Initially, I intended to write an Essay for the GRF contest on de Sitter subregions, which I have been fascinated by for quite some time now. The first thing I had in mind was to work on a bit with these subregions and see if I could come up with something, but noticing that the natural direction of progression went to doing double Wick rotations and stuff, I decided I wanted to ease on doing something original. Let me explain why I came to this decision.
The whole thing with dS/CFT is that we don't know what to do with subregions, and instead of having nice entanglement entropy like we have in AdS/CFT with the Ryu-Takayanagi formula, we instead have to deal with a non-unitary CFT, and work with transition matrices instead of density matrices. The result of this is that instead of having something like
\[S=-\mathrm{Tr} \;\rho \log \rho \in \mathbb{R}\;,\]
we have to deal with something like
\[\mathcal{S}=-\mathrm{Tr}\;\tau \log \tau \in \mathbb{C}\;,\]
which is ugly for two reasons: (1) complex-valued entanglement entropy is indicative of extremal surfaces that are timelike in nature rather than spacelike, and (2) this also implies a non-trivial set of information theoretic things. For instance, in AdS/CFT, Ryu-Takayanagi with corrections is the FLM formula, which in turn implies an equivalence of bulk and boundary relative entropies from the JLMS formula:
\[S(\rho _{A}|\sigma _{A})=S(\rho _{a}|\sigma _{a})\;,\]
where $\rho _{A}$ are density matrices associated to the boundary subregion $A$ and $\rho _{a}$ are density matrices associated to bulk subregions $a$ -- but how to look at something like this in dS/CFT is not entirely clear, since we have to deal with transition matrices; for that matter, what to even naively expect of subregion duality is not clear (except Narayan's geometric works, which are pretty good in having some intuition with this). One idea is to use the first law of entanglement with perturbed states $\rho \to \rho +\delta \rho $, which also works for transition matrices from pseudo-modular Hamiltonians so that we have something like
\[S(\tau +\delta \tau )-S(\tau )\sim \langle \widetilde{K}_{\tau +\delta \tau }\rangle -\langle \widetilde{K}_{\tau }\rangle +O(\delta \tau ^{2})\;,\]
and do something similar to Dong, Harlow and Wall's works in AdS/CFT in dS/CFT, which is something I am currently working on. But the thing about double Wick rotations is that at least for me, it does not seem empirical enough; the basic idea is that going from Poincare AdS path
\[ds^{2}=\frac{l^{2}_{\text{AdS}}}{z^{2}}\left(-dt^{2}+dz^{2}+\sum _{a=1}^{D-2}dx^{a}dx^{a} \right)\;,\]
to Euclidean AdS and double Wick rotating this by
\[z\longrightarrow i\eta \;, \;\;\;\;\; l_{\text{AdS}}\longrightarrow -il_{\text{dS}}\;\]
to relate to the dS planar slicing (here I set $l_{\text{dS}}=1$)
\[ds^{2}=\frac{1}{\eta ^{2}}\left(-d\eta ^{2}+\sum _{a=1}^{D-1}dx^{a}dx^{a} \right)\;.\]
From this, one can find the timelike entanglement entropy and correspondingly subregions (at least in the Hartman-Maldacena fashion). I am trying to get a better feel for the more ``canonical" side of things, and this seems to be a little too straightforward for my liking.
So instead, I am writing a Tom Banks-inspired Essay in which I am basically presenting a few of my thoughts on how some things in dS/CFT could be resolved, although these are presented in a very straightforward way and is not meant to be precise whatsoever. One of the other things I was thinking of adding in the Essay is on asymptotic quantization and CFT partition-like functionals obtained in the $\mu \to 0$ limit (where $\mu $ is some deformation parameter; essentially tells us how the rescaling of the metric $g$ takes us to different slices, and more precisely is the deformation parameter attributed to $T\overline{T}$-deformations in doing Cauchy slice holography), although my remarks in that section are not very clear and I am yet to work on it.
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