156. Again about condensed matter applications of AdS-CFT
Save This Post as PDFPeter Horava has recently released a somewhat mysterious paper about non-relativistic Yang-Mills theories. Since I was unable to understand why one should be ultimately interested to consider such exotic beasts (that is, non-relativistic YM theories), I decided to postpone the reading until understanding will eventually appear due to background work of my brain 
Understanding of the motivation came after I stumbled upon Balasubramanian-McGreevy PRL paper and Son’s paper on AdS/cold atoms correspondence and realized that what those people ultimately want is to construct a kind of AdS/CFT applicable to condensed matter systems (we have recently discussed AdS/CFT applicability to condensed matter systems). More accurately, the ultimate goal is to find (if it even exists, of course) a gravity dual to a non-relativistic system with conformal invariance (I don’t want to say “described by CFT”, because CFT means conformal field theory, that is, relativistic theory by definition; non-relativistic limit of a field theory is given by quantum mechanics).
One example of non-relativistic systems with scaling symmetries is the theories with quantum critical points. The latter means that a second order phase transition is achieved by changing couplings at zero temperature. Often, correlators of observables in theories with quantum critical points scale differently along x and t directions – so effective relativistic invariance near the critical point is broken (for details, see Sachdev’s book on quantum phase transitions).
Another example of a non-relativistic system with scaling invariance is so called cold fermions at unitarity. What is it, exactly? Consider non-interacting fermions
(1)
and add a source coupled to the dimer 
, so that
.(2)
The theory (2) is called “fermions at unitarity” since s-wave scattering cross section between two fermions saturates the unitarity bound. This theory is UV complete in the physically interesting situation 
. Another interesting feature of the theory is that the Lagrangian (2) (as well as the Lagrangian (1) describing free fermions) is symmetric w.r.t. the Schroedinger group – a non-relativistic analogue of the conformal group (there is in particular one special conformal transformation in the group).
Ultimately, one is looking for higher-dimensional gravity system described by some non-trivial metric that would have a similar Schroedinger (well, at least Galilean) group symmetry at the boundary. If we are able to find one, we can analyze it and try to figure out what kind of universality class does it correspond to on the non-relativistic quantum mechanical side. In particular, we may expect to find new universality classes theories with quantum critical point may belong to, and this, I think, may very well be the only physical purpose of discussing non-Abelian YM theories.
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