In AdS/CFT the bulk geometry is AdS spacetime, the flat limit of AdS is taking to the radius of AdS to infinity. By taking this limit can one get the holography dual in flat spacetime from AdS/CFT, or obtain some information about the field theory dual to flat space?
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$\begingroup$ I really don't know but suspect that you'll get stringy S-matrix. Though have little idea what would it be in strong coupling. $\endgroup$– OONCommented Jan 12, 2017 at 13:24
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$\begingroup$ Check the answer (refs in it) to this question physics.stackexchange.com/q/52748 $\endgroup$– OONCommented Jan 12, 2017 at 13:33
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$\begingroup$ As @OON mentioned, there has been work on deriving flat space scattering amplitudes from AdS/CFT. There is some controversy about whether you can get the full S-matrix from the flat space limit, see for example arxiv.org/abs/1611.05906v2 and arxiv.org/abs/1106.3553. I don't know if these problems are settled yet though. $\endgroup$– David MCommented Jan 13, 2017 at 0:45
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First, the $R \rightarrow \infty$ limit is subtle in $AdS_5$ spacetime, because the symmetry group of stringy $\sigma$-model on $AdS_5$ is $PSU(2,2 \vert 4)$, while the symmetry of flat space $\sigma$-model is super-Poincare group. One has to appropriately scale the generators of $SU(2,2 \vert 4)$ before taking the limit in order to get the flat space.
Second, the dictionary of $AdS/CFT$ correspondence states that the 't Hooft parameter of the gauge theory $\lambda=g_{YM}^2 N$ is dual to $R^4/\alpha^{\prime 2}$ on the $AdS$ side. Thus, the limit of infinite radius is the extremely strong coupling limit of the gauge theory. The interesting limit corresponding (presumably) to weakly coupled SYM theory is $R \rightarrow 0$ limit which was extensively studied using Pure Spinor formalism.
These (and many other) topics are discussed in Superstrings in $AdS$ by Mazzucato.
answered Jan 13, 2017 at 10:44