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 For the interested reader, everything from the two conferences has been archived here.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

Dear Reader(s),

It has been some time since my last post. Just as I was gearing up with a series of different articles and essays – and generally getting into a good pattern of regular blogging – Covid-19 broached the shores of the UK. With the outbreak, life for all was turned upside down. In addition to having to adjust to the unsettling reality of a global pandemic and its implications – I usually respond to such world events with lots of studying and research as a way to get my head around what is objectively happening – one of the practical consequences was a series of changes to daily life and usual routines. As a person with Asperger’s, these types of changes means it can take time for me to re-establish solid ground. I’m sure everyone reading this will be able to sympathise one way or another, given that the global pandemic has affected us all in different ways. For some, the pandemic will have had a greater impact than for others, whether directly in terms of illness or economically or otherwise. People have lost loved ones; there are stories of vulnerable individuals struggling for support or being taken advantage; we are now seeing unemployment rates increasing; among a list of other things. I once wrote about the concept of rational compassion as an alternative to empathy, and it seems pertinent. In addition to the public health crisis and the suffering that has caused, there seems no shortage of economic suffering for many. It is unfortunate, I think, in a very critical and objective way, that rational debate and dialogue has often seemed to be overtaken by extreme political rhetoric and ideology on both sides of the spectrum. I know in the UK and elsewhere, important debates about managing public health and economic health have often become reduced to two ideological positions and what is essentially a false moral dichotomy. It shouldn’t be so surprised, I suppose, since wearing a mask and social distancing – two science and evidenced-based policies that require some semblance of social reason and rational compassion – have become politicised and subject to the irrational. We of course have the far-left and the far-right, both of which ultimately seem to fold into one another as the spectrum of extremes takes on the image of the ouroboros (you can decide which is the head and which is the tail); but it is not just the extreme polarisation, but also the interior of these poles. Perhaps the moral is as it always was: it is no secret that the condition of the human being is one of struggle for reason and rationality; as I am currently writing in a series of essays on the enlightenment and the anthropology of (re-)enchantment, the enlightenment was a historical moment that resulted in the culmination of developments since Plato and beyond. Enlightenment disenchantment was by no means total, such is evidenced surely in taking a systems view of contemporary patterns and trends.

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, ‘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.

Thinking About the Strong Constraint in Double Field Theory

I’ve been thinking a lot lately about the strong (or section) constraint in Double Field Theory. In this post, I want to talk a bit about this constraint.

Before doing so, perhaps a lightning review of some other aspects of DFT might be beneficial, particularly in contextualising why the condition appears in the process of developing the formalism.

One of the important facets of DFT is the unification of B-field gauge transformations and diffeomorphisms acting on the spacetime manifold {M}. The result is a generalisation of diffeomorphisms acting on the doubled space {P} [1]. This doubled space is not too difficult to conceptualise from the outset. Think, for instance, how from the perspective of a fully constructed closed string theory, the closed string field theory on a torus is naturally doubled. But in DFT, as we advance the formalism, things of course become more complicated.

Simply put, from a first principle construction of DFT, two motivations are present from the start: 1) to make T-duality manifest and 2), extend the spacetime action for massless fields. For 2), the low-energy effective action that we want to extend is famously,

\displaystyle  S_{MS} =  \int d^{D}x \sqrt{-g}e^{-2\phi}[R + 4(\partial \phi)^{2} - \frac{1}{12}H^{ijk}H_{ijk} + \frac{1}{4} \alpha^{\prime} R^{ijkl}R_{ijkl} + ...] \ (1)

In reformulating the low-energy effective action, the tools we use begin with the fact that the coordinates in DFT are doubled such that {X^{M} = (\tilde{x}_{i}, x^{i})}. Given that the full closed string theory is rather complicated – i.e., the field arguments are doubled and we would have infinite fields, so Lagrangian wouldn’t be trivial – this motivates from the start to restrict our focus to a subset of fields. Naturally, we choose the massless sector, with the motivation to obtain a description of spacetime where the gravity-field {g_{ij}}, Kalb-Ramond field {b_{ij}} and the dilaton {\phi} are manifest. We also work with a generalised metric which rediscovers the Buscher rules such that we write, mixing {g} and {b}-fields,

\displaystyle \mathcal{H}_{MN}(E) = \begin{pmatrix} g^{ij} & -g^{ik}b_{kj} \\ b_{ik}g^{kj} & g_{ij}-b_{ik}g^{kl}b_{lj} \\ \end{pmatrix} \in O(D,D) \ (2)

Here we define {E = g_{ij} + b_{ij}}.

Most importantly, to find the analogue for {S_{MS}} formulated in an O(D,D) covariant fashion (O(D,D) is the T-duality group), a lot of the problems we need to solve and the issues we generally face are deeply suggestive of constructions in generalised geometry. We will talk a little bit about generalised geometry later.

1. The Strong Constraint

Where does the strong constraint enter into this picture? From the cursory introduction provided above, one of the quickest and most direct ways to reaching a discussion about the strong constraint follows [1]. Relative to the discussion in this paper, beginning with the standard sigma model action, where the general background metric {g_{ij}} and the Kalb-Ramond field {b_{ij}} are manifest, our entry point proceeds from the author’s review of the first quantised theory of the string and the obtaining of the oscillator expansion for the zero modes. Important for this post is what turns out to be the definition of the derivatives from the oscillator expansions,

\displaystyle  D_{i} = \partial_{i} - E_{ik}\bar{\partial}^{k}, \\ D \equiv g^{ij}D_{j}

\displaystyle  \bar{D}_{i} = \partial_{i} + E_{ki}\bar{\partial}^{k}, \\ \bar{D} \equiv g^{ij}\bar{D}_{j} \ (3)

Famously, as can be reviewed in any standard string theory textbook (for instance, see [2]), in the first quantised theory we find the Virasoro operators with zero mode quantum numbers,

\displaystyle  L_{0} = \frac{1}{2}\alpha^{i}_{0}g_{ij} \alpha^{j}_{0} + (N-1)

\displaystyle \bar{L}_{0} = \frac{1}{2}\bar{\alpha}^{i}_{0}g_{ij}\bar{\alpha}^{j}_{0} + (\bar{N} - 1) \ (4)

In (2) {N} and {\bar{N}} are the number operators. Importantly, in string theory, from the Virasoro operators we come to find the level matching constraint that matches left and right-moving excitations. This is an unavoidable constraint in closed string theory that demands the following,

\displaystyle  L_{0} - \bar{L}_{0} = 0 \ (5)

As this is one of the fundamental constraints of string theory, it follows that all states in the closed string spectrum must satisfy the condition defined in (5). The complete derivation can be found in [2].

So far everything discussed can be reviewed from the view of standard string theory. What we now want to do is use the definition of the derivatives in (3) and express (5) as,

\displaystyle  L_{0} - \bar{L}_{0} = N - \bar{N} - \frac{1}{4}(D^{i}G_{ij}D^{j} - \bar{D}^{i}G_{ij}\bar{D}^{j})

\displaystyle  = N - \bar{N} - \frac{1}{4}(D^{i}D_{i} - \bar{D}^{i}\bar{D}_{i}) \ (6)

After some working, using in particular the derivative definitions and the definitions of the background fields (see an earlier discussion in [1]), one can show that

\displaystyle  \frac{1}{2}(D^{i}D_{i} - \bar{D}^{i}\bar{D}_{i}) = -2\partial^{i}\tilde{\partial}_{i} \ (7)

In (7) we use the convention as established in [1] to denote ~ as relating to the dual coordinates. Notice, then, that what remains is a relatively simple contraction between normal and dual derivatives. What is significant about (7) is that we can now express the fundamental string theory constraint (5) as a constraint on the number operators. Since {L_{0} - \bar{L}_{0} = 0} for all states of the theory we find,

\displaystyle  N - \bar{N} = \frac{1}{2}(-2\partial_{i}\tilde{\partial}^{i} = -\partial^{i}\tilde{\partial}_{i} \equiv \partial \cdot \bar{\partial} = p_{i}\omega^{i} \ (8)

So we see that from the number operators we have constraints involving differential operators. But what is this telling us? In short, it basically depends on the fields we use for the closed string theory. The fields that are arguably most natural to include are of a first quantised state expressed in the sum,

\displaystyle  \sum_{p,\omega} e_{ij}(p,\omega) \alpha^{i}_{-1}\bar{\alpha}^{j}_{-1}c_{1}\bar{c}_{1} |p, \omega \rangle

\displaystyle  \sum_{p,\omega} d (p,\omega) (c_{1}C_{-1} - \bar{c}_{1}\bar{c}_{-1}) |p, \omega \rangle \ (9)

Where we have momentum space wavefunctions {e_{ij}} and {d (p,\omega)}. Furthermore, in the first line, {e_{ij}} denotes the fluctuating field {h_{ij} + b_{ij}}. The {c} terms are ghosts. So what we observe in (9) is matter and ghost fields acting on a a vacuum with momentum and winding.

Here comes the crucial part: given {N = \bar{N} = 0} it follows that the fields, which, to make explicit depend on normal and dual coordinates, {e_{ij}(x, \tilde{x})} and {d(x,\tilde{x})} are required to satisfy,

\displaystyle  \partial \cdot \tilde{\partial} e_{ij} (x, \tilde{x}) = \partial \cdot \tilde{\partial} d(x, \tilde{x}) = 0 \ (10)

This is weak version of the strong or section constraint, a fundamental constraint in DFT for which we can go on to define an action. What it says is that every field of the massless sector must be annihilated by the differential operator {\partial \cdot \tilde{\partial}}.

This constraint (10), when developed further, turns out to actually be very strong. When we proceed to further generalise in our first principle construction of DFT, first with the study of {O(D,D)} transformations and then eventually the construction of {O(D,D)} invariant actions, we come to see that not only all fields and gauge parameters must satisfy the constraint {\partial \cdot \tilde{\partial}}. But this constraint is deepened, in a sense, to an even stronger version that includes the product of two fields.

The argument is detailed and something we’ll discuss in length another time, with the updated definition that {\partial \cdot \tilde{\partial}} annihilates all fields and all products of fields. That is, if we let {A_{i}(x, \tilde{x})} be in general fields or gauge parameters annihilated by the constraint {\partial^{M}\partial_{M}}, we now require all products {A_{i}A_{j}} are killed such that,

\displaystyle  \partial_{M}A_{i}\partial^{M}A_{j} = 0, \forall i,j \ (11)

Here {\partial_{M}A_{i}\partial^{M}} is an {O(D,D)} scalar. Formally, the result (11) is the strong {O(D,D)} constraint. What, finally, makes this condition so strong is that, from one perspective, it kills half of the fields of the theory and we in fact lose a lot of physics! In full string theory the doubled coordinates are physical. Effectively, however, the above statement ultimately implies that our fields only depend on the real space-time coordinates, due to a theorem in {O(D,D)} in which there is always some duality frame {(\tilde{x}^{\prime}_{i},\tilde{x}^{\prime})} in which the fields do not depend on {\tilde{x}^{\prime}_{i}}. So we only have dependence on half of the coordinates.

There is maybe another way to understand or motivate these statements. In the standard formulation of DFT, what we come to find is the appearance of the generalised Lie derivative. It is essentially unavoidable. The basic reason has to do with how, in pursuing the construction of the {O(D,D)} invariant action as highlighted at start, which includes the generalised metric {\mathcal{H}}, we find that the conventional Lie derivative is not applicable. It is not applicable in this set-up because, even when using trivial gauge parameters, we find that it simply does not vanish. In other words, as can be reviewed in [1], {\mathcal{L}_{\xi} \neq 0}. So the definition of the Lie derivative becomes modified using what we define as the neutral metric {\eta}. Why this is relevant has to do with how, interestingly, from the generalised Lie derivative (or Dorfman bracket) we may then define an infinitesimal transformation that, in general, does not integrate to a group action [3-5]. This means that it does not generate closed transformations.

The convention, as one may anticipate, is to place quite a strong restriction on the space of vector fields and tensors. Indeed, from the fact that DFT is formulated by way of doubling the underlying manifold, we have to use constraints on the manifold to ensure a consistent physical theory. But this restriction, perhaps as it can be viewed more deeply, ultimately demands satisfaction of what we have discussed as the strong constraint or section condition. So it is again, to word it another way, the idea that we have to restrict the space of vectors and tensors for consistency in our formulation that perhaps makes (11) more intuitive.

There is, of course, a lot more to the strong constraint and what it means [5], but as a gentle introduction we have captured some of its most basic implications.

2. Some Nuances and Subtleties

Given a very brief review of the strong constraint, there are some nuances and caveats that we might begin to think about. The first thing to note is that the strong constraint can be relaxed to some degree, and people have started researching weakly constrained versions of DFT. I’m not yet entirely familiar with these attempts and the issues faced, but an obvious example would be the full closed string field theory on a torus, because this is properly doubled from the outset and subject only to the weak level-matching constraint {\tilde{\partial} \cdot \partial = 0}.

I think a more important nuance or caveat worth mentioning is that, as discussed in [3], the strong constraint does not offer a unique solution. That is to say, from what I currently understand, there is no geometrical information that describes the remaining coordinates on which the fields depend. This contributes to, in a sense, an arbitrariness in construction because there is a freedom to choose which submanifold {P} is the base {M} for the generalised geometry {TM}.

In a future post, we’ll discuss more about this lack of uniqueness and other complexities, as well as detail more thorough considerations of the strong constraint. As related to simplified discussion above, the issue is that we can solve the basic consistency constraints that govern the theory by imposing the strong constraint (11). This is what leads to the implied view that DFT is in fact a highly constrained theory despite doubled coordinates, etc. In this approach, we have restriction on coordinate dependence such that, technically, the fields and gauge parameters may only depend on the undoubled slice of the doubled space. We haven’t discussed the technicalities of the doubled space in this post, but that can be laid out another time. The main point being that this solution is controlled. But there are also other solutions, of which I have not yet studied, but where it is understood that the coordinate dependence is no longer restricted (thus truly doubled) at the cost of the shape of geometric structure. At the heart of the matter, some argue [3] that when it comes to this problem of uniqueness the deeper issue is a lack of a bridge between DFT and generalised geometry. This is also a very interesting topic that will be saved for another time.

References

[1] B. Zwiebach, ‘Double Field Theory, T-Duality, and Courant Brackets’ [lecture notes]. 2010. Available from [arXiv:1109.1782v1 [hep-th]].

[2] J. Polchinski, ‘String Theory: An Introduction to the Bosonic String’, Vol. 1. 2005.

[3] L. Freidel, F. J. Rudolph, D. Svoboda, ‘Generalised Kinematics for Double Field Theory’. 2017. [arXiv:1706.07089 [hep-th]].

[4] B. Zwiebach O. Hohm. Towards an invariant geometry of double field theory. 2013. [arXiv:1212.1736v2 [hep-th]].

[5] B. Zwiebach O. Hohm, D. Lust. The Spacetime of Double Field Theory : Review, Remarks and Outlook. 2014. [arXiv:1309.2977v3 [hep-th]].

[6] K. van der Veen, ‘On the Geometry of String Theory’ [thesis]. 2018. Retrieved from [https://pdfs.semanticscholar.org/8c17/53af2fce4d174ca63a955c39ee2fedf37556.pdf]