174
12 Application to Superstring Theory
From these two characteristics, it is not an exaggeration to say that research on string
theory has greatly advanced. In the following, we will mention that there are two
important inverse problems in string theory. The first inverse problem is the problem
of compaction, and the second inverse problem is the holographic principle. 2
12.1.1 Compactification as an Inverse Problem
The first constraint and the second point are related. The spacetime dimension of our
universe is 4, so if it was originally a 10-dimensional spacetime, the 6-dimensional
space must be a very small and compact space. If we adopt what is generally called
a Calabi–Yau manifold as the space, it is known that the resulting 4-dimensional
spacetime theory will be closer to our familiar elementary particle theory. There
are many types of Calabi–Yau manifolds, all of which are mathematically allowed
in perturbative string theory. If one Calabi–Yau manifold is adopted as a compact
6-dimensional internal space, the types, numbers, and symmetry of the remaining
4-dimensional spacetime elementary particles will be determined accordingly.
In this sense, string theory leaves various possibilities. Instead of deriving a single
elementary particle model in 4-dimensional spacetime, it derives many types of
elementary particle models. Since there are a finite number of Calabi-Yau manifolds,
mathematically the number of models is finite, but the number is very large. Of
course, there is an infinite variety of types of quantum field theory that do not
include gravity, and string theory imposes restrictions on quantum field theories
in that sense. However, the restrictions are not very strong, and various elementary
particle models are allowed. 3
Therefore, if we consider string theory as a theory describing our universe, we
need to find a Calabi–Yau manifold that leads to a 4-dimensional model closer to
the standard model of elementary particles. So far, a number of Calabi-Yau varieties
are known that provide particle content and symmetry close to that of the standard
model of elementary particles. However, the research is not complete.
In addition, when deriving a 4-dimensional elementary particle model from string
theory, it is not only a matter of choosing the Calabi–Yau manifold. For example,
D-branes 4 on which strings can end are introduced into string theory. D-branes
of various dimensions can be considered, and by placing them in 10-dimensional
2 These are mutually related, but it is too technical, so here we will just give an overview of each.
3 Of course, further progress in string theory research may prove even more restrictive. There
are two reasons for this. First, the region where the superstring theory as the quantum theory
of gravity is well understood is where the perturbational picture holds, that is, when the coupling
constant of the string is small. So, once the non-perturbative properties is understood, stronger
restrictions could be imposed. Second, quantum gravity theory is not always superstring theory.
In the AdS/CFT correspondence which will be described later, quantum gravity theory is defined
more widely, and in that sense the string theory is extended.
4 See [129] for an introduction to D-branes.
12 Application to Superstring Theory
From these two characteristics, it is not an exaggeration to say that research on string
theory has greatly advanced. In the following, we will mention that there are two
important inverse problems in string theory. The first inverse problem is the problem
of compaction, and the second inverse problem is the holographic principle. 2
12.1.1 Compactification as an Inverse Problem
The first constraint and the second point are related. The spacetime dimension of our
universe is 4, so if it was originally a 10-dimensional spacetime, the 6-dimensional
space must be a very small and compact space. If we adopt what is generally called
a Calabi–Yau manifold as the space, it is known that the resulting 4-dimensional
spacetime theory will be closer to our familiar elementary particle theory. There
are many types of Calabi–Yau manifolds, all of which are mathematically allowed
in perturbative string theory. If one Calabi–Yau manifold is adopted as a compact
6-dimensional internal space, the types, numbers, and symmetry of the remaining
4-dimensional spacetime elementary particles will be determined accordingly.
In this sense, string theory leaves various possibilities. Instead of deriving a single
elementary particle model in 4-dimensional spacetime, it derives many types of
elementary particle models. Since there are a finite number of Calabi-Yau manifolds,
mathematically the number of models is finite, but the number is very large. Of
course, there is an infinite variety of types of quantum field theory that do not
include gravity, and string theory imposes restrictions on quantum field theories
in that sense. However, the restrictions are not very strong, and various elementary
particle models are allowed. 3
Therefore, if we consider string theory as a theory describing our universe, we
need to find a Calabi–Yau manifold that leads to a 4-dimensional model closer to
the standard model of elementary particles. So far, a number of Calabi-Yau varieties
are known that provide particle content and symmetry close to that of the standard
model of elementary particles. However, the research is not complete.
In addition, when deriving a 4-dimensional elementary particle model from string
theory, it is not only a matter of choosing the Calabi–Yau manifold. For example,
D-branes 4 on which strings can end are introduced into string theory. D-branes
of various dimensions can be considered, and by placing them in 10-dimensional
2 These are mutually related, but it is too technical, so here we will just give an overview of each.
3 Of course, further progress in string theory research may prove even more restrictive. There
are two reasons for this. First, the region where the superstring theory as the quantum theory
of gravity is well understood is where the perturbational picture holds, that is, when the coupling
constant of the string is small. So, once the non-perturbative properties is understood, stronger
restrictions could be imposed. Second, quantum gravity theory is not always superstring theory.
In the AdS/CFT correspondence which will be described later, quantum gravity theory is defined
more widely, and in that sense the string theory is extended.
4 See [129] for an introduction to D-branes.