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.
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