Mathematical language of duality

On December 17, 2021December 17, 2021 By Robert C. SmithIn Mathematics

As we’ve discussed at various times on this blog, many of the most important recent developments in string / M-theory are based on duality relations. Physical insight is quite ahead of mathematics in this regard. But, in the last decade or two, mathematics has started to properly formulate a language of duality that, on first look, seems incredibly simple but is ultimately very powerful: namely, the language of categories. In foundational mathematical terms, category theory provides tools to express structures – often very general structures – and their duals in a way that comes out naturally through the concept of a categorical product and coproduct. Below is a very brief summary.

Definition of a category

Let us quickly recall the definition of a category $\mathcal{C}$ . As mentioned in a past post, a category can be constructed for essentially any mathematical object. We can think of a category as a quintessential representation of structure.

Definition 1. A category $\mathcal{C}$ consists of a class of objects, and, for every pair of objects $A,B \in \mathcal{C}$ , a class of morphisms $hom(A,B)$ satisfying the properties:

Each morphism has specified domain and codomain objects. If f is a morphism with domain A and codomain B we write $f: A \rightarrow B$ .

For each $A \in \mathcal{C}$ , there is an identity morphism $id_A \in \text{hom}(A,A)$ such that for every $B \in \mathcal{C}$ we have left-right unit laws:

$f \circ id_A = f \text{for all} f \in \text{hom}(A,B),$

$id_A \circ f = f \text{for all} f \in \text{hom}(B,A).$

For any pair of morphisms f,g with codomain of f equal to codomain of g, there exists a composite morphism $g \circ f$ . The domain of the composite morphism is equal to the domain of f and the codomain is equal to the codomain of g.

In simple terms, a category is just a collection of objects (metric spaces, topological spaces, or whatever) and structure preserving maps between those objects. It is, in a sense, like a deeper generalisation of set theory, except that we can have categories of sets. A simple illustration of a category is as follows

There are two axioms that must be satisfied in the defining a category:

For any $f: A \rightarrow B$ , the composites $1_B f$ and $f1_A$ are equal to f.

Composition is associative and unital. For all $A, B,C,D \in \mathcal{C}$ , $f \in \text{hom}(A,B)$ , $g \in \text{hom}(B,C)$ , and $h \in \text{hom}(C, D)$ , we have $f \circ (h \circ g) = (g \circ f) \circ h$ .

Functors

We can also define a functor, which maps between categories. We define the notion of a functor as corresponding to a mapping that sends the objects and arrows of one category to the objects and arrows in another category in a structure preserving way.

Definition 2. A functor $F$ from $C$ to $D$ is a structure preserving map between categories such that for each object $A$ of $C$ , we have $F(A)$ in $D$ .

For each arrow (morphism) $f: A \rightarrow B$ in $C$ , we have $F(f): F(A) \rightarrow F(B)$ such that $F(g) \circ F(f) = F(g \circ f)$ and $F(Id_A) = Id_{F(A)}$ .

Suppose $f: A \rightarrow C$ is a functor between categories $A$ and $C$ . For purposes of illustration, we’ll call $A$ an indexing category, and let’s suppose it’s a simple one with objects $a_1, a_2, \ \text{and} \ a_3$ :

A functor f out of this category $A$ is simply the choice of three objects and three arrows in the category $C$ such that

where $f(a_1) = c_1$ , $f(a_2) = c_2$ , and $f(a_3) = c_3$ . The image of the arrows in $A$ are the arrows g, k, and h in $C$ where $g = h \circ k$ .

Categorical products

What is very neat and exciting is that we can also define the notion of a categorical product (e.g., a product of two categories). For a long time, it was thought that taking a product between two sets was one of the most fundamental operations in mathematics. But, it turns out, from the definition of a categorical product we can still drill deeper and therefore also capture the essence behind the Cartesian product of sets, the direct product of groups or rings, and the product of topological spaces.

This topic is again quite technical but, in short, a simple definition of a categorical product is as follows:

Definition 3. For any categories $C$ and $D$ , there is a category $C \times D$ , their product, whose

objects are ordered pairs $c,d$ , where c is an object of $C$ and d is an object of $D$ ,

morphisms are ordered pairs with $\pi_1 : C \times D \rightarrow C$ , $\pi_2 : C \times D \rightarrow D$ such that for the other candidate $X$ we define the maps $f: X \rightarrow A$ , $g: X \rightarrow B$ for every unique $h: C \times D$ , and $\pi_1 \circ h = f$ and $\pi_2 \circ h = g$ ,

and in which composition and identities are defined componentwise.

A first glimpse at duality

Now, what is absolutely amazing is how, from the notion of a product of categories (which is like a generalisation of the Cartesian product of ordered sets), the first glimpse of a fundamental mathematical description of duality naturally emerges in the definition of a categorical coproduct.

Let us return to the definition of a categorical product and its diagram in the previous section. We want to think of its coproduct (i.e., the product in the opposite category). We will have the same picture, except all of the arrows will be reversed which is the same as exchanging domain and codomain.

Definition 4. The co-product $C + D$ , $p_1 : C \rightarrow C + B$ , $p_2 : D \rightarrow C +D$ is such that for each $X$ , $f: C \rightarrow X$ , $g: D \rightarrow X$ there exists a unique $h: C + D \rightarrow X$ that makes the diagram commute $h \circ p_1 = f$ and $h \circ p_2 = g$ .

The coproduct naturally takes the form of the category-theoretic dual notion to the categorical product. We can think of this in terms of a mapping from $C$ to $C^{\text{op}}$ .

Definition 5. Let $C$ be any category. The opposite category $C^{\text{op}}$ has

the same objects as in $C$ , and

a morphism $f^{\text{op}}$ in $C^{\text{op}}$ for each a morphism $f \in C$ so that the domain of $f^{\text{op}}$ is defined to be the codomain of f and the codomain of $f^{\text{op}}$ is defined to be the domain of f: i.e., $f^{\text{op}}: X \rightarrow Y \in C^{\text{op}} \leftrightarrow f: Y \rightarrow X \in C$ .

What this means is that, given $C^{\text{op}}$ has the same objects and morphisms as $C$ , the notion of duality in category theory is defined by a reversal of arrows: i.e., each morphism in $C^{\text{op}}$ is pointing in the opposite direction.

The dual of each of the axioms for a category is also an axiom, while the dual of the dual returns the original statement. This is the duality principle in a nutshell.

Reading

[1] E. Riehl, Category theory in context. Dover Publications, 2016. [online].

[2] S. Mac Lane, Category theory for the working mathematician. Springer, 1978. [online].

[3] P. Smith, Category theory: A gentle introduction. [online].

[4] J. Baez, Category theory course. [online].

Polchinski_string theory_tracingcurves blog

Reading Polchinski – My notes on string theory

On December 4, 2021December 5, 2021 By Robert C. SmithIn Physics

When I was in the first year of my undergraduate degree, I started self-studying string theory. At that time, I set myself the task of working through Joe Polchinski’s two volume textbook, String Theory. It was honestly one of the best years of my life, despite the general uncertainty about my academic future at the time. I also began a project of sharing my own notes on this blog, as an extension of my enthusiasm.

Between the time as a first year undergraduate student and the commencement of my Masters the next year, I hadn’t much time to finish converting my notes to latex and uploading them to this blog. Things move fast, and quickly I was on to double field theory, developing my interests in M-theory, and other cool stuff with pressures of producing a thesis. Also, in that time, I had to change my web hosting service. As a result of transferring my blog, the latex broke in all of my old posts (including what string notes I had uploaded) which was a bit frustrating.

I have been intending to reupload what notes I had converted to latex and then to also continue my project for sometime. Now that I have settled into my PhD years, I feel able again to take up the task (of course, as and when the time becomes available). As part of my scholarship, I also feel obliged to participate in science communication. Sharing technical and explanatory notes in maths/physics is probably where I am most capable.

***

I consider my handwritten notes a product of rigorous examination and review, organised around the central motivation to rederive the whole of bosonic and then superstring theory from first principles, or, where appropriate, at least from as close to first principles as possible. They are a thorough companion to Polchinski’s textbook, providing a workthrough of its many pages and subtle details. They also offer a lot of comments from my own perspective, such as when making an interesting observation or offering more background work than presented in Polchinski’s textbook. I have also developed my own preferences, such as emphasising at the outset the notion of string theory as a generalisation point particle theory.

Due to the way I upload latex to my blog nowadays, what I currently plan to do is start from the very beginning and reupload everything that was already posted and then work toward uploading the rest. As the notes were originally written for myself, converting them to latex and organsing them in a presentable way takes quite a bit of time. I like the idea of uploading one batch of notes at a time as individual posts, such as in presenting one lecture note at a time (working page by page through Polchinski). Eventually I will also create a directory on my blog, where the reader may navigate by topic through the complete notes. (I will also file everything under the tag, ‘Notes on String Theory’). They may or may not prove useful for others, but, to be honest, I ultimately doing it because it is something I enjoy.

As for what is to come next: in the first entry we will start with a brief review of the non-relativistic string, before moving to a review of the relativistic free-point particle in a separate note. This will then take us to the first pages of Polchinski and the construction of the Nambu-Goto action.

Generalised supergravity and the dilaton

On November 29, 2021November 29, 2021 By Robert C. SmithIn Physics

I spent sometime in the early autumn months thinking about the cosmological constant problem (CC). This was actually secondary, because my primary note taking focused more on S-duality and manifestly duality invariant actions, non-perturbative corrections, and the dilaton. But my supervisor, Tony, has spent a lot of time thinking about this problem, with one of his big ideas being vacuum energy sequestering, so naturally there is motivation whenever we get the chance. There has also been some renewed interest in the CC problem in the context of generalised double sigma models and double field theory. In general, there is a lot of interesting cosmology to be investigated here.

I’m currently drafting a post on the CC problem from the view of string theory. This will hopefully provide the reader with a thorough introduction. But as a passing comment in this short note, it suffices to say that the role of the CC in string theory is generally mysterious. In standard textbook analysis, one sees that the mystery starts with the massless sector contribution, with the dilaton central to the discussion; but the mysteriousness comes further into focus once the role of dual geometry is investigated and the peculiar change of the CC under duality transformation. Intuitively, I am inclined to think that a piece of the picture is missing.

One idea I find interesting to play with involves adding extra fields. Another idea people play with is redefining the dilaton. An example comes from a breakthrough paper by Tseytlin and Wulff [1].

Admittedly, I wasn’t aware of this paper until my early autumn investigations. Within it, a 30 year old problem is solved using the Green-Schwarz (GS) formulation of supergravity theory. The short version is that, in the standard GS formulation of Type IIB string theory there is a problem with the number of degrees of freedom. The space-time fermions have 32 components. An on-shell condition reduces the degrees of freedom to 16, but it needs to be 8. It was later discovered that kappa-symmetry is present in the theory, which is a non-trivial gauge symmetry, and this symmetry may be used to reduce the remaining 8 degrees of freedom. However, issues remained in proving a number of associated conjectures – that is, until Tseytlin and Wulff formulated generalised type IIB SUGRA on an arbitrary background.

The key observation is that generalised SUGRA is equivalent to standard SUGRA plus an extra vector field. Furthermore, one of the characteristics is that, under generalised T-duality, there is a modification of the dilaton such that a non-linear term is added $\Phi \rightarrow \tilde{\Phi} = \Phi + I \cdot \tilde{x}$ [2]. I think this is quite interesting, and it is something I want to look at more deeply in the future.

Although the context of the calculation is completely different to my own investigations, it is worth noting that this generalised Type IIB theory can be obtained from double field theory. Perhaps not surprisingly, I have seen some pin their hopes that generalised SUGRA could contribute to solving the cosmological constant problem (and potentially also give de Sitter vacua). That seems premature, from my vantange; but in any case it is an interesting bit of work by Tseytlin, Wulff, and others.

References

[1] Tseytlin, A.A., Wulff, L., \textit{Kappa-symmetry of superstring sigma model and generalized 10d supergravity equations}. (2016). [arXiv:1605.04884 [hep-th]].

[2] Tseytlin, A.A., et al, Scale invariance of the $\eta$-deformed $AdS5 \times S5$ superstring, T-duality, and modified type II equations. (2016). [arXiv:1511.05795 [hep-th]].

(n-1)-thoughts, n=5: Freedom of speech, university statement on free speech, the late Steven Weinberg, and delayed autism research

On November 10, 2021November 11, 2021 By Robert C. SmithIn Philosophy and General Reading

Freedom of speech

Outside of science, one of my favourite things to study as a hobby is history. I also deeply enjoy and appreciate philosophy. One thing I’ve learned in my time studying history and philosophy is that, when judged alongside the human character (insofar that we may establish such a generalisation), democracy is a system that perhaps shouldn’t work but somehow functions in miraculous ways. The miraculous part of democracy is that, as a system, it is generally stable despite or, perhaps, because of multiple competing forces. How it stablises despite so many pressure points, is a very interesting question of political theory and systems theory. Admittedly, it is naive to think in the following way, but there are times when I am pulled to consider a newtonian, mechanical view of social systems and their configuration. In the context of social discourse, think of how a view or movement based on certain ideas and arguements often seems to evoke an equal, opposite view. Look at the social world as a distant observer might, and notice the pattern that oftentimes there is a movement and then a reaction. In conditions of increasing polarisation, concepts and ideas – viewpoints – can become extremised and so too do their opposite. If a person is not left, then they must be right. There are a number of books on political polarisation, including some that take a science view of bias, and they all hint toward combinations of structural, cognitive, and psychological factors.

I often look at the social world as the absence of reason. This might be a bit too classical and critical enlightenment, but in many ways I think we’ve lost touch with the concept of subtlety in the rational process: that there is nuance and subtlty to concepts and to formulating rigorously researched ideas about complicated topics. For example, am I a ‘climate denier’? No. But am I skeptical of a lot of the hysteria around climate change? Yes. (I think, for example, of anti-modern movements or those that organise themselves under the notion of Deep Ecology). Does this mean I completely reject climate science, or that I completely reject the notion of climate change, although in places I may be sceptical? No. It seems that in the world of concepts and of human ideas, more often than not views become extremised and concepts are taken to their ideological boundaries where irrationality transforms into unreason. We see it all the time, not just in politics where formally it is accepted that designations of left and right, along with their associated bias, may clash in debate without much objectivity. To me, it is an absurdity. But one thing that history has taught me, and, certainly, the history of science, is that it is important to constantly resist getting tied down to bias, prejudice, and the type of knowledge formation that comes with ideology in all its guises. Much of what the history of science teaches is about our utter stupidity as a species in thinking that, in whatever historical period, we may belive to possess all of the answers or have a complete grasp on the truth. It is thus only a matter of pure comedy that we may engage in politics in such a way as thinking ours is the righteous view.

If I may speak honestly, I find a lot about modern politics – by which I mean the nature of its structure and engagement – irrational. I’ve never understood why in modern British democracy we assign the role of secretary of education, for example, to a professional politician with no experience in the field of education. Why is evidence-based, expert driven governance made to seem like a concept associated with some alien-rational, futuristic, scientific utopia? I suppose when contrasted to the system of competing echo chambers known as party politics, the idea of evidenced-based policy appears futuristic. Given that we do not live in anything like a scientific society, I’m not sure an actual scientific society would be structured in such a way that non-experts are allocated important roles in the practice of democratic governance. I mean, what does it say about the prospect of a society predicated on, or at least hoped to be informed by evidence based policy, when professional politicians with pre-established agendas preside both over the evidence and the policy? To me, the hard truth seems to be that all of politics is based on subjectivism and, in some sense, with the loss of the rational process that strives to seek the objective. Discourse instead seems to manifest in ways that formalise false equivalence or the categorical fallacy of inconsistency. For any issue, at least two sides are portrayed as equally valid when there may in fact be asymmetry. In some or many cases, perhaps no two political reductions are even capable of capturing the total complexity of the matter at hand. But with the loss of the objective as a concept that ought to be strived toward, debate is reduced to subjective bias and political prejudice that is symbologic of the postmodern vacuum in which we find ourselves.

Maybe I am just pessimistic. Then again, think of Brexit. Rub away all of the dross and antics, all of the extremisms and prejudiced ideologies that sought to exploit the situation, one will see that there were logical arguments from both sides of the debate. There were arguments from both the left and right-wing to leave the EU, with the former emphasising democratic control and participation in a similar way as the sovereignty argument on the right. Likewise, arguments to Remain were not just a left versus right issue, although, as it is so often today, simplistic narratives tend to rule public discourse and political slogan design. What was most striking about the entire process is that, rarely if ever, one observed a politician or public intellectual change their mind. It’s as though people didn’t engage in debate, but instead focused on shutting the other down. Maybe it is a matter of polarisation in which two sides often emerge as set against each other, and then from there discourse seems to shut down. Or maybe there is something to that old Newtonian idea. What is clear is that there was no collective encircling of an issue (or it was an exception to the rule), no process of gathering information from all sides – taking in new evidence and data – and constantly working through rational arguments (often through a process of changing one’s mind or outlook). This is how a civilised democratic society, armed with science and modern technologies, was meant to function. Or, at least, that’s how I like to imagine it.

This brings me to another thing that history has taught me: a democratic society based on core liberal and enlightenment values is one that requires citizens in deeply fundamental ways to be able to disagree. But the concept of the enlightenment requires something still much deeper – and this relates directly to democracy – that individuals enter into a debate, or disagree, within the frame of reason. Think of it this way: If I disagree with someone about a mathematical matter, it doesn’t make sense that I debate with them outside of mathematics. I pick up a dry wipe marker and explain mathematically why I don’t agree. Debates about freedom of speech in modern western society seem to lose sight of key content within the concept: what gives it so much fundamental import as a social concept is that it is intrinscially rational. If, at my university, an individual was invited to give a talk on why they are sceptical about the interpretation of climate science data, I may or may not agree; but given that their argument is rigorously constructed, well-researched, and rationally presented I support the freedom to present the view. If I don’t agree, and if I think that their argument is logically inconsistent or wrong, then it is up to me to disprove their case. This, to me, is what freedom of speech means. Yes, on some simplistic and practical level one may deduce their right to say anything, as so often this is what the popular debate on free speech seems to imply: for example, I may in this moment conjour some fanciful theory about why alien hamsters control all of human society, and I may provide a provocative argument for why this is true. But freedom of speech isn’t freedom to be unreasonable or freedom to not engage rationally in the arena of rational ideas, should one wish to engage at all; and although one might conflate their right to free speech with an imagined right to be taken seriously, such a view is in fact tantamount to utter stupidity.

Freedom of speech requires responsibility – it requires that one be normatively critical of one’s own view and be capable of exploring openly the thoughtful argument of the other – and it seems we have somehow lost sight of this fact just as we have lost sight of the meaning of constructive debate.

Stephen Fry, one of my favourites, had a fantastic line recently which is paraphrased below: ‘on one side is the new right, promoting a bizarre mixture of Christianity and libertarianism; on the other, the “illiberal liberals”, obsessed with identity politics and complaining about things like cultural appropriation. These tiny factions war above, while the rest of us watch, aghast, from the chasm below. […] It’s a strange paradox, that the liberals are illiberal in their demand for liberality. They are exclusive in their demand for inclusivity. They are homogenous in their demand for heterogeneity. They are somehow un-diverse in their call for diversity — you can be diverse, but not diverse in your opinions and in your language and in your behaviour. And that’s a terrible pity.’

University of Nottingham statement on free speech

As mentioned in a past post, since the start of term I have struggled to keep up with my blog. One thing I meant to write about was the recent statement by my university on freedom of speech. It may have been updated since I read it in the summer, as the university was seeking collaboration and feedback at the time. I should go back and read it again, but my assessment at the time was that it seemed well-balanced. It struck me that, as written, it conveyed the intention to genuinely realise the meaning of inclusivity, diversity, openess, and respect. This is what should come from an institution that seeks to foster learning, intellectual exploration, rational debate, and the wonderful process of formal inquiry in the collective pursuit of truth.

A thought of existential variety

The late Steven Weinberg had a wonderful comment about life and the human condition in his book, The First Three Minutes: ‘The more the universe seems comprehensible,’ he wrote, ‘the more it also seems pointless.’ I’m sympathetic with his view about the god-of-the-gaps. Truth be told, I consider myself agnostic; I don’t know for certain that there isn’t a God and if there is I would be inclined revolt in typical Camus fashion. That needless suffering should exist under the watch of some supreme being is detestable, in my view. So, although not an atheist in the extreme, I’ve always found Weinberg’s reflections reasonable when talking about the absence of God and how science may contribute positively to human meaning. Speaking in an interview, he once reflected: ‘To embrace science is to face the hardships of life—and death—without such comfort’. Pertinently, he continued: ‘We’re going to die, and our loved ones are going to die, and it would be very nice to believe that that was not the end and that we would live beyond the grave and meet those we love again. Living without God is not that easy. And I feel the appeal of religion in that sense.’

I often think that I could be diagnosed with cancer next week and be dead within a month. There is an innate indifference about the human condition, and with that I think a deep human fear of death, as Ernest Becker noted, governs a lot of human social systems. We can of course speak on the grandest scales and describe the precise nature of our cosmic insignificance – that we are not even a speck of dust on the scale of the universe. But even on a microbial and biochemical level, there is much that dictates the course of our lives over which we have no control. We can of course do our best to limit the probability of contracting some horrible disease or illness, and therefore play the percentages. And yet, really good people by the best moral standards, who eat right and live healthy, can contract the most awful of illness. These thoughts may appear morbid, but they describe reality. We’ve each known this indifference and fundamental arbitrariness from birth – catapulted into existence with no choice as to our geography or time in human history, we set forth with the conditions of our lives quite plainly and starkly defined. We can of course choose to fill the gaps – what some philosophers call the god of the gaps – but I’ve never found that a helpful or reasonable idea.

What I have found really important in philosophy, is that one can think in this way and acknowledge the gap without succumbing to nihilism. In an odd way, there is also hope to be found. Human beings are meaning makers, if nothing else. One can discover a cool new mathematical object and dedicate the rest of his/her life to studying it. Why? Because it is interesting, exciting, and contributes to knowledge. Of course an asteroid could crash into the earth and wipe out that knowledge completely, but that doesn’t mean that such knowledge shouldn’t have existed in the first place. There is a fine line between recognising and embracing the arbitrary and meaningless nature of life on the grandest scales, and also creating meaning and enjoyment and pursuing interests – to take care of one another and provide better conditions for those of the future – in revolt of that very reality. I often come back to this thought, because within it is a deeply lovely lesson. As Weinberg put it, the deeper idea is ‘to make peace with a universe that doesn’t care what we do, and take pride in the fact that we care anyway.’

Autism genetic project paused

One last thought. Actually, on this issue there is much to say, but I will limit this entry to a simple expression of disappointment.

It was recently announced that an autism genetics study was paused due to backlash. From what I understand, criticism includes a failure to consult the autism community about the goals of the research and there are concerns that the research could be misused, which I assume to be a concern about eugenics. This is obviously a very complicated issue, and always there are ethical points that need to be considered; but I think the latter is a bit misunderstood and this is probably a failure of scientific communication. The genetics of autism is complex. For example, cystic fibrosis involves a single gene, so it easier to screen for it. And, when screen is done, it is has nothing to do with eugenics. In the case of autism, it is likely that there are multiple genes, if not thousands, such that prenatal screening seems incredably unlikely – not that this was an intended outcome of the research anyway. Furthermore, while I understand some have concerns about eradicating autism as though it were an illness, when, in fact, it also contributes many positive traits, from what I have read the proposed research has no such intentions.

As a person diagnosed with ASD, I am very much supportive of the research. I think that, as with anything, it is best to study and understand a phenomenon as deeply as possible. Indeed, we should strive to have more of a scientific understanding of autism. At the same time, I understand that some may have ethical concerns. In science, we always have to proceed cautiously and thoughtfully. It is important to hold all scientific research to the highest ethical standards, which should be a normative process, and to also think about all possible outcomes and potential future (mis)use; but, in this case, it seems mistrust was largely down to a failure in scientific communication.

*Edited for grammar and clarity.

Doubled diffeomorphisms and the generalised Ricci curvature

On October 7, 2021October 8, 2021 By Robert C. SmithIn Physics

I was asked a question the other week about the idea of doubled diffeomorphisms, such as those found in double field theory. A nice way to approach the concept is to start with dualised linearised gravity [1]. That is to say, we start with a theory considering only the field $h_{ij}(x^{\mu}, x^a, \tilde{x}_a)$ . This field transforms under normal linearised diffeomorphism as

$\delta h_{ij} = \partial_i \epsilon_j + \partial_j \epsilon_i \ \ (1)$

and, under the dual diffeomorphism as

$\tilde{\delta} h_{ij} = \tilde{\partial}_i \tilde{\epsilon}_j + \tilde{\partial}_j \tilde{\epsilon}_i. \ \ (2)$

Now, take the basic Einstein-Hilbert action

$S_{EH} = \frac{1}{2k^2} \int \ \sqrt{-g} \ R, \ \ (3)$

and expand to quadratic order in the fluctuation field $h_{ij}(x) = g_{ij} - \eta_{ij}$ . Just think of standard linearised gravity with the following familiar quadratic action

$S^2_{EH} = \frac{1}{2k^2} \int \ dx \ [\frac{1}{4} h^{ij}\partial^2 h_{ij} - \frac{1}{4} h \partial^2 h + \frac{1}{2} (\partial^i h_{ij})^2 + \frac{1}{2} h \partial_i \partial_j h^{ij}]. \ \ (4)$

This is the Feirz-Pauli action and it is of course invariant under (1). But we want a dualised theory. The naive thing to do, for the field $h(x, \tilde{x})$ , is to add a second collection of tilde dependant terms. In comparison with (4), we also update the integration measure to give

$S^2_{EH} = \frac{1}{2k^2} \int \ dx d\tilde{x} \ [\frac{1}{4} h^{ij}\partial^2 h_{ij} - \frac{1}{4} h \partial^2 h \\ + \frac{1}{2} (\partial^i h_{ij})^2 + \frac{1}{2} h \partial_i \partial_j h^{ij} + \\ \frac{1}{4} h^{ij} \tilde{\partial}^2 h_{ij} - \frac{1}{4} h \tilde{\partial}^2 h \\ + \frac{1}{2} (\tilde{\partial}^i h_{ij})^2 + \frac{1}{2} h \tilde{\partial}_i \tilde{\partial}_j h^{ij}]. \ \ (5)$

If you decompose $x, \tilde{x}$ such that $h_{ij} (x)$ no longer depends on $\tilde{x}$ , then this action simply reduces to linearised Einstein gravity on the coordinate space $x^a.$ Similarly, for the dual theory.

When the doubled action (5) is varied under $\tilde{\delta}$ , the second line is invariant under (2). However, the first line gives

$\tilde{\delta} S = \int [dx d\tilde{x}] [h^{ij} \partial^2 \tilde{\partial}_i \tilde{\epsilon}_j + \partial_i h^{ij} (\partial^k \tilde{\partial}_{k})\tilde{\epsilon}_j \\ - h \partial^2 \tilde{\partial} \tilde{\epsilon} + h(\partial_i \tilde{\partial}^i)\partial_j \tilde{\epsilon}^j \\ + \partial_i h^{ij} \partial^k \tilde{\partial}_j \tilde{\epsilon}_k + (\partial_j \partial_j h^{ij})\tilde{\partial} \tilde{\epsilon}. \ \ (6)$

As one can see, the terms on each line would cancel if the tilde derivatives were replaced by ordinary derivatives. Rearranging and grouping like terms, and then relabelling some indices we find

$\tilde{\delta} S = \int [dx d\tilde{x}] \ [(\tilde{\partial}_j h^{ij})\partial^k (\partial_i \tilde{\epsilon}_k - \partial_k \tilde{\epsilon}_i) \\ + (\partial_i \partial_j h^{ij} - \partial^2h) \tilde{\partial} \tilde{\epsilon} \\ + (\partial^i h_{ij} - \partial_j h)(\partial \tilde{\partial})\tilde{\epsilon}^j. \ \ (7)$

For this to be invariant under the transformation $\tilde{\delta}$ we have to cancel each of the terms. In order to cancel the variation, new fields with new gauge transformations are required. For the first term, a hint comes from the structure of derivatives, namely the fact we have a mixture of tilde and non-tilde derivatives. The Kalb-Ramond b-field mixes derivatives in this way, and, indeed, for the first term to cancel we may add $b_{ij}$ . We denote this inclusion to the action as $S_b$

$S_b = \int [dx d\tilde{x}] \ (\tilde{\partial}_j h^{ij})\partial^k b_{ik}, \\ with \ \ \tilde{\delta}b_{ij} = - (\partial_i \tilde{\epsilon}_j - \partial_j \tilde{\epsilon}_i). \ \ (8)$

The second term can similarly be killed upon introduction of the dilaton $\phi$ . It takes the form

$S_{\phi} = [dx d\tilde{x}] (-2) (\partial_i \partial_j h^{ij} - \partial^2 h) \phi, \ \ \text{with} \ \ \tilde{\delta}\phi = \frac{1}{2}\tilde{\partial} \tilde{\epsilon}. \ \ (9)$

This is quite nice, if you think about it. It is not the full story, because in the complete picture of double field theory we need to add more terms and their are several subtlties. In the naive case of dualised linearised gravity, we find in any case that linearised dual diffeomorphisms for the field $h_{ij}$ requires, naturally and perhaps serendipitously, a Kalb-Ramond gauge field and a dilaton – i.e., the closed string fields for the NS-NS sector.

We are now only left with one term, which is the one with curious structure on the third line in (7). To kill this term, we can observe that the gauge parameter $\tilde{\epsilon}$ satisfies the constraint $\partial \cdot \tilde{\partial} = 0$ derived from the level matching condition. This constraint says that fields and gauge parameters must be annihilated by $\partial \tilde{\partial}$ , and it is fairly easy to find in an analysis of the spectrum in closed string field theory.

So that is one way to attack the remaining term. But what is also interesting, I think, is that it is possible to accomplish the same goal by adding more fields to the theory. This is a non-trivial endeavour, to be sure, as the added fields would need to be invariant under $\delta$ and $\tilde{\delta}$ transformations. Ideally, one would likely want to be able to generalise the added fields to the formal case of the duality invariant theory. But it presents an interesting question.

***

From the perspective of string field theory, double field theory wants to describe a manifestly T-duality invariant theory (we talked about this in a number of past posts). The strategy is to look at the full closed string field theory comprising an infinite number of fields, and instead select to focus on a finite subset of those fields, namely the massless NS-NS sector. So DFT is, at present, very much a truncation of the string spectrum.

As a slight update to notation to match convention, for the massless fields of the NS-NS sector let’s now write the metric $g_{ij}$ , with the b-field $b_{ij}$ and dilaton $\phi$ the same as before. The effective action of this sector is famously

$\displaystyle S_{NS} = \int dX \sqrt{-g} e^{-2\phi} [R + 4(\partial \phi)^2 - \frac{1}{12}H^2] + \text{higher derivative terms}. \ \ (10)$

As one can review in any string textbook, this action is invariant under local gauge transformations: diffeomorphisms and a two-form gauge transformation. The NS-NS field content transforms as

$\displaystyle \delta g_{ij} = L_{\lambda} g_{ij} = \lambda^{k} \partial_k g_{ij} + g_{kj}\partial_i \lambda^k + g_{ik}\partial_i \lambda^k,$

$\displaystyle \delta b_{ij} = L_{\lambda} b_{ij} = \lambda^k \partial_k b_{ij} + b_{kj}\partial_i \lambda^k + b_{ik}\partial_i\lambda^k,$

$\displaystyle \delta \phi = L_{\lambda} \phi = \lambda^k \partial_k \phi. \ \ (11)$

We define the Lie derivative $L_{\lambda}$ along the vector field $\lambda^i$ on an arbitrary vector field $V^i$ such that the Lie bracket takes the form

$\displaystyle L_{\lambda} V^i = [\lambda, V]^i = \lambda^j \partial_j V^i - V^j \partial_j \lambda^i. \ \ (12)$

For the Kalb-Ramond two-form $b_{ij}$ , the gauge transformation is generated by a one-form field $\tilde{\lambda}_i$

$\displaystyle \delta b_{ij} = \partial_i \tilde{\lambda}_j - \partial_j \tilde{\lambda}_i. \ \ (13)$

One way to motivate a discussion on doubled or generalised diffeomorphisms in DFT is to understand that what one wants to do is essentially generalise the action (10). This means that at any time we should be able to recover it. The generalised theory should therefore possess all the same symmetries (with added requirement of manifest invariance under T-duality), including diffeomorphism invariance.

In the generalised metric formulation [2] the DFT action reads

$\displaystyle S_{DFT} = \int d^{2D} X e^{-2d} \mathcal{R}, \ \ (14)$

where

$\displaystyle \mathcal{R} \equiv 4\mathcal{H}^{MN}\partial_M \partial_N d - \partial_M \partial_N \mathcal{H}^{MN} \\ - 4\mathcal{H}^{MN}\partial_{M}d\partial_N d + 4\partial_M \mathcal{H}^{MN} \partial_N d \\ + \frac{1}{8}\mathcal{H}^{MN}\partial_{M}\mathcal{H}^{KL}\partial_{N}\mathcal{H}_{KL} - \frac{1}{2} \mathcal{H}^{MN}\partial_{N}\mathcal{H}^{KL}\partial_{L}\mathcal{H}_{MK}. \ \ (15)$

This action is constructed [2] precisely in such a way that it captures the same dynamics as (10). Here $\mathcal{H}$ is the generalised metric, which combines the metric and b-field into an $O(D,D)$ valued symmetric tensor such that

$\displaystyle \mathcal{H}^{MN}\eta_{ML}\mathcal{H}^{LK} = \eta^{NK}, \ \ (16)$

where $\eta$ is the $O(D,D)$ metric. We spoke quite a bit about the generalised metric and the role of $O(D,D)$ in a past post (see this link also for further definitions, recalling for instance the T-duality transformation group is $O(D,D; \mathbb{R})$ , which is discretised to $O(D,D; \mathbb{Z})$ . If $O(D,D)$ is broken to the discrete $O(D,D;\mathbb{Z})$ , then one can interepret the transformation as acting on the background torus on which DFT has been defined). Also note that in (15) $d$ is the generalised dilaton. In the background independent formulation of DFT [5], $e^{-2d}$ is shown to be a generalised density such that the dilaton $\phi$ with the determinant of the undoubled metric $g = \det g_{ij}$ on the whole space is combined into an $O(D,D)$ singlet $d$ establishing the identity $\sqrt{-g}e^{-2\phi} = e^{-2d}$ . We’ll talk a bit more about this later.

There are a number of important characteristics built into the definition of the generalised Ricci (15). Firstly, it is contructed to be an $O(D,D)$ scalar. One can show that the action (14) possesses manifest global $O(D,D)$ symmetry

$\displaystyle \mathcal{H}^{MN} \rightarrow \mathcal{H}^{LK}M_{L}^{M}M_{K}^{N} \ \ \text{and} \ \ X^{M} \rightarrow X^{N}M_{N}^{M}, \ (17)$

where $M_{L}^{K}$ is a constant tensor which leaves $\eta^{MN}$ invariant such that

$\displaystyle \eta^{LK} M_{L}^{M} M_{K}^{N} = \eta^{MN}. \ \ (18)$

Importantly, given $O(D,D)$ extends to a global symmetry, we may define this under the notion of generalised diffeomorphisms. Unlike with the supergravity action (10), which is invariant under the gauge transformations (11) and (12), in DFT the metric and b-field are combined into a single object $\mathcal{H}$ . So the obvious task, then, is to find a way to combine the diffeomorphisms and two-form gauge transformation in the form of some generalised gauge transformation. This is really the thrust of the entire story.

To see how this works, as a brief review, we define some doubled space $\mathbb{R}^{2D}.$ To give a description of this doubled space, all we need to start is some notion of a differential manifold with the condition that we have a linear transformation of the coordinates $X^{\prime} = hX$ , where $h \in O(D,D)$ (similar to the transformation we defined in the post linked above). We will include the generalised dilaton $d$ and we also include the generalised metric $\mathcal{H}$ , although we can keep this generic in definition should we like. For $\mathcal{H}$ we require only that it satisfies the $O(D,D)$ constraint $\mathcal{H}^{-1} = \eta \mathcal{H} \eta$ , where, from past discussion, one will recall $\eta$ is the $0(D,D)$ metric. It transforms $\mathcal{H}^{\prime}(X^{\prime}) = h^{t}\mathcal{H}(X)h$ . We now have everything we need.

Definition 1. A doubled space $\mathbb{R}^{2D}(\mathcal{H},d)$ is a space equipped with the following:

1) A positive symmetric $2D \times 2D-\text{matrix}$ field $\mathcal{H}$ , which is the generalized metric. This metric must satisfy the above conditions and transform covariantly under $O(D,D).$

2) A generalised dilaton scalar $d$ , which is a $2D$ scalar density such that $d = \phi - \frac{1}{2} \ln \det h$ (we’ll show this in a moment).

a) The generalised dilaton is related to the standard dilaton as already described above.

With this definition, we can then advance to define the notion of an $O(D,D)$ module, generalised vectors and vector fields, and so on. To keep our discussion short, the point is that in defining an $O(D,D)$ vector we may combine from before the vector $\lambda^i$ and one-form $\tilde{\lambda}_i$ as generalised gauge parameters

$\displaystyle \xi^M = (\tilde{\lambda}_i, \lambda^i). \ \ (19)$

One can see how this is done in [2,3]. In short, the combination of the gauge transformations into the general gauge transformation with parameter $\xi^M$ is defined under the action of a generalised Lie derivative. The result is simply given here as

$\displaystyle \mathcal{L}_{\xi}A_M \equiv \xi^P \partial_P A_M + (\partial_M \xi^P - \partial^P \xi_M)A_p,$

$\displaystyle \mathcal{L}_{\xi}B^M \equiv \xi^P \partial_P B^M + (\partial^M \xi_P - \partial_P \xi^M)B^p. \ \ (20)$

From this definition, where, it should be said, $A$ and $B$ are generalised vectors, we can eventually write the generalised Lie derivative of $\mathcal{H}$ and $d$ .

$\displaystyle \mathcal{L}_{\xi} \mathcal{H}_{MN} = \xi^P \partial_P \mathcal{H}_{MN} + (\partial_M \xi^P - \partial^P \xi_M)\mathcal{H}_{PN} + (\partial_N \xi^P - \partial^P \xi_N)\mathcal{H}_{MP},$

$\displaystyle \mathcal{L}_{\xi}(e^{-2d}) = \partial_M(\xi^M e^{-2d}). \ \ (21)$

What we see is that, indeed, the generalised dilaton, which we may think of as an $O(D,D)$ singlet, transforms as a density. This means we may think of it as a generalised density. It can also be shown that the Lie derivative of the $O(D,D)$ metric $\eta$ vanishes and therefore the metric is preserved.

What we want, for the purposes of this post, is the generalised Lie derivative of the generalised scalar curvature (15). What we find is that, indeed, it transforms as a scalar provided that the definition of (15) includes the full combination of terms.

$\displaystyle \mathcal{L}_{\xi} \mathcal{R} = \xi^M \partial_M \mathcal{R}.$ (22)

Or, looking at the action (14) as a whole, the subtlety is that the generalised dilaton forms part of the integration measure. The action does not possess manifest generalised diffeomorphism invariance in the typical sense that we might think about it, but it is constructed precisely in such a way that

$\displaystyle \mathcal{L}_{\xi}(e^{-2 d})\mathcal{R} = \partial_I (\xi^{I} e^{-2d}\mathcal{R}) \ \ (26)$

vanishes in the action integral (due to being a total derivative). So we find (14) does indeed remain invariant.

As a brief aside, from the transformations of the generalised metric and the dilaton, we can define an algebra [4]

$\displaystyle [\mathcal{L}_{\xi_1}, \mathcal{L}_{\xi_2}] = \mathcal{L}_{\xi_1} \mathcal{L}_{\xi_2} - \mathcal{L}_{\xi_2} \mathcal{L}_{\xi_1} = \mathcal{L}_{[\mathcal{L}_{\xi_1} \mathcal{L}_{\xi_2}]_C}, \ \ (23)$

where we find first glimpse at the presence of the Courant bracket. Provided the strong $O(D,D)$ constraint of DFT is imposed

$\displaystyle \partial_N A_I \partial^{N} A^J = 0 \ \forall \ i,j, \ \ (24)$

then the Courant bracket governs this algebra such that

$\displaystyle [\mathcal{L}_{\xi_1}, \mathcal{L}_{\xi_2}]^{M}_{C} = \xi_{1}^{N}\partial_{N}\xi_{2}^{M} - \frac{1}{2}\xi_{1N}\partial^{M}\xi_{2}^{N} - (\xi_1 \leftrightarrow \xi_2). \\ (25)$

An important caveat or subtlety about this algebra is that it does not satisfy the Jacobi identity. This means that the generalised diffeomorphisms do not form a Lie algebroid. But nothing fatal comes from this fact for the reason that, whilst we may like to satisfy the Jacobi identity, the gauge transformation leaves all the fields invariant that fulfil the strong $O(D,D)$ constraint.

In closing, recall that DFT starts with the low-energy effective theory as a motivation. It is good, then, that a solution of (24) is to set $\tilde{\partial} = 0$ giving (10). The Ricci scalar is the only diffeomorphism invariant object in Riemannian geometry that can be constructed only from the metric with no more than two derivatives. In DFT, we have an action constructed only from the generalised metric and doubled dilaton with their derivatives.

References

[1] Hull, C.M., and Zweibach, B., Double field theory. (2009). [arXiv:0904.4664 [hep-th]].

[2] Hohm, O., Hull C.M., and Zwiebach, B., Generalized metric formulationof double field theory. JHEP, 08:008, 2010. [arXiv:1006.4823 [hep-th]].

[3] Zwiebach, B., Double field theory, T-duality, and Courant brackets. [arXiv:1109.1782 [hep-th]].

[4] Hull, C.M., and Zwiebach, B., The gauge algebra of double field theory and courant brackets. Journal of High Energy Physics, 2009(09):090–090, Sep 2009. [arXiv:0908.1792 [hep-th]].

[5] Hohm, H., Hull, C.M., and Zwiebach, B., Background independent actionfor double field theory. Journal of High Energy Physics, 2010(7), Jul 2010. [arXiv:1003.5027 [hep-th]].

Statement $\Sigma$	Dual statement $\Sigma^{\star}$
$f : a \rightarrow b$	$f : b \rightarrow a$
$a = \text{dom} f$	$a = \text{cod} f$
$i = 1_a$	$i = 1_a$
$h = g \circ f$	$h = f \circ g$
$f$ is monic	$f$ is epis
$u$ is a right inverse of $h$	$u$ is a left inverse of $h$
$f$ is invertible	$f$ is invertible
t is a terminal object	t is an initial object