# Introduction to Mirror Symmetry in Aspects of Topological

## Transcript Of Introduction to Mirror Symmetry in Aspects of Topological

IMPERIAL COLLEGE LONDON DEPARTMENT OF PHYSICS

Introduction to Mirror Symmetry in Aspects of Topological String Theory

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 string theory shows there is duality called mirror symmetry, which implies there is an isomorphism between two string theories under the compactifications of two topologically different internal manifolds. By twisting the topological string theory in two methods, the twisted theories named π΄-model and π΅ -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 KΓ€hlar Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.4 Holonomy Group of KΓ€hlar Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . .20 2.5 πΜ -cohomology Groups and Hodge Numbers . . . . . . . . . . . . . . . . . . . . . . 21 2.6 Calabi-Yau Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .24 2.7 Moduli Spaces of Calabi-Yau Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . 26

3 Topological String Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.1 Superspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .29 3.2 Supersymmetric Nonlinear Sigma Model . . . . . . . . . . . . . . . . . . . . . . . . .31 3.3 R-Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.4 Twisting Supersymmetric Sigma Model . . . . . . . . . . . . . . . . . . . . . . . . . .34 3.5 Cohomological Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .36 3.6 π΄-model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .40 3.7 π΅-model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .43 3.8 Coupling to Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .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 Einsteinβs general theory of relativity and the quantum field theory with the standard model. However these two theories cannot be unified together properly to become βthe theory of everythingβ. There are several attempts of such great grand unification, and superstring theory 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 particles are considered to be the vibration modes of the 1-dimensional string moving in the spacetime, and the fundamental fermions are the super partners of these bosons under the supersymmetry. 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 universe should be in a spacetime with 10 or 11 dimensions (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 peopleβs 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, π΄-model and π΅-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 π΅-model to the KΓ€hlar structure on π΄-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 worldsheet 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 π1,3 Γ π6, where π1,3 is the 4-dimensional Minkowski spacetime manifold and π6 is the 6-dimensionl compact manifold for the spatial compactification. There are some requirements on π6. Firstly, the compact manifold should be a vacuum solution of the Einsteinβs field equation, which also means the manifold should be Ricci-flat. On the other hand, π6 is also required to preserve some supersymmetries rather than break them all [18][19], so it implies that the manifold needs to be a KΓ€hlar manifold. All the requirements leads that manifold π6 should be a compact Ricci-flat 6-dimensional KΓ€hlar 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 KΓ€hlar 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 βπ. A complex function : βπ β β is holomorphic, if π = π1 + ππ2 satisfies the Cauchy-Riemann relations for each π§π = π₯π + ππ¦π,

ππ1 = ππ2 , ππ₯π ππ¦π

ππ2 = β ππ1 ππ₯π ππ¦π

(2.1.1)

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

(i)

M is a topological space.

(ii) M is provided with a family of pairs {(ππ, ππ)}. (iii) {ππ} is a family of open sets which covers M . The map ππ is a homeomorphism

from ππ to an open subset π of βπ. [Hence, M is even dimensional.]

(iv) Given ππ and ππ such that ππ β© ππ β β , the map πππ = ππ β ππβ1 from ππ(ππ β© ππ) to ππ(ππ β© ππ) is holomorphic.

The number π is the complex dimension of π, and is denoted as dimβ π = π, and its real dimension is 2π. 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 π on π by using the coordinate π§π = π(π) = π₯π + ππ¦π in a chart (π, π), where 1 β€ π β€ π. Then the

tangent space πππ of complex manifold π is naturally spanned by 2π vectors

π

ππ

π

{ππ₯1 , β¦ , ππ₯π , ππ¦1 , β¦ , ππ¦π}

(2.1.2)

and its co-tangent space ππβπ is spanned by

9

{ππ₯1, β¦ , ππ₯π, ππ¦1, β¦ , ππ¦π }.

(2.1.3)

Let us define another 2π basic vecoters:

π 1π

π

ππ§π β‘ 2 (ππ₯π β π ππ¦π)

π 1π

π

ππ§Μ π β‘ 2 (ππ₯π + π ππ¦π),

and corresponding 2π one-forms are defined as:

ππ§π β‘ ππ₯π + πππ¦π ππ§Μ π β‘ ππ₯π β πππ¦π

(2.1.4.a) (2.1.4.b)

(2.1.5.a) (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 tensor field of type (1,1) on point π of a complex manifold π, π½π βΆ πππ β πππ such that

π

π

π½π (ππ₯π) = ππ¦π ,

π

π

π½π (ππ¦π) = β ππ₯π.

(2.1.6)

Note that

π½π2 = βπ ππππ

(2.1.7)

π½π is the almost complex structure of π at point π,and it corresponds to the multiplication of Β±π [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

Introduction to Mirror Symmetry in Aspects of Topological String Theory

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 string theory shows there is duality called mirror symmetry, which implies there is an isomorphism between two string theories under the compactifications of two topologically different internal manifolds. By twisting the topological string theory in two methods, the twisted theories named π΄-model and π΅ -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 KΓ€hlar Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.4 Holonomy Group of KΓ€hlar Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . .20 2.5 πΜ -cohomology Groups and Hodge Numbers . . . . . . . . . . . . . . . . . . . . . . 21 2.6 Calabi-Yau Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .24 2.7 Moduli Spaces of Calabi-Yau Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . 26

3 Topological String Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.1 Superspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .29 3.2 Supersymmetric Nonlinear Sigma Model . . . . . . . . . . . . . . . . . . . . . . . . .31 3.3 R-Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.4 Twisting Supersymmetric Sigma Model . . . . . . . . . . . . . . . . . . . . . . . . . .34 3.5 Cohomological Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .36 3.6 π΄-model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .40 3.7 π΅-model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .43 3.8 Coupling to Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .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 Einsteinβs general theory of relativity and the quantum field theory with the standard model. However these two theories cannot be unified together properly to become βthe theory of everythingβ. There are several attempts of such great grand unification, and superstring theory 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 particles are considered to be the vibration modes of the 1-dimensional string moving in the spacetime, and the fundamental fermions are the super partners of these bosons under the supersymmetry. 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 universe should be in a spacetime with 10 or 11 dimensions (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 peopleβs 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, π΄-model and π΅-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 π΅-model to the KΓ€hlar structure on π΄-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 worldsheet 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 π1,3 Γ π6, where π1,3 is the 4-dimensional Minkowski spacetime manifold and π6 is the 6-dimensionl compact manifold for the spatial compactification. There are some requirements on π6. Firstly, the compact manifold should be a vacuum solution of the Einsteinβs field equation, which also means the manifold should be Ricci-flat. On the other hand, π6 is also required to preserve some supersymmetries rather than break them all [18][19], so it implies that the manifold needs to be a KΓ€hlar manifold. All the requirements leads that manifold π6 should be a compact Ricci-flat 6-dimensional KΓ€hlar 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 KΓ€hlar 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 βπ. A complex function : βπ β β is holomorphic, if π = π1 + ππ2 satisfies the Cauchy-Riemann relations for each π§π = π₯π + ππ¦π,

ππ1 = ππ2 , ππ₯π ππ¦π

ππ2 = β ππ1 ππ₯π ππ¦π

(2.1.1)

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

(i)

M is a topological space.

(ii) M is provided with a family of pairs {(ππ, ππ)}. (iii) {ππ} is a family of open sets which covers M . The map ππ is a homeomorphism

from ππ to an open subset π of βπ. [Hence, M is even dimensional.]

(iv) Given ππ and ππ such that ππ β© ππ β β , the map πππ = ππ β ππβ1 from ππ(ππ β© ππ) to ππ(ππ β© ππ) is holomorphic.

The number π is the complex dimension of π, and is denoted as dimβ π = π, and its real dimension is 2π. 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 π on π by using the coordinate π§π = π(π) = π₯π + ππ¦π in a chart (π, π), where 1 β€ π β€ π. Then the

tangent space πππ of complex manifold π is naturally spanned by 2π vectors

π

ππ

π

{ππ₯1 , β¦ , ππ₯π , ππ¦1 , β¦ , ππ¦π}

(2.1.2)

and its co-tangent space ππβπ is spanned by

9

{ππ₯1, β¦ , ππ₯π, ππ¦1, β¦ , ππ¦π }.

(2.1.3)

Let us define another 2π basic vecoters:

π 1π

π

ππ§π β‘ 2 (ππ₯π β π ππ¦π)

π 1π

π

ππ§Μ π β‘ 2 (ππ₯π + π ππ¦π),

and corresponding 2π one-forms are defined as:

ππ§π β‘ ππ₯π + πππ¦π ππ§Μ π β‘ ππ₯π β πππ¦π

(2.1.4.a) (2.1.4.b)

(2.1.5.a) (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 tensor field of type (1,1) on point π of a complex manifold π, π½π βΆ πππ β πππ such that

π

π

π½π (ππ₯π) = ππ¦π ,

π

π

π½π (ππ¦π) = β ππ₯π.

(2.1.6)

Note that

π½π2 = βπ ππππ

(2.1.7)

π½π is the almost complex structure of π at point π,and it corresponds to the multiplication of Β±π [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