Physics Diary

Metastable de Sitter Solutions in 2-dimensions

To obtain stable de Sitter solutions in string theory that also avoid the Swampland is no easy task. It is difficult for many reasons. To give one example, from 10-dimensional Type IIA* and Type IIB* theories, one finds de Sitter related solutions but they come with ghosts that give a negative cosmological constant. Full de Sitter spacetime is a Lorentzian manifold that is the coset space of Lorent groups, dS^d \simeq O(d,1)/O(d-1,1), the cosmology of which, as many readers will likely be familiar, is based on a positive cosmological constant. Hence, in this example, we obtain the wrong sign. Many other examples can be cited in reference to the difficulty we currently face.

To speak generally, it has gotten to the point historically that one possibility being considered is that string theory may simply conspire against de Sitter space – that is, there is some deep incompatibility between our leading theory of quantum gravity and de Sitter vacua. The No-de Sitter Conjecture is an example of an attempt to formalise such a logical possibility, motivating beyond other things the need to understand better the structure of vacua in string theory. This conjecture is by no means rigorous, but it is supported by the fact that historically de Sitter solutions have been elusive. Having said that, one must be cautious about any claims regarding de Sitter constructions, either from the perspective of obtaining de Sitter solutions or from arguments claiming de Sitter belongs to the Swampland. Our present theory is still very much incomplete and there is a lot of work to be done, and I think it is fair to suggest that the fate of de Sitter in string theory is still uncertain. As an expression of opinion, I would currently say that it is more likely de Sitter will come from something much more fundamental than some of the current strategies being proposed. It should also be kept in mind that there may be a larger issue here. Quantum gravity in de Sitter is fundamentally difficult beyond pure string theory reasons. Indeed, it is possible that de Sitter is simply unstable in quantum gravity – that it is a space that simply cannot exist quantum mechanically. For instance, think of complications in QFT in curved space. Indeed, there is a lengthy list of authoritative papers that one may cite when it comes purely to perturbative quantum gravity (see, for example, this page for further references).

Image: de Sitter geometry courtesy of ncatlab.org

Given the sociology and history, when a new paper appears claiming to have found de Sitter solutions, one takes notice. That is precisely what happened last week when Miguel Montero, Thomas Van Riet, and Gerben Venken uploaded to the archive their recent work claiming to have found metastable de Sitter solutions in lower dimensions. More precisely, these parametrically controlled solutions appear when compactifying from 4-dimensions to 2-dimensions, particularly as a result of some clever work that invokes abelian p-1-form gauge fields to stabilise the runaway potential giving in general dS_{d-p} \times S^{p} solutions. However, not all solutions are stable with controlled saddle points. Indeed, the authors find D - p > 2 solutions to be generically unstable. On the other hand, the instability in the homogeneous mode disappears when D - p = 2. In this 2-dimensional case, the solutions relate to the near horizon dS_2 \times S^2 geometry of Nariai black holes. It is also worth pointing out that these solutions are not in string theory, but the work highlights some interesting implications and raises some important questions in quantum gravity more generally, including also when it comes to the Swampland.

If I am not mistaken, I think I remember seeing similar kinds of ideas in studies of dS / CFT where people look into cases of rotating Nariai black holes maximally large at dS_2 \times S^2. The geometry is generally interesting because in that some Nariai constructions with quintessence generalise quasi-de Sitter solutions, utilising for example decompositions from 4-dimemensions to things like 2-dimensional dilaton gravity theory (this may make one recall SYK models), one will often find descriptions of the geometric space as ds^2 = \Gamma(\theta) [-d\tau^2 +\cosh^2 \frac{\tau}{l} d\psi^2 + \alpha(\theta) d\theta^{2}] + \gamma(\theta) (d\phi + k \tanh \frac{\tau}{l} d\psi)^2 which we can think of in terms of an S^2 fibered over a dS_2 base space. I kind of see it as a bit of a playground useful to experiment with and probe. In section 2 of the de Sitter paper, one can read a short overview of Nariai black hole solutions of Einstein-Maxwell theory as a precursor to the main study.

I’ve only had time to skim through the Montero et al. paper, so I still need to give it careful attention. But on quick glance I noticed a number of interesting calculations, not least the derived inequality (p -1) \mid \frac{V^{\prime}}{V} \mid \leq \mid \frac{f^{\prime}}{f} \mid that states how, in order for a dS_{d-p} \times S^{p} saddle point to exist, this constraint that relates the gauge kinetic function of the p−1-form field and the potential must be satisfied. This is an example of one of a handful of points to which I would like to go back and think about more deeply. Additionally, it seems an open question whether, if we can obtain controlled lower-dimensional de Sitter solutions from runaway potentials, do such approaches and constructions fully escape the Swampland? It is worth reading their paper with that in mind.

As alluded at the outset, one should always approach any claims about de Sitter with a healthy dose of scepticism. At the same time, van Riet is a physicist who I admire, because he is one of a number championing the need for greater mathematical rigour in string theory and quantum gravity moving forward. So, for me, this paper immediately comes with some weight and authority. If one thing is certain today, at least from my current vantage, it is the need for thoughtful caution and careful mathematical analysis; and it appears that on a few occasions the authors also stress this point in their analysis.

Standard
Physics Diary

Generalised Geometry, Non-commutativity, and Emergence

The project I am working on for my dissertation has to do with the notion of emergent de Sitter space. Of course an ongoing problem in string theory concerns whether asymptotic de Sitter spacetime can exist as a solution, and needless to say this question serves as one motivation for the research. With what appears to be the collapse of KKLT (this is something I will write about, as from my current perspective, the list of complaints against KKLT have not yet seemed to be satisfactorily answered), this academic year I wanted to start picking at the question of perturbative string de Sitter vacua from a different line of attack (or at least explore the possibility). Often, for instance, we approach de Sitter constructions by way of a classical supergravity approach with fluxes, non-geometry, or KKLT-like constructions which add quantum effects to stablise the moduli. One could also look at an alternative to compactification altogether and invoke the braneworld formalism. But, as it is, I’ve not been entirely satisfied with existing programmes and attempts. So the question over the autumn months, as we approached the winter break, concerned whether there was anything else clever that we can think of or take inspiration from. I’m not comfortable in divulging too much at this time, not least until we have something solid. Having said that, in this post let’s talk about some of the cool and fun frontier mathematical tools relevant to my current research.

For my project the focus is on a number of important concepts, including generalised and non-commutative geometry. Within this, we may also ask questions like whether spacetime – and therefore geometry – is emergent. Sometimes in popular talks, one will hear the question framed another way: ‘is spacetime dead?’ But before getting ahead of ourselves, we may start with a very well known and familiar concept in string theory, namely T-duality. Indeed, one motivation to study generalised geometry relates to T-duality, particularly as T-duality expresses how a string experiences geometry. For example, one will likely be familiar with how, in string theory, if we consider the propagation of a string in spacetime in which one spatial dimension is curled up into a circle, the idea is that when we compactify a dimension (in this case on a circle) we modify the string mass spectrum. Less abstractly, take some 10-dimensional string theory and then compactify on a circle $S^{1}$ of radius $R$. The string moves along the circle with the momenta quantised such that p = n / R (n \in \mathbb{Z}). When compactifying the 10th dimension we obtain for the compactified direction, \displaystyle X_{(s)}^{d} (\tau, \sigma + 2\pi) = X_{(s)}^{d}(\tau, \sigma) + 2 \pi \omega R, where we now have winding modes. This is because, as one will learn from any string theory textbook, the string winds around the circle with coordinate X. We can thus write the statement \delta X = 2\pi R m (m \in \mathbb{Z}). In this basic example T-duality is the statement R \rightarrow \frac{\alpha^{\prime}}{R} with n \longleftrightarrow m. The winding modes that appear are of course a deeply stringy phenomenon. And what is interesting is the question of the generalisation of T-duality. Moreover, how might we think of string geometry in such a way that T-duality is a natural symmetry? Generalised geometry was largely motivated by this duality property, such as in the work by Nigel Hitchin. The basic mathematical statement is that the tangent bundle TM of a manifold M is doubled in the sum of the tangent and co-tangent bundle TM \oplus T \star M. In this formalism we also replace the Lie bracket with a Courant bracket, which we may write as something of the form [X + \xi, Y + \eta]_{C} = [X, Y] + L_{X} \eta - L_{Y}\xi - \frac{1}{2} d(i_{X} \eta - i_{Y}\xi) such that X \xi, Y + \eta \in \Gamma (TM \oplus T \star M). In physics, there is also motivation to ask about the geometry of spacetime in which strings propagate. For instance, the existence of winding modes and the nature in which T-duality connects these winding modes to momentum hints that perhaps the fundamental geometry of spacetime should be doubled. This idea serves as one motivation for the study and development of Double Field Theory, which is something the great Barton Zwiebach has been working on in recent years and which uses the SO(d,d) invariant formalism (see his lecture notes).

Additionally, in these areas of thinking, one will often also come across notions like non-geometry or fuzzy geometry. Sometimes these words seem used interchangably, but we should be careful about their meaning. For instance, non-geometry possess a number of characteristics that contribute to its formal definition, one being that it refers strictly to non-Riemannian geometry. Furthermore, we are also speaking of non-geometry as non-commutative geometry [X_{i}, X_{J}] \approx \mathcal{O}(l_{s}) as well as non-associative geometry [X_{i}, X_{J}] X_{k} \approx (l_{s}). One of many possible ways to approach the concept in this regard is to think quantum mechanically. If General Relativity is a very good approximation at long distances, in which we may think of smooth and continuous manifolds; at the smallest scale – such as the string scale – there are important hints that our typical understanding of geometry breaks down.

We will spend a lot of time on this blog discussing technicalities. For now, I just want to highlight some of the different formalisms and tools. In taking a larger view, one thing that is interesting is how there are many similarities between non-commutative and non-associative algebra and generalised geometry, fuzzy geometry, and finally ideas of emergence and a generalised quantum mechanics, although a precise formulation of their relation remains lacking. But this is the arena, if you will, which I think we might be able to make some progress.

As for my research, the main point of this post is to note that these are the sorts of formalisms and tools that I am currently learning. The thing about string theory is that it allows for is no sharp distinction between matter and geometry. Then to think about emergent space – that spacetime is an emergent phenomena – this infers the idea of emergent geometry, and so now we are also starting to slowly challenge present comforts about such established concepts as locality. When we think about emergent geometry we might also think of the structure of perturbative string vacua and ultimately about de Sitter space as a solution that escapes the Swampland. There is a long way to go, but right now I think in general there is an interesting line of attack.

For the engaged reader, although dated the opening article by Michael Douglas in this set of notes from the 2002 summer school at the Clay Mathematics Institute may serve an engaging introduction or overview. A basic introduction to some of the topics described in this post can also be found for instance in this set of notes by Erik Plauschinn on non-geometric backgrounds. Non-commutative (non-associative) geometry is covered as well as things like doubled geometry / field theory. Likewise, I think this paper on non-associative gravity in string theory by Plauschinn and Ralph Blumenhagen offers a fairly good entry to some key ideas. Dieter Lüst also has some fairly accessible lecture notes that offer a glance at strings and (non)-geometry, while Mariana Graña’s lecture notes on generalised geometry are a bit more detailed but serve as a basic entry. Then there are Harold Steinacker’s notes on emergent geometry from matrix models and on non-commutative geometry in relation to matrix models. Finally, there are these lecture notes by Maxim Zabzine on generalised complex geometry and supersymmetry. This is by no means comprehensive, but these links should at least help one get their feet wet.

Maybe in one of the next posts I will spend some time with a thorough discussion on non-commutativity or why it is a motivation of Double Field Theory to make T-duality manifest (and its importance).

**Cover Image: Study of Curve Folding [http://pr2014.aaschool.ac.uk/EMERGENT-TECHNOLOGIES/Curved-Folding-Workshop]

Standard
Physics Diary

A Year in Review

Hello everyone! Today’s post will be different than the usual string theory focused engagements. Normally I would be planning to write a new entry explaining a piece of computation, uploading a string note from one of my notebooks, or organising an essay on an important physics topic. However, I have been so busy with my research that there just has not been enough hours in the day to maintain a constant flow of posts. This should change soon, and I am happy that I already have a lot written and waiting to be edited. The real difficulty has to do with the fact that I don’t like clickbait articles and I have no interest in providing watered down popular guides. The goal is to contribute to making complex subjects like quantum gravity accessible without losing conceptual and technical detail, given that accessibility here also implies an engaged reader wanting to study and understand the subject at hand. The thing about string theory is that it demands one’s full attention. If I am to maintain a research and general string theory blog I would prefer that every entry, whether based on textbook content or frontier research questions, is reasonably substantial and certainly thorough so that it may be beneficial to the reader. I’ve had some great feedback on my articles and notes so far, which I have found both affirming and motivating. I’ve also received some nice feedback about the odd personal post. Slowly over time the process of sharing more personal updates and keeping a personal physics diary is something with which I have become more comfortable. I thought that in this post it might be nice at the turn of the calendar to write about the last year – a year in review of sorts.

It is actually fitting that I would write such a post on this of all days. It so happens that this morning I received a formal PhD offer! Moving on to a PhD is a reality to which I have been orientating myself for some time. But no matter how many times I have thought about it and have tried to prepare for it, especially in terms of my current research focus, when I received my offer this morning it still felt as though everything was happen very quickly. Objectively, I suppose a lot has happened at a relatively rapid pace. Just last year I was an undergraduate being academically accelerated to a full-time research year. I am now only a few months into that research year, planning my MRes dissertation and celebrating the fact that I have been formally offered a PhD position in quantum gravity.

In the time between first arriving at the University of Nottingham for the start of the 2018/19 academic year and the present, I couldn’t possibly list all of the things that I have studied. It has been intense. It started with a complete and comprehensive review of string theory, rederiving the whole of bosonic string theory for my own notes before moving on to superstring theory. In just that time I also taught myself conformal field theory, I had to brush up on quantum field theory, and I had to learn an assortment of important tools ranging from BRST quantisation and the Faddeev–Popov approach to computing string scattering amplitudes, learning about string compactifications, and then trying to cram everything I could about orbifolds and D-branes. Then, as we pushed further into 2019 I moved to superstrings and string geometry while also learning long lists of other physical concepts and mathematical tools in addition to continually working to sharpen my existing knowledge. But what stands out the most about the last year is the Swampland – in fact, I think for me it was the year of the Swampland. It is an absolutely fascinating space of research and I very much enjoyed my time in Spain attending a PhD summer school reviewing things like the Distance and Weak Gravity Conjectures. What also stands out from that first arrival at university to the present time is learning Calabi-Yau manifolds and related geometry; pure spinors, which I started studies while at the summer school in Spain; and then last autumn having to catch up on advanced gravity theory and the braneworld formalism. And now here we are with all of this stuff and more as I work to learn non-commutative geometry and contemplate the nuances of string de Sitter solutions.

Although this really only scratches the surface of an entire year, these descriptions provide some sense of range. I am by no means an expert in any of it, to be frank. Going back to some old calculations I often need to remind myself of certain first principle definitions, like when recently uploading my CFT notes. Typically, it seems like a day is equivalent to a week, as there is just so much to learn. Of the material covered so far and of all the concepts, tools, and theorems studied I can say that one thing I’ve learned is that, in terms of a definitive and coherent theme to research in string / M-theory, in this field every day there seems a new mountain that needs to be climbed. A few days ago it was more to do with gauge-gravity duality and matrix models, which I am learning. Today, it is ‘axilaton’ models. No matter how many mountains one seems to climb, the nature of frontier physics research seems to guarantee tomorrow there will be many new summits.

The last year has also been a momentous one for me personally. Having Asperger’s (Autism Spectrum Condition), which has been described clinically as severe, I experience many challenges in basic day to day life. This also includes assistance with functioning and needs. Now being a formal member of university also presents many additional challenges. What I can that also stands out about 2019 is that I am also ever so grateful to be at a school with tremendous support staff. Being able to participate in a formal academic institution, thanks to the support I receive, has opened up so many new opportunities that would otherwise not have been possible. It has probably been one of my greatest years. Growing up with little support, perhaps this story will serve as an example of how important proper support is for people like myself – or anyone for that matter – to succeed. It reminds me that in the future I would like to write more about living with Asperger’s. It is something I have tried to write about in the past, however successfully or unsuccessfully. But I think the message is also more general – everyone needs support to be themselves, to pursue their interests, and ultimately to self-actualised in a healthy and positive way.

Thinking about the future is something I find difficult. In moving from the past to the present, I’m not quite sure how to project forward in time. What I know for certain is that the next year should be a productive one, given the current trajectory. At present I am planning my MRes dissertation and thinking about possible PhD projects, with the troubles of string de Sitter vacua very much an interest. As I have written before, non-perturbative theory seems to be my point of entry into string research, instead of computing scattering amplitudes or focusing on SCFTs for example. I am thankful to be working under my brilliant professor, Tony Padilla, who is encouraging in this regard and also with my other interests, such as for instance exploring non-geometric backgrounds and matrix models. Every discussion we have is a stimulating, and I enjoy going to his office every week with new ideas to share. Non-geometry will be a lasting topic. One motivation for it, of course, has to do with the no-go theorems for supergravity, which, in turn, relates to questions about the sigma-model prescription that gives geometric vacua. Moreover, there are a number of suggestions in string / M-theory that a perturbative string vacua will not be geometric in the typical sense. Instead, it will be non-geometric. What this means, and to explore some of the mathematical/physical intuition as to why we might think about non-geometric vacua, I think this would make for a terrific future post. Additionally, if a further consequence of strings is that geometry and gravity may even be emergent concepts, and that there is some hint at possibly the idea of non-commutative (non-associative) theory of gravity, then I think another principle of direction is to try and investigate how these are related. We could also ask about how non-perturbative vacua and non-geometric vacua are related, if in fact a formal relation may be defined.

These sorts of topics and questions I suspect will define much of my research year in 2020. But, then again, with each new mountain there sometimes also appears an exotic new valley waiting to be explored. I think I shall take it day by day.

Standard
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 has 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 pure theory 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 how I’ve spent my time in the last 12 months or so.

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 must start 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. It is also a channel that allows me to drive ever closer to the foundations of string theory and numerous relevant pertinent questions.

So the good news is that, in the sea of frontier physics and with endlessly interesting possible research topics, I have managed to 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. I will reserve a separate article for a detailed explanation. 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 this academic year. An example of possible research activity 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, prior to meeting my professor.

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’.

Standard
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.

Standard