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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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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Stringy Things

Literature: Duality Symmetric String and the Doubled Formalism

When it comes to a T-duality invariant formulation of string theory, there are two primary actions that are useful to study as a point of entry. The first is Tseytlin’s non-covariant action. It is found in his formulation of the duality symmetric string, which presents a stringy extension of the Floreanini-Jackiw Lagrangians for chiral fields. In fact, for the sigma model action in this formulation, one can directly reproduce the Floreanini-Jackiw Lagrangians for antichiral and chiral scalar fields. The caveat is that, although we have explicit O(D,D) invariance, which is important because ultimately we want T-duality to be a manifest symmetry, we lose manifest Lorentz covariance on the string worldsheet. What one finds is that we must impose local Lorentz invariance on-shell, and from this there are some interesting things to observe about the constraints imposed at the operator level.

The main papers to study are Tseytlin’s 1990/91 works listed below. Unfortunately there is no pre-print available, so these now classic string papers remain buried behind a paywall:
1) Tseytlin, ‘Duality Symmetric Formulation of String World Sheet Dynamics
2) Tseytlin, ‘Duality Symmetric Closed String Theory and Interacting Chiral Scalars

For Hull’s doubled formalism, on the other hand, we have manifest 2-dimensional invariance. In both cases the worldsheet action is formulated such that both the string coordinates and their duals are on equal footing, hence one thinks of the coordinates being doubled. However, one advantage in Hull’s formulation is that there is a priori doubling of the string coordinates in the target space. Here, O(D,D) invariance is effectively built in as a principle of construction. This is because for the covariant double sigma model action, the target space may be written as R^{1, d-1} \otimes T^{2D}, in which we have a non-compact spacetime and a doubled torus. From the torus identifications we have manifest GL(2D; Z) symmetry. Then after imposing what we define as the self-duality constraint of the theory, which contains an O(D,D) metric, invariance of the theory reduces directly to O(D,D; \mathbb{Z}).

    1. Hull, ‘Doubled Geometry and T-Folds
    2. Hull, ‘Geometry for Non-geometric Backgrounds
    3. Hull and Reid-Edwards, ‘Non-geometric backgrounds, doubled geometry and generalised T-duality

What is neat about the two formulations is that, turning off interactions, they are found to be equivalent on a classical and quantum level. It is quite fun to work through them both and prove their equivalence, as it comes down to the constraints we must impose in both formulations.

I think the doubled formalism (following Hull) for sigma models is most interesting on a general level. I’m still not comfortable with different subtleties in the construction, for example the doubled torus fibration background or choice of polarisation from T-duality. The latter is especially curious. But, in the course of the last two weeks, things are finally beginning to clarify and I look forward to writing more about it in time.

Related to the above, I thought I’d share three other supplementary papers that I’ve found to be generally helpful:

1) Berman, Blair, Malek, and Perry, ‘O(D,D) Geometry of String Theory
2) Berman and Thompson, ‘Duality Symmetric String and M-theory
3) Thompson, ‘T-duality Invariant Approaches to String Theory

There are of course many other papers, including stuff I’ve been studying on general double sigma models and relatedly the Pasti, Sorokin and Tonin method. But those listed above should be a good start for anyone with an itch of curiosity.

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