H-cobordisms, Manifolds and Poincare

 The h-cobordism theorem is essentially the following result: Let $M$ be a simply connected $N-$cobordism (with $N\geq 6$) between $V_{0}^{N-1}$ and $V_{1}^{N-1}$. Then, $M\xrightarrow{\;\cong\;}V_{0}^{N-1}\times [0, 1]$. Then, if $M$ is a contractible manifold, one has $M\xrightarrow{\;\cong\;}\mathbb{D}^{N}$. We can prove this as follows: Let $\mathfrak{G}$ be an embedding of $\mathbb{D}^{N}$ into $M$ and identify the interior $\mathrm{Int}(\mathbb{D})$, for which $M-\mathfrak{G}\left(\mathrm{Int}(\mathbb{D}) \right)$ is a cobordism $\partial M\Longleftrightarrow \mathbb{S}^{N-1}$. If we piece these sections back, we would have $M$ from $\mathbb{D}^{N-1}$ and the cobordism $M-\mathfrak{G}\left(\mathrm{Int}\left(\mathbb{D} \right) \right)\equiv \mathcal{B}$. The following pushout diagram shows this decomposition:

Here, $\mathcal{B}$ is simply connected (due to homotopy cofiber sequence $\mathbb{S}^{N-1}\to \mathcal{B}\to M$), and therefore, one can has $\mathbb{S}^{N-1}\xrightarrow{\;H\cong\;}\mathcal{B}$, where $A\xrightarrow{\;H\cong\;}B$ denotes homotopic equivalence. Now, one can use the h-cobordism result from theorem to identify that $\mathcal{B}\xrightarrow{\;\cong\;} \mathbb{S}^{N-1}\times [0, 1]$. Gluing $\partial \mathbb{D}$ back, one sees that $M\xrightarrow{\;\cong\;}\mathbb{D}^{N-1}$, completing the proof. Then, we note the follows: Given a map $\mathfrak{G}$ between $\mathbb{S}^{N-1}$, one can identify that there exists a homeomorphism $\mathfrak{F}$ between $\mathbb{D}^{N}$:

Then, the Poincare conjecture can be rewritten into the follows: An $N-$manifold that is homotopy equivalent to the $\mathbb{S}^{N}$ is also homeomorphic to $\mathbb{S}^{N}$. As you can guess at this point, the proof of this is somewhat straightforward in terms of the decomposition of the primes $\mathbb{D}^{N}_{\pm }$ and $\mathbb{S}^{N-1}_{\pm }$ under the assumption of the h-cobordism theorem. The proof is as follows. The first diagram shows the decomposition of $M=\mathbb{D}_{0}^{N}\sqcup \mathcal{B}\sqcup \mathbb{D}_{1}^{N}$ by identifying the boundaries of the disks and $M-\mathfrak{G}\left(\mathrm{Int}\left(\mathbb{D}_{0}^{N}\sqcup \mathbb{D}_{1}^{N} \right) \right)\equiv \mathcal{B}$. Next, using the Alexandrov trick to induce $\mathfrak{F}$ on $\mathbb{D}^{N}$ from $\mathfrak{G}$ homeomorphism induced on $\mathbb{S}$, we would get the second diagram below:

From the second diagram where we used the Alexandrov trick, we see that there is a homeomorphism between $\mathbb{S}^{N}$ and $M$ by identifying the maps of the $N-$disks. However, in using the h-cobordism theorem, one has to be sure that the inclusion maps \textit{indeed} have the homotopy equivalence nature. This can be found as a lemma: 

Lemma: If $M$ is homotpic to $\mathbb{S}^{N}$ and $\mathcal{B}$ is a cobordism between $\mathbb{S}_{0}^{N-1}$ and $\mathbb{S}_{1}^{N-1}$ as obtained from the subtraction of the $N-$disk images $M-\mathfrak{G}\left(\mathrm{Int}\left(\mathbb{D}_{0}^{N}\sqcup \mathbb{D}_{1}^{N} \right) \right)$, then $\mathbb{S}_{0}^{N-1}\hookrightarrow \mathcal{B}$ is a homotopy equivalence.

Due to the above lemma, one can use the h-cobordism theorem to show that there is a homeomorphism between $\mathbb{S}^{N}$ and $M$, concluding our proof. This proof works (with several subtleties) in $N\geq 6$. This discussion is taken from my notes. However, the Ricci flow approach is much more interesting, since it has a lot to do with the finite-time existence of singularities. Based on the nature of prime decomposition, the works of Hamilton and others, and most importantly Perelman, showed that one can do surgery and continue Ricci flow. To illustrate this, consider the neckpinch characterized by the nature of a pullback being a shrinking cylinder soliton. To prove Poincare from Ricci flow, one has to show the finite time existence of the connected sum decomposition, which forms the basis of Perelman's work.

Bulk Physics, Algebras and All That: Part One

 A set of notes, titled Bulk Physics, Algebras and All That, which I have been writing from a couple of weeks has been almost completed. This set of notes is split into three parts -- part one, where bulk reconstruction and aspects of subregion duality are discussed; part two, where most of the focus is on QES and things like their relation to the information problem in AdS/CFT is presented; and part three, where some of the recent things I learnt about JT gravity and SYK model are discussed. Part three still needs some work, but as of now part one has been completed. There are some omissions and refinements to take into account in future revisions, due to which for now, here is part one.

Bulk Physics, Algebras and All That -- Part One: Bulk Reconstruction and Subregions

Review paper on de Sitter and Holography

A review paper on de Sitter and holography that I have been collaborating on with Aayush Verma has been completed. While this is a preliminary version and we may not arXiv it immediately (perhaps after some more refinements), the present state of the paper justifies to an extent the title. It is based on discussions on the general aspects of de Sitter, holographic entanglement entropy from usual dS/CFT and bit threads, the recent works by Suvrat et al on the Hilbert space of de Sitter quantum gravity, the work on algebra of observables in de Sitter space by Chandrasekharan, Longo, Pennington and Witten, some brief aspects of Cauchy slice holography by Wall et al, and finally a mention of bulk reconstruction in the sense of dS/CFT. In upcoming revisions, we will elaborate on some aspects and include further works, such as cosmic ER=EPR by Cotler and Strominger, dS/dS, and Balasubramaniam et al's works on de Sitter space.

Revering Musings on de Sitter and Holography

U-boats

The history of naval warfare is perhaps one of the most intriguing things, and I am not hesitant to admit that I am obsessed with reading about these things. And one cannot talk about naval warfare and not talk about U-boats (or U-boots as they were called in the Kriegsmarine). So this post is to share a bit on the effectiveness of U-boats.

Keep in mind that after WW1 Germany was forced to minimize military forces under the Treaty of Versailles. And then Hitler came along, and here Germany saw a massive expansion in the armed forces, particularly interestingly, the Luftwaffe and the Kriegsmarine. The latter developed powerful dreadnoughts like the Bismarck (lost tied to Operation Rheinubung, which also involved Prinz Eugen), Tripitz, Lutzow, and many others. But the particularly interesting class of developments were in the family of U-boats, which had two essential armaments -- a deck gun, which I think was usually an 8.8 SK/C-35, and around 14 torpedos. To quote Winston Churchill is to best describe how effective these were in raiding shipping lanes in Wolfpack operations: ``the only thing that really frightened me during the [second world] war was the U-boats peril". Eventually, of course, the effectiveness declined because of anti-submarine warfare (ASW) like Hedgehog (which had bow-forward throwing mortars), Depth charges (these were usually the more effective ones, more below) and the Leigh light among others. One more thing that happened in this period was the cracking of the Enigma from U-boat wrecks, which was a major turning point in the war.

From a purely psychological perspective, there were a number of pressures on U-boat crews. More often than not, they would be subject to hours of constant depth charging, which was a technique that turned out to be quite effective. This was because the Depth charges, thrown to the stern-backward would not have to necessarily impact the U-boat, or even detonate anywhere close to it; it only sufficed to explode in a certain radius for the U-boat to have significant damage, usually damaging ballasts, steering, sonar or the entire U-boat altogether. And since U-boats had to recharge batteries, they would have to surface, which they usually did at night in the early war periods. However, after aerial spotting and bombings of U-boats became superior, this was pushed to the day time. Before the Allies developed an effective countermeasure to U-boats (which initially did not work so well), U-boat campaigns were very successful. They referred to the time period around July, 1940 as the First Happy Time, or Die Gluckliche Zeit. There is a Second Happy Time as well, however I won't discuss this.

For that matter, U-boats in WW1 were also recognized as a threat -- however, they did not operate at the scale seen in WW2. The sinking of Lusitania and the Arabic significantly increased pressure on the American-German relations, and at that point the predictability of a move from Britain was not unseen. In the August of 1915, the Baralong incidents took place, which in a single line can be summarized by ``... [unofficially] ... take no prisoners from U-boats". While this was seen as a British war crime, many say that the provocation of the U-boats was not minor, since attacking Lusitania was in itself a crime, although the German claim was that this was justified since Lusitania could also be classified as an auxiliary cruiser. Joining the dots, it should be clear that there are many aspects of historic value involved when discussing just the one topic of U-boats. Of course, there are many other historically significant cases, like Battle of Jutland, Operation Rheinubung, and many others. In fact, the Battle of Jutland is one of three actions in a major war that involved battleships, after Battle of Yellow Sea and most notably, the Battle of Tsushima, which is famous for Admiral Togo's strategic ``U-turn", which crossed-the-T of Admiral Rozhestvensky's fleet. Tsushima is also famous for Admiral Togo's famous quote, ``Weather today fine, but high waves". However, more on these later. 

Botez Gambit! And two new papers

 I have been somewhat obsessed with chess recently, after successfully winning my tenth game day before yesterday. However, often unsuccessfully, I have found myself making the mistake of trying the Botez Gambit. However, I was successful in ensuring that every time my Queen was lost, the opponent's psychology would be at stake. Partly since the opponent may overestimate the value of the Queen in the particular scenario we would be in, due to which I would be able to either also take off the board the opponent's Queen, or ruin the endgame for the opponent altogether. I will add to this post some of my musings and a particularly interesting experience (so far) to this post once I get time. Also, on today's arXiv, we have the following two nice papers:

Thermal Bekenstein-Hawking entropy from the worldsheet

What if Quantum Gravity is "just'' Quantum Information Theory?

Holographic Entanglement Entropy without Holography

 Today, in the morning, I was learning how to do the Botez Gambit without making it a fail, when (after failing to make good of it in the last 30 seconds) I opened arXiv and saw a paper by Marolf and collaborators. The paper has to do with the notion of Ryu-Takayanagi and Hubeny-Rangamani-Takayanagi without AdS/CFT. While this particular title seems to not be very new (since bit threads are independent of holography up to a certain extent), the paper in itself is an amazing read, and unless Engelhardt and Liu's work on Algebraic ER=EPR comes out this month (which I hope it does), this would be the read of the month. Pasterski and collaborators also had a paper, but I am not into celestial holography, so I cannot say anything about it. 

Algebras and Hilbert spaces from gravitational path integrals: Understanding Ryu-Takayanagi/HRT as entropy without invoking holography

Extremal surfaces in de Sitter

 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.