Stringy Things

Gravitational waves, cosmic strings, and recent NANOGrav results

There has been intriguing news in the past week regarding gravitational waves and cosmic strings. I have yet to write about cosmic strings on this blog, a consequence of a lack of time more than anything else; but it is certainly on the list of things I want to cover, especially considering that I would like to write some papers in this area in the future. A very nice introductory article was written some time ago by my professor, Ed Copeland, which I recommend. One of my favourite papers on cosmic superstrings was in fact co-authored by Ed Copeland and Joe Polchinski alongside Robert C. Myers. They also wrote a paper together on macroscopic fundamental and Dirichlet strings, which should provide ample background material. Although I am inclined to say that I come more from the maths side of string research than the cosmology side, cosmic strings are super cool. Thinking of them now has me remember when I first arrived at Nottingham as an undergraduate, it was around the time I first met Prof. Copeland. I was sharing with him my enthusiasm for Joe Polchinski’s textbooks, when Ed shared with me that he had written with Joe on a few occasions. I recall racing home to read the papers they had written together. The next time I spoke with Ed, I had a printed copy of their paper on cosmic superstrings and told him how much I enjoyed reading it! I was only beginning to study strings in a thorough and rigorous way at that time, but his papers stimulated my interests greatly.

I will think of drafting a detailed technical essay in time, but one way to think of cosmic strings is under the Nambu-Goto approximation which describes them as one-dimensional objects. The idea is that these hypothetical objects may have formed early on in the universe, particularly while it was expanding and cooling down. If we take a model of hybrid inflation as an example, we have two scalar fields: $\psi$, which is the inflation field with a flat potential satisfying slow-roll conditions, and a more dynamical scalar $\phi$ whose mass depends on $\psi$. One argument is that, as the universe cooled, spacetime may have cracked – to give a sense of intuition think of a fissure in the ice of a frozen lake or something similar. This crack is what we term a topological defect. In a hybrid inflation model, inflation ends not so much as the slow-roll approximation breaks down but when $\phi$ becomes tachyonic, such that its mass squared becomes negative. One can study tachyons in a direct and wonderfully illuminating way in the spectrum of the bosonic string, but the main point here is that they are highly unstable. So the hybrid model signals an instability, and it is this instability where a phase transition can occur in which such topological defects can form. These are cosmic strings.

Many field-theory models predict the existence of cosmic strings, and experimental evidence of their existence would be truly extraordinary. It would mean a lot of things, not least direct experimental evidence in support of string theory. One of the brilliant properties of cosmic strings is that, like the ordinary string, they may interact and form loops. In the cosmic string case, when two strings intersect or when a single string crosses itself, intercommutation can lead to the formation of a closed loop. These loops can oscillate in different ways depending on the dynamics, but if a single loop is large enough it is very likely to meet another string and reconnect, hence we have a network of cosmic strings which would have evolved in time as the Universe expanded. It should be noted that a mathematical description of these string interactions can very insofar that, for instance, in the context of brane cosmology – think also of Brane-worlds – we have F-strings, which in certain scenarios can grow to cosmological scales. These are cosmic superstrings, and unlike standard cosmic strings they interact (and form loops) probabilistically. What is cool is that, in this string network, the loops oscillate and radiate energy, shrinking and eventually decaying. One such form of radiation is gravitational radiation, and it is through the dissipation of energy from the oscillating loops that we may describe gravitational wave emission. In the formalism, there are two primary concepts, kinks and cusps, which describe the strongest bursts of gravitational waves by the string.

To date there has been no direct evidence for the existence of cosmic strings, but last week a number of papers emerged speculating on a recent report provided by The North American Nanohertz Observatory for Gravitational Waves (NANOGrav). Pierre Auclair, a PhD candidate at Laboratoire Astroparticule et Cosmologie (APC), gave a talk for us last Friday at the Centre For Astronomy And Particle Theory to go over his research on cosmic strings, and toward the end of his presentation he mentioned some of the excitement stirring in response to this NANOGrav report (attached is a screenshot from his lecture slides, it lists a number of notable papers to appear on the archives in recent days). Right now a lot of study is being put into obtaining bounds on the string tension $G\mu$ from gravitational waves detectors (NANOGrav, LIGO, and Virgo), and experiments are searching for individual bursts in the context of stochastic backgrounds. When it comes to the NANOGrav report, the initial view by a number of researchers is that study yields strong evidence for the presence of a stochastic common-spectrum process across the 45 pulsars analysed. It is a 12.5-year data set, and while a conclusive statement on the physical origin of the signal is not immediately obtainable, with a number of qualifications required to ensure the detection of a gravitational wave signal, some have already begun to argue it is reasonable to interpret the data in terms of a stochastic gravitational-wave background emitted by a cosmic-string network.

It is exciting news, to be sure. But just as Auclair advised – and as is generally a reasonable principle to follow – one should proceed tentatively and with caution toward any claims about evidence for the existence of cosmic strings.

Currently I am preparing to submit my thesis for the end of the month, so I haven’t had a moment to properly dig into the report. A letter by Simone Blasi, Vedran Brdar, and Kai Schmitz speculating on the connection to cosmic strings can be found here. It will be great to write about all of this in more technical detail when there is time, and to also watch closely for further reports.

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Disenchantment and the anthropology of (re-)enchantment

I recently read an interesting essay by Egil Asprem entitled Dialectics of Darkness. Its original purpose was to serve as a review of The Myth of Disenchantment: Magic, Modernity, and the Birth of the Human Sciences by Jason Josephson-Storm. I have yet to read Josephson-Storm’s book, so I shall have to reserve comment for another time. But I am certainly already familiar with its main subject and the history behind it, which is one reason I found great interest in navigating Asprem’s essay on the enlightenment (and thus, too, the notion of enlightenment reason) and the anthropology of active (re-)enchantment.

Additionally, following the publication of Asprem’s work, a number of other short essays and articles appeared directly in response. I list them as follows, The Enchanted World Today by Josephson-Storm with a reply by Asprem; The Reemergence of Magical Beliefs by Adam Possamai; and, finally, Models of (Re-)Enchantment by Dafydd Mills Daniel.

The latter article by Daniel offers what I think to be a decent and certainly interesting reading of Bertrand Russell and Richard Dawkins, particularly their nuanced and deeply considerate approaches to naturalist philosophy as well as their attempts to satisfy the demands for ethical norms rooted in a naturalistic model. It is no secret that I enjoy a lot of Russell’s writing, and a short disclaimer would highlight at this point that Russell’s essay A Free Man’s Worship (referenced by Daniel) is perhaps one of my favourite pieces of humanist literature. However, while I think a review of the contents of Daniel’s contribution could, in itself, be the focus of an entire essay, I will save a few comments for the end.

In reading the essay by Asprem, and then the follow-up by Josephson-Storm with a reply by Asprem, one thing struck me in particular. Up to this point, I’ve tended to see the enlightenment not as some cultural totality or as a total cultural shift in a particular moment of time but as a historical process. From d’Alembert and Descartes to Leibniz, Pascal, and Newton (to name a few enlightenment thinkers) – I think there is also a kernel of insight to be retrieved from their respective notes on this issue. Indeed, for many notable enlightenment thinkers, not least Kant, there was no such enlightenment as a historical period that completely extinguished enchantment; it was instead perceived as an ongoing process of social, psychological or spiritual development in human history. (In fact, as an aside, I would be inclined to argue that the enlightenment philosophes are generally distinguishable by the very nature of their confrontation with the dichotomy between process vs. substance metaphysics, a point that I think is relevant here). The philosophes were or can be read as an attempt to formally describe this process and capture its positive implications. Indeed, I think for many enlightenment scholars this view would not be received contentiously. And so, I am inclined to perhaps warn against the view that the enlightenment should be seen as a period of total cultural disenchantment that may or may not have eventually regressed to an unfolding process of (re-)enchantment over time.

Moreover, an investigation into the objective validity of reason and of scientific knowledge discloses, I think, a sort of naivety that sometimes saturates our thinking with regards to the idea of the historical realisation of cultural enlightenment. In the essays cited above, Newton is mentioned because for all his mathematical and scientific genius he also studied alchemy. But when the enlightenment is seen as a process, which too must exist or manifest in given history with its own established domain of concepts and prejudices, the weight of this contradiction becomes more measurable. To generalise my complaint: it is no secret that many enlightenment thinkers, even some members of the radical enlightenment (as some scholars distinguish), maintained belief (personal or via organised religion) in God whilst championing secularised knowledge and humanistic values. However, I’m not convinced this should be seen as a failure or interpreted in the context of (re-)enchantment. Even today, I don’t think it is entirely false to say that some members of the scientific community maintain a belief in the superstitious, supernatural, or the divine. Famously, there were many significant and famous modern physicists who also carried superstitions beliefs or artefact beliefs in myth. Taking a broader view, we may objectively perceive and criticise such logical inconsistency, and perhaps for the benefit of reason take lesson from their example. One lesson to recognise is that myth – or perhaps its remnants depending on how we parameterise the theory – may persist in very organised or established ways as historical legacy or artefact. It is not at all controversial to say that human bias and prejudice may continue to exist despite evidence against whatever belief; and it would seem very appropriate to look at these issues in their sociohistorical context in order to establish as nuanced a view as possible.

So from my own reading and studies, my interpretation of the enlightenment project is as an ensemble of concepts not necessarily unique in category but realised uniquely in time. In a sense, my view has been shaped around the idea of the enlightenment as a unique realisation of concepts, the genesis of which dates back and through such pre-Socratic scholars as Anaxagoras and Democritus, Thucydides and the The Mytilenaean Debate, and then eventually the philosophical considerations of Plato onward. Such concepts include, in modern language, basic ideas of reason as set against myth and political realism.

In many of the grandfathers and fathers of modern science we see this much more in terms of a general shaping of epistemology, however much residues of myth and enchantment may be found (from one philosopher to the next), given that human history is saturated in the perpetuation of prejudice. Although such a course of discussion requires a fuller essay in itself, what I am trying to say, in different words, is how the enlightenment may be viewed as a certain continuation in the historical generation of ideas and that epistemology is perhaps the best site to study its development. In philosophy, particularly or especially philosophy of the subject, this may be expressed by way of a study of the genesis of the modern subject, which some trace as far back as Homer’s Odyssey. Perhaps more insightful is Bertrand Russell’s study of knowledge in The Problem’s of Philosophy (1912) in which, rather than considerations of metaphysics, epistemology is brought directly into focus. From this view – namely, from the study of epistemology – the genesis of well-known enlightenment values and ideas appear in different forms, under different guises, and through manipulations of different frameworks in the very seeds of philosophical thought in classical antiquity forward.

Plato’s Allegory of the Cave can very much be read as an investigation into epistemology not so dissimilar to the enquiry of enlightenment philosophes into the possibility of knowledge – rational and scientific or otherwise. The leading question for the enlightenment may be stated, ‘What even constitutes knowledge, let alone rational and objective knowledge?’. As a profound site of investigation, often ignored or not taken serious enough, it is one that can be traced back to pre-Socratic study which was, I am inclined to argue, eventually refocused with advent of the first Medieval universities and their systematic introductions of Aristotelian language, then in the humanist renaissance of the 14th and 15th centuries, and finally in the 18th century philosophes.

On the other hand, although the enlightenment project did not emerge simply out of nothing, there is something unique about it which continues to draw serious interest and consideration. In a sense, I think it may also be viewed a lexicalized concept. As such, it is unique in what it represents as a particular unification of ideas and values realised in such a way as to be epistemologically revolutionary. Taking this view, I think we can also begin to delineate different epistemologies and their significance with respect to the prospect of enlightenment knowledge.

Arguably, one of the last great philosophical efforts to answer the fundamental question of the status of knowledge and the possibility of knowing – and, really, the status and legitimacy of abstract concepts – was in the work of Kant. Interestingly, I think it can be strongly argued that Kant’s investigations, and likewise also Hume’s skepticism in which Kant seeks to overcome, are much more relevant to philosophical systems of knowledge than scientific systems of knowledge. There is, at a point, a divergence between traditional philosophical epistemologies and scientific epistemologies. For Kant, and also Hume, neither seem to be able to do justice to the unique epistemological domain of scientific enquiry, which, I think was clearly realised in the 20th century to be very different than the general philosophical domain of enquiry. Although, in my opinion, Kant comes extremely close in places, I would be inclined to expand that, at least in a Platonic sense of conceptual space, scientific knowledge can be cleanly and clearly differentiated from purely subjective reasoning, and that while Kant’s response to Hume’s skepticism is not entirely satisfactory, the latter can be overcome through explanation within scientific systems of reasoning. But with these matters put to one side, the deeper point is that I think one can see clearly this delineation of epistemologies not only in the progression of modern science but also in what it has to say about the prospects of reason and human rationality.

Mention has already been made of Russell. It would be terrific to write more about his works in both a critical and non-critical way, because his 1912 investigations are some of the best when it comes to late-modern encircling of these differences in epistemology by way of fairly systematic investigations into the nature of knowledge. That said, I think some of the most illuminating sites of reflection can also be found in the writings of many of the great 20th century physicists, who concerned themselves with such longstanding historical debates. Einstein, to offer one example, is noted to have spent time thinking about epistemological questions and engaging with debates on the nature and status of knowledge.

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These essays, especially the one by Asprem which ignites a wonderful chain of contributions and perspectives, are undoubtedly stimulating. The one thing that stands out to me, given the above reflections, is the risk that one may easily take a viewpoint that is too binary, lacking the complexity of a systems view of human society and the general types of behaviour it fosters. Disenchantment and re-enchantment seem less like fundamental processes than epiphenomena. In many ways, I think the antimony often defined between disenchantment and enchantment can be broken down into very basic elements of the epistemological study of reality versus appearance, from which Russell for example delineates two forms of knowledge: by description and by direct acquaintance. The latter is very much akin to the best of existential phenomenology of the 19th and 20th centuries, in which intimate description of the phenomenal world of direct experience and sense data is given. Often these movements neglect the abstract and theoretical or are simply unable to conceptualise it. Russell, on the other hand, is able to substantiate the validity of logicism and logical reasoning – indeed, we may even extend his analysis today to theoretical knowledge – whilst maintaining ample recognition of the experiential and the phenomenological. Preservation of the recognition of the type of knowledge by direct acquaintance is important in terms of secularised ethical and moral foundations; but doing so while maintaining a concept of the objective and not regressing to subjectivism is no easy task. Much of contemporary post-modern philosophy, for instance, sinks into the muddied confusion of pure subjectivism and at a great cost.

But if we decide that the disenchantment-enchantment model is not satisfactory, what I want to say is that, as I have been alluding, perhaps the more fundamental site of enquiry is the study of epistemology, from which any and all discrepancy between disenchantment and (re)-enchantment may emerge. And, in few words, I think this and the paragraph immediately above describe why Daniel’s essay touches on something very important in his reference to the compelling arguments by Russell and also Dawkins, respectively. Intentionally or not, they both present fairly convincing approaches principled, firstly, on the foundations of knowledge and the validity of objective knowledge. From this, and as modern science would also indicate, (re-)enchantment is reduced to the domain of cognitive human bias, dogma and superstition; the persistence of myth played out in daily human life has its roots here, just as the violence and irrational ideologies that define the contemporary political domain are often a projection of the unreasonable on the basis of the prejudiced nature of the subject’s interaction with the world. If, as some scholars describe, the contemporary political domain may be generally depicted as a polarised space for the practice of bias and prejudice, with the establishment of echo chambers and irrational subjective pursuits of ideological ends as opposed to rational, disclosing, truth-giving processes – I take it from the view of Russell that such a space is merely the continuation pathological epistemologies.

What is also significant about both Russell and Dawkins is that, rather than completely rejecting the human existential inclination to search for meaning, it is acknowledged and reformulated positively. They argue that there is no deeper source of meaning than that which is naturally disclosed within the epistemological domain of science, and that through science and its many lessons the human need and thirst for meaning may be quenched, even in the face of our own cosmic insignificance. Unlike romanticism for example, in which meaning and inspiration is deduced purely subjectively and with emphasis on the primacy of the individual, which completely prefigures the notion of the subject by neglecting the objective; what we see in the better parts of Russell and Dawkins is a positive, evolving notion of enlightenment meaning-giving process that in many ways may begin to answer Camus’ deep (and certainly valid) conundrum.

For these reasons, I agree and sympathise with Daniel’s assertion that, in many ways, Russell and Dawkins successfully carve a path a forward, transcending the pitfalls of the romantics so often tied to (re-)enchantment and anti-modern movements, whilst preserving the existential depth of what it means to be human and in search for meaning. Through this lens, I think the picture of total enlightenment disenchantment from the perspective of cultural anthropology becomes something of a myth. Allow me to explain.

In certain strands of contemporary philosophy, the projection of some complete realisation of reason and the crystallisation of rational society (such as in Weber’s construction) would seem to rely in some way on a view of cultural enlightenment as a sort of final development. In that sense, it too would seem predicated in places on the myth of cultural enlightenment and hence the achievement of solid rational outcome. But I would argue that history has witnessed neither, and even the best examples of contemporary society fail satisfy the demands of both concepts.

Furthermore, many of the critical philosophes of the 20th century, most of which were rooted in or indebted to the enlightenment, placed great importance on reason, its historical genesis, and the ongoing struggle in its realisation. That is to say that the genesis of the modern subject was a central point of focus, and with this focus many provocative debates on knowledge and reason may be found. Crucially, the concept of enlightenment reason is not perceived as a given. The concept of enlightenment reason may have historically crystallised in a unique way – or at least some framework was formalised to better describe it – and hence concepts of rational society may have begun to spring forth. But we learn in the critical philosophies that the parameters in which reason and notions of rational (thus disenchanted) society may be historically realised can be more or less pathological, and that generally in the social and communicative domain it is reason’s absence that continually defines humanity’s historical struggle. In Weber’s construction, then, one could argue that the concept of reason is essentially utilised in a less than rational way. There is, in other words, an ongoing classical distinction between form and content, and their lack of synthesis, that I would argue underlies much of the struggle for reason that continues to the present day.

Such a viewpoint reinforces the idea – indeed, the acute observation – that we do not presently live in a rational, scientific society. Evidence of this can easily be found in the very structure of contemporary debates and the issues they concern. Instead, it would seem much more akin to a society that uses notions of reason or quasi-systems of reason and science at its convenience, without complete subscription to its logical and rational demands. So, in a way, I think there is a deeper truth to Josephson-Storm’s study. I would say that some enlightenment disenchantment has been achieved but only up to a point within a particular epistemological domain that exists within broader social-pathological and enchanted parameters. I think the subtlety and nuance of such a viewpoint carries forward what may have been deemed the radical enlightenment based largely on the assessment that, following a lexicalized concept of the enlightenment, the reality of the process of enlightenment reason is much more akin to a struggle for reason and for a future rational society against the forces of its absence.

One last comment, to conclude this already lengthy engagement. If the enlightenment is seen more as a unique configuration of concepts and ideas, as part of a larger history, which triggered a process (against myth, prejudice, etc.) in the development of reason, science and ultimately fundamental secularised values – from this point of view, reason and human rationality may be perceived within the scope of a theory of society that recognises how, and in what way, such important concepts must be socially fostered. The notions of disenchantment and enchantment, if the binary is correct to construct, discloses a tremendous conflict: namely, the legacy of historical and cognitive biases, in addition to general irrational human tendencies and inclinations which reject the objective. That a society may, in recent time, promote itself as disenchanted only to then be said to have regressed to (re)-enchantment and myth – or only for (re)-enchantment and myth to continue propagate – would seem one of the central themes of Dialectic of Enlightenment. At the same time,  modern science continues to push the boundaries of human thought, and its special epistemological domain of enquiry is generally irrefutable. The influence and demands of enlightenment reason continues to challenge, even scientists, to normatively check one’s biases and to continue to struggle for a clearer recognition of objective knowledge within the historical context of constraints of that knowledge at any given time.

However, in that the promise of enlightenment reason – the promise of reason and human rationality – may exist and yet simultaneously be folded into a human social world of continued and renewed enchantment – and hence, myth and the irrational confluence with the rational – this is akin to acknowledging that differentiated spheres of society may each be affected differently. It is this fragile and precarious existence of reason and its unrelenting possibility of betrayal that seems to be one of the essential features of today’s social world, so much so that in continued enchantment reason can take on the appearance of a disfigured form that is, in fact, absent of any rational content.

The struggle is to see reason and unreason, solid as the ground beneath one’s feet or as the material objects in one’s daily life. Even those who believe they command reason often, in their certainty, fall guilty of its opposite. It is notable that most major cultures and religions to have crystallised in human history possess a concept of good and bad, in moral philosophical terms; light and dark in religious language; or reason and unreason, in epistemological terms. With no exception, none have reconciled these ideas however much one may faithfully believe the contrary. This is as close to an objective view that may be accessed, and almost always whatever lesson one may wish to glean such fleeting objectivity can quickly turns subjective through the simple demand of interpretation. This was as much a struggle for Plato as Aristotle and the 18th century philosophes. In the modern lens, it was as much a struggle for Kant and Hegel as Adorno or alternatively the opposing attempt to formulate the post-modern.

One thing I can speak to is that in mathematics, ideally, we follow the systematic through to the result, and then we ascertain whether the logic is true or not. But this space of concepts and of thought would seem different to the world of social occupations, in which concepts – like policy – can be reasonable, unreasonable, or both simultaneously. This is why there is no realised fundamental moral theory, because the space of concepts is saturated in the subjective stuff of daily human life. The point is not to say that the objective is in accessible, but oftentimes its fleeting and precarious nature cannot be trusted in the eyes of human beings. Even when solid objects are attempted to formed in words, such as God, or in symbols, such as peace, one can easily feign through solid and rigid representations the opposite of its conceptual substance. People have killed in the name of religion or the idea of a just politics without any awareness of the indignation of the contradiction. I think here, too, Russell and Dawkins serve important lessons and insight as we continue to reflect on the importance of the enlightenment and its realisation.

**Image: Projection of the Enlightenment by Anshu Kumar.

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As I noted the other day, there were a number of interesting talks at String Math 2020. I would really like to write about them all, but as I am short on time I want to spend a brief moment thinking about one talk in particular. Robbert Dijkgraaf’s presentation, ‘The Unreasonable Effectiveness of String Theory in Mathematics‘, I found to be enjoyable even though it was not the most technical or substantive. In some sense, I received it more as a philosophical essay – a sort of status report to motivate. I share it here because, what Dijkgraaf generally encircles, especially toward the end, is very much the topic of my thesis and the focus of my forthcoming PhD years. Additionally, while it may have aimed to inspire and motivate string theorists, the structure of the talk is such that a general audience may also extract much wonder and stimulation.

One can see that, whilst, certainly in my view, mathematics is a platonic science, Dijkgraaf wants to establish early on the unavoidable and unmistakable connection between fundamental physics and pure mathematics. So he starts his presentation by ruminating on this deep relationship. Eugene Wigner’s ‘The Unreasonable Effectiveness of Mathematics in the Natural Sciences‘ comes to mind almost immediately (indeed inspiring the title of the talk) in addition to past reflections by many intellectual giants. The historical evidence and examples are overwhelming as to the power of mathematics to speak the language of reality; at the same time, physics exists in this large space of concepts. It is their overlap – the platonic nature and rigid structures of mathematics and the systematic intuition of physics with its ability to capture nature’s phenomena – that in fundamental science suggests deep ideas of unity and synthesis. On this point, Dijkgraaf uses the example of the basic and humble derivative, highlighting the many perspectives it fosters to show that the mathematical and physical use of the concept is broad. The point is to say that there exists a large space of interpretations about even such a basic conceptual tool. The derivative has both physical meaning and interpretation as well as purely mathematical meaning. These many perspectives – similar, I suppose, to Feynman’s notion of a hierarchy of concepts – offer in totality a wealth of insight.

A better example may be the dictionary between the formalism of gauge field terminology and that of bundle terminology. On the one hand, we have physicists studying Feynman diagrams and fundamental particles. On the other hand, we have mathematicians studying and calculating deep things in topology and index theory. Historically, for some time the two did not discuss or collaborate despite their connection. In fact, there was a time when maths generally turned inward and physics seemed to reject the intensifying need of higher mathematical requirements (it seems some in physics still express this rejection). As Dijkgraaf tells it, there was little to no interaction or cross-engagement, and thus there was no mathematical physics dictionary if you will. For those that absolutely despise the increasingly mathematical nature of frontier physics, one may have no problem with such separation or disconnection. But such an attitude is not good or healthy for science. We see progress in science when the two sides talk: for instance, when physicists finally realised the use of index theory. The examples are endless, to be sure, with analogies continuing in the case of the path integral formalism and category theory as Dijkgraaf highlights.

In addition to discussing the connection between maths and physics, there is a related discussion between truth and beauty. For Dijkgraaf, he wants to feature this idea (and rightly so): namely, the two kinds of beauty we may argue to exist in the language of fundamental mathematical physics, the universal and the exceptional. There is so much to be said here, but I will save that for another time!

I will not spoil any more of the talk, only to say that the concept of emergence once again appears as well as the technical idea of ‘doing geometry without geometry’. Readers of this blog will know that what Dijkgraaf is referring to is what we have discussed in the past as generalised geometry and non-geometry. As these concepts reside at the heart of my current research, we will talk about them a lot more.

To conclude, I want to leave the reader with the following playful thought with respect to the viewpoint Dijkgraaf shares. If, for a moment, we look at string theory as the synthesis between geometry and algebra, I was thinking playfully toward the end of the talk that there is something reminiscent of the Hegelian aufhebung in this picture – i.e., the unity of deeply important conceptual spaces in the form of quantum geometry, as he puts it. In the physical and purely mathematical sense, from whatever side one advances, the analogy is finely shaped. From a mathematical physics point of view, it sounded to me that Dijkgraaf was seeking some description of synthesis-as-unification-for-higher-conceptualisation. I suppose it depends on who you ask, but I take Dijkgraaf’s point that string theory would very much seem to motivate this idea.

Stringy Things

String Math 2020

String Math 2020 has been taking place this week. Due to the global pandemic, the dates for the annual conference were moved back a month with everything now taking place online. So far there have been some interesting talks and points of discussion. Edward Witten was at his best yesterday, delivering a brilliant talk on the volumes of supermoduli spaces. It was exceptional, so much so I look forward to going back and listening to it again.

I’ve been quite busy with my thesis and things, so I missed a few presentations from earlier in the week. As today is the final day, I’m going to take some time this afternoon to listen to Soheyla Feyzbakhsh’s talk in algebraic geometry – it will focus mainly on S-duality and curve counting, as discussed in this paper [arxiv.org/abs/2007.03037]. The live stream will be made available here.

Unfortunately, for the same reasons as above I also missed a number of important talks from Strings 2020 earlier in the month. A lot of the feature topics were certainly predictable or foreseeable, with the black hole informational paradox, AdS/CFT, and JT gravity being an example. All good stuff, to be honest. After my final thesis calculations, I want to go back and listen to Ashoke Sen’s talk on D-instanton perturbation theory, as well as the respective presentations by Cumrum Vafa (latest on the Swampland) and Clay Cordova (higher-group symmetries). They’re not concerned with my current focus – generally, I would like to see more in the doubled formalism and non-perturbative theory – but interesting nonetheless. I will probably also catch up a bit on the recent developments regarding replica wormholes etc ;). For the interested reader, everything from the two conferences has been archived here.

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Stringy Things

Propagators for the dual symmetric string and a familiar identity

If I could count all the moments so far when faced with a puzzle or question and it was appropriate to say, ‘it’s in my Polchinski!’ or ‘I should double-check my Polchinski!’. Use of the possessive here should be taken as an endearing reference to the role Joe Polchinski’s textbooks have played in one’s life. They’re like a trusty companion.

A great example comes from the other day. Actually, the story begins with Tseytlin’s first principle construction of the dual symmetric string, which serves as the basis of some work I am doing more generally in the doubled formalism. In one of the papers I have been reading that generalises from Tseytlin there are references made about the propagators in this dual symmetric formulation of string theory, followed by an assortment of assumptions including one about $z \rightarrow 0$ regularisation such that $\bar{\partial} z^{-1} = \pi \delta^{(2)}(z)$. When I first read this over, it wasn’t immediately obvious to me from where this identity originated; I was much more focused on the actual propagators and some of the important generalisations of the construction, so I sort of left it as something to be returned to. Then in the last week I was reminded of it again, so I went back quickly and I realised how silly it was of me to not immediately recognise why this identity is true. The equation from above just comes from eqn. (2.1.24) in Polchinski, $\partial \bar{\partial} \ln \mid z \mid^2 = 2\pi \delta^{(2)} (z,\bar{z})$. Furthermore, one may have also thought equivalently of eqn. (2.5.8) in the context of bc CFT, which was what I first recalled (it also happens to be the subject of problem 2.1 at the end of the chapter).

There are a few ways to verify $\partial \bar{\partial} \ln \mid z \mid^{2} = \partial \bar{z}^{-1} = \bar{\partial} z^{-1} = 2 \pi \delta^{2} (z, \bar{z})$. One direct way is to take the terms to the left of the first equality, noting that $\partial \bar{\partial} \ln \mid z \mid^{2} = 0$ if $z \neq 0$. What we want to do is integrate this over some region $R$ in the complex plane, using divergence theorem given in eqn. (2.1.9) which states $\int_R d^{2}z (\partial_z v^z + \partial_{\bar{z}} v^{\bar{z}}) = i \oint_{\partial R} (v^{z} d\bar{z} - v^{\bar{z}} dz)$, where the contour integral circles $R$ counterclockwise.

For the holomorphic case, using the test function $f(z)$,

$\int_{R} d^{2}z \ \partial \bar{\partial} \ \ln \mid z \mid^{2} \ f(z) \ (1)$

From derivative properties we see $\partial \bar{\partial} \ln \mid z \mid^{2} = \partial \bar{\partial} (\ln z + \ln \bar{z}) = \bar{\partial} z^{1}$. Taking this fact into account and then also finally invoking divergence theorem,

$= \int_{R} d^{2}z \ \bar{\partial} \ z^{-1} \ f(z)$
$=-i \ \oint_{\partial R} \ dz \ z^{-1}$
$= 2 \pi f(0) \ (2)$

Where we have used the fundamental result in complex analysis that the contour integral of $z^{-1}$ is $2\pi i$. The same procedure can also be used for the antiholomorphic case. Hence, $\int \ d^2z \ \partial \bar{\partial} \ln \mid z \mid^{2} \ f(z, \bar{z}) = 2\pi f(0,0)$, which therefore gives us $\partial \bar{\partial} \ln \mid z \mid^{2} = 2 \pi \delta^{2} (z, \bar{z})$.

As an aside, thinking of this reminds me of how I’ve been wanting to go back and update whatever notes I have so far uploaded to this blog as part of my ‘Reading Polchinski’ series, which I started writing in my first undergraduate year. I still like the idea of uploading my hundreds of pages of notes on Polchinski’s textbooks and formatting them into a pedagogical blog series, because there are so many subtleties and nuances that are fun to think through. I think I now also have a better sense of how I want to continue formatting the online version of the notes and communicate them, so after my thesis I intend to return to the project :).

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