13 - AdS/CMT from the top down

Summary

In the previous chapters, we focussed mainly on the phenomenological bottom-up models. Here the gravitational bulk theory in four or five dimensions is phenomenologically put together in a similar way to Ginzburg–Landau theory. The actual Lagrangian of the boundary theory thereby remains completely unknown. In addition, it is not clear either whether the bulk theory is a well-defined and selfconsistent quantum gravity theory. The advantage of these bottom-up models is that the gravity theory is relatively simple and one is free to add new ingredients in order to realise different behaviours in the boundary theory. However, to make sure that the phenomena found are self-consistent and/or to understand the dynamics in terms of the dual field theory more fully, it is necessary to find an explicit system in string theory where both the field theory and the exact dual gravity theory are known. Instead of the bottom-up approach, this calls for a top-down approach that starts directly from string/M-theory. The canonical example is the seminal construction by Maldacena, and the generalisations which were subsequently discovered share the property that the action of the dual field theory can be directly identified, including its weakly coupled limit. Since string theory is thought to be a fully consistent quantum theory, this guarantees that any phenomenon described by a top-down theory is physical.

The disadvantage of the top-down approach is that it is technically much more involved. There are far more fields in the gravity theory, often including whole infinite Kaluza–Klein towers that represent the additional dimensions of string theory. In practice one therefore resorts to a consistent truncation of this full top-down theory. This reduces the number of fields, but in such a way that the solution is still guaranteed to be a solution of the full theory. There is an important caveat, deserving special emphasis: a stable solution in a consistent truncation may turn out to be unstable in the full theory where all the truncated field fluctuations are reinstated. Although it will be ignored henceforth, one should be aware of this potential source of trouble.