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12 Application to Superstring Theory
invariant under conformal transformation. The N = 4 supersymmetric Yang–Mills
theory is an example of CFT.
In the first example by Maldacena, the gravity side is a quantum gravity
theory (i.e., superstring theory), but the classical limit and low energy region
are often considered. This is because in that area, the quantum gravity theory
becomes a classical Einstein theory (in higher-dimensional spacetimes with some
supersymmetries), and Lagrangians are well known and easy to handle. It is known
that taking such a limit on the gravity side is equivalent to taking the following
two limits in the Yang–Mills theory on the CFT side. First you need the large N
limit, where N is the gauge symmetry SU (N) of the Yang-Mills theory. By taking
N to infinity, the gravity side becomes classical (quantum effects can be ignored).
Next, the strong coupling limit is needed. This is called the large λ limit, where
λ ≡ Ng 2 is called the ’t Hooft coupling constant, and g is the coupling constant of
the Yang–Mills theory. By making λ infinite, on the gravitational side, the higherorder derivatives acting on the fields can be ignored.
There are two important open questions of the holographic principle. The first is
to give a proof of the example which Maldacena originally proposed. The second
is to give a condition on what quantum field theory on the boundary side can allow
a gravitational description. These problems are, of course, thought to be related
to each other, but the first example of Maldacena was suggested by some limit
of string theory, while the second problem is recognized as an issue that needs
to be discussed beyond superstring theory. In particular, as the second problem is
elucidated, we will gain important insights on whether string theory is the only
method for quantum theory of gravity. The difficulty in solving these problems is
that, as mentioned above, the Yang–Mills theory, which is the gauge theory side, has
a strong coupling, so that it is not possible to use the usual perturbation method, so
the research approach is difficult.
Against this background, various examples of the AdS/CFT correspondence
have been built. The AdS/CFT correspondence is generally called “gauge/gravity
duality.” The main early examples were derived from string theory, by deforming
the examples proposed by Maldacena. Furthermore, many examples have been
published that go beyond the derivation from string theory. Those examples have
properties that are expected to be common to the gauge and gravity sides (such as
symmetries).
In the standard model of elementary particles known to us, the part responsible
for the strong force is written by quantum chromodynamics (QCD), which is
based on SU (3) Yang–Mills theory. It is known that various hadron pictures appear
at low energies due to its strong coupling. Solving QCD is important for indicating
possible deviation of the experimental data from the standard model. Moreover, as
the best studied example of quantum theory of strongly coupled fields, it has great
mathematical significance and many researchers are attracted. There is a research
field that applies the gauge/gravity correspondence to this QCD and performs
various calculations on the gravity side to evaluate hadron physical quantities. This
field is called holographic QCD.
12 Application to Superstring Theory
invariant under conformal transformation. The N = 4 supersymmetric Yang–Mills
theory is an example of CFT.
In the first example by Maldacena, the gravity side is a quantum gravity
theory (i.e., superstring theory), but the classical limit and low energy region
are often considered. This is because in that area, the quantum gravity theory
becomes a classical Einstein theory (in higher-dimensional spacetimes with some
supersymmetries), and Lagrangians are well known and easy to handle. It is known
that taking such a limit on the gravity side is equivalent to taking the following
two limits in the Yang–Mills theory on the CFT side. First you need the large N
limit, where N is the gauge symmetry SU (N) of the Yang-Mills theory. By taking
N to infinity, the gravity side becomes classical (quantum effects can be ignored).
Next, the strong coupling limit is needed. This is called the large λ limit, where
λ ≡ Ng 2 is called the ’t Hooft coupling constant, and g is the coupling constant of
the Yang–Mills theory. By making λ infinite, on the gravitational side, the higherorder derivatives acting on the fields can be ignored.
There are two important open questions of the holographic principle. The first is
to give a proof of the example which Maldacena originally proposed. The second
is to give a condition on what quantum field theory on the boundary side can allow
a gravitational description. These problems are, of course, thought to be related
to each other, but the first example of Maldacena was suggested by some limit
of string theory, while the second problem is recognized as an issue that needs
to be discussed beyond superstring theory. In particular, as the second problem is
elucidated, we will gain important insights on whether string theory is the only
method for quantum theory of gravity. The difficulty in solving these problems is
that, as mentioned above, the Yang–Mills theory, which is the gauge theory side, has
a strong coupling, so that it is not possible to use the usual perturbation method, so
the research approach is difficult.
Against this background, various examples of the AdS/CFT correspondence
have been built. The AdS/CFT correspondence is generally called “gauge/gravity
duality.” The main early examples were derived from string theory, by deforming
the examples proposed by Maldacena. Furthermore, many examples have been
published that go beyond the derivation from string theory. Those examples have
properties that are expected to be common to the gauge and gravity sides (such as
symmetries).
In the standard model of elementary particles known to us, the part responsible
for the strong force is written by quantum chromodynamics (QCD), which is
based on SU (3) Yang–Mills theory. It is known that various hadron pictures appear
at low energies due to its strong coupling. Solving QCD is important for indicating
possible deviation of the experimental data from the standard model. Moreover, as
the best studied example of quantum theory of strongly coupled fields, it has great
mathematical significance and many researchers are attracted. There is a research
field that applies the gauge/gravity correspondence to this QCD and performs
various calculations on the gravity side to evaluate hadron physical quantities. This
field is called holographic QCD.