Thoughts on de Sitter

In the famous 1997 paper by Maldacena [hep-th/9711200] introducing AdS/CFT (which also happens to be the most cited paper in hep-th), the notion of duality between bulk quantum gravity and a CFT on the boundary was found. In some sense, this can be viewed as the equivalence between the gravitational partition function $Z_{\text{grav}}$ and the CFT partition function $Z_{\text{CFT}}$. Some interesting things have also come out of this, most prominently (in my opinion) the calculation of holographic entanglement entropy by Ryu-Takayanagi [hep-th/0603001] and Hubeny-Rangamani-Takayanagi [0705.0016]. Since the inception of AdS/CFT itself, however, people have looked for a dS/CFT correspondence -- a holographic duality between some field theory on the conformal boundary $\mathcal{I}^{+}$ and the bulk gravitating region described by the $D$-dimensional de Sitter space. See for instance, the original paper putting forward dS/CFT by Strominger: [hep-th/0106113]. However, this is something that seems too much of a long-shot, namely, for the following reasons:

• Holography at $\mathcal{I}^{+}$ means that the partition function $Z_{\text{CFT}}$ requires non-unitarity, which in some sense can be imagined by the lack of a minimal spacelike geodesic separating two distinct points on the boundary. Therefore, any holographic entanglement entropy proposal would define ``pseudo-entropy" and not something as straightforward as the Ryu-Takayanagi formula. Which of course, one can put up with, but at least for a superficial reason I am petty enough to say that the ``actual" dS/CFT correspondence (if one exists at all) gives a slightly more satisfying description of entanglement entropy than the timelike entanglement entropy - pseudo-holographic entropy equivalence by Harper et al [2210.09457]. 
• Far too many proposals. Again, it might just be that I am being petty with multiple descriptions of holography in de Sitter space, but if a proper description of holography did exist at all, it should be somewhat like AdS/CFT in the sense that the duality is concrete. Right now we have multiple descriptions -- such as global dS/CFT, static patch holography (which is a good enough thing due to things like entanglement entropy calculations and so on), half-de Sitter (introduced just recently by KRST, [2306.07575]), and recently, Cauchy slice holography, which I will talk about in a different post. Perhaps the answer is one of two things: either, that such a correspondence simply does not exist, or that a different kind of holography is to be considered, which would be holography of information from asymptotic quantization. 
• Finally, because it seems out of bound from a quantum gravity description without supposing things. Which holographic proposal would be have to pay attention to from, say, canonical quantum gravity? The answer for me, seems to be -- none. I have talked below. 

What is this aspect of quantum gravity I mentioned that does not work for holography? I cannot make sense of this as of yet, due to some reasons that will be apparent soon. However, if one builds holography in the sense of the GPKW relation,

\[\Psi [g]\sim Z[g]\;,\]

where $\Psi [g]$ satisfies the Hamiltonian and momentum constraints (referred to as a Wheeler-DeWitt state, satisfying the annihilation of the Hamiltonian constraint or the Wheeler-DeWitt equation) and $Z[g]$ is a CFT partition function in dS. Now, here is the question: if one assumes this to be true (which is, for well-known case of the Hartle-Hawking state), what is the Hilbert space of dS quantum gravity? One could suppose to do Cauchy slice holography [2204.00591], which for AdS/CFT works amazingly -- you start from the CFT partition function and use a $T\overline{T}$-deformation to move into a bulk Cauchy slice, and identify holographic duality between $\Psi [g]$ and the deformed partition function $Z^{\Sigma }[g]$. Note a thing above, which is that I have not mentioned the dependence of $\Psi [g]$ or $Z[g]$ on matter contributions -- if we consider the case of gravity coupled to a massive scalar field, one would have in those contributions as well, but for the sake of discussion I have not included it. For de Sitter, this deformation from the boundary $\mathcal{I}^{+}$ is not clear, at least for me. But even if one does identify these things, which holographic proposal would we derive? Static patch? Global dS/CFT? This is not clear for me yet. But even if this is clear, this does not answer the question, ``what is the Hilbert space of de Sitter quantum gravity?" This was answered just recently (discussions on which I was fortunate to hear before the paper was arXived at lunch at ICTS) in a paper by Suvrat and collaborators [2303.16315], which is a nice result, and this uses asymptotic quantization, which is to work with late-time slices (essentially by identifying a conformal factor $\Omega $ in the metric so as to identify an intrinsic ``clock" or York time) instead of $\mathcal{I}^{+}$. This has a nice result, that in some sense one has a proper identification of the WDW states in de Sitter. However, more on this later.