 휂 − 푖휁. Then we can easily find that the of the 푞-form 휔 is

푑휔 = 푑휂 + 푖푑휁. (2.2.1)

We can also define the complex 푞-form in the holomorphic and anti-holomorphic bases, which

푞 ℂ would be more helpful for the later description. Let the 푞-form 휔 ∈ Ω푝(푀) (푞 ≤ 2푚) and

ℂ 푟, 푠 be positive such that 푟 + 푠 = 푞. Then we take 푞 vectors 푉푖 ∈ 푇푝푀 (1 ≤ 푖 ≤ 푞 )

+ − + in either 푇푝푀 or 푇푝푀 spaces. If 휔(푉1, … , 푉푞) = 0 unless 푟 of 푉푖 are in 푇푝푀 and 푠 of

− 푟,푠 푉푖 are in 푇푝푀 , 휔 is said to be an (푟, 푠)-form or a form of bidegree (푟, 푠), 휔 ∈ Ω푝 (푀) =

푞 ℂ Ω푝(푀) . Then we can see that the bases of an (푟, 푠)-form are just 푟 holomorphic basic one-forms 푑푧 and 푠 anti-holomorphic basic one-forms 푑푧̅, which can be found in (2.1.5).

Therefore the (푟, 푠)-form 휔 can be written as:

1 휔 = 휔 푑푧휇1 ∧ … ∧ 푑푧휇푟 ∧ 푑푧̅휈1 ∧ … ∧ 푑푧̅휈푠. (2.2.2) 푟! 푠! 휇1…휇푟휈̅1…휈̅푠

All the components in the form are totally anti-symmetric in 휇 and 휈 respectively.

A complex 푞-form 휔 can be uniquely decomposed by disjoint (푟, 푠)-forms:

(푟,푠) 휔 = ∑ 휔 (2.2.3) 푟+푠=푞 where 휔(푟,푠) ∈ Ω푟,푠, and its set can also be decomposed as:

푞 ℂ 푟,푠 Ω (푀) = ⨁ Ω (푀). (2.2.4) 푟+푠=푞

Then the exterior derivative of an (푟, 푠)-form can be found by changing the coordinate system to

13 the (anti-)holomorphic one on (2.2.1):

1 휕 휕 푑휔 = ( 휔 푑푧휆 + 휔 푑푧̅휆) 푟! 푠! 휕푧휆 휇1…휇푟휈̅1…휈̅푠 휕푧̅휆 휇1…휇푟휈̅1…휈̅푠 (2.2.5) × (푑푧휇1 ∧ … ∧ 푑푧휇푟 ∧ 푑푧̅휈1 ∧ … ∧ 푑푧̅휈푠).

Here we can see that the new complex (푞 + 1)-form 푑휔 is actually a mixture of an (푟 +

1, 푠)-form and an (푟, 푠 + 1)-form. Then we can separate the exterior derivative operator into two parts:

푑 = 휕 + 휕̅ (2.2.6) where 휕 is the operator mapping Ω푟,푠(푀) into Ω푟+1,푠(푀), and 휕̅ is the one mapping Ω푟,푠(푀) into Ω푟,푠+1(푀), and both of operators are call the Dolbeault operators [8], and because all three operators in (2.2.6) are , we can easily find the following relation:

휕휕휔 = 휕̅휕̅휔 = (휕̅휕 + 휕휕̅)휔 = 0 (2.2.7)

The de Rham cohomology in a complex manifold is similar to the one for real manifolds, but with respect to each Dolbeault operator. The 휕̅-closed (푟, 푠)-form, i.e. 휔 ∈ Ω푟,푠(푀), 휕̅휔 = 0, is called

푟,푠 ̅ 푟,푠 the (푟, 푠)-cocycle denoted by 푍휕̅ (푀), and the 휕-exact (푟, 푠)-forms, i.e. 휔 ∈ Ω (푀), 휂 ∈

푟,푠−1 ̅ 푟,푠 Ω (푀), 휕휂 = 휔, is called the (푟, 푠)-coboundary denoted by 퐵휕̅ (푀). Then the corresponding cohomology group is

푟,푠 푟,푠 푟,푠 퐻휕̅ (푀) ≡ 푍휕̅ (푀)/퐵휕̅ (푀) (2.2.8) which is a complex vector space and is called the (푟, 푠)th 휕̅-cohomology group. The de Rham cohomology for 휕 is extended in the same way as the one for 휕̅ as shown above.

14 2.3 Kählar Manifolds

For a complex manifold 푀 we can extend the metric 푔 to act on the complex tangent spaces

ℂ ℂ 푔: 푇푝푀 × 푇푝푀 → ℂ. (2.3.1)

ℂ Taking two vectors 푍 = 푋 + 푖푌 and 푊 = 푈 + 푖푉 on the same tangent space 푇푝푀 at point 푝, we can have

푔푝(푍, 푊) = 푔푝(푋, 푈) − 푔푝(푌, 푉) + 푖[푔푝(푋, 푉) + 푔푝(푌, 푈)]. (2.3.2)

For the (anti-)holomorphic coordinates, the components of the metric are simply 푔휇휈(푝) =

휕 휕 휕 휕 푔 ( , ), 푔 (푝) = 푔 ( , ) and etc. . 푝 휕푧휇 휕푧휈 휇휈̅ 푝 휕푧휇 휕푧̅휈

Before we give the definition of the Kählar manifolds, we need first to define what a Hermitian manifold is. If the Riemannian metric 푔 of a complex manifold 푀 satisfies the following restriction:

푔푝(퐽푝푋, 퐽푝푌) = 푔푝(푋, 푌). (2.3.3)

ℂ where point 푝 ∈ 푀 , vectors 푋, 푌 ∈ 푇푝푀 and 퐽푝 is the almost complex structure of the manifold, we say the metric 푔 is a Hermitian metric, and the pair (푀, 푔) we called Hermitian manifold. It can be proven that the components of the metric with holomorphic indices will vanish

휕 휕 휕 휕 푔 = 푔 ( , ) = 푔 (퐽 , 퐽 ) = −푔 = 0, (2.3.4) 휇휈 휕푧휇 휕푧휈 푝 휕푧휇 푝 휕푧휈 휇휈

and it is same for 푔휇̅휈̅. Therefore the Hermitian metric only takes the form with mixed indices:

휇 휈 휇 휈 푔 = 푔휇휈̅푑푧 ⊗ 푑푧̅ + 푔휇̅휈푑푧̅ ⊗ 푑푧 (2.3.5)

15 Hence we can find the of a Hermitian manifold through its metric 푔. The connection on the manifold can be found by its definition, for example:

휕 휕 ∇ = Γ휆 (푧) (2.3.6) 휇 휕푧휈 휇휈 휕푧휆

휆 where ∇휇 is the covariant derivative and Γ 휇휈 is the connection. We can obtain the connection

휆̅ with pure anti-holomorphic indices Γ 휇̅휈̅, in the same way. As the manifold should have the

metric compatibility, we need relations that ∇휅푔휇휈̅ = ∇휅̅푔휇휈̅ = 0 to be true. Therefore we can find the connection can be written as

휆 휆휎̅ 휆̅ 휆̅휎 Γ 휇휈 = 푔 휕휇푔휎̅휈 Γ 휇̅휈̅ = 푔 휕휇̅ 푔̅휈휎, (2.3.7) and the connections with mixed indices are all vanished. These connections are called the

Hermitian connection. Because the only non-vanished connections are the two with pure indices, we can see that a holomorphic tangent space on a Hermitian manifold can only be parallel transported in holomorphic directions, and it is also true for the anti-holomorphic cases. The

Hermitian manifolds are not necessarily torsion free, similar to the connections, the only

휆 휆̅ non-vanishing torsion are also with pure indices, 푇 휇휈 and 푇 휇̅휈̅. Then for the of the manifold, we can use the definition the Riemannian curvature :

푅(푋, 푌)푍 = ∇푋∇푌푍 − ∇푌∇푋푍 − ∇[푋,푌]푍 (2.3.8)

휅 휅 휅̅ 휅̅ to find that the only non-vanishing components are 푅 휆휇̅휈, 푅 휆휇휈̅, 푅 휆̅휇̅휈, 푅 휆̅휇휈̅. Due to the symmetry of switching the last two indices of the tensor, the only two independent components of

휅 휅̅ the Riemannian curvature tensor are 푅 휆휇̅휈, and 푅 휆̅휇휈̅, and they are the complex conjugate of each other. Then we can find the tensors can be written as:

16 휅 휅 휉̅휅 푅 휆휇̅휈 = 휕휇̅ Γ 휈휆 = 휕휇̅ 푔 휕휈푔휆휉̅ (2.3.9.a)

휅̅ 휅̅ 휉휅̅ 푅 휆̅휇휈̅ = 휕휇Γ 휈̅휆̅ = 휕휇푔 휕휈̅푔휉휆̅. (2.3.9.b)

By contracting the first upper and the first lower indices of the Riemann tensor, we can define a anti-symmetric rank 2 tensor:

휅 휉̅휅 ℜ휇휈̅ ≡ 푅 휅휇휈̅ = −휕휈̅(푔 휕휇푔휅휉̅ ) = −휕휈̅휕휇 log 퐺 (2.3.10)

where 퐺 ≡ det(푔휇휈̅) = √푔. We define the Ricci form by

휇 휈 ̅ ℜ ≡ 푖ℜ휇휈̅푑푧 ∧ 푑푧̅ = 푖휕휕 log 퐺, (2.3.11)

1 and it is a real form. We can find ℜ is also a closed form, i.e. 푑ℜ ∝ 푑휕휕̅ log 퐺 = − 푑2(휕 − 2

휕̅) log 퐺 = 0, but it does not imply it is also an exact form. ℜ also defines a the a non-trivial

2 element called the first , i.e. 푐1(푀) ≡ [ℜ/2휋] ∈ 퐻 (푀; ℝ) [20]. It is also the reason

why we say a Ricci flat Kählar manifold has a vanishing first Chern class, i.e. ℜ = 0 ⇒ 푐1(푀) =

0.

Then we also need to give a definition of another concept, Kählar form. In a Hermitian manifold

(푀, 푔), we define a tensor field Ω at point 푝 ∈ 푀, such that

Ω푝(푋, 푌) = 푔푝(퐽푝푋, 푌) 푋, 푌 ∈ 푇푝푀. (2.3.12)

We can find that it is an anti-symmetric tensor field acting on two vectors to give a number, so it is a 2-form field, and we call it Kählar form. We can extend the domain of the Kählar form from

ℂ 푇푝푀 to 푇푃푀 , it shows that it would be a (1,1)-form:

17 휕 휕 휕 휕 휕 휕 Ω ( , ) = 푔 (퐽 , ) = 푖푔 = − Ω ( , ) (2.3.13) 휕푧휇 휕푧̅휈 푝 푝 휕푧휇 휕푧̅휈 휇휈̅ 휕푧̅휈 휕푧휇

where the cases with pure indices will also vanish just like the metric, i.e. 푔휇휈 = 푔휇̅̅휈 = 0. Then we can write Ω

휇 휈 휈 휇 휇 휈 Ω = 푖푔휇휈̅푑푧 ⊗ 푑푧̅ − 푖푔휈̅휇푑푧̅ ⊗ 푑푧 = 푖푔휇휈̅푑푧 ∧ 푑푧̅ , (2.3.14)

휇 or Ω = −퐽휇휈̅푑푧 ∧ 푑푧휈̅ (2.3.15)

휆̅ as 퐽휇휈̅ = 푔휇휆̅ 퐽 ̅휈 = −푖푔휇휈̅, and it is easy to show that Ω is a real form by finding that its complex conjugate is itself.

On a Hermitian manifold with complex dimension 푚 and the Kählar form Ω, we have a nowhere vanishing 2푚-form,

Ω⏟ ∧ … ∧ Ω = Ω푚 . 푚

To prove this, we can use the orthonormal basis:

Ω( 푒̂푖 , 퐽푒푗̂ ) = 훿푖푗 Ω(푒̂푖 , 푒푗̂ ) = Ω(퐽푒̂푖 , 퐽푒푗̂ ) = 0 (2.3.16)

Then it follows that

Ω⏟ ∧ … ∧ Ω (푒1̂ , 퐽푒1̂ , … , 푒̂푚, 퐽푒̂푚) 푚

= ∑ Ω(푒̂ , 퐽푒̂ ) … Ω(푒̂ , 퐽푒̂ ) 푃1 푃1 푃푚 푃푚 (2.3.17) 푝

= 푚! Ω(푒1̂ , 퐽푒1̂ ) … Ω(푒̂푚, 퐽푒̂푚) = 푚! where 푃 is the permutation of 푚 objects. It shows that Ω푚 is a nowhere vanishing real form on the Hermitian manifold and it serves as the volume form. Therefore it again shows that the

18 manifold we are talking about is an orientable manifold, and it fits the fact that the compact internal space should be orientable. Then by using this non-vanishing top form, we can decompose it into two separate parts: a holomorphic volume 푚-form and an anti-holomorphic volume

푚-form, which will be unique and non-vanishing on a Calabi-Yau manifold.

After introducing Hermitian manifold and Kählar form, we can give the definition of the Kählar manifold: Kählar manifold is the Hermitian manifold (푀, 푔) with a closed Kählar form, which means 푑Ω = 0, and the metric satisfying above condition is called Kählar metric. Then we extend the condition:

̅ 휇 휈 푑Ω = (휕 + 휕)푖푔휇휈̅ 푑푧 ∧ 푑푧̅

휆 휇 휈 휆̅ 휇 휈 = 푖휕휆푔휇휈̅푑푧 ∧ 푑푧 ∧ 푑푧̅ + 푖휕휆̅푔휇휈̅푑푧 ∧ 푑푧 ∧ 푑푧̅ (2.3.18) 1 = 푖(휕 푔 − 휕 푔 )푑푧휆 ∧ 푑푧휇 ∧ 푑푧̅휈 2 휆 휇휈̅ 휇 휆휈̅

1 휆 휇 휈 + 푖(휕̅푔 − 휕 푔̅ )푑푧̅ ∧ 푑푧 ∧ 푑푧̅ = 0 2 휆 휇휈̅ 휈̅ 휆휇 where we can find the relations:

휕휆푔휇휈̅ = 휕휇푔휆휈̅ 휕휆̅푔휇휈̅ = 휕휈̅푔휆̅휇. (2.3.19)

Therefore for a given chart (푈푖, 휑푖) on a Kählar manifold 푀, one can write the components of the metric 푔 in the following form:

푔휇휈̅ = 휕휇휕휈̅풦푖 (2.3.20)

where 풦푖 ∈ ℱ(푈푖) is called the Kählar potential of the metric. As the metric can be expressed differently in different chart, the Kählar potential can only be written in the form of (2.3.20)

19 locally rather than globally. The potential is also not unique, because we can change the potential by adding (anti-)holomorphic functions without changing its metric. Then on a compact Kählar manifold without boundary, the form Ω푚 (the wedge product over 푚 Kählar forms) is closed but not exact.

2.4 Holonomy Group of Kählar Manifolds

We can recall that the holonomy group of a Riemannian manifold 푀 of dimℝ 푀 = 푚 is a subgroup of 푂(푚). Then because the Kählar manifolds that we have studied in this topic is also orientable which means the holonomy group should keep the orientation of the parallel transported vectors, the holonomy group of Kählar manifolds with 푚 real dimensions should be a subgroup of 푆푂(푚). Furthermore it follows from the index structure of the connection of the Kählar manifolds that under parallel transport elements of 푇푀+ and 푇푀− do not mix, and the lengths of the vectors should be preserved under the parallel transports. Hence the holonomy group of the

Kählar manifolds should be a subgroup of 푈(푛) where 푛 is the complex dimension of the manifold. The elements of 푇푀+ and 푇푀− transform in 푈(푛) respectively. Then we may recall the definition of the holonomy group, which is the transformation of a vector on the manifold after a parallel transports around a loop and should also be a Lie group. Now we may take a parallel transport around an infinitesimal loop on the manifold, so we can find the infinitesimal transformation of the vector, which is also corresponding to the of the holonomy

group, 픲+ (푛) ⊕ 픲−(푛). Assuming there is a parallel transport of a holomorphic vector 푉 around an infinitesimal loop enclosed by the sides 훿푙 and 휖푘̅ , we can find the change on the vector can be expressed as

20 푖 푘̅ 푙 푖 푗 훿푉 = 휖 훿 푅 푗푘̅ 푙푉 . (2.4.1)

Due to the index structure of the Riemann tensor, only the effects from the loops with mixed

푘̅ 푙 푖 indices are non-vanishing in (2.4.1). The matrix 휖 훿 푅 푗푘̅ 푙 should be the element of the Lie

푘̅ 푙 algebra 픲(푛). If we take the trace of the matrix, it will become 휖 훿 픑푘̅ 푙 which is just proportional to the Ricci form. Then it can generates a 픲(1) part decomposed from the Lie algebra 픲(푛), so we can have the decomposition 픲(푛) ≃ 픲(1) ⊕ 픰픲(푛)[9], where 픰픲(푛) implies

that the holonomy group on a Ricci-flat Kählar manifold 푀 with dimℂ 푀 = 푛 is a subgroup of

푆푈(푛) [36].

2.5 흏̅-cohomology Groups and Hodge Numbers

Reminding that the (푟, 푠)th 휕̅ -cohomology group is defined by (2.2.8). Similarly to the

푟,푠 cohomology group of the real manifolds, 퐻휕̅ (푀) is a complex vector space. The element

푟,푠 ̅ [휔] ∈ 퐻휕̅ (푀) is an equivalence class of 휕-closed forms of bidegree (푟, 푠) differing the form

휔 by a 휕̅-exact form:

[휔] = {휂 ∈ Ω푟,푠(푀)|휕̅휂 = 0, 휔 − 휂 = 휕̅휓, 휓 ∈ Ω푟,푠−1(푀)} . (2.5.1)

One reason why we need to study the 휕̅-cohomology group is that the groups can measure some topological properties of the complex manifolds.

On a Hermitian manifold 푀 with dimℂ 푀 = 푚, a complex Hodge star ⋆ is a mapping:

Ω푟,푠(푀) → Ω푚−푟,푚−푠, and we define the inner product as

21 (훼, 훽) ≡ ∫ 훼 ∧ ⋆̅ 훽 (2.5.2) 푀 where 훼, 훽 ∈ Ω푟,푠(푀) and ⋆̅ is the complex conjugate of ⋆ satisfying that ⋆̅ 훽 ≡̅⋆̅̅훽̅ =⋆ 훽̅. The inner product helps us to define the adjoint Dolbeault operators. Like the adjoint exterior derivative 푑†, the adjoint Dolbeault operators are defined as 휕†: Ω푟,푠(푀) → Ω푟−1,푠(푀) and

휕̅†: Ω푟,푠(푀) → Ω푟,푠−1(푀) such that

(훼, 휕훽) = (휕†훼, 훽) (훼, 휕̅훽) = (휕̅†훼, 훽). (2.5.3)

We can see that 푑† = 휕† + 휕̅†, and both of the adjoint Dolbeault operators are also nilpotent, i.e.

† 2 ̅† 2 (휕 ) = (휕 ) = 0. Then we can define the corresponding Laplacians ∆휕 and ∆휕̅ on the

Hermitian manifolds

† 2 † † ∆휕≡ (휕 + 휕 ) = 휕휕 + 휕 휕 (2.5.4.a)

̅ ̅† 2 ̅ ̅† ̅† ̅ ∆휕̅ ≡ (휕 + 휕 ) = 휕휕 + 휕 휕. (2.5.4.b)

̅ An (푟, 푠)-form 휔 is called a 휕-harmonic form if ∆휕휔 = 0, and it is called the 휕-harmonic form

† if ∆휕̅ 휔 = 0. The 휕-harmonic form is also 휕-closed and 휕-co-closed, i.e. 휕휔 = 휕 휔 = 0, and it has the same relation for the 휕̅-harmonic forms. Furthermore, we can have the Hodge’s theorem in the complex version, and the (푟, 푠)-forms can be decomposed into three orthogonal terms:

푟,푠 ̅ 푟,푠−1 ̅† 푟,푠+1 푟,푠 Ω (푀) = 휕Ω (푀) ⊕ 휕 Ω (푀) ⊕ 퐻푎푟푚휕̅ (푀) (2.5.5)

푟,푠 ̅ where 퐻푎푟푚휕̅ (푀) is the set of 휕-harmonic (푟, 푠)-forms. We can express a (푟, 푠)-forms as

휔 = 휕̅훼 + 휕̅†훽 + 훾 (2.5.6)

푟,푠−1 푟,푠+1 푟,푠 where 훼 ∈ Ω (푀) , 훽 ∈ Ω (푀) and 훾 ∈ 퐻푎푟푚휕̅ (푀) . Then the (푟, 푠) -forms on a

22 Hermitian manifold can also decomposed in the holomorphic way.

In the Hermitian manifold, we define the Laplacians ∆휕 and ∆휕̅ separately, and they are also

different indeed. However, if the manifold is a Kählar manifold, the Laplacians ∆휕 and ∆휕̅ are the same [17].

∆= 2∆휕= 2∆휕̅ (2.5.7)

̅ 푟,푠 푟,푠 We define the complex dimension of 휕-cohomology group 퐻휕̅ (푀) as the Hodge number ℎ , and we can construct a Hodge diamond for the cohomology groups of all possible bidegrees. For a

Kählar manifold with 2 complex dimension, we can build the Hodge diamond as

ℎ2,2 ℎ2,1 ℎ1,2 ℎ2,0 ℎ1,1 ℎ0,2 (2.5.8) ℎ1,0 ℎ0,1 ℎ0,0

2 There are 9 Hodge numbers for this manifold. If we have dimℂ 푀 = 푚, there would be (푚 + 1) of them. However, the Hodge numbers are not all independent to each other, and we have two relations for the numbers if the manifold is Kählar:

ℎ푟,푠 = ℎ푠,푟 (2.5.9.a)

ℎ푟,푠 = ℎ푚−푟,푚−푠 (2.5.9.b)

The relation (2.5.9.a) is due to the fact that the relation in (2.5.7) make the Laplacians on the

Kählar manifold are the same as its complex conjugates. Therefore for any harmonic (푟, 푠)-form,

23 there exists a corresponding (푠, 푟)-form and vice versa, so the Hodge numbers are the same for opposite bidegrees. Then the relation (2.5.9.b) is due to the Poincare duality, which shows that the

푚−푟,푚−푠 푟,푠 cohomology group 퐻휕̅ (푀) is the dual vector space to 퐻휕̅ (푀), and hence they have the same dimensions [8].

The relations in (2.5.9) make the Hodge diamond of the Kählar manifolds symmetric in both vertical and horizontal directions. It allows us to use less independent numbers to parameterize the

Hodge diamond. For example, we can just use 4 Hodge numbers in (2.5.8) instead of 9 to parameterize the diamond. Furthermore, the Hodge numbers in the Kählar manifolds also have a close relation with the Betti numbers. This relation makes the Hodge diamond be able to describe some topological properties of the Kählar manifolds, and it is not true for just general Hermitian

manifold. The theorem describing this relation is: Let 푀 be a Kähler manifold with dimℂ 푀 =

푚 and 휕푀 = ∅. Then the Betti numbers 푏푝 (1 ≤ 푝 ≤ 2푚) satisfy

푝 푠,푟 푏 = ∑ ℎ (2.5.10.a) 푟+푠=푝

푏2푝−1 is even (1 ≤ 푝 ≤ 푚) (2.5.10.b)

푏2푝 ≥ 1 (1 ≤ 푝 ≤ 푚). (2.5.10.c)

Then we can find the Euler characteristic of the Kähler manifold is

푝 푝 푟+푠 푟,푠 휒 = ∑(−1) 푏 = ∑(−1) ℎ (2.5.11) 푝 푟,푠

24 2.6 Calabi-Yau Manifolds

A 푚 dimensional Calabi-Yau manifold is defined to be a compact, complex, Kähler manifold which has 푆푈(푚) holonomy, where 푚 is the complex dimension of the manifold. Also, the previous sections showed that the 푆푈(푚) holonomy implies the corresponding Kähler manifold should be Ricci-flat, and therefore it has a vanishing first Chern class. Chern classes are topological invariants of a manifold. Specifically, we can get the classes through Taylor expansion of the Chern form

푖ℛ 푐(푀) = 1 + ∑ 푐 (푀) = det (1 + ) 푗 2휋 푗 (2.6.1) 푖ℛ 푖ℛ 푖ℛ 푖ℛ 2 = 1 + tr + tr ( ∧ − 2 (tr ) ) + ⋯ 2휋 2휋 2휋 2휋

푘 푖 푗̅ where ℛ is the matrix valued curvature 2-form, i.e. ℛ = 푅 푙푖푗̅푑푧 ∧ 푑푧̅ . Then we can find the

first Chern class 푐1(푀) = tr 푖ℛ/ 2휋 is equal to [픑\2휋]. Furthermore, due to the Ricci-flat condition, there is a unique nowhere vanishing holomorphic (푚, 0)-form on the Calabi-Yau manifold which implies the Hodge number ℎ푚,0 = 1 . Also, by using this non-vanishing

(푚, 0)-form, we can make a isomorphism mapping (0, 푝)-forms to (0, 푚 − 푝)-forms, where

1 ≤ 푝 ≤ 푚. Then the 푆푈(푚) holonomy group does not allow a (0, 푝)-form to be harmonic, which means the corresponding coholonomy is trivial, ℎ0,푝 = 0. It makes the Hodge diamond of a Cababi-Yau three-fold even simpler, we can see:

25 1 0 0 0 ℎ1,1 0 1 ℎ2,1 ℎ1,2 1 (2.6.2) 0 ℎ2,2 0 0 0 1

where there is only two independent Hodge number in the diamond due to the vertical and horizontal symmetry relations, and we can just look at ℎ1,1 and ℎ2,1 . Then the Euler characteristic of the Calabi-Yau manifold is

휒 = 2(ℎ1,1 − ℎ2,1) (2.6.3)

2.7 Moduli Spaces of Calabi-Yau Manifolds

According to the Yau’s theorem, The Calabi-Yau manifold is a Ricci-flat Kähler manifold. Thus it

seems that we are able to perturb the metric 푔휇휈 of the manifold to a new metric 푔휇휈 + 훿푔휇휈 without changing the Ricci flat condition,

푅휇휈(푔) = 0 ⟹ 푅휇휈(푔 + 훿푔) = 0. (2.7.1)

Apparently we can easily achieve such a perturbation by just changing the coordinate system, but these are not the case we are interesting about. We can impose a coordinate condition to the system to fix the choice of coordinate, and we can write the condition as

휇 ∇ 훿푔휇휈 = 0, (2.7.2) which is pretty similar to fixing a gauge. Then we can re-express the perturbation (2.7.1) together

26 with the coordinate condition as:

휌 휌 휎 ∇ ∇휌훿푔휇휈 − 2푅휇 휈 훿푔휌휎 = 0 (2.7.3)

By using the (anti-)holomorphic coordinate system, we can see that the perturbation can be

applied by 훿푔푖푗 in pure indices and 훿푔푖푗̅ in mixed indices, because of the index structure of the metric and the Riemann tensor of the Kähler manifold. Then we can look at these two ways of perturbation:

(1) 훿푔푖푗̅: Under the metric perturbation with mixed indices, it remains the original index structure

of 푔 and keeps the metric still Hermitian. The condition in (2.7.3) with 훿푔푖푗̅ is equivalent to

(∆훿푔)푖푗̅ = 0, and we view 훿푔푖푗̅ as the components of a (1,1)-form. Such (1,1)-forms also

correspond to the changes of the Kähler form Ω and therefore the Kähler class [Ω] of the

manifold. Because the (1,1)-form is also harmonic, it is uniquely associated to an element of

1,1 1,1 퐻휕̅ (푀). Hence we can expand 훿푔푖푗̅ in the basis of real (1,1)-forms with dimension ℎ :

ℎ1,1 훼 훼 훼 훿푔푖푗̅ = ∑ 푡̃ 푏푖푗̅ 푡̃ ∈ ℝ (2.7.4) 훼=1

According to Yau’s theorem, there is a Ricci-flat Kähler metric for any [Ω + 훿Ω]. Hence, the

perturbation with mixed indices can make the new metric still be Kähler metric and also

deform the Kähler structure (which is parameterized by the Kähler class) at the same time.

Therefore, we say the perturbation expressed in (2.7.4) is the Kähler of the

Calabi-Yau manifold.

27 (2) 훿푔푖푗: Under the metric perturbation with pure indices, it will not keep the Hermitian structure

in the manifold and the Kähler metric is also not preserved. The condition (2.7.2) with 훿푔푖푗

is equivalent to

푖 ̅ ̅† ̅† ̅ 푖 ∆휕̅ 훿푔 = (휕휕 + 휕 휕)훿푔 (2.7.5)

푖 푖 푗̅ 푖푘̅ 푗̅ where 훿푔 = 훿푔 푗̅푑푧̅ = 푔 훿푔푘̅ 푗̅푑푧̅ . It is a harmonic (1,0)-form associated to an element

1,0 of cohomology group 퐻휕̅ (푀). As the transformation makes the metric not Kähler any more,

we need to change the coordinate system so that the metric becomes Kähler again. However,

we cannot use a holomorphic transformation to remove the pure-index perturbation, because

such transformation does not change type of the indices. In this step, it seems that this

coordinate transformation could violate the fixed coordinate condition (2.7.2) we have

mentioned before, but it is not (the new metric cannot be obtained from the old one by just a

diffeomophism). Then there is a unique non-vanishing holomorphic (3,0)-form Ω (here is

not the Kähler form) in the Calabi-Yau manifold, and we can use it to define an isomorphism

1,0 2,1 between 퐻휕̅ (푀) and 퐻 (푀) by defining the complex (2,1)-forms

푘 푖 푗 푙 Ω푖푗푘훿푔푙̅ 푑푧 ∧ 푑푧 ∧ 푑푧̅ (2.7.6)

which is still harmonic. Then we can expand the (2,1)-form in a basis with dimension ℎ2,1,

such that

ℎ2,1 푘 훼 훼 훼 (2.7.7) Ω푖푗푘훿푔푙̅ = ∑ 푡 푏푖푗푙̅ 푡 ∈ ℂ 훼=1

which we called the complex structure moduli space, and it tells the complex structure

deformation of the manifold.

28 3 Topological String Theory

After discussing the Calabi-Yau compactification of the superstring theory, a branch of the string theory will be introduced, and it is called the topological string theory. Rough speaking, it is a

“simplified” version of the string theory, as it only discusses how the topological properties affect the theory without caring the exact form of metric of the manifolds (for both worldsheet and the target space). As mentioned in the last chapter, mostly the Calabi-Yau manifold has both Kählar structure and complex structure. One reason that we introduce the topological string theory is that we can twist it in two different ways to obtained two models called 퐴-model and 퐵-model, and these models will depend on the Kählar structure and the complex structure of the target

Calabi-Yau space respectively. It may link two models to the two moduli spaces separately.

In this chapter, the supersymmetric 푁 = (1,1) non-linear sigma model and some corresponding symmetry will be introduced. Then two ways of twisting the theory, 퐴-twist and

퐵-twist, will be given. Some concepts of cohomological field theory will be introduced as well

[35], so that we could illustrate the two theories obtained by two way of twisting, 퐴-model and

퐵-model [23][24] [25][26][27]. At the end, how the theories couples to the gravity to become string theories will be briefly talked.

3.1 Superspaces

Supersymmetry makes the symmetry between bosons and fermions, and d its generators usually transform as 1/2 fermions under . For an 푁 = 1 supersymmetric theory, it has one such supersymmetry generator, which is called , for each bosonic dimension. An

29 푁 = 푝 > 1 supersymmetric theory has 푝 for each dimension, and the number 푝 here is the multiple of 2, such as 2, 4, 8, and so on (maximal number is 32). For the case

푁 = (2,2), it is the 푁 = 2 supersymmetry theory on the complex manifold, as the Lorentz group for each fundamental spin-1/2 supersymmetry generators can be split up into two components which transform with opposite charges (holomorphic and anti-holomorphic). Its makes the two fundamental supercharges become two irreducible positive charges and two irreducible negative charges. To describe each theory in two dimensions, we need to use superspaces, four fermionic coordinates 휃±, 휃̅± (in ) such that under Lorentz transformation 푆푂(2) =

푈(1) if original coordinate 푧 ↦ 푒푖훼푧 then

푖훼 푖훼 ± ± 휃± ↦ 푒 2 휃±, 휃̅± ↦ 푒 2 휃̅± (3.1.1)

Therefore we can have a superfield which we can expend to have both bosonic and fermionic parts rather than just bosonic part. By taking the Taylor expansion on superfeild Φ with respect to the fermionic variables, we have

± ± + − Φ(푧, 푧̅, 휃 , 휃̅ ) = 휙(푧, 푧̅) + 휓+(푧, 푧̅)휃 + 휓−(푧, 푧̅)휃 + ⋯. (3.1.2)

Then we can have an integral such that

푆 = ∫ 푑2푧푑4휃 풦(Φ푖, Φ̅ 푖) (3.1.3) where 풦 is the Kähler potential. The potential, as mentioned before, is only locally defined and non-unique. By integral the potential over the all the fermionic coordinate, we can get the the corresponding Lagrangean density.

30

Then in the 푁 = (2,2) theory, the supercharges of it are 푄± and 푄̅± are all nilpotent, i.e.

2 ̅ 푄± = 푄± = 0, and they have the anti-commutation relations that:

{푄±, 푄̅±} = 퐻 ± 푃, (3.1.4) where 퐻 and 푃 are the Hamiltonian and operator respectively. Then the rest of anti-commutations relations are all vanishing.

3.2 Supersymmetric Nonlinear Sigma Model

We first need to know what a Supersymmetric nonlinear sigma model with 푁 = (2,2) in two dimensions is defined as [5]. It maps a Riemann surface Σ (which is also the worldsheet of the string) to Riemannian manifold (which is the target space) 푀 with the metric 푔 such that

Φ: Σ → 푀. As the target space 푀 we discuss in this dissertation is a complex manifold, we need to use the coordinate 푧, 푧̅ on the worldsheet Σ and (anti-)holomorphic coordinate 휙푖 = 휙̅̅̅푖̅ on

푀 (but we also denote 휙퐼 as the real coordinates for convenience), where locally we can describe Φ in a function 휙푖(푧, 푧̅). Then Let 퐾 and 퐾̅ be the canonical and anti-canonical line bundles on the Riemann surface Σ (which are the bundle of one-forms of type (1,0) and (0,1) respectively), and 퐾1/2 and 퐾̅1/2 are the square roots of these bundles (the square roots here roughly means that the transformation acting on 퐾1/2 is also the square root of the same transformation on 퐾, and it also corresponds to the of fermions). Let 푇푀 be the complexified tangent bundle of 푀, and it can be decomposed as 푇푀 = 푇푀+ ⊕ 푇푀−. Then the

1/2 ∗ + fermi fields of the model are 휓+ and 휓−, where 휓+ can be projected in 퐾 ⊗ Φ (푇푀 ) and

1/2 ∗ − 푖 푖̅ 1/2 퐾 ⊗ Φ (푇푀 ) denoted as 휓+ and 휓+ respectively and 휓− can be projected in 퐾̅ ⊗

31 ∗ + 1/2 ∗ − 푖 푖̅ Φ (푇푀 ) and 퐾̅ ⊗ Φ (푇푀 ) denoted as 휓− and 휓− respectively. Then the action is written as

1 푆 = 2푡 ∫ 푑2푧 ( 푔 휕 휙퐼휕 휙퐽 + 푖휓푖̅ 퐷 휓푖 푔 + 푖휓푖̅ 퐷 휓푖 푔 2 퐼퐽 푧 푧̅ − 푧 − 푖푖̅ + 푧̅ + 푖푖̅ Σ (3.2.1) 푖 푖̅ 푗 푗̅ + 푅푖푖푗̅ 푗̅휓+휓+휓−휓−)

2 푘̅ where 푑푧 is −푖푑푧 ∧ 푑푧̅, 푡 is the , 푅푖푖푗̅ 푗̅ = 푔푖푘̅ 푅 푖푗̅ 푗̅ is the Riemann tensor of

̅ 1/2 ∗ the target space 푀 and 퐷푧̅ is the 휕 operator on 퐾 ⊗ Φ (푇푀) constructed by using the pullback of the Levi-Civita connection on 푇푀. It can be expressed as

휕 휕휙푗 퐷 휓푖 = 휓푖 + Γ푖 휓푘 (3.2.2) 푧 + 휕푧̅ + 휕푧̅ 푗푘 +

푖 where Γ 푗푘 is the affine connection of the target space 푀 in holomorphic indices. Then it is

̅1/2 ∗ 푖̅ similar for 퐷푧 (휕 operator on 퐾 ⊗ Φ (푇푀) with Levi-Civita connection Γ 푗̅푘 on 푀)

Then the supersymmetries of the model can be generated by infinitesimal transformations:

푖 푖 푖 훿휙 = 푖훼−휓+ + 푖훼+휓− (3.2.3.a)

푖̅ 푖̅ 푖̅ 훿휙 = 푖훼̃−휓+ + 푖훼̃+휓− (3.2.3.b)

푖 푖 푗 푖 푚 훿휓+ = −훼̃−휕푧휙 − 푖훼+휓−Γ 푗푚휓+ (3.2.3.c)

푖̅ 푖̅ 푗̅ 푖̅ 푚̅ 훿휓+ = −훼−휕푧휙 − 푖훼̃+휓−Γ 푗̅푚̅ 휓+ (3.2.3.d)

푖 푖 푗 푖 푚 훿휓− = −훼̃+휕푧̅휙 − 푖훼−휓+Γ 푗푚휓− (3.2.3.e)

푖̅ 푖̅ 푗̅ 푖̅ 푚̅ 훿휓− = −훼+휕푧̅휙 − 푖훼̃−휓+Γ 푗̅푚̅ 휓− (3.2.3.f)

where 훼± and 훼̃± are the infinitesimal fermionic parameters (훼− and 훼̃− are in sections of

1/2 1/2 퐾̅ , then 훼+ and 훼̃+ are in sections of 퐾 ).

32

3.3 R-Symmetries

In 2 dimensions, the Lorentz group is 푆푂(2) = 푈(1). Then as mentioned before, the group for 2 fundamental supercharges can be spilt up into two 푈(1) associating to 휃 and 휃̅ with plus

notations and minus notations, and we denote these two groups as 푈(1)푅 and 푈(1)퐿 (In section

3.1, it was said to be in positive and negative charge), and the corresponding symmetries under these group transformations are called the R-symmetries. Then we make two new

R-symmetries groups from the 푈(1)푅/퐿, and this modification is made by twisting the worldsheet

Lorentz group either by vector or axial symmetries:

푈(1)푉 = 푈(1)퐿 + 푈(1)푅, 푈(1)퐴 = 푈(1)퐿 − 푈(1)푅 (3.3.1)

The 퐴-model, we will discuss in later sections, can be obtained by twisting with 푈(1)푉 connection, and it can be considered on any Kähler manifold. Then the 퐵-model is obtained by

twisting with 푈(1)퐴 connection, and it is only well defined on a Calabi-Yau manifold. Then the

action of the vector symmetry 푈(1)푉 and axial symmetry 푈(1)퐴 on the fermionic coordinate can be expressed as

+ + −푖훼 + 푖훼 + − − −푖훼 − 푖훼 − 푅푉(훼): (휃 , 휃̅ ) ↦ (푒 휃 , 푒 휃̅ ), (휃 , 휃̅ ) ↦ (푒 휃 , 푒 휃̅ ) (3.3.2.a)

+ + −푖훽 + 푖훽 + − − 푖훽 − −푖훽 − 푅퐴(훽): (휃 , 휃̅ ) ↦ (푒 휃 , 푒 휃̅ ), (휃 , 휃̅ ) ↦ (푒 휃 , 푒 휃̅ ) (3.3.2.b)

Then the transformation can be applied to the superfield Φ:

푖훼퐹푉 ± ± 푖훼푞푉 −푖훼 ± 푖훼 ± 푅푉 = 푒 : Φ(푥, 휃 , 휃̅ ) ↦ 푒 Φ(푥, 푒 휃 , 푒 휃̅ ) (3.3.3.a)

푖훽퐹퐴 ± ± 푖훽푞퐴 ∓푖훽 ± ±푖훽 ± 푅퐴 = 푒 : Φ(푥, 휃 , 휃̅ ) ↦ 푒 Φ(푥, 푒 휃 , 푒 휃̅ ) (3.3.3.b)

33 where 퐹푉 and 퐹퐴 are the operators which generate the transformations, and 푞푉 and 푞퐴 are the corresponding Noether charges of the each R-symmetry. Let 푀 be the generator of the two-dimensional Lorentz transformation 푆푂(1,1), and we can then take the 푥0 direction to be

′0 1 −푖푥 and remain the 푥 same as before. Then the new generator 푀퐸 = 푖푀 can be obtained,

and it generates a compact Euclidean rotation group which is isomorphic to 푈(1)퐸. Then we can find the commutation relations between the supersymmetry algebras and the generators:

[푀퐸, 푄∓] = ∓푄±, [푀퐸, 푄̅±] = ∓푄̅±, (3.3.4.a)

[퐹푉, 푄±] = −푄±, [퐹푉, 푄̅±] = 푄̅±, (3.3.4.b)

[퐹퐴, 푄±] = ∓푄±, [퐹퐴, 푄̅±] = ±푄̅±, (3.3.4.c)

It shows that 푄± and 푄̅± have the opposite charges under the 푅-symmetries.

3.4 Twisting Supersymmetric Sigma Model

The twisting here is equivalent to changing the Euclidean rotation group 푈(1)퐸 by the generators of the 푈(1) R-symmetry groups, and it defines the new generators of the Euclidean rotation

′ group 푈(1)퐸′ as 푀퐸 = 푀퐸 + 푅. This twisting can make some supercharge operator scalar with respect to the new Euclidean rotation group [23].

Then the 푁 = (2.2) theory can be twisted in 2 different ways. The first case is that instead of

푖 푖̅ 1/2 ∗ + 1/2 ∗ − taking 휓+ and 휓+ to be the sections of 퐾 ⊗ Φ (푇푀 ) and 퐾 ⊗ Φ (푇푀 ) respectively, we can take them to be sections of Φ∗(푇푀+) and 퐾 ⊗ Φ∗(푇푀−) respectively, and

푖 푖̅ this kind of twist is called a + twits. Then the second case is that we twist 휓+ and 휓+ in the

푖 ∗ + 푖̅ way that 휓+ become the sections of 퐾 ⊗ Φ (푇푀 ) and 휓+ becomes the sections of

34 ∗ − 푖 푖̅ Φ (푇푀 ), and this is called the – twist. Similarly, we can also twist 휓− and 휓− in such ways.

푖 푖̅ ∗ + Under the + twist, 휓− and 휓− will be taken to be in the sections of Φ (푇푀 ) and 퐾̅ ⊗

Φ∗(푇푀−) respectively. Then under the – twist, they will be twisted to be the sections of

퐾̅ ⊗ Φ∗(푇푀+) and Φ∗(푇푀−) respectively. It can be found that the kinetic terms of the

푖̅ 푖 fermions of the Lagrangean, e.g. 휓−퐷푧휓−푔푖푖̅ , do not change under each twist, even though we

푖 푖̅ have twisted 휓± and 휓± from terms to the -like terms. Then we can have the

퐴-twist and 퐵-twist by combining ± twists in different ways. The 퐴-twist is the kind of twist in

which we do the − twist for 휓+ and do the + twist for 휓−, and it is denoted as (−, +). Then

the 퐵-twist is an orthogonal twist to the first one, and it is doing the + twist for both 휓+ and

휓−, which is denoted as (+, +). There is also a variant that is possible, and that is to twist only

휓+ or only 휓− and leave the other untwisted. These are called the half-twisting.

The 퐴- and the 퐵-twists, as mentioned before, change the original Euclidean rotation group

푈(1)퐸 into two new groups such that

퐴-twist: 푀퐸′ = 푀퐸 + 퐹푉 (3.4.1.a)

퐵-twist: 푀퐸′ = 푀퐸 + 퐹퐴 (3.4.1.b)

We denote the generator of the Euclidean group modified by the 퐴-twist as 푀퐴 and denote the

one modified by the 퐵-twist as 푀퐵. Then we can find the new commutation relations that

35 [푀퐴, 푄+] = −2푄+ [푀퐵, 푄+] = −2푄+ (3.4.2.a)

[푀퐴, 푄−] = 0 [푀퐵, 푄−] = 2푄+ (3.4.2.b)

[푀퐴, 푄̅+] = 0 [푀퐵, 푄̅+] = 0 (3.4.2.c)

[푀퐴, 푄̅−] = 2푄+ [푀퐵, 푄̅−] = 0 (3.4.2.d)

In the relations above, one can find that some supersymmetry operators become scalar, which is what we expected before, and the others becomes like spin-1 spinors. Then we can define

푄퐴 = 푄− + 푄̅+ (3.4.3.a)

푄퐵 = 푄̅− + 푄̅+ (3.4.3.b)

where 푄퐴 and 푄퐵 are scalar, nilpotent operators which can be used to define two different cohomological theories, which are called the 퐴-model and the 퐵-model respectively ( Two

models can also be defined by (+, −) and (−, −) twists, but it will leave 푄̅퐴 and 푄̅퐵 scalar and nilpotent. ). Furthermore, we call an operator 휙 a chiral operator or (푐, 푐) operator if 휙 is

푄퐵-closed:

[푄퐵, 휙] = 0 (3.4.4) and similarly, we call the operator 휙 a twisted chiral operator or (푎, 푐) operator if

[푄퐴, 휙] = 0 (3.4.5)

3.5 Cohomological Field Theory

As mentioned in section 3.4, both of the 퐴-model and 퐵-model are Topological cohomological field theory, so the concepts of cohomological field theory is needed to be introduced before we

36 go to the 퐴- and 퐵- models parts. cohomological field theories are the field theories that possess a very special type of symmetry. There are serval requirements needed to define a cohomological theory. Firstly, the fermionic symmetry operator 푄 should squares to zero, i.e. 푄2 = 0, which is exactly satisfied by theory we are interested in. Then the second property is that the physical

operators 푂푖 in the theory should be closed under the action of the operator 푄.

{푄, 푂푖} = 0 (3.5.1)

This relation is also called that the operator is 푄-closed. Both the first and the second properties are corresponding to the BRST symmetry and . Then third requirement is that theory needs the 푄-symmtry to be not spontaneously broken, which means the vacuum is symmetric.

Then there will be an equivalence relation that:

푂푖~푂푖 + {푄, Λ} (3.5.2) where Λ is an arbitrary operator function. As the vacuum is symmetric, the vacuum should satisfy that

푄|0⟩ = 0. (3.5.3)

Then by using this relation we can find that the expectation value of the 푄-exact operator {푄, Λ} should be zero:

⟨ | { } | ⟩ 0 푂푖1 … 푂푖푗 푄, Λ 푂푖푗+1 … 푂푖푛 0 (3.5.4) ⟨ | | ⟩ = 0 푂푖1 … 푂푖푗 (푄Λ − Λ푄)푂푖푗+1 … 푂푖푛 0 and each term will vanishes separately

37 ⟨ | ( ) | ⟩ 0 푂푖1 … 푂푖푗 푄Λ 푂푖푗+1 … 푂푖푛 0

⟨ | | ⟩ = ± 0 푂푖1 … 푄푂푖푗Λ푂푖푗+1 … 푂푖푛 0 (3.5.5)

⟨ | | ⟩ = ± 0 푄푂푖1 … 푂푖푗 Λ푂푖푗+1 … 푂푖푛 0 = 0.

In this property, we can find that the physical operators are 푄-cohomology class through the

equivalence relation. Then the final requirement is that the energy-momentum tensor 푇휇휈 of the theory should not only be 푄-closed but also 푄-exact:

훿푆 푇 ≡ = {푄, 퐺 } (3.5.6) 휇휈 훿ℎ휇휈 휇휈

휇휈 where 퐺휇휈 is some operator, and ℎ is the metric. A direct consequence of this last property is that the correlation functions do not depend on the metric, and the proof is that

훿 훿 ⟨푂 … 푂 ⟩ = (∫ 퐷휙푂 … 푂 푒푖푆[휙]) 훿ℎ휇휈 1 푛 훿ℎ휇휈 1 푛 훿푆[휙] = 푖 ∫ 퐷휙푂 … 푂 푒푖푆[휙] (3.5.7) 1 푛 훿ℎ휇휈

= 푖⟨푂1 … 푂푛{푄, 퐺휇휈}⟩ = 0 where 퐷휙 is the measure of the path integral, and the 푄 symmetry acting on the last line shows the metric independence. Then there is a practical way to ensure (3.5.6) which is to use a 푄-exact

Lagrangean:

퐿 = {푄, 푉} (3.5.8) where 푉 is an operator. We can find this kind of Lagrangean make the calculation easier as it makes correlation function independent of the ’ constant. It can be proven by put the

Planck’s constant back to the quantum measure:

38 푖 exp {푄, ∫ 푉} . (3.5.9) ℏ 푀 and consequently the same method in (3.5.7) can be used to show that

푑 ⟨푂 … 푂 ⟩ = 0. (3.5.10) 푑ℏ 1 푛

Therefore one can find that the independence of ℏ can make the calculation exactly in a classical limit.

Recall the equation (3.5.6), we can find the momentum operator by integrating the energy-momentum tensor field over a spatial hypersurface.

푃휇 = {푄, 퐺휇} (3.5.11)

where 퐺휇 is a fermionic operator. Then we may consider the operator

(1) (0) 푂휇 = 푖{퐺휇, 푂 } (3.5.12) where 푂(0) is a scalar physical 푄-closed operator, i.e.{푄, 푂(0)} = 0. By doing some calculations

We can find that

푑 푂(0) = 푖[푃 , 푂(0)] = {푄, 푂(1)} (3.5.13) 푑푥휇 휇 휇 where we can write momentum operator in the form in (3.5.11) and then use the Jacobi identity to

(1) obtain the result of (3.5.13). Then we can define the operator 푂휇 to be a component of a

(1) (1) 휇 one-form operator 푂 = 푂휇 푑푥 , so that we can rewrite (3.5.13) as

푑푂(0) = {푄, 푂(1)}. (3.5.14)

Then we can find that the integration of this equation over a closed curve 훾 ⊂ 푀 is zero by using

39 the Stoke’s theorem. We can repeat the above method again and again so that we can obtain a whole tower of 푝-form operators:

{푄, 푂(0)} = 0

{푄, 푂(1)} = 푑푂(0)

{푄, 푂(2)} = 푑푂(1)

….

{푄, 푂(푛)} = 푑푂(푛−1)

0 = 푑푂(푛) (3.5.15)

The integrals of 푂(푝), where 1 ≤ 푝 ≤ 푛, over a 푝-dimensional submanifold of 푀 give us large classes of non-local physical operators. As the “top-form” operator needs to be integrated over the whole manifold, by using the Stoke’s theorem again, we will have

(푛) . {푄, ∫푀 푂 } = 0 (3.5.14)

휇 (푛) 휇 which implies that we are free to add terms 푡 푂휇 for any coupling constant 푡 to the

Lagrangean without breaking the fact that the theory is cohomological.

3.6 푨-model

푖 푖̅ ∗ + ∗ − In the 퐴-model, we regard 휓− and 휓+ as the sections of Φ (푇푀 ) and Φ (푇푀 ) and regard

푖 푖̅ ∗ + ∗ − 푖 the 휓+ and 휓− as the section of 퐾 ⊗ Φ (푇푀 ) and 퐾̅ ⊗ Φ (푇푀 ), and the scalar fields 휙 and 휙푖̅ are still the same as the untwisted theory. To classify the twisted fermionic fields, we express the field in a new way:

40 푖 푖 ∗ + 휓+ ≡ 휓푧 ∈ 퐾 ⊗ Φ (푇푀 ) (3.6.1.a)

푖 푖 ∗ + 휓− ≡ 휒 ∈ Φ (푇푀 ) (3.6.1.b)

푖̅ 푖̅ ∗ − 휓+ ≡ 휒 ∈ Φ (푇푀 ) (3.6.1.c)

푖̅ 푖̅ ∗ − 휓− ≡ 휓푧̅ ∈ 퐾̅ ⊗ Φ (푇푀 ) (3.6.1.d)

In term of these renamed variables the action is

1 푆 = 2푡 ∫ 푑2푧 ( 푔 휕 휙퐼휕 휙퐽 + 푖휓푖̅ 퐷 휒푖푔 + 푖휓푖 퐷 휒푖̅푔 2 퐼퐽 푧 푧̅ 푧̅ 푧 푖푖̅ 푧 푧̅ 푖푖̅ Σ (3.6.2) 푖 푖̅ 푗 푗 푗̅ + 푅푖푖푗̅ 푗̅휓푧휒 휓−휒 휓푧̅ )

Then we act the fermionic symmetry 훿 = 휖−̅ 푄̅+ + 휖+푄− on the fields so that we can have the following relations

푖 푖 푖̅ 푖̅ 훿휙 = 휖+휒 훿휙 = 휖−̅ 휒 (3.6.3.a)

푖 푖 푖 푗 푘 푖 훿휓푧 = 2푖휖−̅ 휕푧휙 + 휖+Γ 푗푘휓푧 휒 훿휒 = 0 (3.6.3.b)

푖̅ 푖̅ 푖̅ 푗̅ 푘̅ 푖̅ 훿휓푧̅ = −2푖휖+̅ 휕푧̅휙 + 휖−̅ Γ 푗̅푘휓푧̅ 휒 훿휒 = 0 (3.6.3.c)

As mentioned before, 퐴-model is the topological cohomological theory, we can write the

Lagrangean in the form of {푄퐴, 푉} and we can also find that the theory is independent of 푡.

However, the action of theory is actually rewritten as

′ 2 ∗ 푆 = 푖푡 ∫푑 푧{푄퐴, 푉} + 푡 ∫Φ (Ω) (3.6.4) Σ Σ where

푖 푖̅ 푖 푖̅ 푉 = 푔푖푖̅(휓푧휕푧̅휙 + 휕푧휙 휓푧̅) (3.6.5)

41 푖 푗 and Ω is the Kähler form of the target space, i.e. Ω = −푖푔푖푗̅푑푧 ∧ 푑푧̅ .

∗ 푖 푖̅ 푖 푖̅ 푡 ∫Φ (Ω) = 2푡푔푖푖̅(휕푧휙 휕푧̅휙 − 휕푧̅휙 휕푧휙 ) (3.6.6) Σ

The Lagrangean can be partly written in the 푄-exact form, and the other term is related to the

Kähler structure of the target manifold. On the second term of the action, we can make the pullback Φ∗ on the Kähler form Ω back act on the worldsheet, and then it can be found that the it becomes important that Ω is a closed form so that the integral will only depend on the

cohomology class of Φ(Σ). We denote the cohomology class as 훽 ∈ 퐻2(푀), and the integral of the Kähler form over it is written as Ω ∙ 훽. Then it shows that

∫Φ∗(Ω) = ∫ Ω = Ω ∙ 훽 ≥ 0. (3.6.7) Σ Φ(Σ)

If an anti-symmetric tensor field such as the Kalb–Ramond field, i.e. the B-field, is non-vanishing,

we can replace the original real Kähler form Ω by a complexified Kähler form Ω푐 = Ω + 푖퐵 =

푖 푗 (푏푖푗̅ + 푖푔푖푗̅)푑푧 ∧ 푑푧̅ .

Then we can find the correlation relations of the 퐴-model can be written as

−푖푡 Ω∙훽 −푖푡{푄퐴,∫ 푉} ⟨푂1 … 푂푛⟩ = 푒 ∫ 퐷휙퐷휒퐷휓푒 푂1 … 푄푛 (3.6.8) 푀 where 퐷휙, 퐷휒 and 퐷휓 are the corresponding measures of the path integral. We can find that the correlation function of these physical operators does not have dependence of the metric on the term 푉, but it only depends on the metric of target space 푀 via the Kähler form Ω from the term 푒−푖푡 Ω∙훽 by using the similar method in (3.5.7). That is to say, the theory does not depend on any structure which appears only in 푉, but it depend on the structure in rest term on the action

(For this case, it is the second term in (3.6.4).). Therefore it can be found that the 퐴-model is

42 independent of the choice of the complex structure, but it clearly depends on the choice of the

Kähler class of the target space. It means that the theory is “half-topological” with respect to the target space, as it depends on “half ” of the moduli of the Calabi-Yau manifold. In conclusion, the

퐴-model theory only topologically depends on the Kähler moduli of the Calabi-Yau target space.

Furthermore, by taking the derivative with respect to the coupling constant 푡 and using the

similar method in (3.5.10), the second factor in (3.6.10) ( the 푄퐴-exact part ) is also independent of the choice of 푡 so that we can calculate it exactly by taking the classical 푡 → ∞ limit, and therefore we can somehow reduce the theory to a weak coupling theory.

3.7 푩-model

푖 In the 퐵-model, we will do the same method as for the 퐴-model. For this model, we twist 휓+

푖 ∗ + ∗ − and 휓− to be the section of 퐾̅ ⊗ Φ (푇푀 ) and 퐾 ⊗ Φ (푇푀 ) respectively, and we also

푖̅ 푖̅ ∗ − 푖 푖̅ regard both 휓+ and 휓− as the section of Φ (푇푀 ). Then the field 휙 and 휙 remain the same as before.

푖 ∗ + 휓+ ∈ 퐾 ⊗ Φ (푇푀 ) (3.7.1.a)

푖 ∗ + 휓− ∈ 퐾̅ ⊗ Φ (푇푀 ) (3.7.1.b)

푖̅ ∗ − 휓+ ∈ Φ (푇푀 ) (3.7.1.c)

푖̅ ∗ − 휓− ∈ Φ (푇푀 ) (3.7.1.d)

푖̅ One may find that in the 퐵-model, the new twisted scalar fields 휓± are both the space-time

(1,0)-forms, which is slightly different from the case in the 퐴-model. Therefore, the new can be chosen in a more convenient way, and we rename all new fields as:

43 푖̅ 푖̅ 푖̅ 휂 = 휓+ + 휓− (3.7.2.a)

푗̅ 푗̅ 휃푖 = 푔푖푗̅(휓+ − 휓−) (3.7.2.b)

푖 푖 휌푧 = 휓+ (3.7.2.c)

푖 푖 휌푧̅ = 휓− (3.7.2.d)

One of the reason that we introduce 휃 field with lower holomorphic indices is that it may lead to

푖̅ 푖̅ 푗̅ 푘̅ some simpler expressions, e.g. {푄퐵, 휃푖} = 0, but {푄퐵, 휃 } = −2Γ 푗̅푘휂 휃 . Then we can also write the action of the 퐵-model in the rename terms.

1 푆 = 푡 ∫ 푑2푧 ( 푔 휕 휙퐼휕 휙퐽 + 푖휂푖̅(퐷 휌푖 + 퐷 휌푖 )푔 + 푖휃 (퐷 휌푖 2 퐼퐽 푧 푧̅ 푧 푧̅ 푧̅ 푧 푖푖̅ 푖 푧̅ 푧 Σ (3.7.3) 푖 푖 푗 푖̅ 푘푗̅ − 퐷푧휌푧̅) + 푅푖푖푗̅ 푗̅휌푧휌푧̅ 휂 휃푘푔 )

As the scalar supercharge operator in 퐵-model is defined as 푄퐵 = 푄̅− + 푄̅+, we can also find the

supersymmetry transformation 훿 = 휖푄̅ ̅+ + 휖푄̅ ̅− by setting 휖+̅ = −휖−̅ = 휖̅ to be constant and

휖± = 0 and the following relations:

훿휙푖̅ = 휖휂̅ 푖̅ (3.7.4.a)

푖 푖̅ 훿휙 = 훿휃푖 = 훿휂 = 0 (3.7.4.b)

푖 푖 훿휌휇 = ±푖휖휕̅ 휇휙 (3.7.4.c)

The action of the 퐵-model can also be rewritten partly in the form of {푄퐵, 푉} which is in the same way as what we do in the 퐴-model section.

푆 = 푡 ∫{푄퐵, 푉} + 푡푊 (3.7.5) Σ where

44 푖 푗̅ 푖 푗̅ 푉 = 푔푖푗̅(휌푧휕푧̅휙 + 휌푧̅휕푧휙 ) (3.7.6) and

푖 푖 푖 푗 푖̅ 푘푗̅ 푊 = ∫ (−휃푖퐷휌 − 푅푖푖푗̅ 푗̅휌 ∧ 휌 휂 휃푘푔 ) . (3.7.7) Σ 2

퐷 in the equation (3.7.7) is the exterior derivative on the worldsheet Σ, and ∧ is the wedge product. It can be found that the term 푊 in the action is anti-symmetric in the exchange of the holomorphic and anti-holomorphic 푧 indices, and it can be written as a differential (1,1)-form.

Then integral of such a form over the two-dimensional worldsheet Σ is independent of the metric,

so the only metric dependence of the action is in the term {푄퐵, 푉}. Furthermore the variations of

푊 with respect to the cohomology class of the Kähler form Ω on the target space 푀 are

푄퐵-exact, so the metric dependence on the Kähler structure of the Calabi-Yau target space is also vanishing. However, by looking through the anti-commutation relations between the 퐵-model scalar supercharge operator and the untwisted original scalar field 휙, we can find the asymmetry in the relation,

푖 푖̅ 푖̅ {푄퐵, 휙 } = 0 {푄퐵, 휙 } = −휂 , (3.7.8) so that it implies the theory depends on the choice of the complex structure of the target space [24].

It shows that the 퐵-model is also “half-topological” with respect to the target space, and the theory only topologically depends on the complex structure moduli of the Calabi-Yau target space.

Furthermore for the 푡-dependence, we can use the same way as we did in the 퐴-model

introduction part to show that the 푄퐵-exact term of the action is also independent of the coupling constant 푡. On the other hand, one can find that the term 푊 in (3.7.7) is linear in 휃, but the other

45 term, i.e. the 푄퐵-exact term has not 휃-dependence. Therefore we can remove the 푡-dependence by redefining 휃 → 휃/푡 so that the term 푡푊 in the action is changed to be just 푊. Then the theory becomes independent of the coupling constant 푡 except for factors that come from the

휃 -dependence of the observables. Due to the trivial 푡 -dependence of the 퐵 -model, the calculations can be taken for large 푡 limit.

3.8 Coupling to Gravity

So far the 퐴- and the 퐵-model introduced are both still in topological field theory, in which the metric of the theory are not dynamical. To make the models be string theories, we need to couple them to the gravity, which means the worldsheet theory should not only involve the path integral over the maps Φ to the target space and their fermionic partners, but also a path integral over the

metric ℎ휇휈 of the worldsheet. Then we call such theory the topological string theory. There are several things needed to achieve such coupling. Firstly, of course the Lagrangean of the theory should be rewritten in a covariant way by changing the flat metrics to the dynamical ones, and the covariant derivatives and the factor of √det ℎ should be introduced as well. Furthermore an

Einstein-Hilbert term also needs to be introduced to act like the kinetic term of the metric field, and the new term should still preserve the symmetries of the original theories. Finally, the theory should be able to be integrated over space of all metrics. The first two steps are relatively straightforward to achieve, because changing the Lagrangean of theory may not lead to a large change of the properties of the theory. However, integrating over the space of all metric may result in some difficulties. Even though we have a metric independence of the theory, it is not a correct way to just integrate the partition function over space of all metrics and divide the results by a

46 volume of the topological “gauge group”. Therefore an alternative way of integrating is first to do the integration over all conformally equivalent metrics and to do the integral over the remaining finite-dimensional moduli space of the worldsheet [24]. It may not be explained further as the theories treated as the cohomological field theories could give the sufficient explanation for the next chapter.

47 4 Mirror Symmetry

After discussing the two models of the topological string theory, we may talk about the mirror symmetry, which will make two twisted models isomorphic to each other by exchanging the two topological structures. Besides the mirror symmetry can also link the different types of superstring theories as well. This kind of duality relation will make it possible to calculate one theory by doing the calculation on the other, which is similar to the cases for T-duality and S-duality. For example some calculation on 퐴-model will be way more difficult than the one in 퐵-model, but the mirror symmetry conjecture state it would be possible to just obtain the result of 퐴-model by just calculate the dual case in 퐵-model. Such symmetry relation between two internal spaces may lead a new understanding of the string theory to the physicists nowadays.

In this chapter, we will firstly give some brief introduction to the mirror symmetry and show how this type of duality relates two string theories with totally different topological properties [33][34].

Then we will discuss the mirror symmetry in different aspects and some applications of mirror symmetry will be given as well [25[28][32].

4.1 Brief Introduction to Mirror Symmetry

Generally speaking, mirror symmetry is a conjecture that there are pairs of Calabi-Yau manifolds with different topological properties that imply the same superconformal field theory, and we call such pair of Calabi-Yau manifold a mirror pair.

SCFT(푋) ≃ SCFT(푋̂) (4.1.1) where 푋 and 푋̂ are the mirror pair. In superstring theory, the mirror symmetry lead two type-II

48 theories to have mirror duality, which make an isomorphism between the type-IIA theory in

3-dimensional Calabi-Yau manifold 푋 and the type-IIB theory in the mirror manifold 푋̂ and

vice versa (while in the Calabi-Yau mirror pairs with even dimℂ 푋 the mirror symmetry will lead a self-duality on both type-IIA and type-IIB superstring theories). Meanwhile, there is another duality relation between two type-II theories, which is called the T-duality, a duality between different target spaces, and the simplest case of such duality is to lead a equivalence between one theory in a compact internal manifold with a circle of radius 푅 and the other with radius 1/푅

( in the ). Then there is a conjecture that there is connection between mirror symmetry and T-duality, which is called the SYZ conjecture proposed by Strominger, Yau and

Zaslow in 1996 [4][16]. Furthermore, in the topological string theory, the mirror symmetry also implies that the 퐴-model in target space 푋 is isomorphic to the 퐵-model in the mirror manifold

푋̂ and vice versa.

In the aspect of Hodge numbers, the mirror symmetry implies an extra symmetry relation on the

Hodge diamond for Calabi-Yau manifolds such that

ℎ푟,푠(푋) = ℎ푛−푟,푠(푋̂) (4.1.2) where 푛 is the complex dimension of the Calabi-Yau manifolds, 0 ≤ 푟, 푠 ≤ 푛, and 푋 and 푋̂ are the mirror pairs. For the 푛 = 3 case, we can find the mirror manifold through relation, but for the manifold that ℎ2,1 = 0, there is not mirror symmetry since that the Hodge number ℎ1,1 of a

Calabi-Yau manifold must be positive number as shown in (2.5.10.c). By looking at the

Hodge diamond of the Calabi-Yau threefold, we can find the symmetry switches the only two

49 independent Hodge numbers to each other, i.e. ℎ1,1(푋) = ℎ2,1(푋̂) and ℎ2,1(푋) = ℎ1,1(푋̂). As the mirror symmetry exchange the dimensions of two cohomology groups, it would lead the isomorphism that 퐻1,1(푋) ≃ 퐻2,1(푋̂) and 퐻2,1(푋) ≃ 퐻1,1(푋̂). By using the knowledge in the moduli space section, we can find that the Kähler moduli space and the complex structure moduli space are exchanged under the mirror symmetry for the Calabi-Yau manifolds. Such moduli exchange can be understood in a different direction. In the topological string theory, we have found that the 퐴-model has the topological dependence of the only Kähler moduli on the target space (퐴-model may have some other topological dependence of the worldsheet of the strings), and the 퐵-model only has the topological dependence of the complex structure moduli of the target space. Then the mirror symmetry exchanges the two moduli to lead the equivalence between two models.

퐴-model on 푋 ⟷ Kähler moduli of 푋

Mirror symmetry ↕ (푋, 푋̂) (4.1.3)

퐵-model on 푋̂ ⟷ complex moduli of 푋̂

Consequently, the triple products of the coholomogical fields of each model will also have an isomorphism under the mirror symmetry, and that will lead a kind of Yukawa coupling equivalence relation between two theories.

4.2 Mirror Symmetry in Aspects of T-duality

Let’s consider the simplest case for T-duality (which is also the mirror symmetry), in which the

1 compact manifold is just a circle of radius 푅, i.e. 푀10 = 푀9 × 푆 , and such way of hiding extra dimension is called the Kaluza-Klein compactification. Then for the coordinate in the 10th

50 dimension 푋9, we will have the following relation:

푋9(휏, 휎 + 2휋) = 푋9(휏, 휎) + 2휋푅푤 (4.2.1) where 휏 and 휎 are the coordinate on the worldsheet of the string, and 푤 is the wind number which is the number of times that the string winds around on 푆1. The number 푤 is a of the spectrum of the physical state, and there is another quantum number for the theory, which is corresponding to the momentum of the center of of the string going around 푆1 and is denoted as 푛. Then the contribution to the worldsheet energy of the state form these two quantum numbers is [6]

푛 2 퐸 = (푤푅)2 + ( ) . (4.2.2) 푛,푤 푅

We can find the value 퐸푛,푤 here is invariant under the transformation that 푅 ⟷ 1/푅. Taking the inverse of the radius of the compact 푆1 as the new radius leads the symmetry, and consequently the wind number and the momentum quantum number are exchanged. This is the T-duality for the circle.

Then we can talk about the T-duality for a rectangular 2-torus 푇2. Such compact manifold is the simplest Calabi-Yau manifold, and it is also the only case for one-dimensional Calabi-Yau manifold. The Hodge numbers of 푇2 are all just one, so the mirror manifold of a 2-torus is just

another 2-torus. Taking the two radius of the torus as 푅1 and 푅2, we can define

퐴 = 푖푅1푅2 휏 = 푖푅2/푅1 (4.2.3) which characterizing the Kähler structure and the complex structure of the manifold. The T-duality

51 applying on such manifold will lead the transformation of 푅1 → 1/푅1 to be symmetric under exchanging 퐴 ↔ 휏, an exchange between the Kähler moduli and the complex structure moduli.

The exchange between two moduli spaces in such mirror symmetry may naturally lead to the exchange of 퐴-model and 퐵-model. 푇2 is a trivial Cablabi-Yau manifold with only 1 complex dimension, and it has genus 1 which can lead us to still be able to look at the one-loop free energy

̅ 퐹1 of the theory. The 퐵-model at one loop computes the inverse of the determinant of the 휕 operator acting on 푇2, and the determinant is the Deddkind 휂 function:

∞ 휂(푞) = 푞1/24 ∏(1 − 푞푛) (4.2.4) 푛=1 where 푞 = 푒2휋푖휏 . As the mirror symmetry implies the 퐵-model in target space of 푇2 is isomorphic to the 퐴-model in the mirror manifold, another 푇2, the 퐴-model will be able to have the calculation with 휂 function but with an exchange 휏 ↔ 퐴. The factor 푞 will be related to

푒−퐴 by the mirror symmetry, and the coefficient of 푒−푛퐴 will counts maps which wrap the torus over itself 푛 times.[25]

4.3 Yukawa Couplings in Mirror Symmetry

As mentioned in the mirror symmetry introduction section, the cohomology class of the mirror pairs can be linked by the mirror symmetry, and it leads an isomorphism between 퐻1,1 and 퐻2,1 for the triple products, which could determine a Yukawa coupling of the theories. The fields in the

interaction are considered in the fundamental (248) representation of 퐸8 which is the gauge group of the uncompactified . Under the compactification of the Calabi-Yau

internal space with 푆푈(3) holonomy group, the group 퐸8 will be break into a 퐸6 group for the

52 4-dimensional Minkowshi spacetime and a 푆푈(3) group, i.e. 퐸8 ⊂ 퐸6 × 푆푈(3). Therefore we have the decomposition 248 = (27,3) ⊕ (27̅̅̅̅, 3̅) ⊕ (1,8) ⊕ (78,1), where the 4-dimensional

field transform as 27 and 27̅̅̅̅ of 퐸6 (which are also the fundamental representation of

퐸6 and its dual) and the zero modes transform in 3 and 3̅ of the group 푆푈(3). Then we can find there are two kind of Yukawa coupling for the fields, which are ⟨273⟩ and ⟨27̅̅̅̅3⟩, and the first one is in the form that

0(27) 0 휅푎푏푐 (푋) ≡ 휅푎푏푐(푋) = ∫ℎ푎 ∧ ℎ푏 ∧ ℎ푐, (4.3.1) X

1,1 where ℎ푎 ∈ 퐻휕̅ (푋) and the index 푎, 푏 and 푐 are in the range from 1 to the Hodge number

ℎ1,1. Then the second kind of Yukawa coupling is that

(27̅̅̅̅) 푖 푗 푘 휅훼훽훾 (푋) ≡ 휅̅훼훽훾(푋) = ∫Ω ∧ 푏훼 ∧ 푏훽 ∧ 푏훾 Ω푖푗푘 (4.3.2) X where Ω is the unique non-vanishing holomorphic (3,0)-form on the Calabi-Yau manifold,

푖 푖푗푘 푙̅ 2,1 푏훼 = Ω (푏훼)푗푘푙̅ 푑푧 such that 푏훼 ∈ 퐻휕̅ (푋), and the index 훼, 훽 and 훾 are in the range from 1 to ℎ2,1. We can find that the coupling in (4.3.1) in purely topological and the one in (4.3.2) depends on the complex structure through Ω. There are one-to-one correspondence relations between the field and the moduli: 27 ⟷ Kähler moduli and 27̅̅̅̅ ↔ complex structure moduli.

Under the mirror symmetry, the full ⟨273⟩ couplings on the manifold 푋 depend on the Kähler moduli in such a way that the full ⟨27̅̅̅̅3⟩ couplings on the mirror manifold 푋̂ depend on the complex structure moduli. [28][32]

In the 퐴-model the Yukawa three-point correlation function is computed as

53 ℂ 2휋푖 ∫훽 Ω 0 푒 ⟨ℎ푎, ℎ푏, ℎ푐⟩ = 휅푎푏푐(푋) + ∑ 푛훽 ∫ ℎ푎 ∫ ℎ푏 ∫ ℎ푐 ℂ (4.3.3) 2휋푖 ∫훽 Ω 훽≠0 훽 훽 훽 1 − 푒

ℂ where 푛훽 is the number and 훽 ∈ 퐻2(푋; ℤ) and Ω is the complexified Kähler form.

Then the corresponding dual case in 퐵-model have the three point correlation defined as

⟨푏훼, 푏훽, 푏훾⟩ = ∫ Ω ∧ (∇푏훼 ∇푏훽 ∇푏훾 Ω) (4.3.4) 푋

where Ω is still the holomorphic 3-form on 푋 and ∇푏훼 is the Gauss-Manin connection taking a

(푟, 푠) class to a (푟 + 1, 푠 − 1) class.

54 5 Conclusion

We has discussed how the compactification of string theory required the internal space to be

3-dimensional Calabi-Yau manifold and also discussed the holonomy group, cohomology class and the two moduli spaces of such manifolds. Furthermore the two twisted 푁 = (2,2) topological string theories, 퐴-model and 퐵-model were also introduced and we showed that they are totally determined by the Kähler structure and the complex structure of the target space respectively. Then, the mirror symmetry conjecture were illustrated and it makes the mirror pairs by exchanging the two topological structures and makes an isomorphism between two string theories, including two twisted models. Some applications of the mirror symmetry were briefly introduced in the end of the final chapter.

55 References

[1] Yau S T. On the ricci curvature of a compact kähler manifold and the complex monge‐ampé re equation, I[J]. Communications on pure and , 1978, 31(3): 339-411.

[2] Greene B R, Plesser M R. Duality in Calabi-Yau moduli space[J]. B, 1990,

338(1): 15-37.

[3] Candelas P, Lynker M, Schimmrigk R. Calabi-Yau manifolds in weighted P4[J]. Nuclear

Physics B, 1990, 341(2): 383-402.

[4] Strominger A, Yau S T, Zaslow E. Mirror symmetry is T-duality[J]. arXiv preprint hep-th/9606040, 1996.

[5] Witten E. Mirror manifolds and topological field theory[J]. arXiv preprint hep-th/9112056,

1991.

[6] Becker K, Becker M, Schwarz J H, et al. String Theory and M-Theory: A Modern

Introduction[J]. PhT, 2008, 61(5): 57.

[7] Green M B, Schwarz J H, Witten E. Superstring theory: volume 2, loop amplitudes, anomalies and [M]. Cambridge university press, 2012.

[8] Nakahara M. Geometry, and physics[M]. CRC Press, 2003.

[9] Greene B. String theory on Calabi-Yau manifolds[J]. arXiv preprint hep-th/9702155, 1997.

[10] Font A, Theisen S. Introduction to string compactification[M]//Geometric and Topological

Methods for Quantum Field Theory. Springer, Berlin, Heidelberg, 2005: 101-181.

[11] Hori K, Thomas R, Katz S, et al. Mirror symmetry[M]. American Mathematical Soc., 2003.

[12] Gurrieri S, Louis J, Micu A, et al. Mirror symmetry in generalized Calabi–Yau compactifications[J]. Nuclear Physics B, 2003, 654(1-2): 61-113.

56 [13] Graña M, Triendl H. String theory compactifications[M]//String Theory Compactifications.

Springer, Cham, 2017: 1-74.

[14] Newlander A, Nirenberg L. Complex analytic coordinates in almost complex manifolds[J].

Annals of Mathematics, 1957: 391-404.

[15] Polchinski J. String theory: Volume 1, an introduction to the bosonic string[M]. Cambridge university press, 1998.

[16] Gross M. Mirror symmetry and the Strominger-Yau-Zaslow conjecture[J]. arXiv preprint arXiv:1212.4220, 2012.

[17] Schwartz L, Narasimhan M S. Lectures on complex analytic manifolds[M]. Bombay: Tata

Institute of Fundamental Research, 1955.

[18] Zumino B. Supersymmetry and Kähler manifolds[J]. Physics Letters B, 1979, 87(3):

203-206.

[19] Van Holten J W. Kaehler manifolds and supersymmetry[J]. arXiv preprint hep-th/0309094,

2003.

[20] Chern S. Characteristic classes of Hermitian manifolds[J]. Annals of Mathematics, 1946:

85-121.

[21] Yau S T. Calabi's conjecture and some new results in [J]. Proceedings of the National Academy of Sciences, 1977, 74(5): 1798-1799.

[22] Calabi E. On Kähler manifolds with vanishing canonical class[C]//Algebraic geometry and topology. A symposium in honor of S. Lefschetz. 1957, 12: 78-89.

[23] Klemm A. Introduction in Topological String Theory on Calabi-Yau manifolds[J]. 2003.

[24] Vonk M. A mini-course on topological strings[J]. arXiv preprint hep-th/0504147, 2005.

57 [25] Neitzke A, Vafa C. Topological strings and their physical applications[J]. arXiv preprint hep-th/0410178, 2004.

[26] Alim M. Lectures on mirror symmetry and topological string theory[J]. arXiv preprint arXiv:1207.0496, 2012.

[27] Witten E. Topological sigma models[J]. Communications in , 1988,

118(3): 411-449.

[28] Greene B R, Plesser M R. An introduction to mirror manifolds[J]. AMS IP STUDIES IN

ADVANCED MATHEMATICS, 1998, 9: 1-30.

[29] Joyce D D. Compact manifolds with special holonomy[M]. Oxford University Press on

Demand, 2000.

[30] Wells R O N, García-Prada O. Differential analysis on complex manifolds[M]. Berlin,

Heidelberg, New York: Springer, 1980.

[31] Candelas P. Lectures on complex manifolds[M]//Superstrings and grand unification. 1988.

[32] Candelas P, Xenia C, Green P S, et al. A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory[J]. Nuclear Physics B, 1991, 359(1): 21-74.

[33] Hosono S, Klemm A, Theisen S. Lectures on mirror symmetry[M]//Integrable models and strings. Springer, Berlin, Heidelberg, 1994: 235-280.

[34] Hori K, Vafa C. Mirror symmetry[J]. arXiv preprint hep-th/0002222, 2000.

[35] Witten E. Introduction to cohomological field theories[J]. International Journal of Modern

Physics A, 1991, 6(16): 2775-2792.

[36] Larfors M, Lukas A, Ruehle F. Calabi-Yau manifolds and SU (3) structure[J]. Journal of High

Energy Physics, 2019, 2019(1): 171.

58