String geometry & phenomenology
Orbifold, orientifold, and conifold
1. Orbifold: In mathematics,orbifold is a topological space which locally looks like the quotient space of a Euclidean space under the linear action of a finite group. Physically, it describes an object that can be globally written as an orbit space M/G, where M is a manifold or a theory, G is an isometric group or symmetries.
2. Orientifold: Orientifold is the generalization of orbifold. It produces unoriented strings. The most essential example is that type I string theory can be obtained by type IIB string theory through orientifold projection.
3. Conifold: In string theory, conifold is the generalization of manifold because it can contain conical singularity. Compactifying a theory, the base is usually a 5-dimensional real manifold, since the consideration of conifolds, which are complex 3-dimensional (real 6-dimensional) spaces.
2. Orientifold: Orientifold is the generalization of orbifold. It produces unoriented strings. The most essential example is that type I string theory can be obtained by type IIB string theory through orientifold projection.
3. Conifold: In string theory, conifold is the generalization of manifold because it can contain conical singularity. Compactifying a theory, the base is usually a 5-dimensional real manifold, since the consideration of conifolds, which are complex 3-dimensional (real 6-dimensional) spaces.
Fig 1.1 The deformation/resolution of a conifold
Calabi-Yau manifold
Mathematically, "Calabi-Yau manifold" is a complex manifold. More exactly, Calabi-Yau manifolds have some characteristics as the following:
- They are compact Kähler manifolds.
- First Chern class = 0
- Ricci curvature = 0 (Ricci flatness)
- SU(n) holonomy
General speaking, the most important one is Calabi-Yau 3-manifold, which is a 6-real dimensional (3-complex dimensional ) manifold, K3 surface is a quintessential example.
- They are compact Kähler manifolds.
- First Chern class = 0
- Ricci curvature = 0 (Ricci flatness)
- SU(n) holonomy
General speaking, the most important one is Calabi-Yau 3-manifold, which is a 6-real dimensional (3-complex dimensional ) manifold, K3 surface is a quintessential example.
Fig 1.2 Calabi-Yau 3-manifold (2D projection)
During the second superstring revolution, Calabi-Yau manifold has been used in superstring theory. According to superstring theory, there are 6 invisible extra dimensional space hidden in the extreme microscopic scale, and the shape of the compact space is exactly Calabi-Yau 3-manifold.
On the other hand, "mirror symmetry" implies that these Calabi-Yau space are existed in the spacetime everywhere. This is the quintessence of the combination of mathematical product and physical theory.
Fig 1.3 According to mirror symmetry and the illustration of Kaluza-Klein theory, Calabi-Yau space existed in spacetime everywhere.
String Field Theory
String field theory (SFT) is the dynamics of relativistic strings, and described by quantum field theory.
There are three common types of SFT:
- Open string field theory (Cubic string field theory): On the basis of BRST invariance, it is assumed that the string interaction is divided into two strings, and the half of the two strings are overlapped and overlapped, while the other two are combined into the third string. So, in line with the product of the law, which is a string of non-commutative geometry of the classic example. In addition, since the two strings produce a third string by interaction, Witten argues that the interaction value can be expressed in the cubic order of the string field, so the theory can be called the cubic string field theory.
- Vacuum string field theory: The vacuum string field theory begins with one of Sen's conjectures, which assumes that the final stable vacuum is a closed-string vacuum, so the final state does not have a D-brane presented. The theory of vacuum string field theory describes the final state of D-brane decay, which is also the focal field.
- Boundary string field theory: It is considered that the D-film can be regarded as a coherent state on the boundary of the conformal field, and this coherent state can be deduced from the closed-string theory, which means that there is a duality between the open and closed strings. The boundary string theory further argues that the tachyon is a boundary operator of the conformal field theory on the boundary of the discs of the string, and if the quantum renormalization group is concerned, the renormalization group flow can be regarded as a tachyon condensation. If the conformal field corresponding to its spatial and temporal coordinates, the D-brane will gradually decay into the low-dimensional membrane in the coordinate direction. The result is in line with Sen's conjecture. However, the current boundary-string theory is almost confined to the description of the strings, for the closed-string theory has not yet satisfied with the answer.
There are three common types of SFT:
- Open string field theory (Cubic string field theory): On the basis of BRST invariance, it is assumed that the string interaction is divided into two strings, and the half of the two strings are overlapped and overlapped, while the other two are combined into the third string. So, in line with the product of the law, which is a string of non-commutative geometry of the classic example. In addition, since the two strings produce a third string by interaction, Witten argues that the interaction value can be expressed in the cubic order of the string field, so the theory can be called the cubic string field theory.
- Vacuum string field theory: The vacuum string field theory begins with one of Sen's conjectures, which assumes that the final stable vacuum is a closed-string vacuum, so the final state does not have a D-brane presented. The theory of vacuum string field theory describes the final state of D-brane decay, which is also the focal field.
- Boundary string field theory: It is considered that the D-film can be regarded as a coherent state on the boundary of the conformal field, and this coherent state can be deduced from the closed-string theory, which means that there is a duality between the open and closed strings. The boundary string theory further argues that the tachyon is a boundary operator of the conformal field theory on the boundary of the discs of the string, and if the quantum renormalization group is concerned, the renormalization group flow can be regarded as a tachyon condensation. If the conformal field corresponding to its spatial and temporal coordinates, the D-brane will gradually decay into the low-dimensional membrane in the coordinate direction. The result is in line with Sen's conjecture. However, the current boundary-string theory is almost confined to the description of the strings, for the closed-string theory has not yet satisfied with the answer.
F-theory
F-theory is a 12-dimensional theory discovered by Vafa. Compactifying the theory on a 2-torus, one can obtain 10-dimensional type IIB superstring theory with SL(2,Z) S-duality.
On the other hand, one can compactify F-theory on certain manifolds such as K3 manifold through "elliptic fibration". Then we will get more and more dualities.
As we will discuss in next section, the amount of stable vacua can also derived by F-theory compactification on Calabi-Yau 4-manifold.
On the other hand, one can compactify F-theory on certain manifolds such as K3 manifold through "elliptic fibration". Then we will get more and more dualities.
As we will discuss in next section, the amount of stable vacua can also derived by F-theory compactification on Calabi-Yau 4-manifold.
In effect, there are plenty of ways to obtain dualities or theories through compactification. For instance, one can obtain N=1 supersymmetry by compactifying M-theory on "G2-manifold".