Finally, I have managed to scrape together some time to type in some problems related to the Strings 2024 conference and turn it into a mini-edition of Bulk Physics, Algebras and All That. I most likely will continue on this in the next edition, but for now, this is a light discussion on some interesting stringy problems. This is also the debut of the $\textsf{string theory}$ category for this blog.
Bulk Physics, Algebras and All That Part Three: Strings Edition -I
[From my Twitter post about this.] Strings 2024 ended. It was an amazing conference and a lot of good progress was made. Amazing work to everyone involved, and great works in several good directions like chaos theory, QIT, de Sitter, etc. (as usual celestial holography gets cut out.) Some of the talks:
The first talk was by Miguel Monterro on string compactifications, which was a review talk on supersymmetric vacua, swampland constraints, non-SUSY string vacua among others. The next one was by Wiesner on bottom-top proof of the emergent string and dependence of species on higher-derivative corrections to the Einstein-Hilbert action. I then listened to Chang’s talk (skipping Figuiredo’s talk) on supercharge “Q” cohomologes and fortuitous states and near-BPS black holes. Next was Collier’s talk on 2D dS as a matrix model was fascinating. It was based on his work with Beatrix, Victor and Lorenz on Virasoro minimal string in 2023, and has directions I am interested in reading more on. Blommaert then had a talk on the gravity dual of DSSYK and fake DSSYK temperature relations to real temperature. This was followed by Stanford, Maxfield, Turiaci, Malda, Lorenz and Lin’s discussion on JT gravity, which was a dope review.
Next day was kicked off by Juan’s talk on BFSS conjecture, followed by a soft theorems talk that I didn’t attend. Then, Cho had a talk on nonworldsheet string backgrounds, followed by a review by Yin and Erler on SFT. I skipped Budzik’s twisted holography talk and listen to Mahajan’s talk on non-perturbative minimal super string duality with matrix integrals. There was then a gong show with Tourkine, Zhong, Tamargo, Biggs, Delgado (on bordisms group which was fascinating), Gesteau, Guo, Ji, Kundu, Levine, Lin, Parihar and Priyadarshi. Next day, Palti talked about emergent kinetic terms in string theory, and an ML talk on CY geometry that I skipped. Norris had a great talk on dS vacua, which I have to review again. I skipped most of the next observational talks except for van Riet’s talk, and skipped the Townhall (on postdoc applications) and the AI talk by Kaplan. The next day (Thu) I skipped the talk by Duffin and attended Casini’s talk on the ABJ anomaly and U(1) symmetry.
I skipped all the bootstrap talks unfortunately, but was pleasantly greeted with Wong’s talk on 3D gravity and random ensemble of approx CFTS. Next, Vardhan had a talk I did not quite understand, but was followed up by Faulkner’s gravitational algebras talk, which was great and is timely for me, since I’m working on algebras. The last session yesterday was by Chris Akers and Dan Jafferis, which was great but I had internet issues. Today started with Nameki’s talk on generalized symmetries which I did not get either, and had to skip Dumitrescu’s QCD talk. Hansen had an interesting talk on bootstrapping Virasoro-Shapiro amplitudes in AdS which I have to review again.
I also skipped Bobev’s talk on M2 branes. Yonekura had an interesting talk on non-SUSY branes in heterotic string theory, followed by Minwalla’s talk on large J+E holographic CFTs. I couldn’t attend Dabholkar’s talk on stringy quantum entanglement entropy, nor Beiras’ talk on topological strings. Zhiboedov had a talk on the future of strings, and and the outro was Hirosi and Andy’s discussion on 100 string problems, which was very good and had de Sitter comments. I asked about analytic continuation, but unfortunately my internet connection dropped out as he was answering. On an all, it was a great conference, and a particularly better improvement over Strings 2023 in light of non-stringy talks. Already feeling nostalgic and missing the talks, and this is how amazing these talks are. David Gross + Ahmed Almheiri’s comments were really touching to hear, especially Ahmed’s joke on the UV index being in Planck units. Can’t wait for Strings 2025@NYUAD in Jan 2025, and since Ahmed +Suvrat+Eva et al are hosting, won’t be surprised if it is as good and even better than this one. Thank you everyone @CERN for this wonderful event.
The May 31, 2013 El Reno tornado is considered one of the most powerful tornadoes to hit the midwest, crossing US-81. With the highest width of 2.6 miles, it was recorded by several storm chasers, and among them, was the group TWISTEX (Tactical Weather-Instrumented Sampling in/near Tornadoes Experiment), which was founded and led by Tim Samaras. The crew at the time of El Reno consisted of Tim Samaras, Paul Samaras and Carl Young. A brief summary of the events is as follows. At around 6 pm, the development of a tornado wall cloud was imminent, and at 6:03 pm, the ground circulations coincided and a tornadic system of multiple sub-vortices and circulations had developed south of El Reno. With increasing intensity (which initially categorized the tornado to EF3), the system developed over 2.6 miles, and over a span of 25 minutes had a course across the Canadian County, OK. The exact categorization of the tornado is controversial, and is typically associated to the EF5 scale.
One of the characteristics that the El Reno tornado is infamously associated with is that of quick course changing. For instance, the tornado changed directions quickly and cut across the I-40, and would go on for a total span of $\sim $ 40 minutes, stretching over a span of around 17 miles. Another prominent characteristic attributed to El Reno is that the usual clear inside the cell of precipitations around a tornado was not present, and rather the outermost circulations of the tornado itself was a cell of precipitation. Unfortunately, the TWISTEX team was not travelling in their usual chasing-suited heavy lined vehicle, but rather in a Chevrolet with little modifications to suit it to the chase, and some of the first casualties of El Reno was that of Tim Samaras, Paul Samaras, and Carl Young. In Reed Timmer's video of the same tornado [1, 2], the Dominator 2's hood was badly damaged by overthrown lines, and it is suggested that had it not been that they stopped due to the overthrown lines on the road, they would have likely run into the same abrupt-direction changing incident that claimed TWISTEX. While usually the vehicles would be less susceptible to such situations while intercepting from the weaker side, the internal sub-vortices impacted the team. I recommend the paper by NOAA on the aerial survey of the El Reno tornado [3].
Associated to this mesmerizing but dangerous tornado was also the characteristic ``walking dead man" vortices, which can be seen forming and dissipating in the footage of the El Reno tornado [4].
References
[1] Reed Timmer et al, Tornado Chasers S2E11: Nemesis, Part 1. Tornado Chasers, S2 Episode 11: "Nemesis, Part 1" 4K [May 31, 2013]
[2] Reed Timmer et al, Tornado Chasers S2E12: Nemesis, Part 2. Tornado Chasers, S2 Episode 12: "Nemesis, Part 2" 4K [May 31, 2013]
[3] Wakimoto et al, Monthly Weather Review -NOAA. Aerial Damage Survey of the 2013 El Reno Tornado Combined with Mobile Radar Data. Aerial Damage Survey of the 2013 El Reno Tornado Combined with Mobile Radar Data [2016]
[4] Markus Pfister, Multivortex-Tornado south of El Reno, May 31, 2013
[From my Twitter thread]: Maybe it is just time to reboot dS holography altogether. No more double Wick rotations, start from ground up and tell me what for the love of god the FLM corrections to entanglement entropy are. My *educated guess* is that static path is the description where we should start from. We get a similar Ryu-Takayanagi-like formula, for which we have to find semiclassical corrections. Leaving technicalities, we’ll assume there is an FLM-type formula. And here is where I note an interesting point. Pseudo-entropy formalism for EE in dS implies that the relevant quantities like the modular Hamiltonian are also pseudo-valued, and while there seem to exist ways to describe pseudo-relative entropy with $K=-\log \tau $ ($\tau $ is a transition matrix) and have a first law of entanglement, in static patch this doesn’t seem to be the case, so finding a JLMS formula becomes neater. Intuitively as well modular flows make sense, which is a good check mathematically. Subsequently subregion-duality would make sense. In retrospect, bulk reconstruction is a slightly unintuitive thing, and perhaps this should be addressed earlier. On the other hand, maybe a mathematical approach could be better suited, starting from operator algebras that Chandrasekharan, Penington, Longo and Witten did. Whether subregion-subalgebra in static patch dS follow on those lines is not entirely clear to me, but if it does follow a Liu-Leutheusser-type formulation with type II_1 algebras, that would go a long way in clearing what exactly dS holography means. One interesting thing here is that opposed to type II_\infty algebras in crossed product AdS/CFT one has a nicer vN algebraic setting, and it would be good to ask how the entropy of subspaces/subalgebras (I’m not sure how to know which in dS yet) corresponds to bulk wedges. Maybe some corresponding notion of entanglement wedges could be derived? Will update the thread with the next set of comments.
Something I am working right now is with the crossed product construction. The idea that the crossed product construction has a physical meaning in AdS/CFT should be very obvious. This is primarily for two reasons: (1) there already exists a well-defined operator algebra for which outer automorphisms group can be defined, and (2) the Hamiltonian for $\mathcal{N}=4$ SYM goes like $H\sim N/g^{2}$, and has a central operator $N\times U$ defined as the difference of the Hamiltonian with the expectation value in the large $N$ limit. The first point is rather interesting and more or less provides most of the motivation needed for the crossed product construction, since time translations generated by the Hamiltonian form a group of outer automorphisms of the left and right copies of the thermofield double state, which we will discuss better in the next subsection. The second point is interesting from a more formal standpoint, where we divide the difference $H-\langle H\rangle $ by $N$ and obtain a well-defined large $N$ limit. This observation leads to the construction presented in 2112.12828, where the canonical ensemble crossed product construction was presented. We will highlight the general idea below.
Start by recalling that the group of automorphisms of an algebra $\mathcal{A}$ generated by a self-adjoint operator $\mathcal{O}$ is defined as
\[e^{i\mathcal{O}s}\mathsf{a}e^{-i\mathcal{O}s}\in \mathcal{A}\]
for some $\mathsf{a}\in \mathcal{A}$ and $s\in \mathbb{R}$. If $e^{i\mathcal{O}s}\in \mathcal{A}$, then the group of automorphisms generated this way are said to be an ``inner" group of automorphisms, and ``outer" otherwise. We will stick with Takesaki's original convention and define our convention so that the group of automorphisms of $\mathcal{A}$ are Aut$[\mathcal{A}]$, inner automorphisms of $\mathcal{A}$ as Int$[\mathcal{A}]$ and outer automorphisms as Out$[\mathcal{A}]$. Recall that a left Haar measure is a non-zero Radon measure $\mu $ so that for Borel sets $\mathcal{B}\subset G$ and $g\in G$,
\[ \mu \left(g\mathcal{B} \right)=\mu \left(\mathcal{B} \right)\;.\]
Let $\alpha :G\to $Aut$[\mathcal{A}]$. Then, the covariant representation of the triplet $(G, \mathcal{A}, \alpha )$ is a pair $(u, \pi )$ of a unitary representation $u:G\to \mathcal{U}(\mathcal{H})$ to the unitary group of $\mathcal{H}$ and $\pi :\mathcal{A}\to \mathcal{B}(\mathcal{H})$ (i.e. the set of bounded operators on $\mathcal{H}$) so that the covariance condition is satisfied:
\[u(g)\pi (\mathsf{a})u(g)^{*}=\pi \left(\alpha _{g}(\mathsf{a}) \right)\;.\]
We can then formally approach the crossed product as follows: taking the von Neumann algebra $\mathcal{A}$ acting on a Hilbert space $\mathcal{H}$ composing the triplet, the von Neumann algebra on $L^{2}(\mathcal{H}, G)$ generated by $u(G)$ and $\pi _{\alpha (\mathcal{A})}$ is the crossed product algebra $\mathcal{A}\rtimes G$ w.r.t $\alpha $. This may seem more complicated than necessary, but the physical picture can be found by noting that the crossed product of the type III$_{1}$ algebra $\mathcal{A}$ by Aut$[\mathcal{A}]$ is a type II$_{\infty }$ algebra $\mathcal{A}\rtimes $Aut$[\mathcal{A}]$. P.S. Go watch Dune Part 2.
Suppose we have 1. e4 e5 2. Nf3 Nc6 3. Bb5, which is the Ruy-Lopez opening. Black has a number of options here, but among the variations I have played against, the two most common are a6 and Nf6, the former being Morphy variation and the latter being Berlin defense. After a6, Bxc6 is an option but I rule it out because it does not improve any position. A note here is that after Bxc6 black should take like dxc6 to prevent Nxe5, after which Qd4 is a fork. Nf3 here is an option, and a queen trade may ensue after which white is not quite very better. So after a6, 4. Ba4 is logical, and after Be7 5. O-O, white is better. Here black may try to support the center and chase the bishop away with b5, after which Bb3 and white is again better. In Sicilian positions where black plays like 1. e4 c5 2. Nf3 Nc6, you necessarily have to exchange on c6 because after a6 4. Ba4 b5 5. Bb3 c4 simply traps the bishop -- this Rossolimo line is very interesting and leads to a lot of nice open positions. Going back to the mainline, after Bb3 black could castle, after which c3 is a very good move intending to support the center and possibly aim for expansion. The old Steinitz defense is a very good counter to the Ruy-Lopez, typically going for closed positions, which I typically find a little inconvenient.
One reason why Ruy-Lopez is so fascinating but complicated is because once the game deviates from the mainline, it becomes easy for black to take any opportunity to get a better position. Against fianchetto options on b7 the Ruy-Lopez does not change, because for instance say we have 1. e4 b6 2. Nf3 Bb7 3. Bc4. Then, black could take on e4, but while it seems like a free pawn, Bxf7+!! is a brilliant move, after which we force black to take and fork the king and the bishop on e4, so the line goes 3. Bc4 Bxe4 4. Bxf7+!! Kxf7 5. Ng5+! Ke8 6. Nxe4 and black has little development, no right to castling and structural weaknesses due to the open light squared diagonal along e4 to a8. Against Sicilian, there are again many options. After 1. e4 c5 2. Nf3 Nc6, we have the previously seen Bb5 line where we take on c6 and damage black's pawn structure and expand with d4 and O-O later. On the other hand, there are also lines where we have 2. Nf3 d6, after which we play the same 3. Bb5+ nonetheless, forcing either a similar Bxc6 position or bishop trade after black plays Bd7. One line that I found very interesting was something like the Berlin defense with delayed d4 after Nxe4, which usually would be refuted by Re1. The anti-Berlin with d3 immediately after Nf6 is also an option, and is the line that Gukesh played against Alireza Firouzja recently. Typically, a second option that Berlin defense allows is a delayed capture on e4 after something like Be7 O-O a6 Ba4 b5 Bb3 Nxe4, which has an interesting line which usually ends in a draw after d4/d3. One of the plays that I don't quite understand how to tackle are positions where black has already played Nf6 -- the Petrov's defense -- where lines along 1. e4 e5 2. Nf3 Nc6 3. Nxe5 is met with Nc6, called the Stafford gambit. This is a line I play as black, but as white my prep sticks with bxc6 4. Nc3 Bb4, with a kind of reversed Spanish variation from black. 5. Bd3 seems like an obvious logical move also preparing the kingside for castling short. A trick beginners/intermediate players play is to instead play Nxe4, and if white plays a dumb move like h4 black can play Qe7, attacking on Ne5. In case white moves the knight anywhere, Nc3 discovered check simply wins the queen, but in general Petrov's defense heads into a Stafford gambit. Against the Caro-Kann my prep goes for the exchange variation after 1. e4 c6 2. Nf3 d5 3. exd6 cxd6 4. Bb5+ following a similar line as against Sicilian with d6. As black, the Zhuravlev countergambit is natural for Ruy-Lopez players, but we will discuss more on black reversed-Spanish variations in a next post.
Let $\tilde{F}$ be a field (called the algebraic closure of $F$) obtained by adjoining all roots of polynomials over $F$. The maximal abelian quotient of $\mathrm{Gal}\left(\tilde{F}/F \right)$ is the Galois group of maximal abelian extension $F'$ of $F$, which is the largest subfield of $\tilde{F}$ whose Galois group is abelian. Setting $F=\mathbb{Q}$, from the Kronecker-Weber theorem, the maximal extension $\mathbb{Q}'$ of $\mathbb{Q}$ is found by adjoining $\mathbb{Q}$ to all roots of unity. Let $\zeta _{N}$ be a fixed primitive $N$th root of unity and $\mathbb{Q}\left(\zeta _{N} \right)$ be the $N$th cyclotomic field. Then, there exists an isomorphism
\[\left(\mathbb{Z}/N \right)^{\times}\cong \mathrm{Gal}\left(\mathbb{Q}\left(\zeta _{N}\right)/\mathbb{Q} \right)\;.\]
The Kronecker-Weber theorem is then the statement that every finite abelian extension of $\mathbb{Q}$ is contained in a cyclotomic field. This has some interesting properties, particularly those linking to p-adic numbers (which caught my attention thanks to Optimized Fuzzball discussing p-adic AdS/CFT), which I will write about in a later post. I am also preparing a Lichess study on Ruy-Lopez, which I will link sometime soon.
Here is a helpful tip I am giving you for free: if you use Overleaf for collaborative works (which is ok), I would strongly recommend switching to Dropbox+GitHub for better version control and ease of working. Before I started using GitHub, most of the works were on Overleaf, but it is clear that working on it is not too good. When collaborating with another person (or people or aliens or whatever your colleagues are), it is very important that all the edits are perfectly synced-up, which Overleaf does not do offline on a free plan. Here is how you setup your collaboration folder.
Suppose you (say X) are working with Y. Then, setup a GitHub repository with a folder of the TeX, bbl, bib and whatever files you want. Then, add Y as a collaborator to the project. I would strongly recommend using Sublime Merge for version control. Now, to whatever the initial state of the files are (denoted by $|\Omega \rangle $), whenever you make a change (or depending upon when you want to merge files), in the gui, FIRST commit, then if Y makes any changes he commits, he will then PUSH his commits onto the master branch, which you have to PULL -- ONLY AFTER YOU FIRST COMMIT, then MERGE the files, and after you obtain some merged commit (without any issues left to be resolved), only then will you PUSH to some state $|\Omega '\rangle $. This goes on back and forth, but for your ease, I strongly advise turning on email notifications for the GitHub commits so that the other person (or people or aliens or whatever your colleagues are) can pull and merge. There will be many issues, and particularly some may be a little more complicated if there are remote branches. So, a way of making sure that no information is lost (because let's face it, information during collaboration is more easily lost than the black hole information problem), people also sync the GitHub repo folder via Dropbox, so that there is a detailed synced-up version history. In case the other person (or people or aliens or whatever your colleagues are) messes something up, the version history can be consulted, but bear in mind that this needs your machine to be online.
So, to summarise, setup a GH repo, invite collaborator(s), use version control apps such as Sublime Merge, setup email notifications, sync the repo to Dropbox, ask the other person (or people or aliens or whatever your colleagues are) to clone the repo, edit, THEN pull, THEN merge, THEN push, and watch Dune Part 2 when you can.
The GRF Essay that I wrote with Aayush is now online, which you can see at the following link -- we have submitted to arXiv as well, which should be out in a couple of days. While the musings are based off from the de Sitter review we wrote, this essay should convince you that de Sitter quantum gravity is enigmatic to its core.
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.
Most of the time in AdS/CFT the bulk is the more complicated story, although in the general discussion of holography the bulk and the boundary both are somewhat mischievous. A result that I am working with is that the type III and type II attribution of the boundary algebras reflect the algebraic-ification of algebras $\mathcal{A}(\mathcal{U})$ associated to a bulk bounded region $\mathcal{U}$, in the following sense. One works from the basis of Haag-Kastler setting, where we attribute to these $\mathcal{U}\subset \Sigma $ in a globally hyperbolic manifold $(M, g)$. We do not wish to work in a non-holographic theory for the sake of this article, although being in a non-holographic theory allows some amount of ease with things like the split property. It should be clear immediately that this can be extended to bulk subregions, for which the algebra is type III as Liu-Leutheusser showed. In this framework, when $\mathcal{A}(\widetilde{R})$ (for some boundary subregion $\widetilde{R}$) is a type II algebra, one can associate the notion of density matrices and entanglement entropy; the bulk dual $\mathcal{A}(R)$ ($R$ being the bulk subregion dual to $\widetilde{R}$) would then have a plausible definition of generalized entropy. This is a very general argument; one can define entropy for $\widetilde{R}$, $\mathcal{S}(\widetilde{R})$ and attribute to it some generalized entropy, and establish a neat relation with the relative entropy. Formally, in type III von Neumann entropy is not well-defined, but nonetheless one can describe the relative entropy of some semiclassical state $|\hat{\Phi }\rangle $ with $|\hat{\Psi }\rangle $ a cyclic and separating state, set with $\delta A_{X}=0$, which looks something like
\[S(\hat{\Phi }|\hat{\Psi })=-S(\hat{\Phi })-\langle \ln \rho _{\hat{\Psi }}\rangle _{\hat{\Phi }}\;,\]
which can be simplified into a better form by identifying the modular Hamiltonian. In type II, this is no longer merely ``formal", and allows us to elaborately identify the generalized entropy of the subregion. Of course, to work with bulk algebras more formally is certainly interesting, since usually the bulk is the more complicated half of AdS/CFT. A very vague idea of what I had initially is to essentially do this in terms of LCQFTs to see what happens to the split property established at the bifurcation surface $X$. While I did make some good-ish progress, I never got to formalizing it. However, in between, I am working on some set of notes on LCQFTs, which might be a good read once completed.
P.S. More Hans Niemann debate in the chess world from the St. Louis club. Apparently SLCC did not invite him for their tournaments because of some ``inappropriate behaviour", which Niemann apologized for but is now claiming is just more drama. Catch up with Niemann's video on their letter to him here: I have fully addressed all of the claims made by the STL Chess club in their letter, addressed their private letter to me and given context to my history with the Club. Let's make one detail absolutely clear, I received 0 invitations to STL Events, before I regretably caused..."
A paper that came to my attention from a Twitter feud on LQG is this paper by Smolin, resulting in me adding a new category to this blog, called ``Loops":
[1608.02932] Holographic relations in loop quantum gravity
This is interesting for two reasons. Firstly, it invokes categorial holography as Smolin terms it, which was introduced by Crane in his paper called Categorial Physics, and secondly, it uses spin networks in an interesting way. The idea is to use punctured 2-surfaces and attribute to them a Hilbert space, and eventually attributing to quantum spinnet a ``bulk" by extending from the punctures. Essentially, stating that there is a map $\mathcal{T}^{p}: \mathcal{H}_{\mathcal{B};j, i}\to \mathcal{H}_{\Sigma ; j, i}$ as noted in the paper. However, I must say something here; one may not need to invoke these constructions in the first place and instead approach a deformation perspective, such as Cauchy slice holography or in general, holography of information if canonical quantum gravity is all we wanted. The feud seems to be very unnecessary and for that matter I don't think there is any need to point fingers saying ``string theory is just conjecture" or ``LQG is just conjecture". If such holography is emergent (as it seems above, but I am no expert) in LQG, then it is very fascinating. Calumny is a dangerous thing and as is the nature of politicising quantum gravity.
I was asked what papers I have on my library so far. Since I found it hard to try and point out each paper, I have uploaded a part of the library to the github repository. Hopefully, these papers are the significant advances that one just beginning to work in hep-th can refer to, and other papers that I have not included will be mentioned later. I do not mean any copyright infringement if such a thing happens. If there is the possibility of that, I will instead replace the directory with arXiv links, which should be better. For now, enjoy :)