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
Philosophy and General Reading

Review: Bertrand Russell’s ‘In Praise of Idleness’

In Praise of Idleness and Other EssaysIn Praise of Idleness and Other Essays by Bertrand Russell
My rating: 4 of 5 stars

View all my reviews

To some, or perhaps to many, it may seem a radical idea: idleness. But for the great British logician, mathematician, and Nobel laureate Bertrand Russell, idleness is seen as a historically rooted concept which ties intimately together the bonds of labour, leisure, and the prospect of human rationality. Or, at least that is my reading of his famously titled composition, ‘In Praise of Idleness’.

So, what does Russell mean by ‘idleness’? In some sense, it infers a socially organised definition of time that is economically independent of professional labour, in which one may instead expend their energy to fulfil personally meaningful pursuits. This could be, for example, a time for a person to explore painting or to explore a scientific pursuit or any number of interests. In some bodies of literature, such projects are called ‘existential projects’ to convey the personalisation of their meaning in one’s life. One may also call them ‘special interests’. In this sense, one can think of idleness simply as being the economically independent pursuit of a subject, activity, or quality for no other reason than it evokes the state of personal interest. Study for study sake, or a painter to paint without the pressure of starving – these are the sorts of examples that Russell evokes.

Russell_In Praise of Idleness

Idleness should thus not infer or be confused with one’s being disinclined to work or with simplistic views pertaining to individual laziness. Idleness should also not be seen as ‘the root of all evil’, as the idiom would have it. If we are to follow Russell’s arguments, idleness has substantial roots in positive human traits, such as curiosity, exploration, and invention. We also read how the notion of idleness is based on ethical, moral and empirical economic arguments. For Russell, social consumption can mean something very different, both existentially and socially, and thus humanistically. He also speaks of economic production and the way in which work and leisure cycles could generally mean something altogether more philosophically transformed in conception, particularly in terms of the meaning of leisure and its tradition and practical cultural configuration.

It is interesting to consider, on that note, how for thousands of years human beings have established traditions of celebrating different sorts of festivals – Judeo-Christian, Pagan, and so on. Think, for instance, of midwinter festivals based on the solstice or on religious themes. With these traditions follows also a deep historical relation between festivity and work. The festival represents, to frame it in terms of economic history, an interruption of daily labour cycles, with its concept rooted primarily in principles of free time for enjoyment [1]. Thinking of this, it is also interesting to recall that, using Christmas as an example, it was during the Victorian era that a formal socioeconomic relation developed between festivity, worker rights, and the commercial profit motive, particularly as middle-class families were afforded time off work with the financial means for surplus consumption. But if festivity and leisure – or idleness – are intricately related with labour by their very definition, and thus with economics, Russell’s account would seem to carry a certain diametric opposition to work patterns that exhaust the possibility of what he describes as energetic leisure.

In this sense, I read Russell’s essay as having some classical enlightenment motivation. Thoughtfulness – indeed, the time to practice thought and to explore intellectually – this seems a theme to Russell’s social philosophical view of which an advanced and aspiring rational society should strive to achieve. In other words, if idleness is a positive human experience, one which supports or fosters the individual subject to flourish rationally and, perhaps, self-actualise existentially, Russell ties this concept with the possibility of continued self-education and self-betterment, among other things. At the same time, while he celebrates the concept and experience of idleness, he also laments the loss of its broader social-economic and cultural realisation. It is argued that leisure time is expunged of idleness much as in the present-day example of Christmas, which is hyper-commercialised and seemingly increasingly filled with passive entertainments, as active energies are instead exhausted by work, intensely driven consumer cycles, and various other contemporary social behavioural patterns rigidified in such a way to maintain systemic mores. Russell’s arguments are based on traditional views of social-economic class structures, and he seems to suggest that the logic of social economy has been skewed; contemporary societies have in some ways lost sight of the meaningful idea of social production and the social purpose of consumption that may foster a more enlightened and rational society.

For these reasons, we read how with more energetic and thoughtful leisure one would then be better able to enjoy pleasures in which it was possible to take an active part. The central thrust of Russell’s argument in this regard is not so different than in present-day concepts of economic democracy and automation, in which in advanced technological society it is argued individuals should be increasingly afforded the freedom from necessary labour in order to pursue the many positive possibilities that life has to offer, including education and learning.

Reading his essay, I was reminded of a few historical examples. Think, for instance, about the development and evolution of writing and of our early mathematical ideas – a history that is intimately entwined with the genesis of civilisation. A good example comes from the ancient Babylonians. To Russell’s larger point, the early development of mathematics, much like writing, can be seen to be owed to the economic development of agriculture; because with agriculture one result was increased freedom from the precariousness of sustenance living in which people were then allowed more free time, with greater access to resources. As new technologies were conceived, and human pursuit was increasingly freed from the limits of basic survival to expand beyond that which was unavailable to hunter-gathers, the time available to explore, experiment with, and create things like writing became possible. The study of mathematics could also be pursued and formalised.

Indeed, to offer another example, the entire history of physics is riddled with such stories, like Michael Faraday playing with his magnets on a park bench in London or Issac Newton watching apples fall from trees, contemplating the nature of gravity. To the point of anthropologists and biologists who study human play, as another example that we may interpret in the frame of idleness, there is an argument to be made that what Russell is describing is in fact a fundamental biological and cognitive feature of universal human experience that is very much tied to inventiveness.

At this point, we may enter into various complex social, economic, and political arguments. Instead, as there are already many terrific reviews of Russell’s essay, both positive and critical, to close this discussion I instead want to focus on two things that struck me when recently rereading ‘In Praise of Idleness’. One playful thought was the potentially interesting applications in relation to a physics of society and of human beings, particularly regarding energetics. This has to do with the study of energy under transformation, and one may think of such transformation particularly between the individual and their labour under the fairly universal economic notion of the work-leisure trade-off. For the author, he argues that there is a sort of fetishisation of labour, especially manual labour, and he seems to want to argue that how we use labour energy is not efficient or optimised in the best ways. From the standpoint of a physics of humans and of society, it would be fascinating to see if some of his ideas are quantitatively grounded.

There are also many interesting economic points of consideration. First, it is worth noting that the contents of ‘In Praise of Idleness’ remain quite relevant today, given the resurgence of the idea of a shortened work week, especially in the UK and Europe. Some would argue that there is empirical evidence and many qualitative arguments about why the current configuration of work hours is not optimised for the benefit of both productivity and well-being [3, 4], supporting his view. Take a quantitative and qualitative view: work hours, commuting time, modern pressures of digital communication in which it is well studied that people also now routinely answer work emails in their leisure time – all of this and more matches data that substantiates the claim of an emerging culture of longer working hours [5]. Are the effects, psychological or otherwise, just as Russell observed or predicted?

On the other hand, inasmuch that the philosophical idea of idleness is tied with the economic argument of a shortened work week, how economically substantiated and viable is his argument? Some examples are as follows. If as a general rule of labour economics working more hours correlates with higher hourly rates of pay, and if as a general rule from a behavioural perspective higher rates of pay are one motivation for people in their social and economic life, then one may ask whether an economic conception of idleness is realistic. For instance, if the introduction of a shortened work week were to correspond to a cut in pay, would people be dissuaded to pursue the possibility of increased free time for the benefit of obtaining greater earnings? As this is a question about human behaviour and behavioural regulators, and hence agency, it is not so easy to model. Having said that, we observed major strikes by German steel workers in 2018 that saw them secure the right to work less at the cost of a drop in weekly earnings – although this also came with flexibility where workers may work longer hours if they choose. Perhaps agency and choice matter in this discussion.

Another point one may consider is that some economists argue that a shortened work week will likely result in an increase in earnings differentials and inequality. If, in general, those who work longer hours have higher hourly earnings than those who work shorter hours, then one would expect increased disparity in the earnings structure. Additionally, in a UK study of the public sector, a shortened work week was approximated to cost upwards of £45 billion, depending on some modelling assumptions including no increase in productivity [6].

For these reasons, when it comes to recent debates in the UK, should a shortened work week be considered some studies have shown that this reduction in time would need to be matched with an increase in productivity during work hours. There are some empirical examples where businesses that trialled shortened work hours saw productivity remain as it was or effectively increase. Although the sample is small, the argument here is that work hours – maximal output of energy during those hours – is better optimised and maintained when shortened and focused. This ties into arguments about the inefficiency of work hours within the current model – that, in the sense of Russell’s energy economics, maximum productivity and the maximum time of energetic labour – i.e., maximum labour hours – do not contradict an increase in leisure. This is partly why I think a physical theory would be interesting, if we could even construct the appropriate Hamiltonian. In empirical sociology, observations of phenomena like ’empty labour’ may also serve as an illustration of what some interpret as the outdated nature of present economic values and of modern conceptions of work [7]. Do these types of studies offer clues or evidence as to how and why economy may be reconfigured in ways in which Russell seems to indicate? It would furthermore be interesting to learn, in using separation theorem or something similar in the study of labour economics [8], whether energetic leisure serves as a positive argument in the utility function of the individual.

The problem when it comes to these sorts of economic ideas and debates is that, in many cases, we require much more accurate modelling. Current mainstream economics is quite inept at understanding the reality of human behaviour. If one considers the likes of Paul Romer’s contentions on macroeconomics (as well as notable research by many other contemporary economists), it is not controversial to say that the current economic model and its established ideas may be challenged quantitatively and qualitatively [9]. From what I can see at the present time, some arguments are emerging about the need for an interdisciplinary theory. Much like a physics of society, in which it has been suggested that a physical theory of society will not achieve systematic and objective clarity without an interdisciplinary form of research [2], in economics agent-based models are issuing similar demands. If the challenge of an objective economics is to look for the cause of instabilities inside the system, some argue that this means that what one inevitably comes up against are the details of human decision making, which, in principle, drives one toward the randomisation of decisions based on both rational and irrational processes. But it also seems more than that: it’s about thinking systemically – not just about economic models in the abstract sense but also the incentive structure and the problem that economics faces in terms of an orientation of ethics. A trivial example is as follows: if a model fosters the pathology of a simplified self-preservation worldview, and if I am one of the only two bakers in town, am I not incentivised in some way to run the other baker out of town by whatever means justified by that very principle of my own preservation? The point to be drilled into is that in social-economic modelling, simplified arguments and narratives about agents engaging in free or purely voluntary trade can, and often do, end up moralising what are otherwise deeply systemic issues. I think, in certain respects, this takes us some way toward the message in Russell’s essay about realistic economic models.

Given the transformation of the incentive structure, perhaps energetic and thoughtful leisure would be realised as an important feature of a healthy system. In terms of Russell’s arguments, framed in a systems way, the benefits would be in reducing the social deficit of reason by maximising the subject’s energetic capabilities to reason, in which education may then be ‘carried farther than it usually is at present’, fostering the provision of ‘tastes which would enable a man to use leisure intelligently’. As I read it, his argument implies the enlightenment ideal that the individual would be better scientifically informed (eg., against myths); they would potentially be better politically informed about policies and more engaged when fulfilling their democratic duties; they would make thoughtful economic decisions; and, perhaps ideally, they would approach social debates with greater consideration and in greater awareness of their own biases.

References

[1] Josef Pieper, 1999, ‘In tune with the world‘. St. Augustines Press.

[2] Guido Caldarelli, Sarah Wolf, Yamir Moreno, ‘Physics of humans, physics for society’. Nature Physics Volume 14, p. 870. DOI:10.1038/s41567-018-0266-x.

[3] Will Stronge and Aidan Harper (ed.), ‘Report: The Shorter Work Week’ [http://autonomy.work/wp-content/uploads/2019/01/Shorter-working-week-final.pdf]

[4] Lord Skidelsky, ‘Report: How to achieve shorter working hours’ [https://progressiveeconomyforum.com/wp-content/uploads/2019/08/PEF_Skidelsky_How_to_achieve_shorter_working_hours.pdf]

[5] Peter Kuhn and Fernando Lozano, ‘The Expanding Workweek? Understanding Trends in Long Work Hours among U.S. Men, 1979-2006’. Journal of Labor Economics, 26 (2) April 2008: 311-43.

[6] Centre for Policy Studies, ‘The Costs of a Four-Day Week to the Public Sector’ [https://www.cps.org.uk/research/the-costs-of-a-four-day-week-to-the-public-sector]

[7] Roland Paulsen, 2014, ‘Empty Labor: Idleness and Workplace Resistance’. Cambridge University Press.

[8] Daron Acemoglu and David Autor, ‘Lectures in Labour Economics’ [https://economics.mit.edu/files/4689]

[9] Paul Romer, 2016, ‘The Trouble with Macroeconomics’. [paulromer.net/the-trouble-with-macro/].

**Cover image: ‘Woman Reading in a Landscape’ by Jean-Baptiste-Camille Corot.

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
Stringy Things

Notes on String Theory: Ward Identities, Noether’s Theorem, and OPEs

1.1. Example 1

In the last entry we derived both the quantum version of Noether’s theorem and the Ward identity given in Polchinski’s textbook. This means we obtained the idea of the existence of conserved currents and how Ward identities in general constrain the operator products of these same currents. Let us now elaborate on some examples. The solutions to these examples are given in Polchinski (p.43); however, a more detailed review of the computation and of some of the key concepts will be provided below.

We start with the simplest example, where we once again invoke the theory of free massless scalars. Following Polchinski, the idea is that we want to perform a simple spacetime translation {\delta X^{\mu} = \epsilon a^{\mu}}. The action will be left invariant under worldsheet symmetry. But as what we want to derive is the current, given what we have been discussing, this means we should pay special attention to the fact that we are required to add {\rho (\sigma)} to the above translation. Recall that we defined {\rho(\sigma)} in our derivation of the Ward identity. The important point to note is that, again, the action is still invariant and from past discussions we already understand {\rho(\sigma)} has a compact or finite support. From this set-up, let us now rewrite the action for massless scalars,

\displaystyle  S_{P} = \frac{1}{4\pi \alpha^{\prime}} \int d^2\sigma \partial X^{\mu}\partial X_{\mu} \ \ (1)

When we vary (1) we obtain the following,

\displaystyle  \delta S = \frac{\epsilon a_{\mu}}{2\pi \alpha^{\prime}} \int d^2 \sigma \partial^{a}X^{\mu} \partial_{a}\rho \ \ (2)

There is a factor of 2 from varying {\partial X} that gives us a reduced denominator. We have also used the identity stated in Polchinski’s textbook, namely {\delta X^{\mu}(\sigma) = \epsilon \rho(\sigma) a^{\mu}}, where we can treat {\epsilon} and {a^{\mu}} as constants and therefore pull them in front of the integral.

Before we can move forward, there is something we have to remember. Recall the path integral formulation from our last discussion, where we found the variation to be proportional to the gradient. The result is written again below for convenience,

\displaystyle  [d\phi^{\prime}]e^{-S[\phi^{\prime}]} = [d\phi]e^{-S[\phi]}[1 + \frac{i\epsilon}{2\pi} \int d^2\sigma J^{a}\partial_{a}\rho + \mathcal{O}(\epsilon)^2] \ \ (3)

If there is no contribution from the metric, then the measure in brackets becomes {- \delta S}. What this tells us is that the variation must be something like,

\displaystyle  \delta S = -\frac{i\epsilon}{2\pi} \int d^2\sigma J^{a}\partial_{a}\rho \ \ (4)

Now notice, in computing both (2) and (4) we may establish the following interesting relation,

\displaystyle  \partial^{a}X_{a} \partial_{a} \rho \frac{\epsilon a_{\mu}}{2\pi \alpha^{\prime}} = -\frac{i\epsilon}{2\pi}J^{a} \partial_{a}\rho \ \ (5)

The first step is to simplify. Immediately, we can see that we can cancel the {\partial_{a}\rho} terms on both sides,

\displaystyle \partial_{a}X^{\mu} \frac{\epsilon a_{\mu}}{2\pi \alpha^{\prime}} = -\frac{i\epsilon}{2\pi}J_{a} \ \ (6)

This still leaves us with a bit of a mess. What we need to do is recall another useful fact. In the last section we studied the invariance of the path integral under change of variables, which, at the time, enabled us to obtain Noether’s theorem as an operator equation. Explicitly put, we had something of the general form { \frac{\epsilon}{2\pi i} \int d^{d} \sigma \sqrt{g} \rho(\sigma) \langle \nabla_{a}J^{a}(\sigma) ... \rangle}. Notice that we have all of the ingredients. Given the Noether current is,

\displaystyle J_{a} = a_{\mu}J_{a}^{\mu}

We may substitute for {J_{a}} in (6) and then work through the obvious cancellations that appear, including a cancellation of signs. Once this is done, we go on to obtain the following expression for the current,

\displaystyle J_{a}^{\mu} = \frac{i}{\alpha^{\prime}} \partial_{a}X^{\mu} \ \ (7)

Which is precisely what Polchinski gives in eqn. (2.3.13) on p.43 of his textbook. Automatically, we can see our currents are conserved. And, of course, we are free to switch to holomorphic and antiholomorphic indices and we can do so with relative ease,

\displaystyle J_{a}^{\mu} = \frac{1}{\alpha^{\prime}}\partial X^{\mu}

\displaystyle  \bar{J}_{a}^{\mu} = \frac{1}{\alpha^{\prime}}\bar{\partial} X^{\mu} \ \ (8)

In the manner indicated above, we have successfully constructed the current following a spacetime translation. For this example the goal is to now use an operator to check the Ward identity and see if the overall logic is sound. What we require in the process are the appropriate residues, and to find these we will need to compute the OPEs. So to test some of the ideas from earlier discussions in the context of the given example.

Recall the formula for OPEs given the reverse of the sum of subtractions, namely the sum of contractions, as described in (11) of this post. In this formula recall that we have two operators which are normal ordered, : \mathcal{F}: and : \mathcal{G}: . These are arbitrary functionals of X and typically the range of X is non-compact.

Now, in Polchinski’s first example we consider the case where {\mathcal{F} = J_{a}^{\mu}} and {\mathcal{G} = e^{ikX(z, \bar{z})}}. In other words, we want to compute the product of the current and the exponential operator. As the product is normal ordered, there are no singularities and the classical equations of motion are satisfied. Instead, the singularities are produced from the contractions, or, in this case, the cross-contractions as {z \rightarrow z_{0}}. Furthermore, as a sort of empirical rule, it can be said that the most singular term in {\frac{1}{z - z_{0}}} comes from the most cross-contractions. And we should recall that we compute the cross-contractions by hitting our operators with {\delta / \delta X^{\mu}_{\mathcal{F}}} and {\delta / \delta X^{\mu}_{\mathcal{G}}}, respectively. Hence, from the master formula for cross-contractions,

\displaystyle  : \frac{i}{\alpha^{\prime}} :\partial X^\mu(z): :e^{ik  X(z_0,{\bar z_{0}})}: = \exp [- \frac{\alpha^{\prime}}{2} \int d^2 z_1 d^2 z_2 \ln \mid z_{12}\mid^2 \frac{\delta}{\delta X_{\mathcal{F}}^{\mu}(z_1, \bar{z}_1)} \frac{\delta}{\delta X_{\mathcal{G} \mu}(z_2, \bar {z}_2)}] \ \ :\frac{i}{\alpha^{\prime}} \partial X^\mu(z) e^{i k X(z_{0},\bar{z}_{0})}:

\displaystyle  = : \frac{i}{\alpha^{\prime}} \partial X^{\mu}(z)  e^{i k X(z_{0},\bar{z}_{0})} : - \frac{i}{2} : \int d^2 z_1 d^2 z_2 \ln \mid z_{12} \mid^2 \frac{\delta(\partial X^{\mu}(z))}{\delta X_{\mathcal{F}}^{\mu}(z_1, \bar{z}_1)} \frac{\delta  ( e^{i k X(z_0, \bar{z}_0)})}{\delta X_{\mathcal{G} \mu}(z_2, \bar{z}_2)} \ \ (9)

Note that for {\mathcal{G}}, which, in this case is {e^{ikX}}, it is an eigenfunctional of {\delta / \delta X_{\mathcal{G}} (z_{2}, \bar{z}_{2})}. Likewise, for for {\delta / \delta X_{\mathcal{F}}} we will end up with a delta function,

\displaystyle  = : \frac{i}{\alpha^{\prime}} \partial X^{\mu}(z)  e^{i k X(z_{0},\bar{z}_{0})} : - \frac{i}{2} : \int d^2 z_1 d^2 z_2 \ln \mid z_{12} \mid^2 \partial (\delta^{\mu}_{\alpha} \delta^2(z_1, z)) i k^{\alpha} \delta^2(z_2, z_0)  e^{i k X(z_{0}, \bar{z}_0)} : \ \ (10)

Now, we can pull out the ik^{\mu} which flips the sign,

\displaystyle = : \frac{i}{\alpha^{\prime}} \partial X^{\mu}(z)  e^{i k X(z_{0},\bar{z}_{0})}: + \frac{k^{\mu}}{2} : \partial  (\int d^2 z_1 d^2 z_2 \ln \mid z_{12} \mid^2 \delta^2(z_1, z) \delta^2(z_2, z_{0})  e^{i k X(z_{0},\bar{z}_0)}) : \ \ (11)

Notice that we have delta functions inside the integrand, so we are left with,

\displaystyle  : \frac{i}{\alpha^{\prime}} \partial X^{\mu}(z)  e^{i k X(z_{0},\bar{z}_{0})} : + \frac{k^{\mu}}{2} : \partial (  \ln \mid z - z_{0} \mid^2 e^{i k X(z_0,\bar{z}_0)}) :

\displaystyle = : \frac{i}{\alpha^{\prime}} \partial X^{\mu}(z)  e^{i k X(z_{0},\bar{z}_{0})} :  + \frac{k^{\mu}}{2 (z - z_{0})}  e^{i k X(z_0,\bar{z}_0)} \ \ (12)

And so we obtain the following result,

\displaystyle \frac{i}{\alpha^{\prime}} : \partial_{a}X^{\mu} : :e^{ikX}: \sim \frac{k^{\mu}}{2 (z - z_{0})} : e^{ikX}: \ \ (13)

Where {\sim} means the most singular pieces. We can also perform the same calculations for the antiholomorphic term,

\displaystyle \frac{i}{\alpha^{\prime}} : \bar{\partial}X^{\mu}: : e^{ikX}: \sim \frac{k^{\mu}}{2(\bar{z} - \bar{z}_{0})} :e^{ikX}: \ \ (14)

As Polchinski notes and as we see, the OPE is in agreement with the Ward identity. But we can still carry on a bit further. To conclude this example, recall explicitly to mind the Ward identity and our residues. Switching back to the holomorphic case and evaluating the LHS of (14) notice we find, picking out only the residues,

\displaystyle  \frac{i}{\alpha^{\prime}} : \partial X^{\mu} e^{ikX}: = (\frac{k^{\mu}}{2} + \frac{k^{\mu}}{2}) G = k^{\mu} G \ \ (15)

So, we have that it must be equal to {k^{\mu}} times the operator {\mathcal{G}} from above. Now, given that {\mathcal{G} = \mathcal{A}}, the right-hand side of the Ward identity tells us that,

\displaystyle  k^{\mu} \mathcal{A} = \frac{1}{i \epsilon} \delta \mathcal{A} \ \ (16)

And, again, from the Ward identity we can see in (15) that with a bit of algebra the variation of the operator must be,

\displaystyle  \delta \mathcal{A} = ik^{\mu}\epsilon \mathcal{A} \ \ (17)

Where we are assuming the variation is only in one direction. Interestingly, as an aside, what is actually happening are the following transformation properties,

\displaystyle  \mathcal{A} = e^{ikX} \rightarrow e^{ikX + ik^{\mu}\epsilon}

\displaystyle  X^{\mu} \rightarrow X^{\mu} + \epsilon \ \ (18)

1.2. Example 2

In the first example we considered a spacetime translation. We can now look to the second example in Polchinski’s textbook, where we want to consider a worldsheet translation, particularly how the {a} of the {\sigma} coordinates transforms as {\delta \sigma^{a} = \epsilon v^{a}}. Here {v^{a}} is a constant vector. It follows that from the action for free massless scalars is invariant under this transformation, with the above symmetry clearly understood given {X} is a scalar and how {\delta \sigma^{a}} does not change the measure of integration. And so, just as in the first example, what we want to do is investigate the construction of the conserved current as a result of this worldsheet symmetry transformation and then test the Ward identity.

The first step is to note that because we are dealing with a scalar theory we may write explicitly,

\displaystyle  \sigma^{a} \rightarrow \sigma^{\prime a} = \sigma + \epsilon v^{a} \ \ (19)

Where, for any worldsheet symmetry transformation, the scalar fields simply transform as follows,

\displaystyle X^{\prime \mu}(\sigma^{\prime}) = X^{\mu}(\sigma) \ \ (20)

From which it also follows that,

\displaystyle  X^{\prime \mu}(\sigma + \delta \sigma) = X^{\mu}(\sigma) \implies X^{\prime \mu}(\sigma) = X^{\mu}(\sigma - \delta \sigma) \ \ (21)

Where we should recognise that in brackets on the left-hand side of the first equality, {\sigma + \delta \sigma = \sigma^{\prime}}.

Of course, like the first example, we’re interested in how operators transform. And so we want to consider,

\displaystyle  \delta X^{\mu}(\sigma) = X^{\prime \mu}(\sigma) - X^{\mu}(\sigma) = X^{\prime \mu} (\sigma^{a} - \epsilon v^{a}) - X^{\mu}(\sigma) \ \ (22)

Expanding and only keep the 1st terms, what we end up with is precisely an expression for how our operators transform,

\displaystyle  \delta X^{\mu}(\sigma) = -\epsilon(\sigma) v^{a}\partial_{a}X^{\mu} \ \ (23)

Now, what we want to do is check with the Ward identity. So, like before, let’s start by varying the action and then build from there,

\displaystyle  \delta S = \delta [\frac{1}{4\pi \alpha^{\prime}}\int d^2\sigma \partial^{a}X^{\mu}\partial_{a}X_{\mu}]

\displaystyle = \frac{1}{2\pi \alpha^{\prime}} \int d^2\sigma \partial^{a}X^{\mu}\partial_{a}\delta X_{\mu} \ \ (24)

Where {\delta X_{\mu} = -\epsilon(\sigma)v^{a}\partial_{a}X_{\mu}}. The implication is as follows. From (24) we can substitute for {\delta X_{\mu}},

\displaystyle \delta S = \frac{1}{2\pi \alpha^{\prime}} \int d^{2}\sigma \partial^{a}X^{\mu}\partial_{a}(-\epsilon(\sigma)v^{a}\partial_{a}X_{\mu})

\displaystyle  = -\frac{\epsilon}{2\pi \alpha^{\prime}} \int d^2\sigma \partial^{a}X^{\mu}\partial_{a} v^{b}\partial_{b}X_{\mu} +  \partial^{c}X^{\mu}\partial_{c}v^{d}\partial_{d}X_{\mu}

\displaystyle = -\frac{\epsilon}{2\pi \alpha^{\prime}} \int d^2\sigma \partial^{a}X^{\mu}\partial_{a} v^{b}\partial_{b}X_{\mu} \ [1] + \partial_{d}(\frac{1}{2}v^{d} \partial^{c}X^{\mu}\partial_{c}X^{\mu}) \ [2] \ \ (25)

Where, for pedagogical purposes, the first and second integrands have been labelled [1] and [2] respectively. The reason is because it will be useful to recall these pieces separately in order to highlight some necessary computational logical and procedure. Before that, however, we should think of the conserved current. It follows, as we have already learned,

\displaystyle -\delta S = \frac{i}{2\pi} \int d^2\sigma \sqrt{-g}J^{a}\partial_{a}\epsilon \ \ (26)

Remember, looking at (2) in a previous entry, we can see clearly that {J^{a}(\sigma)} is the coefficient of {\partial_{a}\rho (\sigma)}. In the first example we become more familiar with this fact. And what Polchinski is referencing in the single passing sentence that he provides prior to eqns. (2.3.15a) and (2.3.15b) is that we need to make contact with this formalism. It is convenient to now reassert the {\rho(\sigma)} term,

\displaystyle = -\frac{\epsilon}{2\pi \alpha^{\prime}} \int d^2\sigma [ (\partial^{a}X^{\mu}\partial_{b} v^{b}X_{\mu}) \partial_{b}(\rho(\sigma)) \ [1] + (\rho(\sigma)) \partial_{c}(\frac{1}{2}v^{c} \partial^{d}X^{\mu}\partial_{d}X^{\mu})] \ [2] \ \ (27)

Now, let’s look at both pieces of (27). Piece [1] above looks fine and, on inspection, seems quite manageable. Piece [2], on the other hand, is not very nice. In taking one step forward, what we can do is integrate the second piece by parts. This has the benefit that we can eliminate the total derivative that arises and eliminate the surface terms. To save space, the result is given below,

\displaystyle  \delta S = \frac{\epsilon}{2 \pi \alpha'}\int d^2\sigma v^b\partial^a X^{\mu} \partial_b X_{\mu} \partial_a - \partial_b (\frac{1}{2}v^b \partial^a X^{\mu} \partial_a  X_\mu)

\displaystyle  =\frac{\epsilon}{2 \pi \alpha'}\int d^2\sigma [v^b(\partial^a X^\mu  \partial_b X_{\mu} -\frac{1}{2}\delta^{a}_{b} \partial_b X^\mu \partial^b  X_\mu) \partial_a] \ \ (28)

How to interpret (28)? Notice something very interesting. We have the stress-energy tensor plus some additional terms outside the small brackets. If we make the appropriate substitution for the stress-energy tensor we therefore obtain,

\displaystyle  \delta S = -\dfrac{\epsilon}{2 \pi}\int d^2\sigma \, (v^c\,T_c^a)\partial_a \ \ (29)

If we bring the constant epsilon back into the integrand, we have an integral over the worldsheet times a derivative of the parameter of an infinitesimal transformation. Whatever is left can be interpreted as a conserved current. Hence, then, if we go back and inspect (2) in this post we come to establish what Polchinski states in eqn. (2.3.15a). Our indices are slightly different up to this point, but this is merely superficial and when we rearrange things we find,

\displaystyle  J^{a} = iv^{b}T_{b}^{a}

And then lowering the index on {J},

\displaystyle  J_{a} = iv^{b}T_{ab} \ \ (30)

This is our conserved current. In certain words, it is natural to anticipate a conserved current on the string worldsheet and also for this current to be related to the stress-energy tensor. Just thinking of the physical picture gives some idea as to why this is a natural expectation. But we are not quite done.

What we want to do, ultimately, is define the stress-energy tensor as an operator with full quantum corrections. But, as we are working in conformal field theory, there is an ambiguity about how we might define it related to normal ordering. Let’s explore this for a moment.

We should think of stringy CFTs by way of how we will define a set of basic operators, and then from this show what is the stress-energy tensor. Moreover, it is a property of the stress-energy tensor and the basic operators we utilise that will give definition to the CFT. In CFT language, it is given that the stress-energy tensor can be written as,

\displaystyle  T_{ab} = \frac{1}{\alpha^{\prime}} : \partial_{a}X^{\mu}\partial_{b}X_{\mu} - \frac{1}{2}\delta_{ab}(\partial X)^2 : \ \ (31)

This is what Polchinski cites in eqn. (2.3.15b). We can still go a step further and discuss the topic of conformal invariance in relation to this definition. For instance, from the principles of conformal invariance, it remains the case that as discussed much earlier in these notes,

\displaystyle  T_{a}^{a} = 0

Which is to say, as we should remember, the stress-energy tensor is traceless. This condition of tracelessness tells us how, if we were to go to holomorphic and antiholomorphic coordinates,

\displaystyle  T_{a}^{a} = 0 \rightarrow T_{z\bar{z}} = 0 = T_{\bar{z}z} \ \ (32)

Where one may recall, also, the non-vanishing parts {T_{zz}} and {T_{\bar{z}\bar{z}}} from an earlier discussion in this collection of notes. It follows that if the stress-energy tensor is, in fact, traceless, we may invoke the conservation of the current such that,

\displaystyle  \nabla^{a}J_{a} = 0 = \nabla^{a} T_{ab} = 0 \ \ (33)

Which is to say that we have full conservation for the full stress-energy tensor. We can write this in terms of holomorphic and antiholomorphic coordinates as expected,

\displaystyle  \bar{\partial}T_{zb} + \partial T_{\bar{z}b} = 0  \ \ (34)

This gives us two choices:

\displaystyle  b = z \implies \partial T_{zz} = 0

\displaystyle  b = \bar{z} \implies \partial T_{\bar{z}\bar{z}} = 0 \ \ (35)

Where, as it was discussed some time ago, {T_{zz} = T(z)} is a holomorphic function and {T_{\bar{z}\bar{z}} = \bar{T}(\bar{z})} is an antiholomorphic function.

It is perhaps quite obvious at this point that we may also write,

\displaystyle  T(z) = -\frac{1}{\alpha^{\prime}} : \partial X^{\mu}\partial X_{\mu}:

\displaystyle  \bar{T}(\bar{z}) = -\frac{1}{\alpha^{\prime}} : \bar{\partial} X^{\mu}\bar{\partial} X_{\mu}: \ \ (36)

Now, returning to our current (30), we can be completely general in our study of the current,

\displaystyle  J_{z} = iv(z)T(z)

\displaystyle  \bar{J}_{\bar{z}} = i\bar{v(z)}\bar{T}(\bar{z}) \ \ (37)

If we have conservation of the current, then the above is the same as,

\displaystyle  \nabla_{a}J^{a} = \bar{\partial}J_{z} + \partial J_{\bar{z}} = 0 \ \ (38)

Which is to say that the new currents are conserved provided {v(z)}, previously considered a constant vector, is holomorophic. Additionally, the current is of course associated with symmetries; but what are these symmetries? They are the conformal transformations.

If, in the bigger picture, what we want to do is find {\delta X} due to symmetries {J_{z} = ivT(z)}, to proceed recall the Ward identity {\mathcal{A} = X^{\mu}}. It follows we need to compute an OPE for the stress-energy tensor with our scalar field (complete computation is given in the Appendix of this chapter, along with other important and useful OPEs),

\displaystyle  :T(z) : :X^{\mu}(z_{0}, \bar{z}_{0}): = -\frac{1}{\alpha^{\prime}} : \partial X\partial X: : X^{\mu}(z_{0}, \bar{z}_{0}) :

\displaystyle  \sim -(\frac{2}{\alpha^{\prime}}) \cdot (-\frac{\alpha^{\prime}}{2}\partial_{z}\ln \mid z - z_{0}\mid^2) : \partial X(z_{0}) :

\displaystyle  T X^{\mu} \sim \frac{1}{z - z_{0}}\partial X(z_{0})

\displaystyle  \bar{T}X^{\mu} \sim \frac{1}{\bar{z} - \bar{z}_{0}}\bar{\partial}X(\bar{z}_{0}) \ \ (39)

Which is what Polchinski states in eqn. (2.4.6). And now we can use the Ward identity and take the residue of the current with the coefficients of the OPE for the holomorphic and antiholomorphic pieces,

\displaystyle  iv(z_{0}) \partial X(z_{0}) + i \bar{v}(z_{0} \bar{\partial} X(\bar{z}_{0}) = \frac{1}{i\epsilon} \delta X \ \ (40)

And so what we find is that, for the current we have constructed, we have a symmetry transformation of the following form,

\displaystyle  \delta X = -iv(z_{0})\partial X(z_{0}) - i\bar{v}(z_{0} \bar{\partial}X(\bar{z}_{0}) \ \ (41)

For {z_{0} \rightarrow z_{0} + \epsilon v(z_{0})}. If we drop {z_{0}} and generalise,

\displaystyle  \delta X^{\mu} = -\epsilon v(z)\partial X - \epsilon\bar{v}(z)\bar{\partial}X \ \ (42)

For {z \rightarrow z + \epsilon v(z)} which is an infinitesimal transformation, where the only constraint is that {z \rightarrow z^{\prime} = f(z)} is holomorphic.

The reason the transformation is so simple,

\displaystyle  \delta X^{\mu} = X^{\prime \mu}(z^{\prime}, \bar{z}^{\prime}) - X^{\mu}(z, \bar{z}) = X^{\mu}(z - \epsilon v, z - \epsilon\bar{v}) - X^{\mu} \ \ (43)

Where, after Taylor expansion,

\displaystyle  \delta X^{\mu} = - \epsilon v\partial X - \epsilon\bar{v}\bar{\partial}X \ \ (44)

It is important to point out that {f(z) = \xi(z)} represents a global rescaling (but can also represent a local rescaling). If {\mid \xi \mid = 1} then we have a simple rotation, and in general no scaling.

To conclude, from the very outset of this chapter, we may also recall to mind that in the context of the conformal group we are working in 2-dimensions. When we ask, ‘what is the analogue of this symmetry in higher dimensions?’, the answer is that in higher dimensions we can construct scale invariance as well. Indeed, in D-dimensions, if you have {y^{\mu} \rightarrow \lambda y^{\mu}} you have additional special conformal transformations.

References

Joseph Polchinski. (2005). ‘String Theory: An Introduction to the Bosonic String’, Vol. 1.

Blumenhagen, R. and Plauschinn. (2009). ‘Introduction to Conformal Field Theory’.

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