Strange But Charming

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

Black Hole Interior in AdS/CFT

Black holes in AdS/CFT are interesting things. One could ask if, following along with the usual bulk-boundary description, there is a way to find CFT operators dual to bulk fields in the exterior and interior of a large AdS-Schwarzschild black hole. The exterior is simple enough; it is our usual HKLL scheme that has to be used. This is following the extrapolate dictionary,

\[\mathcal{O}(t, \Omega )=\lim _{r\to \infty }r^{\Delta }\phi (r, t, \Omega )\;,\]

where $\Delta $ is the usual conformal weight. One could also sit in the Poincare setting for working with the expansion of these operators and modes. The primary idea here is that dual to a bulk field, one can either take a set of local operators that are ``smeared", or a family of nonlocal operators instead. For instance, solving $(\Box -m^{2})\phi =0$, and compressing each set of Bessel functions and denoting the normalizable mode by $\xi _{\omega , k}(t, x, z)$ as in 1211.6767, a nonlocal CFT operator in the Poincare patch looks like

\[\Phi _{\text{CFT}}(t, x, z)=\int \frac{d\omega d^{D-1}k}{(2\pi )^{D}}\;\mathcal{O}_{\omega , k}\xi _{\omega , k}(t, x, z)+\mathcal{O}^{\dagger }\xi ^{*}_{\omega , k}(t, x, z)\;.\]

Then, operators in region II of the Penrose diagram can be written as

\[\phi ^{\text{II}}_{\text{CFT}}(t, x, z)=\int _{\omega >0}\frac{d\omega d^{D-1}k}{(2\pi )^{D}}\; \mathcal{O}_{\omega , k}g^{(1)}_{\omega , k}(t, x, z)+\tilde{\mathcal{O}}_{\omega , k}g^{(2)}_{\omega , k}(t, x, z)+\dots \;.\]

This is obtained from interpolation between operators in the regions I and III. Read more on this in 1211.6767 and 1310.6334. More on this will be detailed in Part Two of my Bulk Physics, Algebras and All That notes, which will come out by tomorrow.

Where is High Energy Physics Going?

I came across this Phys.SE post, to which there is a particular answer that states the following, quoted:

``... but can tell you this. High Energy Physics goes nowhere now as String Theory fails to produce any measurable prediction in two decades. There is no progress in standard model too. Problems left in GR/math ph. are either very difficult or exotic. If I were you I'd choose a field as close to experiment as possible because standard theoretical physics is practically dead. "

My first reaction is best described by three letters: lol. However, I thought I would expand a little more on this, while listening to Style by Taylor Swift.

The statement that hep-th goes nowhere as far as string theory is concerned because it ``fails to produce any measurable prediction in two decades" is an absurd one. While I am not a string theorist, there have been plenty of developments as far as theoretical hep is concerned. If you want a debatable and non-trivial problem to work on, go for string theory and de Sitter space. Putting aside the part that there ``is no progress in standard model" (since I am not aware of it), there is the next statement that ``problems left in GR/math-ph are either very difficult of exotic". Putting aside that most of the problems in pure GR right now are either things appealing to math.DG or math.AP, or things requiring numerical brute-force computations (see the state of the gr-qc arXiv), I doubt there are many problems in the overlap of gr-qc and hep-th that are ``exotic", defined by the poster as meaning ``... of little importance to physics".

The second aspect of this is that in the face of modern hep-th, math-ph is a very hot topic. At any given time, the math-ph arXiv has at least two papers whose primary listing is hep-th. And for that matter, string theory in the mathematical physics arXiv is very infamous, owing to the fact that most of string theory appeals to a wide range to things like math.RT, math.QA, math.NT and math.DG to name a few. Hep-th does not simply mean the surface level works with AdS/CFT or so, and is a field that has very beautiful things to work with. And finally, as of ``... choose a field as close to experiment ... theoretical physics is practically dead", one may see the hep-th arXiv instead to get a better understanding of how very much alive theoretical physics is. From my side, there are some topics that I would provide as examples:

JT gravity. For instance, today a paper appeared which works with the semiclassical Bousso bound in JT gravity. It has also worked alongside random matrix theory.
Operator algebras. In all sorts of places, most strikingly in AdS/CFT.
Information theoretic aspects with things like pseudo Renyi entropy and non-trivialization of Araki's definition of relative entropy for density matrices into complex valued entanglement entropy.
de Sitter space. For instance, Chandrasekharan, Longo, Pennington and Witten recently worked on static patch algebra of observables and type II$_{1}$ algebras.
Information problem and islands.
Topological QFTs.
Involvements of math.AG (see for instance Kapustin and Witten's paper on the electromagnetic duality and geometric Langlands).

Naval Footage

I found some amazing videos of WW2 footage. Some are remastered and edited, but they are worth watching. Music is kind of annoying.

The Definition of a CFT

Here is the second part in Segal's track titled The Definition of a Conformal Field Theory. Puts things in a nice mathematical perspective that the usual introduction to CFTs does not offer. Besides this there is the Langlands mix of CFTs by Frenkel, but I cannot recommend it to anyone who (like me) are not well established with it.

The Definition of a Conformal Field Theory -- Graeme Segal

Travel Reading!

I am going to be travelling for about two days, and I am not a fan of travelling. But a good thing is that I can take a break from working and instead read a few papers. So here are a few papers that I am sharing for you to also have some nice reading time ;) No specific theme, but a collection of things I want to read. Happy reading.

Algebraic ER=EPR

A paper I have been looking forward to with a lot of excitement since Netta's Strings talk has finally been arXived. It was worked on in conjunction with Hong Liu, who previously worked on algebra in AdS/CFT with Samuel Leutheusser in their subregion-subalgebra and emergent times papers. I had written a bit on algebraic ER=EPR in my notes on bulk reconstruction and subregion duality, where I briefly discussed this based on her Strings talk slides.

[2311.04281] Algebraic ER=EPR and Complexity Transfer

Interestingly, the way I had tried to make an algebraic formulation of a ``strong" No Transmission Principle had a lot to do with the identification of type I and type III algebras. I may not arXiv that draft, but for the sake of it I may archive them here soon. For instance, in the paper by Engelhardt and Liu, the type I statement is that taking the bulk Hilbert space $\mathcal{H}_{bulk}=\mathcal{H}^{Fock}_{R}\otimes \mathcal{H}^{Fock}_{L}$, the boundary algebras $\mathcal{A}_{R}$ and $\mathcal{A}_{L}$ are type I if they are disconnected -- if they are connected, it must be type III (classically connected, or type II if quantum connected). The idea I had was that of a strong NTP, so that if the algebras are type I the bulk duals must be ``independent". The statement of the strong NTP was meant to be a strengthened version of NTP, saying that if the boundary CFTs are type I, they are disconnected and the bulk duals being disconnected should imply that the Cauchy slices are incomplete. I was yet to make this more precise when I saw Engelhardt's Strings talk.

Homotopy et TQFTs

The essential idea of TQFTs is that they are a symmetric monoidal functor $\mathcal{Z}$ from the category of topological spaces, here $n$-bordisms (with a number of technicalities suppressed for now):

\[\mathcal{Z}\;:\; \text{Bord}_{n}\;\longrightarrow \;\text{Vect}_{\mathbb{K}}\;.\]

$\mathcal{Z}$ is functorial w.r.t orientation preserving diffeomorphisms of $\Sigma $, an oriented smooth $D$-manifold and $M$, a $D+1$ manifold. The general approach is by defining a homotopy axiom and an additive axiom, in the sense that one can attribute the above functorial definition. In the TQFTs the homotopy axiom, which has to do with cylinders, is replaced by cobordisms instead -- established by Atiyah in his paper on topological quantum field theories. One can now use this as a starting point and define homotopy quantum field theories (HQFTs) as the following alteration of the above definition: taking $\mathbf{B}$-cobordisms, one can define the symmetric monoidal category $\textbf{Hcobord}(n, \mathbf{B})$. Then, an HQFT is a functor:

\[\mathcal{Z}^{\mathcal{H}}\;:\;\textbf{Hcobord}(n, \textbf{B})\; \longrightarrow \;\text{Vect}_{\mathbb{K}}\;.\]

There are some more aspects about $\text{Vect}_{\mathbb{K}}$ that are of importance, but I do not have an understanding strong enough to explain them.

Nonlinear PDE aspects of Ricci Flow

While in discussion with a colleague, I remembered Terry Tao's excellent paper on the nonlinear PDE aspects of Ricci flow and the Poincare conjecture. See Perelman's proof of the Poincaré conjecture: a nonlinear PDE perspective. See also his slides on the Poincare conjecture proof: Perelman's proof of the Poincaré conjecture. I personally am fascinated by the Sacks-Uhlenbeck theorem, which has to do with a nontrivial $\pi _{2}(M)$ and shows the finite-time existence of singularities. This finds its way in turn into what Hamilton first showed as a kind of finite-time existence of singularities. It is also interesting how the homotopy cobordism theorem aspects also have implications in a fundamental sense.

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.

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