It actually hurts to write this. Danya passed away today. I will not give in to speculations, but Olexandr Bortnyk found him today. Every day, to eat I would watch Danya and Levy videos, and the fact that we won't be getting more of Danya's videos from now on is very depressing. I will miss you forever, Danya.
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.
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..."
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?