On page 131 of these notes, a precise formulation of the AdS/CFT correspondence is given by the GKPW dictionary

$$Z_{\text{grav}}[\phi_{0}^{i};\partial M] = \langle \exp \left( - \frac{1}{\hbar} \sum_{i} \int d^{d}x\ \phi_{0}^{i}(x)O^{i}(x) \right) \rangle_{\text{CFT on } \partial M},$$

where the LHS is the gravitational partition function in asymptotically $AdS_{d+1}$ space.

However, the metric is defined by coordinates

$$ds^{2} = \frac{\ell^{2}}{z^{2}}(dz^{2}+dx^{2})$$

on Euclidean $AdS_{d+1}$.

I fail to see how an asymptotically $AdS_{d+1}$ space is compatible with an Euclidean $AdS_{d+1}$.

Is the metric

$$ds^{2} = \frac{\ell^{2}}{z^{2}}(dz^{2}+dx^{2})$$

for Euclidean $AdS_{d+1}$ just what the asymptotically $AdS_{d+1}$ spacetime asymptotes to at the boundary?

  • $\begingroup$ Do you have any reason at all to suspect the answer to this is not just "Yes."? Physics.SE questions should be potentially useful to a broader audience, we are not a tool to check your understanding of every slightly unclear passage you read in a paper. $\endgroup$ – ACuriousMind Jun 28 '17 at 11:05
  • $\begingroup$ Thanks for the comment. I will keep this in mind. Should I delete my question? $\endgroup$ – nightmarish Jun 28 '17 at 11:14
  • $\begingroup$ Perhaps you could explain further what your confusion is? Are you worried that by changing the signature of the metric you have a space with very different causal structure? $\endgroup$ – gj255 Jun 28 '17 at 11:23
  • $\begingroup$ @ gj255 That's exactly my point of confusion. $\endgroup$ – nightmarish Jun 28 '17 at 12:01
  • $\begingroup$ yes. $AdS_{d+1}$ have negative signature for time coordinates. And by wick rotating we go the Euclidean time. We can see there is no $i$ in the partition function' exponential. So they are the same except one coordinate is wick rotated. So asymptotes must be the same. $\endgroup$ – Hare Krishna Jul 4 '17 at 15:09

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