Today, a paper by K. Narayan came up on arXiv. However, I had the privilege of seeing the draft from two days earlier, and here is a quick glimpse on why these extremal surfaces are important. From AdS/CFT, we know that extremal surfaces are the things that make up the entanglement wedge, where we do all sorts of nice things like entanglement wedge reconstruction, make sense of subregion-subalgebra duality (at least motivationally) and calculate entanglement entropy, etc. One interesting thing is that of phase transitions, where if you vary the area of a two-component boundary subregion, the corresponding entanglement wedges eventually coincide, and you will be able to reconstruct a bulk operator deeper than before. But, such kinds of things only exist because these extremal surfaces have that interpretation. In dS/CFT, this becomes a very subtle question: if one does construct entanglement entropy (which we nicely can), what subreion duality could we possibly construct? Clearly this has to be Lorentzian, i.e. into the bulk from $\mathcal{I}^{+}$ to $\mathcal{I}^{-}$ (without ``turning points" going to the same boundary), due to which we have to make sense of extremal surfaces going timelike. This paper has a very nice description of the extremal surfaces, and I have some thoughts on a possible reconstruction scheme in this scenario. However, it must be noted that, while subregion duality has been previously highlighted in AdS/CFT with the backing of type III$_{1}$ von Neumann algebras associated to the bulk and boundary subregions, I am not sure about the nature of algebras in this case. So, things like JLMS and pure bulk reconstruction schemes do not exist as of yet. However, this is an interesting field, and I think the next big thing will be in dS/CFT.
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