Physics Diary

Update: My Dissertation in Non-perturbative String Theory – Thinking about Emergent Geometry

The week has come where I need to refine and perfect my dissertation topic. There are a number of constraints around my dissertation this year, and, as my professor been teaching me, there is also a degree of necessary pragmatism to which I must heed.

Over the course of the last year, especially since my academic acceleration from an undergraduate degree to an MRes, I have spent most of my time reading as much as I can at the frontier. After a year of reading what I would estimate to be 100s of research papers from all different areas of string / M – theory, as well as across mathematical physics more generally, I have been piecing together as much of the ‘total picture’ as possible. Along the way I also developed several distractions, covered quite a bit of the Swampland, studied the Braneworld formalism, and also started to get a taste of things like noncommutative geometry. All-encompassing, is perhaps one description of my reading.

For me, I often need to start with the endgame and then work from there; so after cramming so much pure theory, learning about what others at the frontier of string / M-theory are thinking, what directions we might take, and what I might be able to do moving forward, I decided that my own research direction starts with nonperturbative theory. It is what I find most challenging and where, currently, I would like to focus my PhD and extended research over the next years.

So the good news is that, in the sea of frontier physics and with endlessly interesting possible research topics, I have managed to increasingly constrain my focus. This is a major success, especially as my tendency is to want to study and write about everything and anything.

And so this year, for my MRes, my main focus is to significantly advance my studies in nonperturbative string theory (and string geometry). The list of possible research projects within this context remains vast. But to constrain my focus further, I have been moving toward and narrowing in on a project in the area of emergent geometry.

One motivation is an idea I find quite tantalising: namely, in quantum gravity, spacetime geometry is an emergent phenomenon. There are many reasons why we think that, given the mounting evidence in string theory, space and time are actually emergent phenomena. The fact is that string theory challenges us to think of geometry in new ways. The implications of the theory alters how we may approach the question of a generalised geometry, which extends beyond the picture we see for instance in General Relativity (GR).

In working on a project that considers the concept of emergent geometry, one of a number of exciting features is that it would also entail working in gauge / gravity duality. The gauge theory I would be working with are matrix models, which means I would get to learn matrix theory which is something I desperately want to study. An example would be to review and then experiment with constructing geometric probes using strings and branes, studying the various affects on the local field theory. Another example would be to experiment with holographic matrix models as a means to probe the emergence of geometry, which, in this case, would come from matrix coordinates.

Having said all of that, my primary research question has not yet been set, as this is something that I will be thinking about and discussing in the next week.

I look forward to writing more about these topics in time.

*Image: Watercube by Marina Lazareva motivated by Scottish mathematical biologist Sir D’arcy Thompson and his famous publication ‘Growth and Form’.

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Physics Diary

Diary: Sick Note and a Minor Research Update

I’ve been sick for a couple of weeks now. It’s that time of year, ‘freshers flu’ abound. At the beginning, when I knew I was coming down with an infection, I made the choice that it was also a good time for a full reset. Often when I am sick I still work on my physics, if capable, or in the very least dabble and then continue to studying new papers. But this time has been different: a complete and total sick break, the meaning of which has also been extended to include a complete and total break from everything in life. That is, no maths problems, no Polchinski textbook questions or random integrals to keep me occupied (ok, I admit, I worked on a few integrals and I am growing increasingly eager to work on some string problems), no Asperger’s appointments or random errands or administration. Just a proper shutdown, slowing life to a halt. Whilst dealing with being ill, I have taken advantage of the time to simply reflect on a busy year or more and to play some Magic (or other games). I also plan to catch up on some films this weekend in addition to the new Swamp Thing series on Netflix.

To be honest, it’s probably the first proper break I’ve had in 24+ months. Before contracting an infection and becoming ill, I was starting to experience mental burnout – the same hard crash and periodic fatigue that is a definite pattern in my life. I could objectively observe – like a narrator – silly errors that I was starting to make and I could see my computation time increasing.

Thankfully, though, I am starting to feel better and I am eager to get back to my studies and to working on a number of projects. For the first month of my MRES, I have spent a large chuck of my time learning the braneworld formalism and picking up some bits in advanced gravity theory that I previously missed. I also spent time starting to think more deeply about string geometry, classical vacua / moduli stabilisation, and to also start digging into the world of non-geometric backgrounds. String geometry has become one of my main research interests, especially non-perturbative effects. M-theory is also of great interest. And in these areas, the question of geometric constraints on string vacua has become an increasingly interesting research question. In truth, there are still a lot of pressing questions and problems in these areas that one day I would like to be able to explore, but it means I first need to rebuild the picture for myself from first principles which takes time. This is how my brain best functions. So that likely means months of pure string geometry, learning M-theory, and studying lots and lots of compactifications.

Meanwhile, in addition to these research activities, for the second month of my MRes I imagine more time will be spent on an ongoing braneworld project alongside my professor and a PhD colleague, Cesc. It has been a lot to cram in a short amount of time, but I am very much looking forward to exploring some stringy questions in this area as well. I’ve also been learning more about black strings and an entirely new, wacky world of physics that includes black shells and black holes as bubble nucleation sites.

A short blog series on braneworlds is an idea that I have been playing with, but I still need to think about some of the logistics. Do I begin the series by building the geometry completely from first principles, or do I dive straight in to Randall-Sundrum? On the theme of the short-term future of this blog, I plan on uploading more string notes from my notebooks and also spending some time working on how the blog is organised (for easier reader navigation). I saw on someone else’s website that they have a section specifically for study notes, so I may do something similar. After creating such a page, I could make a table of contents for my string notes (linking to each entry) and I could also link to other documents I upload, like my notes on pure spinors or on Hodge theory or whatever. It seems a sensible way to organise things.

In the meantime, thanks for reading.

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Physics Diary

String Conferences for the 2019/20 Academic Year

I’ve had to narrow my list of string conferences that I would like to attend during the 2019/20 academic year. In the best case, it is likely this year I will only be able to attend one more in person. So hopefully most, if not all, of the following conferences will be live streamed or recorded.

With the theme of my research this year, I’ve prioritised my list of conferences to be mainly on the topic of string geometry. That said, the Swampland conference in August 2020 organised by Eran Palti, Daniel Kläwer, Irene Valenzuela, and Timo Weigand is also one I have highlighted.

Here’s my current list along with links (where available), in order of date,

‘Geometry and Duality’ (2-6 Dec 2019) at the Max Planck Institute for Gravitational Physics in (Potsdam, Germany).

‘Supergeometry, Supersymmetry, and Quantisation’ (16-19 Dec 2019) at The University of Luxembourg (Esch-sur-Alzette, Luxembourg).

‘Strings and Geometry’ (28 April – 12 May 2020) at Utrecht University (Utrecht, The Netherlands).

‘Strings 2020’ (29 June – 03 July 2020) at the University of Cape Town (Cape Town, South Africa).

‘Eurostrings 2020’ (24 – 28 August) at the University of Oxford (Oxford, England).

‘Mathematical Foundations of the Swampland’ (17 August – 4 Sept 2020) at the Mainz Institute for Theoretical Physics, Johannes Gutenberg University (Mainz, Germany). This is the Swampland conference being organised by Eran Palti alongside Daniel Kläwer, Irene Valenzuela, and Timo Weigand.

I may add or subtract from this list over the coming weeks, depending on ongoing research commitments and a number of other variables. As things stand, it looks to be an exciting year.

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Physics Diary

Navigating the Swampland (25-27 Sept)

There is an intriguing Swampland workshop set to take place this week. The event has been given the title, ‘Navigating the Swampland’, and it will be held at UAM / IFT beginning tomorrow (25 September) and running through to Friday afternoon. For the interested reader, a stream of all the talks should be made available here. For myself, I am planning on streaming a number of talks so hopefully the feed is of good quality.

I remember hearing about the workshop when I was at IFT in the summer, and I remember thinking that the idea behind its programme was interesting, with a lot of the big names currently working on Swampland stuff scheduled to be there. Moreover, the idea behind the event, as far as I understand, is to organise a sort of comprehensive review – or navigation of – the Swampland, which entails collating important results and discussing the status of each conjecture. From this, might further fundamental structures or properties of quantum gravity be found? There is also of course some emphasis on particle physics and broader cosmological implications.

Of a large list topics I will say that Weigand’s presentation on emergent strings based on a recent paper with Lee and Lerche is one of a few already highlighted in bright yellow. One thing I am also curious to learn is whether anyone will be presenting studies of possible consistency constraints on QFTs given different curved backgrounds. I am also interested in some of the talks that will undoubtedly be based on possible additional universal properties of quantum gravity, as well as talks on potentially new insights into universal properties of the Swampland or those that discuss relating the numerous conjectures in a fundamental way.

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Physics Diary

SiftS 2019

SiftS 2019 concluded on Friday. It was an enjoyable two weeks of study and discussion on topics in string theory and holography. Eran Palti and Kyriakos Papadodimas were for me the highlight of the event. This is not meant to take away from others, it is just that Palti and Papadodimas were one of the main reasons for my attending SiftS. I could sit and listen to Papadodimas talk physics for hours. And Palti’s lectures on the Swampland were outstanding, as expected.

If I had one minor personal grief about the summer school as a whole, it’s that there wasn’t enough pure string theory. But it is very likely that I would say this at a number of different engagements, with the exception perhaps of Strings 2019 and String-Math 2019, two of the main string conferences. So it is unfair to make any such complaint formal, and one must also be mindful that while string theory was the theme, the engagement wasn’t necessarily meant to serve pure stringy discussion.

All of this is to say that I am both thrilled and honoured to have had the privilege of attending SiftS 2019. To mark its conclusion, I want to take a moment to congratulate the SiftS organisers for putting together a terrific summer school. I also want to take a moment to thank everyone at the Universidad de Autonomous Madrid for their hospitality and support throughout my stay. My impression of the university before arriving was that it was one of the best in Europe, and I left the campus and the Instituto de Física Teórica UAM/CSIC with the same view. I can say with honesty that I very much look forward to my return at some point in the future.

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Now that SiftS is over for the year, and with the conclusion of my admittedly brief holiday during the weekend, I have returned to my research and studies at the University of Nottingham. There is a lot to discuss and catch up on with Prof. Padilla, with a number of possibly interesting ideas percolating. My return to Nottingham also means that I will start actively blogging again. In addition to covering some interesting topics from SiftS 2019, I am also working on a number of research projects which will be nice to write about in the coming days, weeks, and months. I will also be continuing my series of string notes, where the reader and I are on our way to covering the whole of textbook bosonic and superstring theory. We will start from where we left off, namely an introduction to conformal field theory. (In the background, I am going to continue working on my blog to fix the LaTeX of older posts as a result the move).

With regards to SiftS 2019 in particular. I will not write about all of the lecture series and topics covered. Instead, I will focus on sharing my notes and thoughts from the lectures by Palti on the Swampland and by Papadodimas on the Black Hole Interior. This will serve as a nice opportunity to also reflect on some of their respective papers, and to summarise key arguments.

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Physics Diary

Papers: Holography and the cosmological constant problem, plus a new publication from Zwiebach

[latexpage]

Non-perturbative de Sitter vacua via $alpha^{prime}$ corrections – Barton Zwiebach

Barton Zwiebach has a new paper out. It’s rather lovely.

The paper was written for, and submitted to, the Gravity Research Foundation 2019 Awards for Essays on Gravitation. Zwiebach’s contribution was awarded second place. However, given the content and general parameters of the first place paper, I believe Zwiebach’s contribution could have easily been given the top prize. (Granted, I have my stringy biases).

As for the paper itself, the premise is both subtle and interesting. The stage is set in the context of two-derivative supergravity theories, in which a large – indeed, infinite – number of higher-derivative corrections correlate with the parameter $alpha^{prime}$. One should note that $alpha^{prime}$ is indeed the dimensionful parameter in string theory.

One of the challenges since the 1980s has been to achieve a complete description of these higher-derivatives. It turns out a rather interesting approach may be taken by way of duality covariant ‘stringy’ field variables, as opposed to directly supergravity field variables.

From a quick readthrough, the idea presented in Zwiebach’s paper is to drop all dependence on spatial coordinates, such that only time dependence remains. The time dependent ansatz enables a general analysis on cosmological, purely time-dependent backgrounds on which all the duality invariant corrections relevant to these backgrounds may be classified. Leveraging the duality group $O(d,d, R)$, an $O(d,d, R)$ invariant action is constructed of the form

[ I_{0} equiv int dt e^{-Phi}frac{1}{n}(-dot{Phi}^{2} – frac{1}{8}tr(dot{S}^{2}) ]

Then, Zwiebach includes in the two-derivative action all of the $alpha^{prime}$ corrections. Around this point, the work of Meissner is referenced. Admittedly, I will have to review Meissner’s work moving future. Meanwhile, I rewrite the general duality invariant action from the paper as it is quite nice to look at,

[ I_{0} equiv int dt e^{-Phi}frac{1}{n}(-dot{Phi}^{2} + sum_{k=1}^{infty}(alpha^{prime})^{k+1}c_{k}tr(dot{S}^{2k}) + multi-traces ]

This action, it turns out, encodes the complete $alpha^{prime}$ corrections. Moreover, these corrections are for cosmological backgrounds. The aim is thus to classify the higher-derivative interactions, and then derive general cosmological equations to order of $alpha^{prime}$. Upon deriving these equations of motions, the key point to highlight is how one can then go on to construct non-perturbative de Sitter solutions.

Perhaps it remains to be said that de Sitter vacua are a hot topic in string theory. In the end, Zwiebach finds a non-perturbative de Sitter solution – that is, non-perturbative in $alpha^{prime}$, which is very cool.

After only a few minutes with this paper, I am eager to dig a bit deeper. Needless to say, it is intriguing to think more about the string landscape in the context of its study.

AdS/CFT and the cosmological constant problem – Kyriakos Papadodimas

This 2011 paper by Kyriakos Papadodimas is on the cosmological constant problem in the context of the dual conformal field theory. The preprint version linked is several years old, and so I should preface what follows by stating that I have not yet had a chance to review all of the papers that directly proceed it. Indeed, this paper ends with the allusion of a follow-up, and I look forward to reading it.

Another comment for the sake of historical context, this paper on the CC problem precedes by almost two-years the publication of two important papers by Papadodimas and Suvrat Raju, the focus of which is on the black hole interior and AdS/CFT.

Now, as for the paper linked above, one of the key questions entertained concerns whether holography theory can teach us anything new about the cosmological constant problem. Of course the CC problem – or what is sometimes dramatically referred to as the vacuum catastrophe – represents one of the big questions for 21st century physics. It is one which Prof. Padilla and I sometimes talk about.

In approaching Papadodimas’ paper, the following question may be offered: is it possible that Nature’s mechanism for apparent fine-tuning may reside in a fundamental theory of emergent fields?

In the first section or two, Papadodimas builds an analogue of c.c. fine-tuning in terms of 4-point functions. We come to understand that, in the CFT, 4-point functions can more easily be translated into correlation functions. Using Witt diagrams, which are like Feynman diagrams, one takes the external points to infinity – or, in this case, to the conformal boundary of AdS. (The basics of this approach will most certainly be studied in my series of blog posts on string theory, beginning with the collection of notes on Conformal Field Theory). The advantage, moreover, is that the 4-point functions become Witt diagrams, and, then, through the AdS/CFT correspondence, one can relate these to the correlators on the boundary.

So this is the plan, which, Papadodimas argues, will enable the study of fine-tuning in the bulk in terms of fine-tuning within the dual CFT.

In proceeding his study, Papadodimas gives quite a few comments as to how we might think of fine-tuning as it manifests in the dual CFT. The picture here is really quite interesting; because if, as noted above, we can express fine-tuning in terms of graviton correlation functions, what I have so far failed to mention is that these correlation functions of gravitons are duel correlation functions of the stress-energy tensor on the boundary.

The greater goal – or ideal outcome – would be to study fine-tuning on the boundary, particularly how it is resolved. What we come to learn is how the large N expansion, the concept of which one may be familiar from their study of QFT, plays an important role. Moreover, one of the interesting suggestions is how the $frac{1}{N}$ suppression of correlators, which relates to the large N gauge theories in the t’Hooft limit, gives us early hints of fine-tuning.

Now, one of the questions I had early on pertained to the comment that it would not be difficult to split up a correlator into a sum of terms, and then expose this broken correlator to some interpretation of fine-tuning. That is to say, nothing is stopping us from breaking up the correlator into a sum of terms and then pointing at this object and saying, ‘see we have fine-tuning!’. But Papadodimas is a smart guy, and he already knows this.

To be clear, and to summarise what I am talking about, part of the strategy explained by the author early on includes expanding “the correlators into sums of terms, each of which corresponds to the exchange of certain gauge invariant operators in intermediate channels i.e. into a sum of conformal blocks. While the sum of all these contributions is suppressed by the expected power of 1/N, individual terms in the sum can be parametrically larger, as long as they cancel among themselves” (p.2). These cancelations are what we’re looking for, because, the interpretation of this paper is that they are in a way an expression of bulk fine-tuning in the dual CFT.

Papadodimas quickly extinguished my one concern, however; because we are not arbitrarily breaking up the correlator. He recognised that one may perform such an act artificially, which wouldn’t have much meaning. Instead, he searches for and performs an canonical procedure. I found this very interesting to follow. What we end up with is an expansion in conformal blocks that looks like this,

[ langle tilde{T}(x_{1})tilde{T}(x_{2})tilde{T}(x_{3})tilde{T}(x_{4})rangle_{con} = sum_{A} mid C_{TT}^{A}mid^{2}G_{A}(x_{1}, x_{2}, x_{3}, x_{4}) ]

Then, following a brief review of holography, some comments on hierarchy vs. fine-tuning, and also a study of important cutoffs, the idea of a sort of dual picture emerges, in which the physical description may appear natural in one picture – such as in the $frac{1}{c}$ expansions discussed in the paper – and then finely tuned in the second picture.

There is too much to summarise in such a small space. So I will just pick up on how, in the dual CFT, I found it intriguing that, should the arguments hold, the double operator product expansion of the correlator written above displays partial sums over the conformal blocks that cancel. As alluded early, in the context of the particular CFT, the interpretation is that the conformal blocks appear fine-tuned. But the mechanism behind this exhibited fine-tuning is not fine-tuning as I read it, such that it becomes clear that in large N gauge theories this expansion of the conformal block is natural. The naturalness of the expansion is consistent with what we know of the large $frac{1}{N}$ expansion in the t’Hooft limit.

Papadodimas closes with the explanation that in the context of a CFT with a large central charge, there is a notable c.c. problem in the holographic dual. As I focused on, it is found in this CFT a semblance of fine-tuning in the correlator blocks of the $frac{1}{N}$ expansion. However this may be interpreted, the interesting idea is that: “this fine-tuning may be visible when the correlators are expressed in terms of the exchange of conformal primaries, it may disappear if the correlators are written in terms of the fundamental fields of the underlying QFT” (p.32).

If this is in any way true, it raises some very intriguing questions and raises some very interesting potential pathways for future study, the likes of which Papadodimas spells out in explicit terms. The suggestion of the use of a toy theory, as opposed to a non-perturbative study, makes me think of the SYK model. It will be curious to follow-up on the papers that directly follow.

One last closing comment: I am excited to say that Papadodimas will be one of the lecturers at a special engagement I will be attending this summer at the University of Madrid. The engagement, SIFTS 2019, comprises of two weeks of discussion, review, study and brainstorming on issues in string theory (and fundamental physics broadly). Papadodimas will be lecturing on the black hole interior and AdS/CFT. It shall be brilliant.

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Physics Diary

Some papers I’ve recently read, including a new one from Susskind (12/05/19)

I thought I would experiment with a new type of weekly post. The premise is simple: I collect and describe some of the papers that I have read that are my favourite or that standout for whatever reason. The papers could be from the last calender week or fortnight (we’re working with loosely defined parameters). Or they could simply be papers I read some time ago that have been on my mind as of late.

I imagine these posts will be primarily research based. There will be heavy focus in string theory, and certainly on new research. But I am also one for obscure papers, and for reading across other areas, which means that one should expect an occasional mixture. Within this mixture, also expect some pedagogical literature to be flagged. The papers listed will be old and new. I am also still trying to figure out the balance between specialist and pedagogical language as a blogging principle, so I imagine these posts will be a product of trial and error during the fledgling stage production.

Without further ado, and in no particular order:

Complexity and Newton’s Laws – Leonard Susskind

Last week Leonard Susskind uploaded a new paper to the archive. It is was originally forwarded to me by my Professor, as it relates to some interesting questions which may form the basis of a future research project in holography theory.

In this paper, Susskind follows recent efforts to explore the holographic origin of gravitational attraction with a study of the size-momentum correspondence. We’re working in the SYK model here, which simplifies things rather nicely. Susskind argues that Newton’s laws are a consequence of, or can be retrieved from, this improved version of the size-momentum correspondence.

A theory of gravity on the boundary is something I’ve started thinking about in recent time. So this paper was enjoyable to read. I take inspiration from the Susskind’s efforts here, and also from the surrounding literature, insofar that I have more or less being intuiting my way to the same domain of enquiry. That is always nice. I will have to dig a bit deeper into some of the background literature (like on the CV correspondence, etc.). The notion and treatment of complexity in this paper is also intriguing (I had a time where I was obsessed with complex systems, generally, and I maintain interest in the study of their evolution).

One last thing before moving on: the notion that entropy may behave like observables over a code space is super intriguing.

Modular invariance and orbifolds – Stefan Huber

This paper offers a survey of the some of key tools and ideas pertaining to
modular invariance in string theory. It uses Di Francesco et al., “Conformal Field Theory” (1997), as its main resource. The contents of discussion are also those covered by Polchinski in Volume 1. In any case, the paper offers a useful review of modular transformations on the torus, focusing particularly on the constraints of modular invariance in the context of CFTs defined on the torus.

Lectures on Two-Loop Superstrings – Eric D’Hoker and D.H. Phong

Lecture notes by D’Hoker and Phong from 2002. Though some time has passed, I’ve found these notes helpful. The main attraction is their review, and treatment of, multiloop superstring perturbation theory. The emphasis to start is a first principles construction of a two-loop superstring measure on moduli space. Much of the discussion, and certainly also the techniques on display, are useful to review. The section on the vanishing of the cosmological constant (CC) is interesting, as is the chapter on compactifications and the CC. The subtlties of chiral splitting is something I need to look into more thoroughly. In fact, this paper is filled with facts and assumptions that I need to still need think about.

Graviton Dominance in Ultra-High-Energy Scattering – G. ‘t Hooft

I’ve been thinking a lot about graviton scattering and more generally about the uniqueness of solutions in string theory. There is actually a lengthy story to be told here, including a motivating discussion with my professor, which relates to what is below.

In this paper from 1987, ‘t Hooft studies high-energy scattering of two particles in which the energy is so great that the gravitational field of the particle comes into direct focus. Here, ‘t Hooft describes how this field consists of a “shock wave”. The physics and the calculations are interesting, and I recommend going through it. But the main reason I was thinking about this paper again in recent days can be found on p.62. ‘t Hooft notes that there is a rather striking similarity between the scattering amplitude computed and the well-known Veneziano amplitudes. Anyone familiar with string theory will know about the Veneziano amplitudes or will be on course to become familiar with them. The similarity is most curious, indeed! It is interesting to think about from a number of perspectives.

Anything by ‘t Hooft is brilliant. He’s one of my favourite physicists, and I’ve said before that I hope I will get to meet him one day.

Magic: The Gathering is Turing Complete – Alex Churchill, Stella Biderman, and Austin Herrick

To end with something off track, and also quite fun, a new paper was uploaded to the archive which seeks to argue that optimal play in Magic: The Gathering (MTG) is at least as hard as the Halting Problem. As an avid player, one thing that stands out about Magic is that the gameplay has incredibly high variance. This high variance almost renders the notion of optimal play to be a sort of platonic and not quite attainable concept which one nevertheless continuously strives to achieve. It is generally what makes MTG thrilling. One strives for optimal gameplay through a mixture of logical and well-reasoned decision making. Each choice, or play, tends to matter. Pattern recognition is essential. However, even the best of players, who, at any given time, may be playing one of the more optimally constructed decks given the format and the current meta, will inevitably suffer a series of loses. One reason, as the linked paper argues, is that deterministic outcomes in MTG are essentially non-computable. In some sense, as one necessarily strives for as optimal of gameplay as possible, there is some definite limit to which one can effectively configure a logical and well-reasoned structure for decision making; because, by design, the game is configured to produce a complex and even mildly chaotic system of variables and inconsistencies.

As I have yet to dissect the paper, including the methodology, I will reserve further comment on the author’s study. Given some free time, it will be intriguing to go through it systemically.

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