AdS-CFT correspondence: This name is based on its historical derivation, and it focuses on the first and most detailed example of the correspondence where one side is a conformal field theory (CFT) and the other side is gravity in Anti-de Sitter (AdS) spacetime. I feel this label is somehow outdated in the sense that, as you will see in the following, the correspondence can now be employed for situations which are much more general than a simple CFT and AdS spacetime.
This is not totally true. As we will explain later, instabilities of the gravitational theory could be totally physical and interesting rather than a pathology of the theory.
2.
Small or big does not mean anything in physics. The way this is properly done is by making the coupling dimensionless or, in other words, to compare it with a characteristic quantity of the system with the same mass dimension.
3.
We will not consider here the problem that most of the series as (1.2) are asymptotic series, with finite or even zero radius of convergence. This implies that increasing the number of terms m will not improve the final result after a certain step.
To be fair the duality has never been proven rigorously and mathematically. There are anyway several hints and indications that it might hold. Moreover, in the context of string theory, there are several explicit examples [43–45], where the physical observables can be computed exactly from both sides (the field theory and the gravitational theory) and they perfectly match.
6.
This is evident using the redefinition in Eq. (1.8) \(K\equiv J/T\).
7.
Technically, this is due to our limitations, not those of the duality.
8.
One could always do numerical simulations like lattice computations, Monte Carlo, etc., but as we will see later those techniques present very strong limitations.
9.
For the purposes of these lectures.
10.
To be more precise, this is just a necessary condition to have a well-defined gravity dual description. The question of determining which field theories have a gravity dual and which are all the necessary requirements to have it is largely to be determined. We thank Christopher Rosen for this clarification.
An easy way to understand this is by noticing that
meaning that the central charge controls the two-point function of the stress tensor. The latter is indeed the quantity associated to the transport of energy and momentum in the field theory, and it is therefore counting the numbers of those “active” degrees of freedom. For details have a look at http://www.damtp.cam.ac.uk/user/tong/string/four.pdf. We thank Amadeo Jimenez Alba for this comment.
14.
Yes, it is a superfluid in this sense. Unless you do some tricks [85].
15.
More recently, superconductivity at very high temperature (\({\sim }250\) K) has been obtained for certain materials under extreme pressures [99]. It is not yet clear if such phenomenon can be understood within BCS theory and from a standard electron–phonon interaction [101]. We thank A.Garcia Garcia for pointing this out.
16.
Be careful. There is another class of “strange” materials, which are defined bad metals. The latter violate the Mott-Ioffe-Regel bound [103]. A priori, it is not clear if this is connected or not with the \(T-\)linear resistivity of strange metals.
17.
Fermi liquid theory predicts that in standard metals \(\rho \sim T^2\), which comes from simple arguments related to the electron–electron scattering and the phase space in presence of a Fermi surface. There is no known weakly coupled logic or scattering mechanism which is known to give linear in T resistivity \(\rho \sim T\).
18.
Surprisingly enough, the holographic models [83] are quite close to these values.
19.
One can modify BCS theory into the so-called Eliashberg theory [104]. The resulting theory is still a “boring” mean-field theory of superconductivity but it can describe materials such as Pb or Hg, for which the ratio \(\Delta /T_c\) is higher than the BCS prediction. Therefore, a non-BCS value of the ratio only means that the superconductor is more strongly correlated but not that Fermi liquid theory breaks down as is the case in Cuprates. We thanks A.Garcia Garcia for this comment.
20.
Here we keep a very optimistic view.
21.
To be fair, also all the recent constructions trying to formulate an action using Keldysh–Schwinger techniques do not write the action in transparent variables which could immediately identified from a physical point of view.
22.
I have to mention that non-Hermitian models can be a fresh view on this problem [121].
Author information
Authors and Affiliations
Physics Department, IFT Instituto de Fisica Teorica Madrid, Madrid, Spain
Baggioli, M. (2019). A Strings-Less Introduction to AdS-CFT.
In: Applied Holography. SpringerBriefs in Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-35184-7_1
