# Generalised geometry #1: Generalised tangent bundle

1. Introduction

The motivation for generalised geometry as first formulated in [Hitc03], [Hitc05], and [Gual04] was to combine complex and symplectic manifolds into a single, common framework. In the sense of Hitchin’s formulation, which follows Courant and Dorfman, generalised geometry has deep application in physics since emphasis is placed on adapting description of the physical motion of extended objects (i.e., strings). In this way, one can view generalised geometry as analogous to how traditional geometry is adapted to the physical motion of point-particles. There are also more general forms of generalised geometries, which can be thought of as further extended and adapted geometries to describe higher dimensional objects such as membranes (and hence also M-theory). These notions of geometry, which we can organise under the conceptual umbrella of extended geometries, correlate closely with the study of extended field theories that captures both Double Field Theory (DFT) and Exceptional Field Theory (EFT).

In these notes, interest in generalised geometry begins with the way in which generalised and extended geometry makes manifest hidden symmetries in string / M-theory. In particular, emphasis is on obtaining a deeper understanding and sense of mathematical intuition for the structure of generalised diffeomorphisms and gauge symmetries. The purpose was to then extend this emphasis to a study of the gauge structure of DFT, which is well known to be closely related with generalised geometry but in fact extends beyond it. We won’t get into this last concern in these notes; it is merely stated to make clear the original motivation for reviewing the topics.

Given that generalised geometry inspired the seminal formulations of DFT, it is no coincidence that what we observe in a detailed review of generalised geometry is the way in which the metric and p-form potentials are explicitly combined into a single object that acts on an enlarged space. This enables a description of diffeomorphisms and gauge transformations of the graviton and Kalb-Ramond B-field in a combined way. In fact, one of Hitchin’s motivations for the introduction of generalised geometry was to give a natural geometric meaning to the B-field. As will become clear in a later note, a key observation in this regard is that the automorphism group of the Courant algebroid ${TM \oplus T^{\star}M}$ is the semidirect product of the group of diffeomorphisms and B-field transformations. We will then study the structure of this group.

Remark 1 (Generalised geometry, branes, and SUGRA) Although not a focus of these notes, it is worth mentioning that generalised geometry in the sense of Hitchin is an important framework for studying branes and also T-dualities, including mirror symmetry. It also offers a powerful collection of tools to study Calabi-Yau manifolds, particularly generalised Calabi-Yau, proving important in the search for more realistic flux compactifications.

2. Generalised tangent bundle

The main objects to study on generalised geometry are Courant algebroids. But before we reach this stage, there are two fundamental structures of generalised geometry that we must first define: 1) the generalised tangent bundle and, 2) the Courant bracket. In this note, we introduce the generalised tangent bundle. Then in the following notes we explore the properties of this structure and the related extension of linear algebra to generalised linear algebra. This brings us to finally study the Courant bracket, its properties and symmetries, before we study Courant algebroids and generalised diffeomorphisms.

Definition 1 (Generalised bundle) The generalised tangent bundle is obtained by replacing the standard tangent bundle ${T}$ of a D-dimensional manifold ${M}$ with the following generalised analogue

$\displaystyle E \cong TM \oplus T^{\star}M. \ \ \ \ \ (1)$

The generalised tangent bundle ${E}$ is therefore a direct sum of the tangent bundle ${TM}$ and co-tangent bundle ${T^{\star}M}$. As we will learn, the bundle ${E}$ has a natural symmetric form with respect to which both ${TM}$ and ${T^{\star}M}$ are maximally isotropic.

Remark 2 (Notation) Often in these notes we will use ${E}$ and ${TM \oplus T^{\star}M}$ interchangeably, which should be clear in the given context.

The generalised bundle (1) fits the following exact short sequence

$\displaystyle 0 \longrightarrow T^{\star}M \hookrightarrow E \overset{\rho}{\longrightarrow} TM \longrightarrow 0, \ \ \ \ \ (2)$

which, later on, we’ll see is the sort of sequence that describes an exact Courant algebroid.

Remark 3 (Early comment on Courant algebroids) As we will study in a later entry, it is the view afforded by generalised geometry that the bundle ${E}$ is in fact an extension of ${TM}$ by ${T^{\star}M}$, and so it is a direct example of a Courant algebroid such that, in the exact sequence (2), the Courant algebroid has a symmetric form plus other structure (e.g., the Courant bracket) that makes it isomorphic to ${E}$. This is true for suitable isotropic splittings of the exact sequence, an example of which is called a Dirac structure.

The sections of ${E}$ are non-trivial sections of ${TM \oplus T^{\star}M}$. This means that, unlike in standard geometry and how we typically consider vector fields as sections of ${TM}$ only, we now consider elements of the non-trivial sections

$\displaystyle X = x + \xi, Y = y + \varepsilon, \ x,y \in \Gamma(TM), \ \xi, \varepsilon \in \Gamma(T^{\star}M), \ \ \ \ \ (3)$

where ${x, y}$ are vector parts and ${\xi, \varepsilon}$ 1-form parts.

The set of smooth sections ${C^{\infty}(M)}$ of the bundle ${E}$ are denoted by ${\Gamma(E)}$ such that the set of smooth vector fields is denoted by ${\Gamma(TM)}$ and the set of smooth 1-forms by ${\Gamma(T^{\star}M)}$.

Remark 4 (Sequence and string background fields) For the sequence (2), note that in the map ${\rho}$ there exist sections ${\sigma}$ that are given by rank 2 tensors, which can then be split into symmetric and antisymmetric parts, ${\sigma_{\mu \nu} = g_{\mu \nu} + b_{\mu \nu}}$. The sections of ${E}$ describe infinitesimal symmetries of these fields, as they are encoded in a generalised vector field ${X}$ capturing infinitesimal diffeomorphisms and a 1-form ${\xi}$ describing the b-field gauge symmetry.

References

[Gual04] M. Gualtieri. Generalized complex geometry [PhD thesis]. arXiv: 0401221[math.DG]. [Gua11] Marco Gualtieri. Generalized complex geometry. Ann. of Math. (2), 174(1):75–123, 2011. url: https://annals.math.princeton.edu/wp-content/uploads/annals-v174-n1-p03-s.pdf. [Hitc03] N. Hitchin. Generalized Calabi–Yau manifolds, Quart. J. Math. Oxford Ser. 54 (2003) 281. arXiv: 0209099 [math.DG]. [Hitc05] N. Hitchin. Brackets, forms and invariant functionals. arXiv: 0508618 [math.DG]. [Hitc10] N. Hitchin. Lectures on generalized geometry. arXiv: 1008.0973 [math.DG]. [Rub18] R. Rubio. Generalised geometry: An introduction [lecture notes]. url: https://mat.uab.cat/~rubio/gengeo/Rubio-GenGeo.pdf

# (n-1)-thoughts, n=6: Asperger’s and writing, Lie 2-algebroids, linguistics, and summer reading

Asperger’s, studying, and writing

As a person with autistic spectrum disorder (ASD), I’ve learned that writing plays an important and meaningful role in my life. I write a lot. By ‘a lot’ I mean to define it as a daily activity. Sometimes I will spend my entire morning and afternoon writing. Other times I will be up through night because my urge to write about something has kept me from sleeping. Most often I write about maths and physics, keeping track of my thoughts and ideas, planning essays, or writing about my work. But I also make it a principle of life to read widely. Indeed, I enjoy reading – studying – as much as I enjoy writing, and this often motivates me to write about many other topics. The two go hand-in-hand.

One reason writing has become important for me has to do with how, as a person with Asperger’s, social communication (by which I mean verbal, but of course also entails other forms like sign) is a source of struggle. I don’t often write about my Asperger’s, mainly because I find it a difficult process. It is hard to organise my thoughts about it, and I am never sure what is appropriate to share. In formal language, my Asperger’s is described clinically as high-functioning but severe. A big part of my life is about learning new strategies to cope. Some of the strategies may even be familiar to others without ASD, like learning to talk in front of others in ways that minimise anxiety and stress, or without completely freaking out (what we call in my language ‘red card’ moments). Or, to give another example, we work on finding strategies for the times I am at the office, so my brain doesn’t go into hyperdrive and so I can focus on discussion and also things like writing on the whiteboard. Another thing about my Asperger’s is that it can be hard adjusting to new people and it can be very stressful acclimatising to new environments. I’ve been working with Tony, now my PhD supervisor, for two years or more and I have only recently started to acclimatise and find our engagement a bit easier to manage. Indeed, in the same time I’ve been at the University of Nottingham, it remains an ongoing process adjusting to this new environment and to being on campus. Like with my close friend, Arnold, who, even after seeing him everyday for years, it was often still a challenge for me to engage with him socially and to visit his house. There is a lot to my experience, not just the social aspect of experience, that can be difficult and demanding as well as overwhelming. I also struggle a lot with anxiety and other things, in addition to extreme sensory sensitivity. So I require a lot of time and space for stillness in my own environment, with my own structure and routine – usually in my own space with my books and other comforts – because sensory overload can easily overwhelm.

In my one attempt to write about living with ASD I expressed how it can be difficult to understand cultural meanings as another example. This is a way of describing orientation to many of the ‘codes’ or behavioural routines that normalise in society. For example, I remember when I was a teenager being pressured a lot to establish the same routine economic patterns as others, or blamed because I didn’t have a job or couldn’t maintain one. I find it difficult to compute things like why daily life is the way it is for most individuals or why people behave as they do. What motivates daily behaviour and routine? How do people make decisions or direct the future course of their lives? Science, textbooks, and studying fervently became, at least in part, a survival-based mechanism. There is no instruction manual about humans; or about why history has taken the path it has in the course of human and societal development; or why many arbitrary social customs have come to be the way they are; or why my father acted and behaved the way he did; among many other things that come to be a feature of life. Studying became my way to cope and to understand, and writing became an extension of that. For instance, I studied every aspect of psychology to help better understand my experiences growing up or why, at least in part, people act violently or use violent language. I’ve read and written across most of philosophy; the same for economics, certainly enough to understand the fundamental debates; and also a lot of sociology. At one point I read a lot of political history, with history one of my favourite subjects. While all of this has a purpose in aiding my attempt to try and understand the world I am a part of, it also supports my passion for studying, my focused interests, and provides the stimulation I need.

On top of it all, living with Asperger’s can be quite exhausting. Indeed, one thing that is common for people diagnosed with autism is the experience of a certain type of fatigue, or what, in my house, we call ‘crashes’. These are a daily experience, where I need to put on my headphones and sit in my own (still and comfortable) space for however long it takes to calm my brain. For these reasons, day to day life is often spent in controlled environments, because it helps ease the red card moments, reducing stress and anxiety, and thus also helps combat the amount of crashes.

I think it all adds up in some sort of complicated sum as to why I find writing an important outlet. But even writing has its own difficulties. I remember my teacher, when I was 6 or 7 years old, say that my brain runs faster than my pen. I think this is true. I think of the sluggish pen effect as the difficulty in converting the internal representations of whatever concept or idea into concise written form at the pace I wish to feed ink to paper. So even though I write everyday and have been practising for many years, the usual result of my writing is typically permeated with errors. The process can be disabling and discouraging, to be honest, with many moments of frustration and failure; but, I’ve also learned that when I battle through and produce something I am happy with, the moment of victory is worth so much.

For many personal reasons, I’ve been regularly encouraged to write more and share more on my blog, and this is something I’ve been working toward. I think that, over a couple of years, I’ve grown more and more comfortable sharing essays and technical notes, although perhaps that is especially true in recent months; but I am also practising writing in other ways, like more personally and less formally. Technical writing is much easier than informal discussion, although a definition of the latter still seems somewhat unclear.

So as one step, this is a new blog post format that I may start experimenting with over the coming weeks, in addition to my usual research entries, essays, and technical notes. Although I prefer to keep my blog focused on my maths and physics research, which of course is mainly string related, allowing from time to time the inclusion of the odd bit of academic diversion, I think this (weekly or fortnightly) format of (n-1)-thoughts may be a fun space that allows me to practice writing in different ways, to share disconnected thoughts or random interests, outside of the formal essay or technical structure.

Generalised geometry, higher structures, and some John Baez papers

Another gem by, Urs! In a recent post on higher structures and M-theory, I made a comment recommending that people read Urs Schreiber’s many notes over the years. In my own research, I’ve found them to be invaluable. The most recent example relates, in some ways, to what I also mentioned in that post about how we may motivate the study of higher structures in fundamental physics: namely, how the Kalb-Ramond 2-form can be seen as an example of a higher structure as it is generalised from the gauge potential 1-form. I won’t go into details here, but the other day I was thinking about such generalisations, and I was thinking about Hamiltonian mechanics in the process. As I’ve mentioned before, if I were to teach string theory one day I would take this approach, emphasising at the outset the important generalisation from point particle theory to the extended object of the string.

Thinking of higher structures, I knew there were many connections here, and I was wanting to fill out my notes, for instance from how in generalised geometry the algebraic structure on $TX \otimes T*X$ is a Courant Lie 2-algebroid. Those who study DFT will likely be quite familiar with Courant algebroids, and, certainly from a higher structure perspective this line of study is interesting. I also knew there was an original paper, which I had seen in passing, talking about this and the relation to symplectic manifolds, but I couldn’t find it. Then, bam! As Schreiber notes in a forum reply, ‘Courant Lie 2-algebroids (standard or non-standard) play a role in various guises in 2-dimensional QFT, thanks to the fact that they are in a precise sense the next higher analogue of symplectic manifolds and thus the direct generalization of Hamiltonian mechanics from point particles to strings’.

The part ‘from point particles to strings’ was hyperlinked to an important paper, the very paper I was looking for! The paper is Categorified Symplectic Geometry and the Classical String by John C. Baez, Alexander E. Hoffnung, Christopher L. Rogers. I look forward to working through this.

I also want to highlight several other papers from around the same time by Baez, including one co-authored with Schreiber, that I think are also foundational to the programme:

Categorification co-authored with James Dolan;

Higher-Dimensional Algebra VI: Lie 2-Algebras co-authored with Alissa S. Crans;

Lectures on n-Categories and Cohomology co-authored with Michael Shulman;

and, finally, Higher Gauge Theory co-authored with Urs Schreiber.

My summer holiday is in June this year, as I have a conference in mid-July and then I am scheduled to return back to university 1 August. I think Beth and I are going to spend a week in a North Norfolk, one of our favourite places, which has also sort of become a home for both of us. In anticipation of my break, I’ve started putting together my summer reading list, as I do every year. To be honest, there are so many good books right now, it is difficult to choose.

Although my list isn’t complete, one book that I’m already looking forward to is Jennifer Ackerman’s ‘The Genius of Birds‘. I had this book on my Christmas break reading list but, unfortunately, I didn’t have enough time to get to it.

I recently purchased ‘Explaining Humans: What Science Can Teach Us About Life, Love and Relationships’ by Camilla Pang, and I think I will add this to my list. Camilla has a PhD in biochemistry and, as she also has ASD, my interest in this book is more so about her personal journey coming to grips with the complex world social around her through the lens of science. It sounds, on quick glance, that we’ve come to cope with the world in similar ways and share an interest in understanding human behaviour and development. Having said that, I think there is a bit of a risk that people might read this book and conflate it with some sort of autistic worldview, which is completely incorrect, or, equally incorrect, as a scientific view of human behaviour. Contrary to some reviews, I wouldn’t read Pang’s book looking for a strictly scientific view (else one will be disappointed). I could be wrong, but I think ‘Explaining Humans’ may have potentially been mispromoted, hence some of the confused feedback. I approach this book as I would when reading someone’s memoirs, like ‘Diary of a Young Naturalist‘ by Dara McAnulty, ‘Lab girl‘ by Hope Jahren, or ‘Letters to a Young Scientist‘ by Edward O. Wilson. With topics including the challenges of relationships, learning from mistakes, and navigating the human social world by finding tools in things like game theory and machine learning, my interest is in the fact that this is another author with autism and, for myself, I similarly use textbooks and my studies to understand and manage my experience the world. Even on a purely phenomenological level, it will be interesting.

Another book that I may add is of a completely different tone: namely, Saul David’s ‘Crucible of Hell’. I’ve been enjoying reading about WWII again, and, as noted in this post on Dan Carlin’s podcast series on the events in the Asiatic-Pacific theatre, the battle of Okinawa (and others) I haven’t read much about. A few more books I have been thinking about: Douglas R. Hofstadter’s ‘Gödel, Escher, Bach: an Eternal Golden Braid‘, ‘The Deeper Genome‘ by John Parrington, ‘King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry‘ by Siobhan Roberts, ‘Decoding Schopenhauer’s Metaphysics’ by Bernardo Kastrup, ‘Quantum Computing Since Democritus‘ by Scott Aaronson, Jared Diamond’s ‘Guns, Germs, and Steel: The Fates of Human Society‘, and Daniel Kahneman’s latest ‘Noise: A Flaw in Human Judgement‘. Tough decisions.

Linguistics

I’ve been short on time this week finishing some calculations and working on a paper, prior to receiving my second Covid jab. But the other afternoon I thoroughly enjoyed this article. It’s on the Galilean challenge and its reformulation, wherein discussion unfolds on why there is an emerging distinction between the internalised system of knowledge and the processes that access it.

As alluded a moment ago, a general theory of development has interested me for a long time. For my book published by Springer Nature, a lot of the study and references were originally motivated by this interest. When I last did an extensive read on the topic, there was a lot of progress in developmental models – biology, bio- and neuro-linguistics, child psychology, and so on. The summer when I was writing my book, I had already compiled all of my research and I was running short on time in terms of the writing process (I wrote the book in the span of two weeks). Around the time of my research, if I recall there was discussion in biolinguistics regarding the hypothesis of ‘[t]he fibre tract [as one reason] for the difference in language ability in adults compared to pre-linguistic infants’. I remember noting that interesting ideas were developing, and this is a nice article on that front. What is particularly fascinating, I would say, is how language design appears to maximise computation efficiency, but ‘disregards communicative efficiency‘ [italics mine].

This certainly runs directly counter to common belief, as mentioned in the article, namely the established view that communication is a basic function of language. For a long time, as I understand it, there was belief that there was an experiential component to early language formation; but what current research suggests is that, an experiential component is not fundamental at all. Of course an experiential component plays a role, in some capacity, when it comes to externalisation processes, such as in development of variances in regional accent here in England as an example. I mean, the subject is mediated (to whatever degree) by his/her sociohistorical-cultural circumstances, but, unless I am misunderstanding (I need to read through the research more deeply) language itself is not some purely social construct.

Regarding reference to the evolutionary record, I wonder how the developing view in the article relates to ongoing research concerning, for example, certain species of birds, their migratory paths, and the question of inherited or genetic knowledge. It’s an absolutely fascinating area of study, something I’ve been reading about with my interests in mathematical biology, and of course there is very apt analogy here also with broader developments in microbiology.

One last thing of note from reading the article, as I have written quite a bit about the enlightenment philosophes and the start of modern science, it is notable how they sought to ask the question of language. Descartes’ fundamental enquiry into language – the Cartesian question – remains interesting to this day, and I was delighted to see it referenced at the outset. I recommend reading Descartes’ meditations plus other contributions to the enlightenment philosophes – Kant, Spinoza, Hume, to name a few. There is so much here that remains relevant to our modern history and to the development of the contemporary social world. For a few years I’ve been writing a series of essays on Hegel’s science of logic and his epistemology, which is notably relevant today in my area of work in fundamental maths/physics.

Mental health awareness week

Finally, it’s mental health awareness week in the UK. Often these sort of campaigns can be incredibly superficial, failing to look at root causes or ask fundamental questions about well-being and support, but they don’t have to be. Mental health awareness is something that I’ve always taken seriously, not least because I have experienced many challenges with my own mental health throughout my life. The last time I did research and wrote on the subject, suicide statistics in many leading Western countries were significant. I know, too, that for people with autism, like myself, mental health can present a significant challenge in addition to the other challenges one may face. People with autism are much more likely to die by suicide than the general population, as many cause-specific analyses of mortality for people with autistic spectrum disorder (ASD) indicate. Sometimes these facts are overlooked when we talk about mental health as a society, and often I find it important to highlight. But mental health doesn’t discriminate, it affects all people from all backgrounds, and weeks like this one are a good time to help foster discussion, combat stigma, and to think about mental health in all of its facets.

*Image: ‘Streams of Paint‘ by markchadwickart (CC BY-NC-ND 2.0).

The case of the duality symmetric string is a curious one (in a recent post we began discussing this string in the context of building toward a study of duality symmetric M-theory). In this essay, which may serve as the first of a few on the topic, I want to offer an introduction to some of the characteristic features of the duality symmetric string – what I will also refer to as the doubled string – as well as discuss some of its historical connections. One thing that we will focus on at the outset is the deep connection between this extended formulation of string theory, string field theory (SFT), and the more recent development of double field theory (DFT). Such a connection is prominent not least in how we treat the string fields in constructions in which T-duality is a manifest symmetry. For the purposes of this essay, these constructions may be defined in terms of what are called double sigma models.

To help lay this out, let’s quickly review some history. In the early 1990s, a series of papers appeared by Tseytlin [1,2], Siegel [3, 4], and Duff [5]. In these papers, the important topic of string dualities was explored, particularly the fundamental role target-space duality (T-duality) plays in string theory. T-duality is of course an old subject in string theory, and we have already spoken several times in the past about its key features. Recall, for instance, that the existence of this fundamental symmetry is a direct consequence of the existence of the string as a generalisation of point particle theory. Given how for the closed string in the presence of ${d}$ compact dimensions T-duality interchanges the momentum modes ${k}$ of a string with its winding modes ${w}$ around a compact cycle, one of the deep implications is that in many cases two different geometries for the extra dimensions are found to be physically equivalent.

From the space-time perspective, T-duality is a solution generating symmetry of the low energy equations of motion. However, from a world-sheet point of view, T-duality is a non-perturbative symmetry. The fact that it is an exact symmetry for closed strings suggests, firstly, that one should be able to extend the standard formulation of string theory based famously on the Polyakov action (for review, see the first chapter of Polchinski). The idea is that we may do this at the level of the world-sheet sigma-model Lagrangian density, by which I mean the motivation is to construct a manifestly T-duality invariant formulation of closed string theory on the level of the action, remembering from past discussions that we may capture T-duality transformations under the group $O(D,D,\mathbb{Z})$. When we extend the theory in this way, we find that we are obliged to introduce the compact coordinates ${X}$ and the dual ones ${\tilde{X}}$ in the sigma model, which means we double the string coordinates in the target-space. This gives the name double string theory.

Let’s explain what this all means in clearer terms, as many of these ideas can be sketched cleanly in the context of SFT. In 1992/93, around the same time as the first duality symmetric string papers, field theory emerged as a complete gauge-invariant formulation of string dynamics [6, 7]. This led to the development of a precise spacetime action whose gauge symmetry arguably takes the most elegant possible form [8]. What was observed, furthermore, is how the momentum and winding modes may be treated symmetrically and on equal footing. For instance, let us explicitly denote the compact coordinates ${X^{a}}$ and the non-compact coordinates ${X^{\mu}}$, with ${X^{I} = (X^{a}, X^{\mu})}$. Conventionally, we define the indices such that ${I = 1,...,D}$, ${\mu = 1,...,d}$, and ${a = 1,...,n}$. If the string field gives component fields that depend on momentum ${p^{a}}$ and winding ${w^{a}}$, then in position space we may assign the coordinates ${X^{a}}$ conjugate to the momentum and, as alluded above, new periodic dual coordinates ${\tilde{X}_{a}}$ conjugate to the winding modes.

The key point is as follows: if one attempts to write the complete field theory of closed strings in coordinate space, then as stated the full theory depends naturally on dual coordinates ${X^{a}}$ and ${\tilde{X}_{a}}$. This is also to say that naturally the full phase space of the theory accompanies both the momentum and the winding modes. Or, to phrase it in a slightly different manner, for toroidal compactification there is a zero mode ${X^{a}}$ and ${\tilde{X}_{a}}$, and, as the expansion of a string field provides component fields that depend on both momentum and winding, we come to the statement that the arguments of all fields in such a theory are doubled. For the doubled fields ${\phi(X^{a}, \tilde{X}_{a}, X^{\mu})}$ we may write the following seemingly simple action

$\displaystyle S = \int dX^{a} d\tilde{X}_{a} dX^{\mu} \mathcal{L}(X^{a}, \tilde{X}_{a}, X^{\mu}) \ (1)$.

The Lagrangian in (1) may seem straightforward, but in fact it proves incredibly complicated. One issue has to do with how the physical content of the theory becomes buried underneath unphysical and computationally inaccessible data, with the full closed string field theory comprising an infinite number of fields. This is where DFT may be motivated from first-principles; because, in response, DFT answers this problem by issuing the following simplification strategy: what if we instead choose some finite subset of string fields? An obvious choice for such a subsector of the full theory is the massless sector. In the study of DFT, we may then ask, if for the standard bosonic string the low-energy effective action is famously

$\displaystyle S = \int dX \sqrt{-g} e^{-2\phi} [R + 4(\partial \phi)^2 - \frac{1}{12}H^2] + \text{higher derivative terms}, \ (2)$

what does this action become in the case of doubled coordinates on tori? Is T-duality manifest? What about for non-trivial geometries? Historically, DFT emerged with the aim to answer such questions. In fact, following Nigel Hitchin’s introduction of generalised geometry [9, 10], itself inspired by the existence of T-duality, serious efforts materialised to incorporate this mathematical insight into the study of the target-space geometry in which strings live [11, 12, 13, 14], beginning especially with the study of phase space and invariance of respective Hamiltonians. This culminated in 2009, when Hull and Zwiebach formulated such a T-duality invariant theory explicitly [11], formalising DFT almost two decades after the original duality symmetric string papers. What one finds is a theory constructed on the product manifold ${\mathbb{R}^{d-1,1} \times T^{n}}$ with coordinate space fields ${\phi(X^{\mu}, X^a, \tilde{X}_{a})}$. The torus is doubled, containing the spacetime torus and the torus parameterised by the winding modes, such that ${(X^a, \tilde{X}_{a})}$ are periodic on ${T^{2n}}$. The spectrum for the massless fields is then described in terms of the supergravity limit of string theory.

By taking this approach, DFT has presented fresh insight on T-duality in string theory, leading to the development of deeper connections between frontier theoretical physics and mathematics through the appearance and use of Courant brackets, and by gaining new insight on the deepening role generalised geometry seems to play in string theory.

Much like field theory, the doubled world-sheet theory has also been reinvigorated in the last decade or more. This follows from breakthrough work by Hull [16, 17], who established the doubled formalism to define strings in a class of non-geometric backgrounds known as T-folds. These are non-geometric manifolds where locally geometric regions are patched together such that the transition functions are T-duality transformations.

***

Currently, there are primarily two doubled string actions that we may consider when constructing double sigma models: Tseytlin’s first-principle construction of the duality symmetric string [1, 2, 15] and Hull’s doubled formalism [16, 17]. Both actions satisfy the requirement of T-duality appearing as a manifest symmetry, with the former possessing general non-covariance and the latter possessing general covariance.

Hull’s doubled formalism is interesting for several reasons. In this formulation we have manifest 2-dimensional Lorentz invariance from the outset, and a notable advantage is that there is a priori doubling of the string coordinates in the target space. In other words, both the Tseytlin approach and the Hull approach are formulated such that both the string coordinates and their duals are treated on equal footing. But in Hull’s formulation, ${O(D,D)}$ invariance is effectively built in as a principle of construction. This is because for the covariant double sigma model action, the target space takes the form ${R^{1, d-1} \otimes T^{2n}}$, in which we have a non-compact spacetime and a doubled torus. From the torus identifications we have manifest ${GL(2n; \mathbb{Z})}$ symmetry. Then, after imposing what is defined as the self-duality constraint of the theory, which contains the $O(D,D)$ metric, invariance of the theory reduces directly to $O(D,D; \mathbb{Z})$. In other words, while the doubled formalism starts with a covariant action that involves doubled coordinates, the invariance of this theory under $O(D,D)$ is generated by imposing this self-duality constraint, which, similar to DFT, effectively halves the degrees of freedom and ensures that the remaining fields are physical.

Think of it this way: in Hull’s doubled formalism the essential motivation is to double the torus by then adding ${2n}$ coordinates such that the fibre is ${T^{2n}}$; however, typically the fields depend only on the base coordinates. Finally, the strategy is generally to proceed with a patch-wise splitting ${T^{2n} \rightarrow T^{n} \oplus \tilde{T}^n}$ so that we have demarcated a strictly physical subspace ${T^n}$ and its dual ${\tilde{T}^{n}}$. For a geometric background local patches are glued together with transition functions which include group ${GL(n, \mathbb{Z})}$ valued large diffeomorphisms of the fibre. For the non-geometric case, this is approached by gluing local patches with transition functions that take values in ${GL(n, \mathbb{Z})}$ as well as in the complete T-duality group, such that ${O(D,D,\mathbb{Z})}$ is a subgroup of ${GL(2n, \mathbb{Z})}$ large diffeomorphisms of the doubled torus.

On the other hand, Tseytlin’s first-principle formulation of the duality symmetric string and world-sheet theory for interacting chiral scalars, which presents a direct stringy extension (or stringification) of the Floreanini-Jackiw Lagrangians [31] for chiral fields, does not possess ${O(D,D)}$ by principle of construction. Instead, we find that it emerges rather organically as an intrinsic characteristic of the doubled string, with the caveat being that we lose manifest Lorentz covariance on the string world-sheet. What one finds is that we must instead impose local Lorentz invariance on-shell. The equivalence of the Tseytlin and Hull actions on a classical and quantum level has been shown in [32, 33, 34]. Like DFT, both of these approaches are constructed around the generalised metric $\mathcal{H}_{IJ}$ which we’ll touch on later.

It is no surprise that earlier formulations of the duality symmetric string were a primary reference in the development of DFT. In [1, 2], Tseytlin argues that the existence of the intrinsically stringy winding modes, which appear in the spectrum of the closed string compactified on a torus (created by vertex operators involving both ${X}$ and ${\tilde{X}}$), can result in 2d field theories with interactions indeed involving ${X}$ and ${\tilde{X}}$. Similar models have been explored in statistical mechanics, with the key point in closed string theory being how for fully-fledged local quantum field theories we are required to treat ${X}$ and ${\tilde{X}}$ as independent 2d fields (dual to each other on-shell). An advantage of such an extended formulation of string theory is that we may obtain more vacua than the standard formulation. Furthermore, as one may have guessed, the notion of the duality symmetric string is based on the fact that duality symmetry becomes an off-shell symmetry of the world-sheet action. Thus, T-duality for example may be made manifest in the scattering amplitudes and on the level of the effective action.

To study the construction of the duality symmetric string, we note that directly from 2-dimensional scalar field theory constructed to be symmetric in ${\phi}$ and ${\tilde{\phi}}$, Tseytlin derives the Lagrangian density

$\displaystyle \mathcal{L}_{sym} = \mathcal{L}_{+}(\phi_{+}) + \mathcal{L}_{-}(\phi_{-}) \ (3)$

with

$\displaystyle \mathcal{L}_{\pm}(\phi_{\pm}) = \pm \frac{1}{2}\dot{\phi}_{\pm}\phi^{\prime}_{\pm} - \frac{1}{2} \phi^{\prime 2}_{\pm}. \ (4)$.

Here ${\mathcal{L}_{+}}$ and ${\mathcal{L}_{-}}$ are the Floreanini-Jackiw Lagrangian densities for chiral and anti-chiral fields, with ${\dot{\phi} = \partial /\partial_{\tau}}$ and ${\phi^{\prime} = \partial / \partial_{\sigma}}$. The total Lagrangian ${\mathcal{L}_{sym}}$ is itself constructed so that it is manifestly invariant under the exchange of ${\phi = \frac{1}{\sqrt{2}} (\phi_{+} + \phi_{-})}$ with its Hodge dual ${\tilde{\phi} = \frac{1}{\sqrt{2}} (\phi_{+} - \phi_{-})}$. Directly from the equations of motion one can derive chirality conditions for this theory (for a complete review see also [32, 33, 34]).

For our present purposes it is important to note that the goal for Tseytlin is to realise from 2-dimensional scalar field theory the corresponding formulation of string theory, which indeed proves general enough to incorporate the world-sheet dynamics of the winding sector. Writing the Lagrangian (3) for ${D}$ scalar fields ${X^{I}}$ and with a general background, in the Tseytlin approach we famously obtain the action

$\displaystyle S [e^{a}_{n}, X^{I}] = - \frac{1}{2} \int_{\sum} d^{2}\xi e [ \mathcal{C}^{ab}_{IJ}(\xi) \nabla_{a} X^{I} \nabla_{b} X^{J}]. \ (5)$.

Here ${I, J = 1,...,D}$. We define the coordinates on ${\sum}$ such that ${\xi^{0} \equiv \tau}$ and ${\xi^{1} \equiv \sigma}$. The two-dimensional scalar fields ${X^{I}}$ depend on ${\xi}$ and they are vectors in ${N}$-dimensional target space ${\mathcal{M}}$. The number ${N}$ of embedding coordinates is kept general, because the purpose of this action is to be as generic as possible while minimising assumptions for its construction. We also note that ${C_{IJ}}$ need not necessarily be symmetric and, from the outset, we can treat it completely generically. We also have the zweibein ${e^{a}_{n}}$, where ${e = \det e^{a}_{n}}$. This term appears in the definition of the covariant derivative of the scalar field ${X^{I} : \nabla_{a} X^{I} \equiv e^{a}_{n}\partial_{a} X^{I}}$, where ${a}$ is a flat index and ${n}$ is a curved index.

In its first principle construction, which occupies the earliest sections of [2], one can recover from this generic action (5) the standard manifestly Lorentz invariant sigma model action for strings propagating in a curved background. Furthermore, if we exclude the dilaton for simplicity we may define ${\mathcal{C}^{ab}_{IJ} = T(\eta^{ab}G_{IJ} - \epsilon^{ab}B_{IJ})}$, where we reintroduce explicit notation for the string tension ${T}$, ${G}$ is the metric tensor on the target space, and ${B}$ is the Kalb-Ramond field.

Keeping to a generic analysis with a general ${C}$, after a number of steps one finds that (5) may be rewritten in the following way,

$\displaystyle S = -\frac{1}{2} \int d^{2}\xi e[ \mathbb{C}_{IJ}(\xi) \nabla_{0} X^{I} \nabla_{1} X^{J} + M_{IJ} \nabla_{1} X^{I} \nabla_{1} X_{J}]. \ (6)$.

Here it is conventional to define ${\mathbb{C}_{IJ} = C_{IJ}^{01} + C_{JI}^{10}}$ and ${M_{IJ} = M_{JI} = C^{11}_{IJ}}$. The action is manifestly diffeomorphism ${\xi^{n} \rightarrow \xi^{\prime n}(\xi)}$ and Weyl ${e^{a}_{n} \rightarrow \lambda(\xi)e^{a}_{n}}$ invariant, but it is not manifestly invariant under local Lorentz transformations. Moreover, notice that (6) must be invariant for the finite transformation of the zweibein, because the physical theory should be independent of ${e^{a}_{n}}$. This means that if under such a transformation we have ${e^{a}_{n} \rightarrow e^{\prime a}_{n} = \Lambda^{a}_{b}(\xi)e^{b}_{n}}$, where one may recognise ${\Lambda^{a}_{b}}$ is a Lorentz ${SO(1,1)}$ matrix dependent on ${\xi}$, we also have an induced infinitesimal transformation of the form ${\delta e^{a}_{n} = \omega^{a}_{b}(\xi)e^{b}_{n}}$ with ${\omega_{ab} = - \omega_{ba}}$. Now, substituting ${\omega^{a}_{b}(\xi) = n(\xi)\epsilon^{a}_{b}}$, we obtain

$\displaystyle \delta e^{a}_{n} = n (\xi)\epsilon^{a}_{b}(\xi)e^{b}_{n}. \ (7)$,

however, as stated, the action is not manifestly invariant under such transformations. The requirement of on-shell local Lorentz invariance is fundamental to the entire discussion at this point. As Tseytlin comments in a footnote [2], alternatively we may prefer Siegel’s manifestly Lorentz covariant formulation, but with that we obtain extra fields and gauge symmetries; whereas in extending the Floreanini-Jackiw formulation it is fairly simple to introduce interactions and, ultimately, we find that the condition in the Siegel approach that requires decoupling of the Lagrange multiplier corresponds to what we will review as the Lorentz invariance condition in the Floreanini-Jackiw approach.

For the action (6), a way to attack the requirement of on-shell Lorentz invariance is by seeing in [2] that it demands we satisfy the condition

$\displaystyle \epsilon^{ab} t_{ab} = 0, \text{where} \ t_{a}^{b} \equiv \frac{2}{\epsilon} \frac{\delta S}{\delta e^{a}_{n}}e^{b}_{n}. \ (8)$.

The general idea is that the tree-level string vacua should be assumed to correspond to ${S[X, \tilde{X}, e]}$, which define the Weyl and Lorentz invariant quantum field theory. In performing the background field expansion, we may take the expansion to be near the classical solution of the ${(X, \tilde{X})}$ equations of motion with the trace of the expectation value of the energy-momentum tensor as well as the ${\epsilon^{ab}}$ trace vanishing on-shell. In Tseytlin’s formulation, ${\hat{t}}$ denotes precisely this epsilon trace such that ${\hat{t} = \epsilon^{a}_{b} t_{a}^{b}}$. The vanishing of ${\hat{t}}$ shows local Lorentz invariance. So let us now vary (6) under local Lorentz transformation, which is proportional to the equations of motion

$\displaystyle t^{b}_{a} = - \delta_{a}^{b} [\mathbb{C}_{IJ}(\xi) \nabla_{0} X^{I} \nabla_{1} X^{J} \ + \ M_{IJ} \nabla_{1} X^{I} \nabla_{1} X^{J}] \ + \ \delta_{0}^{b}[C_{IJ}\nabla_{a} X^{I} \nabla_{1} X^{J}] \ + \ \delta_{1}^{b} [C_{IJ}\nabla_{0} X^{I} \nabla_{a} X^{J}] \ + \ 2\delta_{1}^{b}M_{IJ}\nabla_{a}X^{I}\nabla_{1}X^{J}. \ (9)$

This equation for ${t^{b}_{a}}$ is equivalent to equation 4.3 in [2]. In order for the variation of the action to vanish under such a transformation, we derive the condition

$\displaystyle \epsilon^{ab}t_{ab} = 0. \ (10)$

In other words, the condition that must be satisfied to recover local Lorentz invariance depends on the solution of the equations of motion for the zweibein. In fact, one will recognise that what is observed is completely analogous to the standard string theory formulation based on the Polyakov action, where one will recall that the equations of motion for the world-sheet metric determines the vanishing of the energy-momentum tensor [35].

This constraint must be imposed on a classical and quantum level.

The key point is that now we can choose the flat gauge ${e_{n}^{a} = \delta_{n}^{a}}$, thanks to the invariances under diffeomorphisms, Weyl transformations, and finally local Lorentz invariance imposed on-shell. This is crucial for the formulation of the dual symmetric string in that, using the flat gauge for the zweibein, we are effectively performing the analogous procedure as when fixing the conformal gauge in standard string theory. Keeping ${C}$ and ${M}$ constant, we can compute the equations of motion for the field ${X^{I}}$ to give

$\displaystyle \nabla_{1} [e (C_{IJ} \nabla_{0} X^{J} + M_{IJ}\nabla_{1} X^{J}] = 0. \\\ (11)$

In the flat gauge this result becomes

$\displaystyle \partial_{1} [C_{IJ} \partial_{0} \xi^{J} + M_{IJ} \partial_{1} \xi^{J}] = 0. \ (12)$.

From (12) a now famous identity appears, where, in the flat gauge and along the equations of motion for ${\xi^{I}}$, the following constraint on ${C}$ and ${M}$ is obtained [2]:

$\displaystyle C = MC^{-1}M. \ (13)$

One may recognise the tensor structure of (13) in terms of the action of an ${O(D,D,\mathbb{Z})}$ element. The important thing to highlight is that throughout the lengthy calculation to get to this point, ${C}$ and ${M}$ are held constant. (When ${C}$ and ${M}$ are not treated as constant, a number of interesting questions arise which extend beyond the scope of the present discussion). What is also important is that, after rotating ${\xi^{I}}$, the matrix ${C}$ can always be put into diagonal form such that

$\displaystyle C = \ \textbf{diag} \ (1,...,1,-1,...,-1). \ (14)$

It remains to be said that ${C = C^{-1}}$, which means that the constraint (13) defines the indefinite orthogonal group ${O(p,q)}$ of ${N \times N}$ matrices ${M}$ with ${N = p + q}$ in ${\mathbb{R}^{p,q}}$. The inner product may now be written as

$\displaystyle C = MCM, \ (15)$

in which the matrix ${C}$ eventually takes on the explicit definition of an ${O(D,D,\mathbb{R})}$ invariant metric in the 2D target space ${M}$. Although, admittedly, this cursory review has omitted many important and interesting details, the pertinent point in terms of this essay is as follows. The action (6) turns out to describe rather precisely a mixture of ${D}$ chiral ${\xi^{\mu}_{-}}$ and ${D}$ anti-chiral ${\xi^{\mu}_{+}}$ scalars. In demanding local Lorentz invariance and the vanishing of the Lorentz anomaly, this requires that ${p = q = D}$ with ${2D = N}$. In working through the complete logic of the calculation, we observe quite explicitly that inasmuch the requirement of local Lorentz invariance is imposed through the condition (10), this leads one naturally to an interpretation of the matrix ${C}$ as a 2D target space metric with coordinates

$\displaystyle \xi^{I} = (\xi^{\mu}_{-}, \xi^{\mu}_{+}), \ ds^{2} = dX^{I} C_{IJ} d X^{J}, \ I = 1,...,2D, \ \text{and} \ \mu = 1,...,D. \ (16)$

If we make a change of coordinates in the target space, particularly by defining a set of new chiral coordinates, the matrix ${C}$ takes on the off-diagonal form of the ${O(D,D)}$ constant metric ${L}$ typically considered in DFT (for review, see [36]) and elsewhere. The chiral coordinates we define are

$\displaystyle X^{I} = \frac{1}{\sqrt{2}} (X_{+}^{\mu} + X_{-}^{\mu}), \tilde{X}_{I} = \frac{1}{\sqrt{2}} (X_{+}^{\nu} - X_{-}^{\nu}). \ (17)$

In this frame, the matrix ${C}$ is then shown to be

$\displaystyle C_{IJ} = - \Omega_{IJ} = -\begin{pmatrix} 0 & \mathbb{I} \\ \mathbb{I} & 0 \end{pmatrix}. \ (18)$

It follows that the condition (13) transforms into the constraint

$\displaystyle M^{-1} = \Omega^{-1}M\Omega^{-1} \ (19)$

on the symmetric matrix ${M}$, which can be parametrised by a symmetric matrix ${G}$ and an antisymmetric matrix ${B}$. Therefore, remarkably, the symmetric matrix ${M}$ takes the precise form of the generalised metric in which ${M}$ is found to be positive definite.

To conclude, in the chiral coordinates we arrive at a famous form of the Tseytlin action,

$\displaystyle S = \frac{1}{2} \int d^{2}\xi \ e[ \Omega_{IJ} \nabla_{0} X^{I} \nabla_{1} X^{J} - M_{IJ}\nabla_{1} X^{I} \nabla_{1} X^{J}]. \ (20)$

This action is manifestly ${O(D,D)}$ invariant. When ${O(D,D)}$ transformations are applied to (20), we obtain exactly what we would anticipate for the standard string in the sense of T-duality invariance under ${X \rightarrow\tilde{X}}$ and for the generalized metric ${M \rightarrow M^{-1}}$.

For completeness, from the action (6) in arriving at (20), it should be clear that what we are working with is a sigma model for the dual symmetric string. The generalised version of the celebrated action (20) is indeed often written as

$\displaystyle S_{General} = \frac{1}{2} \int d^{2}\xi \ [- (C_{IJ} + \eta_{IJ}) \partial_{0} X^{I} \partial_{1} X^{J} + \mathcal{H}_{IJ} \partial_{1} X^{I} \partial_{1}X^{J})]. \ (21)$

This final action can be argued to be a very natural generalisation for the standard string on a curved background. It not only contains the generalised metric ${\mathcal{H}_{IJ}}$, but also another symmetric metric ${\eta_{IJ}}$ with ${(D,D)}$ signature and an antisymmetric 2-tensor ${C_{IJ}}$. The coordinates are defined ${X^{I} = \{ X^{I}, \tilde{X}_{I} \}}$ with the background fields in general depending on ${X^{I}}$.

***

In the last decade especially, Tseytlin’s formulation has been refocused in various studies concerning the nature of the doubled string and its geometry. One notable example to which we will return in a moment, pre-dates the first primary collection of DFT papers and, in many ways, can be interpreted to give a prediction to DFT. I am refering to the 2008 paper David S. Berman, Neil B. Copland, and Daniel C. Thompson [18], where they investigated the background field equations for the duality symmetric string using an action equivalent to that of Tseytlin’s but constructed in the context of Hull’s doubled formalism. In recent years, a series of publications on doubled sigma models have appeared in connection [19, 20, 21, 22], where in [20] the double sigma model is for example directly related to DFT.

Another example refers directly to both Tseytlin and DFT from a different perspective. In the years after 2009 when Hull and Zwiebach published their important paper formalising DFT, it was recognised that while a deep connection exists between DFT and generalised geometry, with the former locally equivalent to the latter, it does not completely come into contact with its formal mathematical structures. In fact, an open research question remains motivated by the unmistakeable resemblance DFT has with generalised geometry and the formal gap that remains between them. Recent work in mathematics and physics has displayed some promise, suggesting that the use of para-Hermitian and para-Kähler manifolds may be the solution [23, 24, 25]. Related to these efforts is a recent reformulation of string theory under the heading metastring theory [24, 26, 27, 28, 29], which begins, similar to the studies on double sigma models, with a generalised version of the first-principle Tseytlin action for the duality symmetric string. The metastring is therefore a chiral T-duality invariant theory that, in many ways, wants to generalise from DFT and make direct connection with things like Born geometry [26], relying on the consistency of Tseytlin’s formulation.

If a direct consequence of making T-duality manifest is that the winding modes are treated on equal footing with momentum, then for DFT all of these properties are incorporated into one field theory. The result, as mentioned, is a doubled coordinate space. In metastring theory, on the other hand, the target space of the world-sheet formulation is a phase space, much like in Tseytlin’s original construction. The coordinates of this phase space are indeed doubled, but unlike in DFT they are also conjugate such that in this case the dual coordinates are related directly to energy-momentum coordinates. In other words, ${\tilde{X}}$ is now identified with ${p}$. This means that, instead of a physical spacetime formulation, the goal of metastring theory is to construct a sigma model as a phase space formulation of the string and its dynamics.

The implications of metastring theory, as they have so far been conjectured, are intriguing. For example, there have been claims toward obtaining a family of models with a 3+1-dimensional de Sitter spacetime, argued to be realised in the standard tree-level low-energy limit of string theory in the case of a non-trivial anisotropic axion-dilaton background [29]. A key statement here is that, while string theory has purely stringy degrees of freedom (from first principles consider simply the difference between the left and right-moving string modes), these are not captured by standard effective field theory approaches and their spacetime descriptions. Such approaches are usually employed when investigating de Sitter space. In the phase-space formulation of the metastring, these purely stringy degrees of freedom (generally chiral and non-commutating) are argued to be captured explictly. When it comes to the hope of obtaining an effective de Sitter background, one of the major claims in this non-commutative phase-space formalulation is how, in the doubled and generalised geometric description, the effective spacetime action translates directly into the see-saw formula for the cosmological constant. Furthermore, in this cosmic-string-like solution related to the concept of an emergent de Sitter space, it is argued that the metastring leads naturally to an expression of dark energy, represented by a positive cosmological constant to lowest order. Finally, it is argued that the intrinsic stringy non-commutativity provides a vital ingredient for an effective field theory that reproduces to lowest order the sequestering mechanism [29, 30] and thus a radiatively stable vacuum energy.

***

Building from the Tseytlin action (21), this world-sheet theory of chiral bosons not only takes the heterotic string to its maximal logical completion (a point to be discussed another time), the total doubled space that it sees naturally accomodates stringy non-geometries. With the development of DFT and Hull’s doubled formalism in mind, one interesting question that we can ask concerns whether the best features of all of these approaches can be put together under a more general formulation. There is already a lot in Tseytlin’s original first-principle construction, and so one idea is to generalise from this theory. This was one motivation for my MRes thesis. Another question concerns the presence of generalised geometry and finally how, given a completely generalised treatment of the duality symmetric string, how may we extend the ideas toward the study of duality symmetric M-theory, where exceptional field theory seeks to promote the U-duality group to a manifest symmetry of the spacetime action [37, 38].

These comments take us back to the work of Berman et al. [18], who started to point toward the same question of generalisation in their approach that combines Tseytlin’s action with Hull’s doubled formalism. It is a very interesting entry into the ideas described, and it is this paper where my own MRes thesis more or less entered the picture.

Moreover, the approach in my MRes was basically to follow the prescription first adopted by Berman et al; however, the action they used to study the doubled beta-functionals for the interacting chiral boson model was constructed in the case where the background fields depend trivially on the doubled coordinates but non-trivially on the non-compact spacetime coordinates. This means that in their approach the target-space was constructed in terms of a torus fibration ${T^{n}}$ over a base ${N}$. One may think of this as a description of string theory in which the target space is locally a ${T^n}$ bundle, while ${N}$ is some generic base manifold that may be thought of simply as a base space.

While such constructions are important and deserve attention moving forward – we will certainly discuss cases in the future of more complicated bundles, for example – for my MRes the idea was to first strip everything back and generalise the result with minimal assumptions. The first step, for example, was to not demand anything about the dependence of the background fields. What we arrived at was an action of the form

$\displaystyle S_{Maximally \ doubled} = \frac{1}{2} \int d^{2}\sigma [-\mathcal{H}_{AB}(X^{A}) \partial_{1} X^{A} \partial_{1} X^{B} + L_{AB}(X^{A}) \partial_{1}X^{A} \partial_{0} X^{B}], \ (22)$

where ${\mathcal{H}}$ is the generalised metric and we also have a generic 2-tensor ${L}$ (that we continued to treat generically). In doing away with a base-fibre split (we also dropped a topological term, which isn’t so important here), what we have is the sort of action considered originally by Tseytlin. In fact, (22) is the most general doubled action we can write without manifest Lorentz invariance, because it allows us to calculate the background fields in a way in which the fields maintain arbitrary dependence on the full doubled geometry. That is to say, in taking the democratic approach in which everything becomes doubled, we’re ultimately seeking an effective spacetime theory that corresponds to completely generic non-geometric geometries. At the same time, the structure of the action is precisely the sort proposed to lead directly to DFT [20], and it also remains equivalent to the Polyakov action in the standard formulation of string theory.

Due to the fact that there are papers pending on these calculations and associated topics, I will leave more details for future entries and for when they more formally appear on arxiv.

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# ‘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.

# 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.