# Partition Functions in (A)dS/CFT

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I'm trying to understand some aspects of dS/CFT, and I'm running into a little trouble. Any help would be much appreciated.

In arXix:1104.2621, Harlow and Stanford showed that the late-time Hartle-Hawking state in de Sitter space, \begin{align*} \Psi[h]=\int_{dS} Dg_{g|_{\partial dS}=h} \exp(iS(g)) \end{align*} is the analytic continuation of the infrared wavefunction from anti-de Sitter space near the boundary, \begin{align*} \Psi_{\text{IR}}[h]=\lim_{z\to \partial AdS}\int_{AdS}Dg_{g|_{z}=h}\exp(-S(g)). \end{align*} For simplicity I've taken the metric to be the only field in the theory.

Maldacena's formulation of dS/CFT in arXiv:astro-ph/0210603 is that the partition function of the dual CFT is equal to the Hartle-Hawking state at late times, i.e. $Z[h]=\Psi[h]$. Now, if I take $h$ to be the metric on a sphere cross a circle, say, then $\Psi_{\text{IR}}$ computes the finite temperature partition function of the CFT dual to the AdS theory. This seems to imply that the partition function of the dual to the dS theory is just the analytic continuation of the partition function of the dual to the AdS theory.

However, this seems to contradict the only known example of dS/CFT, which is the $SP(n)$ theory dual to Vasiliev theory on $dS_4$, conjectured in arXiv:1108.5735. In this case, the dS/CFT partition function is that of fermions at finite temperature, while the AdS/CFT partition function is that of bosons. These are definitely not analytic continuations of one another under $\Lambda\to -\Lambda$.

Any ideas?

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