Holography is a duality between a d-dimensional field theory, defined by a series of operators \(\mathcal {O}_i\), and a \(d+1\) gravitational theory, described by a collection of dynamical fields \(\phi ^I\) living in a \(d+1\)-dimensional bulk. The first ingredient we have to learn is how to connect these two sides. This section contains more technical details regarding this bridge, which is usually referred to as the Dictionary.
Religion is a culture of faith; science is a culture of doubt.
As already explained, this coincides with the large N and strong coupling limit of the dual field theory. For the rest of these lectures, only this case will be considered.
2.
See [130, 131] for much more information about conformal field theories.
3.
We will be more precise in the following.
4.
Not really! There are several things one has to be careful about it. Coming soon!
5.
To be precise that is a conformal boundary. Fixing \(z=0\) in (2.10), we obtain a metric which is only conformal to Minkowski \(ds^2\,=\, e^{2\,\sigma (z)}\, ds^2_{Mink}\).
6.
In this course, we will only consider the standard quantization procedure. In the case both the leading and subleading terms in the expansion (2.19) correspond to “normalizable” contributions, an alternative quantization scheme can be used (see [134] for details about it). This alternative scheme is connected to a double-trace deformation of the boundary theory [135–137] and it has nice physical interpretations [138–142].
7.
We will see in a while why this is the case. We do not have to believe it. It is a dictionary, not a bible!
8.
Corresponding to assume the equations of motion for the scalar.
9.
This is a common practice that we will encounter often. The equations of motions in the bulk are second-order differential equations, which need two boundary conditions to guarantee a unique solution. One of the boundary condition is always imposed at the boundary and it is usually the regularity of the bulk fields (we will see later how this condition is modified to compute retarded Green’s functions at finite temperature). A second boundary condition is fixed at the UV boundary and it corresponds to the choice of quantization in the dual field theory.
10.
We make use of the formula:
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Authors and Affiliations
Physics Department, IFT Instituto de Fisica Teorica Madrid, Madrid, Spain
Baggioli, M. (2019). A Practical Understanding of the Dictionary.
In: Applied Holography. SpringerBriefs in Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-35184-7_2
