# ‘The unreasonable effectiveness of string theory in mathematics’: Emergence, synthesis, and beauty

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.

# 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.

# 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 :).

# On Recent Events

But I digress. What I meant to say is that these are uncertain times, to be sure. For whatever it may mean to each individual that reads my blog, I hope you’re keeping well.

For my part, I’ve been fortunate to continue my studies and keep working away on stringy things. During the last few months I’ve spent a lot of time catching up on mountains of string study. One thing that comes with academic acceleration from undergrad to post-grad, at least in my case, is that there is a lot of catching up to do in terms of daily research level string physics and computation. It is not too difficult; rather it is the sheer amount literature. I’ve managed to cover so much in a short time, which has been satisfying, and I’m delighted to have reached a point where it seems each day I am growing increasingly comfortable with the bigger research picture and my place within it. The word ‘orientation’ is perhaps fitting; but how I’ve spent my time is probably more than some typical orientation process. It’s not only about going over foundations as well as contemporary literature and trends of thought, I also find great urgency to go back to the earliest and significant historical string papers and build up as much as I can from first-principles – to deepen my understanding and intuition of the issues and where we need to be.

In the last month or so, I’ve also been working on my MRes thesis, mostly thinking through a lot of double sigma model stuff and generally just putting a lot of energy into maintaining focus on this particular project. Of course, as my professor will attest, numerous things are constantly pulling at my attention and slowly I am finding my way. As for my thesis project, it has been enjoyable. Proficiency in double sigma models is important for future work as well, particularly with my interest in generalised geometry sharpening (among a list of questions from which I may entertain), so I’ve managed to properly sink in to the work allowing the occasional distraction: such as, for instance, Ashoke Sen’s deeply interesting paper on string field theory plus a lot of non-geometry stuff. I often tweet samples of thoughts or references, but I will likely start blogging about all this cool stuff as well.

There is much to write about in string theory and quantum gravity, with some absolutely brilliant papers sitting on my desk. I also have a stack of maths papers I would like to discuss at some point, also not at all irrelevant to string theory. All in good time I’m sure.

For now, as a gentle return, I wanted to make one last comment: with the murder of George Floyd and the re-emergence of Black Lives Matter, there has been a lot of discussion again about racism and racial injustice. There is quite a bit of science behind understanding how bias and prejudice plays notable roles in human experience, including in the sort of cognitive processes that operate in the form bigoted and racist attitudes. As many have highlighted, education is certainly one important strategy as a lot of studies indicate the role of environment in relation to subject development. To that end, I’ve seen a lot of people sharing books and important literature on things like the history of slavery and civil rights. When I was young, about the age of 7 or so, I remember studying the history of slavery in the UK as well as slavery in America, including the Underground Railroad and the life of Harriet Tubman (I can’t recall the books we read, but see for instance this biography by Catherine Clinton). This of course also coincided with studying the American Civil War and other events in Europe. Over time I’ve also read a number of books, like Stephen Bronner’s ‘The Bigot’, which formulates the persistance of racism and bigotry as a sort of anti-modernity. It is an interesting philosophical read. Somewhat relatedly, a couple of weeks ago I tweeted about some of my recollections of Olaudah Equiano (extracts of his memoirs have been digitalised by the British library), a former slave and prominent abolitionist in Britain. In the time of the 18th century enlightenment, he was very much a man of letters. There were also a number of other prominent voices during this period, and, if I remember correctly, a key to generating popular repulsion toward slavery was the industrial workers movement of the time, of which I believe Equiano was a part. Among whites, English Quakers were one a notable organised support. A historian will certainly be able to offer many more details. When one studies this history – take the end of the 18th century in Britain for example, where there was popular support for abolition – it is easy to slip into a view that slavery was abolished and that is the end of it. But it was, and continues to be, a messy and complex moral picture. Indeed, even among enlightenment thinkers of the time, there were several notable secular philosophers supporting abolitionism; but it was certainly morally convoluted and not at all universal. It is fair to say that the actual abolition of slavery in the 19th century also did not mean an end of social-racial thinking; in fact, it is well documented how new forms of formalised racial thought emerged, including new theories of formal racial hierarchy and the formalisation of systems of belief based on eugenics. There is an article in the UN Chronicle that summarises a bit of this history. In terms of books, I’ve recently learned of two that sound informative and interesting: George M. Fredrickson’s ‘Racism: A Short History‘ is often cited. I also recently learned of a book by Timothy C. Winegard entitled, ‘The Mosquito: A Human History of Our Deadliest Predator’. This is not a book on epidemiology, nor is it a work of biology, virology, or for that matter even anthropology. It is a popular history book, the sort I tend to try and avoid; but I’ve heard it offers a fairly detailed history of the slave trade as it relates to mosquito-borne diseases during European colonisation.

Of course, one can easily find online a list of important and widely cited books on the topic. These are two new ones that I’ve highlighted for myself. I’m currently putting together what would be my summer reading list, although my break is delayed until after my thesis is submitted (likely during the autumn before my PhD in December or January). That could be a topic for another post :)

# 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 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.