Possible research implications of proof of John Cardy's a-theorem in QFT According to this recent article in Nature magazine, John Cardy's a-theorem may have found a proof.
Question:


*

*What would the possible implications be in relation to further research in QFT? 

*Specifically, what types of QFT's would now be studied more closely?
 A: The possible applications I can think of are in determining the phases of various QFTs. There are tons of applications like that, here are some ideas: 
-- If the solutions to 't Hooft's conditions are too complicated (entail too many fermions such that their contribution to the IR values of $a$ is greater that $a$ in the UV) there must be symmetry breaking, because then we can match the anomalies in other ways (not just massless fermions).
-- If the broken symmetry group were too large there would be too many Nambu-Goldstone bosons and one would have to conclude that the symmetry is (at least partially) unbroken. 
-- Many other applications of this type for theories which are strongly coupled. One could determine the right candidates for the IR physics and so on. See for instance the very recent http://arxiv.org/abs/1111.3402
There is a viable conjecture in three dimensions, by Myers-Sinha. It involves the entanglement entropy across an S^1. It has already been tested in many perturbative models and also in N=2 SUSY gauge theories in three dimensions. I sincerely believe it is correct. Also a similar entanglement entropy in two dimensions gives the central charge $c$ and a similar entanglement entropy in four dimensions gives precisely the $a$-anomaly. So there seems to be a universal story around the entanglement entropy which has not been uncovered yet.  
A: The most straightforward use of the $a$-theorem is to determine what kinds of spontaneous symmetry breaking are possible. For example, in the usual QCD with three light flavors, at high energy one has a theory of fermions and gauge fields and at low energy one has a theory of pions. If you tried the same thing with a different, large enough of number of fermions you would get a violation of $a$ theorem, since the number of pions is essentially quadratic in the number of fermions.  Therefore there can be no spontaneous symmetry breaking with a sufficiently large number of fermions.  For QCD, we knew this already by other arguments, but one could imagine applying this argument to a theory where we had no real control over anything.
A proof in three dimensions would be great, since thats where a lot of hard condensed matter problems live. Actually, who needs a proof, the right conjecture would be great.  But its a little hard to see how you get that since the arguments relies completely on anomalies.
Added
This is just what I (mis)remember from hearing Zohar Komargodski's "On renormalization groups flows in diverse dimensions" which is available here
