[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.
Crossed Product
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.
A Few Ruy-Lopez Sidelines
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.
Kronecker-Weber Theorem
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.
How to setup GitHub and Dropbox for collaboration
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 Canvas of Holography in (A)dS/CFT
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.
Neural Networks Everywhere
- [2402.13321] Rigor with Machine Learning from Field Theory to the Poincaré Conjecture,
- Related to the above paper, [2310.19870] Metric Flows with Neural Networks,
- [1801.03918] Black Holes as Brains: Neural Networks with Area Law Entropy,
- A paper I found on diffusion models and neural nets (not exactly something I am familiar with yet) recommended by p-adic: [2211.14680] A Physics-informed Diffusion Model for High-fidelity Flow Field Reconstruction.
de Sitter Essay
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.
Bulk Ignorance vs Boundary Algebra
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..."
Loopy Ryu-Takayanagi?
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.
Horowitz-Myers AdS soliton conjecture
- SUGRA solutions: $g_{\mu \nu }=\tilde{g}_{\mu \nu }$ is the only solution to $D=10$ type IIB SUGRA, taking $\tilde{g}_{\mu \nu }\times S^{5}_{l}$ for which $\mathcal{E}= 0$, and $\mathcal{E}\geq 0$ in general.
- Geometric solutions: $g_{\mu \nu }=\tilde{g}_{\mu \nu }$ is the only solution to $D=5$ field equations with $\Lambda = -6/l^{2}$ so that $\mathcal{E}=0$, and $\mathcal{E}\geq 0$ in general.
Papers to Read
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 :)
Nielsen-Ninomiya No-go Theorem
An interesting paper which I have been fascinated by recently is the paper by Holger Nielsen and Masao Ninomiya, which shows that there exists a no-go theorem concerning fermion doubling with chiral fermions on a lattice. Naively, the idea is that the Hamiltonian for fermion field has fermion doubling if the following conditions are satisfied: (1) translational invariance is satisfied, (2) the charges are conserved and quantized, and (3) the interaction Hamiltonian is Hermitian and local. In this sense, there is a no-go theorem, the original arguments of which concern homotopy theory. There is also a very good paper by Friedan on a proof of this theorem. There were some arguments concerning if one could be a little more lenient with the hermiticity condition, and impose something like the PT symmetry, see for instance The Nielsen-Ninomiya theorem, PT-invariant non-Hermiticity and single 8-shaped Dirac cone. There is also a very good paper on the approach from loop quantum gravity by Jake Barnett and Lee Smolin, which seems to have some interesting features. Seemingly, there seems to be emphasis on circumventing the Nielsen-Ninomiya theorem in LQG, which seems to be an interesting remark. A part of the theorem's popularity also lies with the motivation it provides to other things with fermions, most famously massive fermions, which is something that David Tong has worked on.
Holography and Theory Journal Club
We're launching a journal club, for which I am the organizer, called Holography and Theory Journal Club, where we will have weekly or bi-weekly (i.e. every other week) discussion meetings or invited talks on recent works in hep-th. We accept the submission of meeting abstracts, for which a forms link is provided in the website, https://sites.google.com/view/htjc. Kindly note that all talks will be by invitation only, whereas paper discussion meeting abstracts are welcome unsolicited. Preferably, the papers are from the hep-th arXiv in the week of submission itself, so that there are no misses of important papers. We will also have a mailing list ready this by Monday, so just drop in an email for participating in the journal club till then!
P.S. Working on a set of notes on Ricci flows in Kahler manifolds.
Black Hole Information Paradox Notes
Notes on the black hole information problem, which I wrote in a very elementary fashion have been complete. I must say that a part of my inspiration to write this was from Aayush's notes, and I wanted to add on to the discussion in a slightly more elaborate way. Keep in mind that this is not a very AdS/CFT oriented thing, so it may seem to be off-topic for the title. However, in the next set of notes, I talk about the Almheiri, Engelhardt, Marolf and Maxfield + Pennington works exclusively. This series of notes, Bulk Physics, Algebras and All That was originally a three-part series. Now, it is an $N$-part series in the large $N$ limit.
Bulk Physics, Algebras and All That -- Part Two: Black Hole Information Problem