It is maybe an underappreciated fact that the old relation between the 2d Wess-Zumino-Witten model on a Lie group $G$ and the 3d $G$-Chern-Simons theory is an example of the holographic principle. See for instance
- Sergei Gukov, Emil Martinec, Gregory Moore, Andrew Strominger, Chern-Simons Gauge Theory and the AdS(3)/CFT(2) Correspondence (arXiv:hep-th/0403225)
and see on the nLab at AdS3-CFT2 and CS-WZW correspondence for more pointers.
Now the 2dWZW/3dCS correspondence is, to a considerable degree of detail, a mathematical theorem. At the heart this says that the geometric quantization (which is the mathematically precise formulation of quantization) of 3d CS over a (punctured) surface $\Sigma$ yields the spaces of states of 3d CS over $\Sigma$ for each choice of polarization of the theory as a projectively flat bundle -- the Hitchin connection -- over the moduli space of conformal structures on $\Sigma$, and that this are the spaces of conformal blocks of the 2d WZW model with worldvolume $\Sigma$.
For a review of what exactly this means a good place to start is around p. 30 of
Another maybe underappreciated fact is that in
- Edward Witten, AdS/CFT Correspondence And Topological Field Theory JHEP 9812:012, 1998, (arXiv:hep-th/9812012)
it is argued that also AdS5-CFT4 holography as well as AdS7-CFT6 holography, at least, are all controled by just the higher Chern-Simons term on the gravity side, at least as far as the conformal blocks on the CFT side are concerned.
Notably this means by AdS7-CFT6 holography that the 6d (2,0)-superconformal QFT on the M5-brane is determined by the 7d Chern-Simons theory inside the 11d Chern-Simons term of 11-dimensional supergravity after KK-reduction on a 4-sphere.
Indeed, this is how Witten proposed to capture the M5-brane theory in
- Edward Witten, Five-Brane Effective Action In M-Theory J. Geom. Phys.22:103-133,1997 (arXiv:hep-th/9610234)
(there for the simple special case of the abelian theory on a single M5): one can formalize what it means to apply geometric quantization in codimension 1 to the 7d Chern-Simons theory to obtain its space of states, and according to Witten's argument this is, holographically, the space of conformal blocks of the 6d conformal field theory on the 5-brane.
It was the desire to give a fully mathematically precise formulation of this "holographic quantization" of the single M5-brane that led to the seminal article
maybe one of the high points of deep mathematics motivated by or found in string/M-theory.
(Incidentally, the abelian 7d Chern-Simons term that Witten considered in the above article is well known to receive quantum corrections from a Green-Schwarz mechanism in 11d sugra and from the "flux quantization condition". The full 7d Chern-Simons term in 11d sugram KK-reduced on the 4-sphere is that of a nonabelian Chern-Simons theory. We wrote that out as an extended prequantum field theory in the articles here. It is natural to expect that the holographic dual of this complete nonabelian Chern-Simons term of 11d sugra is the 6d-theory on the 5-brane with its nonabelian 2-form field. But this remains to be analyzed.)
In order to find more instances of mathematically precise aspects of holography, it also serves to go down in dimension from WZW2/CS3 to the holographic relation between point particle quantum mechanics and the 2d topological string.
A semi-formalization of this is Kontsevich's formal deformation quantization of any Poisson manifold, which was later understood in
- Alberto Cattaneo, Giovanni Felder, Poisson sigma models and deformation quantization, Mod. Phys. Lett. A 16, 179–190 (2001) (hep-th/0102208)
to be secretly the 3-point function of the open topological string propagating on the Poisson manifold as given by the Poisson sigma-model.
Now Kontsevich's formula is rigorous mathematics, and the interpretation by Cattaneo-Felder in terms of BV-quantization is written at the level of rigour at least of mathematical physics. Therefore I would regard this as one more mathematically precise incarnation of the holographic principle.
Notice that there is a slight variant of this holographic quantization of 1d quantum mechanics in terms of 2d topological string given by Gukov-Witten's quantization via the A-model, where the particle with phase space a symplectic manifold is quantized as the boundary theory of the topological A-model string (a limit of the topological Poisson model string).
Recently my student Joost Nuiten has described a rigorous non-perturbative version of this holographic quantization of particles with phase space a Poisson manifold as the boundary theory of the non-perturbative Poisson sigma-model. This is described in the examples section of
He also describes other rigorous instances of holographic quantization. For instance he indicates how the heterotic string is quantized as the boundary theory of the M2-brane ending on an "M9-brane" heterotic boundary spacetime of an 11d Hořava-Witten spacetime. A survey of this is also in the examples-section in the nLab entry on motivic quantization.
The mechanism described there I suspect gives a general mathematical rigorous formalization of holography. But this needs to be further explored.