In the original form of Maldacena's AdS/CFT, the bulk is classical supergravity and the boundary is superconformal field theory in the Maldacena's limit. However, in many applications of AdS/CFT, for example in AdS/CMT, we only consider the bulk is classical gravity and the boundary is CFT, which does not include supersymmetry. My question is where is the supersymmetry, why we need not consider it in most of the applications?


Though AdS/CMT is not my research area, I recall some basic points from the Nastase book, and thus this is the main reference to answer the AdS/CMT part.

This answer is in two parts.

Part I: Answer the AdS/CMT question.

So, in standard condensed matter theory, near a phase transition, there are fixed points that sometimes can exhibit so-called “dynamical scaling”. These are called Lifshitz points.

The Lifshitz scaling is

$$t \rightarrow \lambda^z t$$

$$\vec{x} \rightarrow \lambda \vec{x}$$

where in the above $z$ is the so-called critical exponent.

To describe these points, people, use a phenomenological AdS/CFT approach. In particular, they try to realize the symmetry group geometrically. In the case of $AdS_{d+1}$, the symmetry group $SO(d, 2)$, however, if you relax the condition in your bulk and if you assume that the $AdS/CFT$ holds for general gravity backgrounds, then you are able to describe a $d + 1$-dimensional gravitational background dual to the Lifshitz point;

$$ds_{d+1}^2 = R^2 (- \frac{dt^2}{u^{2z}} + \frac{d \vec{x}^2}{u^2} + \frac{du^2}{u^2})$$

see Nastase's book for these and more formulas and details of course.

Some condensed matter theory guys have been very sceptical with these methods, so far. Just mentioning this as a fact.

As a punchline: SUSY is not there, by assumption, since there is some indication for this pheno approach of the $AdS/CFT$.


Part II: Comments on the "... supersymmetry, why we need not consider it in most of the applications?"

Most of the application and the formal developments of the $AdS/CFT$ do consider SUSY backgrounds.

The reason for that: in a paper Vafa argued that non-SUSY $AdS/CFT$ belongs to the swampland of string theory, unless the theory comes with and infinite tower of Kaluza-Klein modes.

Examples are $AdS/QCD$, Dynamic $AdS/QCD$ and top-down models- like the $D3/D7$, the Sakai-Sugimoto and others, for holographic flavours possess SUSY.

Final point: Even if SUSY is not realised in nature it remains mathematically consistent -and it actually never started as a physical theory- and there is no problem using it in higher dimensional theories. The only thing that needs treatment is to find a mechanism to break/reduce the number of SUSY when you want to go in the $4d$-theory.

Sorry for the long reply.



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