I'm not sure what you have in mind with "mathematically precise"; as with almost anything involving quantum field theory or string theory, there's no rigorous definition of the theories involved in the duality. But, if you grant me their existence, I would say the sharpest statement is still the early formulation by Gubser, Klebanov, and Polyakov and by Witten, i.e. that the partition function of the CFT in the presence of external sources for single-trace operators is the same as that for string theory in AdS with boundary conditions determined by the sources.

The most detailed computational checks of the correspondence are probably those that use integrability to compute anomalous dimensions of operators over the full range from weak to strong coupling. I'm not an expert on this, but I'll point you to one fairly recent paper that contains some of the major references to get you started.

From a more global perspective, though, gauge/gravity duality extends well beyond the original case of ${\cal N}=4$ SYM and $AdS_5 \times S^5$, to any theory that meets the two requirements of having a large-$N$ expansion and a large 't Hooft coupling. The important ideas, again, were mostly there in the very early papers, but I would say they've been put on a somewhat more solid footing. The key point is that the bulk theory is tractable in the case when only a few fields are involved and curvatures are weak. This condition, translated to a statement about the dual field theory, is that most of the single-trace operators acquire very large anomalous dimensions (which is natural in very strongly coupled theories). Recently there has been some progress in formulating a bottom-up argument from the opposite direction, i.e. starting with the assumption that a large-$N$ conformal field theory includes few low-dimensional single-trace operators and arguing that this implies the existence of a bulk dual theory. See this paper of Heemskerk et al..

I don't know if any of this is what you would think of as "mathematics"....

This post imported from StackExchange Physics at 2014-04-01 16:49 (UCT), posted by SE-user Matt Reece