What sorts of mathematical statements are predicted by the AdS/CFT correspondence?


My "understanding" (term used very loosely) is that this correspondence isn't a mathematically rigorous statement, but I can still imagine that certain baby cases might spit out interesting well-defined identities or relationships. I'm very happy for this question to be broadly interpreted.



Mathematical predictions of AdS/CFT
Although it might seem futile, given how far most of the activity on AdS/CFT is from rigorous mathematics, I think this a good question, provided one is happy, for now, with (very) baby versions of this correspondence.


An example of nontrivial mathematical prediction, using a baby version of AdS/CFT (the [Caffarelli-Silvestre extension](https://arxiv.org/abs/math/0608640)) is the conformal invariance of the scaling limit of critical long-range Ising (or $\phi^4$) models in 3D. You can find this discussed in the article
["Conformal Invariance in the Long-Range Ising Model"](https://arxiv.org/abs/1509.00008) by
Paulos, Rychkov, van Rees and Zan. For a precise formulation of the prediction as a mathematical conjecture see, e.g., my article "Towards Three-Dimensional Conformal Probability", [journal version](https://link.springer.com/article/10.1134/S2070046618040015), [preprint version](https://arxiv.org/abs/1511.03180).


Also note that a lot of work has been done as far as making CFT in 2D rigorous, most notably for the Ising model. Carlo mentioned the Ryu-Takayanagi formula. This is perhaps something one could prove (maybe someone did that already, I don't know).


Good to know also:
A mathematical result which has the flavor of a *very* baby AdS/CFT correspondence is the representation of a temperate distribution in $\mathscr{S}'(\mathbb{R}^d)$, using the wavelet transform, as an integral over the Euclidean AdS, i.e., the hyperbolic space $\mathbb{H}^{d+1}$, with the right invariant metric/measure. See for example the book ["Wavelets, an Analysis Tool"](https://global.oup.com/academic/product/wavelets-9780198505211?cc=us&lang=en&) by Holschneider. If one introduces a cut-off hypersuface approaching the conformal boundary in order to restrict the domain of integration in $\mathbb{H}^{d+1}$, one can use this operation as a way of regulating probability measures on $\mathscr{S}'(\mathbb{R}^d)$ (e.g. the Euclidean functional integrals of a CFT on the boundary). One can show these cut-off measures converge weakly to the given probability measure, as the cut-off surface approaches the boundary.


