12.1 Inverse Problems in String Theory
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spacetime, various symmetries can be realized. By placing a good combination
of D-branes, it is possible to reproduce the symmetry of the standard model of
elementary particles and the particle content. Creating a 4-dimensional spacetime
elementary particle model (or quantum field theory) in this way by combining Dbranes in string theory is generally called brane construction. Various methods have
been combined to create a 4-dimensional spacetime elementary particle model.
At present, a method called F-theory compactification has been developed,
in which compaction by Calabi–Yau manifolds from 10 dimensions and combinatrics of the D-branes are approximately integrated. The F-theory is defined
as a 12-dimensional spacetime, and its 2-dimensional compaction reproduces the
type IIB string theory. When the 8-dimensional compaction is performed, an
elementary particle model is created in the remaining 4-dimensional spacetime.
This 8-dimensional compact space employs a complex 4-dimensional Calabi–Yau
manifold. From this standpoint, if the type of complex 4-dimensional Calabi–Yau
manifold is determined, the elementary particle model of 4-dimensional spacetime
is determined, so the type of Calabi–Yau variety needs to be scrutinized.
As you can see, the problem of deriving the standard model of elementary
particles from string theory is an inverse problem. The problem is to select a
manifold necessary for compaction of string theory so that it becomes the standard
model of elementary particles. One of the difficulties with this problem is that the
Calabi-Yau manifold is not well understood (for example, it is difficult to even know
the metric on the manifold). For exploration of compact space, machine learning that
is effective for inverse problems has begun to be used [131–134].
12.1.2 The Holographic Principle as an Inverse Problem
The progress of string theory research for the past 20 years has been carried
by the AdS/CFT correspondence. The AdS/CFT correspondence discovered by
J. Maldacena in 1997 [135–137] is regarded as a concrete realization of the
holographic principle, originally proposed by G. ’tHooft and generalized by
L. Susskind [138].
The holographic principle is that a quantum theory including gravity is equivalent
to a quantum field theory without gravity defined in a lower-dimensional spacetime.
The first example given by Mardacena is the equivalence between type IIB superstring theory on an AdS 5 × S 5 , that is, a 5-dimensional Anti-de Sitter spacetime
and a 5-dimensional sphere, and the N = 4 supersymmetric Yang–Mills theory
on a flat 4-dimensional spacetime. It was suggested by considering a special limit
in the 10-dimensional string theory with D-branes. It is the equivalence of these
two theories: the former is a quantum gravity theory, while the latter is a lowerdimensional quantum field theory without gravity. It is a concrete example of the
holographic principle. Generally, the former is called “gravity side” or “AdS side,”
and the latter is called the “CFT side” or “boundary side” or “gauge theory side.”
CFT is an abbreviation of conformal field theory, which is a field theory that is
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spacetime, various symmetries can be realized. By placing a good combination
of D-branes, it is possible to reproduce the symmetry of the standard model of
elementary particles and the particle content. Creating a 4-dimensional spacetime
elementary particle model (or quantum field theory) in this way by combining Dbranes in string theory is generally called brane construction. Various methods have
been combined to create a 4-dimensional spacetime elementary particle model.
At present, a method called F-theory compactification has been developed,
in which compaction by Calabi–Yau manifolds from 10 dimensions and combinatrics of the D-branes are approximately integrated. The F-theory is defined
as a 12-dimensional spacetime, and its 2-dimensional compaction reproduces the
type IIB string theory. When the 8-dimensional compaction is performed, an
elementary particle model is created in the remaining 4-dimensional spacetime.
This 8-dimensional compact space employs a complex 4-dimensional Calabi–Yau
manifold. From this standpoint, if the type of complex 4-dimensional Calabi–Yau
manifold is determined, the elementary particle model of 4-dimensional spacetime
is determined, so the type of Calabi–Yau variety needs to be scrutinized.
As you can see, the problem of deriving the standard model of elementary
particles from string theory is an inverse problem. The problem is to select a
manifold necessary for compaction of string theory so that it becomes the standard
model of elementary particles. One of the difficulties with this problem is that the
Calabi-Yau manifold is not well understood (for example, it is difficult to even know
the metric on the manifold). For exploration of compact space, machine learning that
is effective for inverse problems has begun to be used [131–134].
12.1.2 The Holographic Principle as an Inverse Problem
The progress of string theory research for the past 20 years has been carried
by the AdS/CFT correspondence. The AdS/CFT correspondence discovered by
J. Maldacena in 1997 [135–137] is regarded as a concrete realization of the
holographic principle, originally proposed by G. ’tHooft and generalized by
L. Susskind [138].
The holographic principle is that a quantum theory including gravity is equivalent
to a quantum field theory without gravity defined in a lower-dimensional spacetime.
The first example given by Mardacena is the equivalence between type IIB superstring theory on an AdS 5 × S 5 , that is, a 5-dimensional Anti-de Sitter spacetime
and a 5-dimensional sphere, and the N = 4 supersymmetric Yang–Mills theory
on a flat 4-dimensional spacetime. It was suggested by considering a special limit
in the 10-dimensional string theory with D-branes. It is the equivalence of these
two theories: the former is a quantum gravity theory, while the latter is a lowerdimensional quantum field theory without gravity. It is a concrete example of the
holographic principle. Generally, the former is called “gravity side” or “AdS side,”
and the latter is called the “CFT side” or “boundary side” or “gauge theory side.”
CFT is an abbreviation of conformal field theory, which is a field theory that is