# Notes on string theory #3: Nambu-Goto action

1. Introduction

I haven’t been keeping up with this as much as I would like, mainly because I have been busy. But I am committed to continuing to reupload many of my notes on Polchinski’s textbooks. It is fun for me to go through it all again in my spare time, and I’ve noticed that since the time of first working through the textbooks there is more I can add to many topics.

It is worth remembering that, in the last note, we reviewed the classical worldline and polynomial action for the relativistic point particle. We also discussed reparameterisation invariance and calculated the equations of motion. In this note, the focus is to construct the first-principle Nambu-Goto action for the relativistic string as given in equations (1.2.9a-1.2.9b) in Polchinski’s textbook.

Often in popular literature and discourse I read descriptions of the string that almost shroud it in mystery. How could the fundamental constituents of matter be described by a bunch of strings? Other times, caricatures of string theory can leave the impression that to view all elementary particles as vibrating strings is somewhat arbitrary. Why not some other type of objects? It is suggestive of a certain arbitrariness to the idea of modelling fundamental particles as strings; but the development of string theory is, in fact, well motivated. Ultimately, all that we’re doing is extending the concept of point particles that we all know and love, and this is first and foremost evidenced in the Nambu-Goto action. But, in terms of the bigger picture, what we see is that in studying the string and its dynamics an entire universe of implications emerge. It reminds me of a great line in David Tong’s lecture notes that is worth paraphrasing: we find that the requirements demanded by the tiny string are so stringent that we are led naturally to a description of how the entire universe moves. On many occasions it is, indeed, like “the tail is wagging the dog”.

As we’re following Polchinski’s textbook, which only covers the Nambu-Goto action in a few words, if the interested reader would like to spend more time studying this action I would recommend ‘String theory and M-theory’ by Katrin Becker, Melanie Becker, and John H. Schwarz, especially the exercises, or for an even more gentle introduction see Barton Zwiebach’s textbook ‘A first course in string theory’.

2. Area functional

To arrive at the Nambu-Goto action, let us first recall from the last note that a p-brane may be described as a p-dimensional object moving through D-dimensional flat spacetime with ${D \geq p}$. If a 0-dimensional point particle (0-brane) traces out a (0+1)-dimensional worldline, it follows that a 1-dimensional string (1-brane) sweeps out a (1+1)-dimensional surface that we call the string worldsheet. And just as we can parameterise the relativistic point particle’s (0+1)-dimensional worldline, we can parameterise the (1+1)-dimensional worldsheet traced by the string. Coming to grips with this idea is the first task.

The main idea is that the worldline of a particle is replaced by the worldsheet ${\Sigma}$, which is a surface embedded into D-dimensional Minkowksi spacetime. Given that the path of a point particle can be described by a single parameter, the proper time ${\tau}$, which multiplied by c, is the Lorentz invariant proper length of the worldline; for strings, we will define the Lorentz invariant proper area of the worldsheet in a completely analogous way. As we’ll see, the first-principle string action is indeed proportional to this proper area.

To start, we see that because the string worldsheet is a (1+1)-dimensional surface, its requires two parameters which we will denote as ${\xi^{1}}$ and ${\xi^{2}}$. We will also limit our present considerations to the case of an open string (we will talk about closed strings in a later note). In order to define the appropriate area functional, we want to sketch a grid on the spacial surface of the string worldsheet with lines of constant ${\xi^{1}}$ and ${\xi^{2}}$; then we want to embed this spatial surface in the background target space.

The target space is the world where the 2-dimensional surface lives. Ultimately, we want to distinguish between the area we parameterise and the actual physical string worldsheet. In order to accomplish this, we define a one-to-one map, which we may call the string map. The purpose of the string map is therefore to take us from the parameter space that we have constructed to the target space in which the physical surface propagates. Indeed, as we’ll see, the string action is in this precise sense defined as a functional of smooth maps.

To construct the string map, we first formalise the notion of area in parameter space, with this parameter space itself defined by the range of the parameters ${\xi^{1}}$ and ${\xi^{2}}$. One can, in principle, view the parameters we have selected as local coordinates on the surface. And so, as emphasised above, we can think of the worldsheet as a physical surface, which is in fact the image of the parameter space under the one-to-one string map written as ${\vec{x}(\xi^{1}, \xi^{2})}$. The parameterised surface can therefore be described by the coordinate functions

$\displaystyle \vec{x}(\xi^1 , \xi^2) = x^1 (\xi^1 , \xi^2), x^2 (\xi^1 , \xi^2), x^3 (\xi^1 , \xi^2). \ \ \ \ \ (1)$

The area to which we want to give mathematical description is more accurately an infinitesimal area element. Since we begin working in a parameter space, and since our very small square is mapped onto the surface in target space, when we map this very small area from the parameter space to the surface we achieve a parallelogram the sides of which may be denoted as ${d\vec{v}_1}$ and ${d\vec{v}_2}$. We can express this as follows:

$\displaystyle d \vec{v}_1 = \frac{\partial \vec{x}}{\partial \xi^1} d\xi^1$

$\displaystyle d \vec{v}_2 = \frac{\partial \vec{x}}{\partial \xi^2}d\xi^2, \ \ \ \ \ (2)$

in which we have defined the rate of variation of the coordinates with respect to the parameters ${\xi}$. If we multiply this rate by the length ${d\xi}$ of the horizontal side of the infinitesimal parallelogram, we get the vector ${d \vec{v}_1}$ that represents this side in the target space.

The main objective is to now compute the area ${dA}$ of this parallelogram.

IMAGE

Since we have already labelled the sides of the infinitesimal area in the parameter space, we simply need to invoke the formula for the area of a parallelogram:

$\displaystyle d^2 A = \mid d\vec{v}_1 \mid \mid d\vec{v}_2 \mid \mid \sin \theta \mid$

$\displaystyle = \mid d\vec{v}_1 \mid \mid d\vec{v}_2 \mid \sqrt{1 - \cos^2 \theta}$

$\displaystyle = \sqrt{\mid d\vec{v}_1 \mid^2 \mid d\vec{v}_2 \mid^2 - \mid d\vec{v}_1 \mid^2 \mid d\vec{v}_2 \mid^2 \cos^2 \theta}. \ \ \ \ \ (3)$

Here ${\theta}$ denotes the angle between the vectors ${dv_1}$ and ${dv_2}$. Written in terms of dot products in which ${(\vec{A} \times \vec{B}) \cdot (\vec{A} \times \vec{B}) = \mid A \mid^2 \mid B \mid^2 - (A \cdot B)^2}$ such that

$\displaystyle (d\vec{v}_1 \times d\vec{v}_2) \cdot (d\vec{v}_1 \times d\vec{v}_2) = (d\vec{v}_1)^2 (d\vec{v})^2 - (d\vec{v}_1 \cdot d\vec{v}_2)^2$

we have

$\displaystyle = \sqrt{(d\vec{v}_1 \cdot d\vec{v}_1) (d\vec{v}_2 \cdot d\vec{v}_2) - (d\vec{v}_1 \cdot d\vec{v}_2)^2}. \ \ \ \ \ (4)$

From this result, notice that we can now substitute for ${d\vec{v}_1}$ and ${d\vec{v}_2}$ using (2). Doing so gives

$\displaystyle dA = \sqrt{(\frac{\partial \vec{x}}{\partial \xi^1} \cdot \frac{\partial \vec{x}}{\partial \xi^1})(\frac{\partial \vec{x}}{\partial \xi^2} \cdot \frac{\partial \vec{x}}{\partial \xi^2}) - (\frac{\partial \vec{x}}{\partial \xi^1} \cdot \frac{\partial \vec{x}}{\partial \xi^2})^2} d\xi^1 d\xi^2. \ \ \ \ \ (5)$

We have now obtained a general expression for the area element of the parameterised spatial surface. Written as the full area functional we have

$\displaystyle A = \int d\xi^1 d\xi^2 \ \sqrt{(\frac{\partial \vec{x}}{\partial \xi^1} \cdot \frac{\partial \vec{x}}{\partial \xi^1})(\frac{\partial \vec{x}}{\partial \xi^2} \cdot \frac{\partial \vec{x}}{\partial \xi^2}) - (\frac{\partial \vec{x}}{\partial \xi^1} \cdot \frac{\partial \vec{x}}{\partial \xi^2})^2}, \ \ \ \ \ (6)$

where the integral extends over the ranges of the parameters ${\xi^{1}}$ and ${\xi^{2}}$. This functional is reparameterisation invariant, which can be quickly verified by reparameterising the surface with tilde parameters ${(\tilde{\xi}^1, \tilde{\xi}^2)}$ that then gives back (6) when ${\tilde{\xi}^1 = \tilde{\xi}^1 (\xi^1)}$ and ${\tilde{\xi}^2 = \tilde{\xi}^2 (\xi^2)}$.

The main issue is that the area functional (6) is not very nice, and reparameterisation invariance is not completely general. We want reparameterisation invariance to be manifest.

3. Induced Metric

Suppose we have some vector ${d\vec{x}}$ on the surface ${\Sigma}$ that we have so far pencilled into the target space. We know that we can describe this surface through the string mapping functions ${\vec{x}(\xi^{1}, \xi^{2})}$. What if we then consider ${d\vec{x}}$ tangent to the surface ${\Sigma}$? We could then let ${ds}$ denote the length of this tangent vector, and hence we could invoke some early idea of a metric on ${\Sigma}$.

Given the vector tangent to the surface, with ${ds}$ its length, we can write

$\displaystyle ds^2 = d\vec{x} \cdot d\vec{x}. \ \ \ \ \ (7)$

But what is ${d\vec{x}}$ in terms of the parameter space coordinates that we constructed? In other words, can we relate ${d\vec{x}}$ with ${\xi^{1}, \xi^{2}}$? This is precisely what our mapping accomplishes such that we can express ${d\vec{x}}$ in terms of partial derivatives and derivatives of ${\xi^{1}, \xi^{2}}$:

$\displaystyle d\vec{x} = \frac{\partial \vec{x}}{\partial \xi^1} d\xi^1 + \frac{\partial \vec{x}}{\partial \xi^2} d\xi^2 = \frac{\partial \vec{x}}{\partial \xi^i} d\xi^i, \ \ \ \ \ (8)$

with the summation convention assumed for the repeated indices over possible values 1 and 2. If we now return to (7) and plug ${d\vec{x}}$ back into our equation for ${ds^2}$ we see that we can now write

$\displaystyle ds^2 = \frac{\partial \vec{x}}{\partial \xi^i} d\xi^i \cdot \frac{\partial \vec{x}}{\partial \xi^j} d\xi^j. \ \ \ \ \ (9)$

But notice something interesting. If we set ${h_{ij}(\xi) = \frac{\partial \vec{x}}{\partial \xi^i} d\xi^i \cdot \frac{\partial \vec{x}}{\partial \xi^j}}$, this means we can write a more simplified equation of the form

$\displaystyle ds^{2} = h_{ij}(\xi) d\xi^i d\xi^j, \ \ \ \ \ (10)$

in which the quantity ${h_{ij}(\xi)}$ is called the induced metric. It is a metric on the target space surface precisely in the sense that, as ${\xi_i}$ play the role of coordinates on ${\Sigma}$, we see that (10) determines distances on this surface. It is said to be induced because it uses the metric on the ambient space in which ${\Sigma}$ lives to determine distances on ${\Sigma}$. More technically, we say that the induced metric is the pullback of the target space metric onto the worldsheet.

A question we can now ask is whether, upon constructing a metric on the target space surface, does this then lead us to an equivalent expression for (6)? Observe that, in matrix form, we have for the induced metric

$\displaystyle h_{ij} = \begin{pmatrix} \frac{\partial \vec{x}}{\partial \xi^1} \cdot \frac{\partial \vec{x}}{\partial \xi^1} & \frac{\partial \vec{x}}{\partial \xi^1} \cdot \frac{\partial \vec{x}}{\partial \xi^2} \\ \frac{\partial \vec{x}}{\partial \xi^2} \cdot \frac{\partial \vec{x}}{\partial \xi^1} & \frac{\partial \vec{x}}{\partial \xi^2} \cdot \frac{\partial \vec{x}}{\partial \xi^2} \\ \end{pmatrix}. \ \ (11)$

What is this telling us? Notice that if you compute the determinant of the matrix ${h_{ij}}$, you find the same quantity that resides under the square root in (6). This is a massive hint that the construction is on the right track. So, let’s substitute the appropriate matrix elements into our earlier expression for the infinitesimal area. This is what we find,

$\displaystyle dA = \sqrt{(h_{11})h_{22} - h_{12}^2} \ d\xi^1 d\xi^2$

$\displaystyle = \sqrt{\det h} \ d\xi^1 d\xi^2$

$\displaystyle \therefore dA = \sqrt{h} \ d\xi^1 d\xi^2, \ \ \ \ \ (12)$

where ${h \equiv \det h_{ij} (\xi)}$. This implies,

$\displaystyle A = \int d\xi^1 d\xi^2 \sqrt{h}. \ \ \ \ \ (13)$

This new way to express the area, ${A}$, is now given in terms of the determinant of the induced metric. And although we are not yet done constructing the Nambu-Goto action, we see quite clearly from (13) why Polchinski says that the action ${S_{NG}}$ in equations (1.2.9a-1.2.9b) in his textbook is proportional to the area of the worldsheet.

4. Reparameterisation invariance

The wonderful thing about this last result (13) is that we can now show manifest reparameterisation invariance in a much simpler way, as it may now be described by way of how the induced metric transforms.

To do this, we invoke a different set of parameters and therefore also a different metric, and then we show that the original vector ${d\vec{x}}$ does not depend on our original parameterisation.

We begin with

$\displaystyle ds^2 = h_{ij}(\xi) d\xi^i d\xi^j = \tilde{h}_{ij}(\tilde{\xi}) d\tilde{\xi}_1 d\tilde{\xi}_2. \ \ \ \ \ (14)$

We then use the chain rule

$\displaystyle ds^{2} = \tilde{h}_{pq}(\tilde{\xi}) \frac{\partial \tilde{\xi}^p}{\partial \xi^i} \frac{\partial \tilde{\xi}^q}{\partial \xi^j} d\xi^i d\xi^j$

$\displaystyle h_{ij}(\xi) = \tilde{h}_{pq} (\tilde{\xi}) \frac{\partial \tilde{\xi}^p}{\partial \xi^i} \frac{\partial \tilde{\xi}^q}{\partial \xi^j}. \ \ \ \ \ (15)$

Next, recall that the change of variable theorem tells us how the integration measure transforms

$\displaystyle d\xi^{1} d \xi^2 = \mid \det \frac{d \xi^i}{d \tilde{\xi}^j} \mid d \tilde{\xi}^1 d \tilde{\xi}^2 = \mid \det M \mid d \tilde{\xi}^1 d \tilde{\xi}^2, \ \ \ \ \ (16)$

where ${M}$ is the matrix defined by ${M_{ij} = \partial \xi^1 / \partial \tilde{\xi}j}$ and similarly

$\displaystyle d\tilde{\xi}^{1} d \tilde{\xi}^2 = \mid \det \frac{d \tilde{\xi}^i}{d \xi^j} \mid d \xi^1 d \xi^2 = \mid \det \tilde{M} \mid d \xi^1 d \xi^2, \ \ \ \ \ (17)$

where ${\tilde{M}}$ is defined by ${\tilde{M}_{ij} = \partial \tilde{\xi}^i / \partial \xi^j}$. Using this and returning to (15) we can rewrite this equation for ${h}$ and ${\tilde{h}}$ such that

$\displaystyle h_{ij}(\xi) = \tilde{h}_{pq} \tilde{M}_{pi}\tilde{M}_{qj} = (\tilde{M}^T)_{ip}\tilde{h}_{pq} \tilde{M}_{qj}. \ \ \ \ \ (18)$

If we denote ${h \equiv \det h_{ij}}$, and if take the determinant of the right-hand side of (18) we find

$\displaystyle h = (\det \tilde{M}^T) \tilde{h} (\det \tilde{M}) = \tilde{g}(\det \tilde{M})^2. \ \ \ \ \ (19)$

Clearly, then, if we take the square root we obtain

$\displaystyle \sqrt{h} = \sqrt{\tilde{h}} \mid \det \tilde{M} \mid, \ \ \ \ \ (20)$

which is the transformation property for the square root of the determinant of the metric.

Finally, we conclude using (16) and (20) with the fact that ${\mid \det M \mid \mid \det \tilde{M} \mid = 1}$ we can show that (13) is reparameterisation invariant

$\displaystyle \int d\xi^1 d\xi^2 \sqrt{h} = \int d\tilde{\xi}^1 d\tilde{\xi}^2 \mid \det M \mid \sqrt{\tilde{h}} \mid \det \tilde{M} \mid = \int d\tilde{\xi}_1 d\tilde{\xi}_2 \sqrt{\tilde{h}}. \ \ \ \ \ (21)$

There is perhaps a much more elegant way to show this proof. But for now, one should focus on how (21) is just a standard metric transformation inasmuch that ${\int d\xi^1 d\xi^2 \sqrt{h}}$ transforms via a Jacobian determinant of ${\xi}$ with respect to ${\tilde{\xi}}$ as ${\int d\tilde{\xi}_1 d\tilde{\xi}_2 \sqrt{\tilde{h}}}$.

5. String propagating in spacetime

Let us now work toward constructing the Nambu-Goto action as it appears in equations (1.2.9a-1.2.9b). Up to this point we have taken the approach of mapping from a parameter space to a target space in which the surface ${\Sigma}$ lives. But we are interested in the case of surfaces in spacetime. These surfaces are obtained by representing in spacetime the history of the string as it propagates, in the same way the worldline of the point particle is described by representing its history.

Spacetime surfaces, such as string worldsheets, are not all that different from the spatial surfaces we considered in the previous sections. Instead of the coordinates ${\xi^{1}}$ and ${\xi^{2}}$, for a relativistic string we should parameterise the string worldsheet in such a way that we account for both the proper time and the string’s spatial extension. Another way to put this is that, if our interest is to consider surfaces in spacetime (the worldsheet traced by the string), we now use ${\tau}$ to denote the proper time and ${\sigma}$ to denote the spacial extension of the surface. Given usual spacetime coordinates, which we write following string theory conventions ${X^{\mu} = (X^0, X^1, ..., X^d)}$, the surface is then described by the mapping functions

$\displaystyle X^{\mu}(\tau, \sigma). \ \ \ \ \ (22)$

Hence, we come to the point emphasised at the outset of this note. The string worldsheet action formally defines the map ${\Sigma : (\tau, \sigma) \mapsto X^{\mu}(\tau, \sigma) \in \mathbb{R}^{1, d-1}}$. If it is still not clear, remember that what we’re working toward is a description of the string worldsheet ${\Sigma}$ as a curved surface embedded in spacetime. This embedding is given by the fields ${X^{\mu}(\tau, \sigma)}$, in which the parameters ${\tau}$ and ${\sigma}$ can be viewed (locally) as coordinates on the worldsheet. So the string map tells us that given some fixed point ${X^{\mu}(\tau, \sigma)}$ in the parameter space, we are performing a direct mapping to a fixed point in spacetime coordinates. Typically we drop the arguments ${(\tau, \sigma)}$ and leave them implicit, with the inverse of the map ${X^{\mu}}$ taking the worldsheet to the parameter space.

It is also worth noting that the functions ${X^{\mu}}$ describe how the string propagates and oscillates through spacetime, while the endpoints of the string are parameterised by ${\tau}$ such that ${\frac{\partial X^{\mu}}{\partial \tau} (\tau, \sigma) \neq 0}$. In our present case, we are considering an open string; but if ${\sigma}$ is periodic then we’d be working with a closed string embedded in the background spacetime.

Getting back to the task at hand: to find the area element we proceed in similar fashion as before, except now we must use relativistic notation. So for the area element we have ${d\tau}$ and ${d\sigma}$ describing the sides of an infinitesimal parallelogram in parameter space. In spacetime, this becomes a quadrilateral area element. We therefore set-up a direct analogue with our expression for ${dA}$ in (4) where we consider the vectors ${dv^{\mu}_{1}}$ and ${dv_{2}^{\mu}}$ spanning the quadrilateral,

$\displaystyle dv^{\mu}_{1} = \frac{\partial X^{\mu}}{\partial \tau} d\tau, \ \ dv^{\mu}_{2} = \frac{\partial X^{\mu}}{\partial \sigma} d\sigma. \ \ \ \ \ (23)$

Notice that we may substitute for ${dv^{\mu}_{1}}$ and ${dv_{2}^{\mu}}$ into (4),

$\displaystyle dA = d\tau d\sigma \sqrt{(\frac{\partial X^{\mu}}{\partial \tau} \frac{\partial X_{\mu}}{\partial \tau})(\frac{\partial X^{\nu}}{\partial \sigma} \frac{\partial X_{\nu}}{\partial \sigma}) - (\frac{\partial X^{\mu}}{\partial \tau} \frac{\partial X_{\mu}}{\partial \sigma})^2}. \ \ \ \ \ (24)$

We now invoke relativistic dot product notation so that we ensure that what we are working with is the proper area. The object under the square root turns out to be negative, but we can switch the sign without violation of any rules. The basic idea is that, for a surface with a timelike vector and a spacelike vector the square root is always positive such that Cauchy-Schwarz inequality flips. This means, ${(\dot{X}^2 \cdot X^{\prime})^2 - (\dot{X})^2 (X^{\prime})^2 > 0}$. We also want to integrate (24). So, putting everything together, we have

$\displaystyle A = \int_{\sum} d\tau d\sigma \sqrt{(\frac{\partial X}{\partial \tau} \cdot \frac{\partial X}{\partial \sigma})^2 - (\frac{\partial X}{\partial \tau} \cdot \frac{\partial X}{\partial \tau})(\frac{\partial X}{\partial \sigma} \cdot \frac{\partial X}{\partial \sigma})}. \ \ \ \ \ (25)$

We can still simplify our expression for the area using the more compact notation, ${\dot{X}^{\mu} \equiv \frac{\partial X^{\mu}}{\partial \tau}}$ and ${X^{\prime \mu} \equiv \frac{\partial X^{\mu}}{\partial \sigma}}$. This means we can write,

$\displaystyle A = \int_{\Sigma} d\tau d\sigma \sqrt{(\dot{X})^2 (X^{\prime})^2 - (\dot{X} \cdot X^{\prime})^2}. \ \ \ \ \ (26)$

Now comes the important part. From (26) there are a few ways we can approach the Nambu-Goto action. The most direct approach is to remember how, inasmuch that we are generalising the point particle action, we may anticipate the existence of some constant of proportionality. Indeed, it is completely reasonable to anticipate an action of the form general form ${S = -T \int dA}$. And this proves to be the case, because it follows that we may write the Nambu-Goto action for the string as

$\displaystyle S_{NG} = -\frac{T_0}{c} \int_{\tau_i}^{\tau_f} d\tau \int_{0}^{\sigma_1} d\sigma \sqrt{(\dot{X} \cdot X^{\prime})^2 - \dot{X}^2 \cdot X^{\prime^2}}, \ \ \ \ \ (27)$

where ${\frac{T_0}{c}}$ is a constant of proportionality to ensure units of action. To explain this, consider the following. Given that the string action is proportional to the proper area of the worldsheet, the area functional has units of length squared. We see this because ${X^{\mu}}$ has unites of length, i.e., ${[X] = L}$, and there are four under the square root. Each term in the square root also has two ${\sigma}$ derivatives and two ${\tau}$ derivatives, with their units cancelling against the derivatives. Since ${S_{NG}}$ must have the units of action ${[S] = \hbar = ML^2/T}$ with ${A}$ having units ${L^2}$, the total proper area must be multiplied by the value ${M/T}$. We know that the string will have a tension, ${T_0}$, which has units of force. We also know that force divided by velocity has the units ${M/T}$; so to ensure units of action the proper area is multiplied by ${T_0 / c}$.

6. Manifest Reparameterisation Invariance of the Nambu-Goto Action

We still shouldn’t be completely satisfied with this early form of the Nambu-Goto action (28). How do we know, for instance, that what we have ended up with is manifestly reparameterisation invariant? It is crucial that the ${S_{NG}}$ action be dependent only on the embedding in spacetime and not the choice of parameterisation.

To explore the action (28) in a deeper way, we first need to invoke the target space Minkowski metric, ${\eta_{\mu \nu}}$, and we should consider a differential line element of the form

$\displaystyle -ds^{2} = dX^{\mu} dX_{\mu} = - \eta_{\mu \nu} dX^{\mu} dX^{\nu}. \ \ \ \ \ (28)$

We may now expand the derivatives acting on ${X}$,

$\displaystyle -ds^{2} = - \eta_{\mu \nu} \frac{\partial X^{\mu}}{\partial \xi^{\alpha}} \frac{\partial X^{\nu}}{\partial \xi^{\beta}} \ d\xi^{\alpha} d\xi^{\beta}, \ \ \ \ \ (29)$

where ${\alpha}$ and ${\beta}$ run from 1 and 2. Similar as before for the spatial surface, we can define an induced metric. In this case, the induced metric on the string worldsheet is given as ${h_{\alpha \beta}}$. It is simply the pullback of the target space Minkowski metric, ${\eta_{\mu \nu}}$. This allows us to define the induced metric as,

$\displaystyle h_{\alpha \beta} = \eta_{\mu \nu} \frac{\partial X^{\mu}}{\partial \xi^{\alpha}} \frac{\partial X^{\nu}}{\partial \xi^{\beta}}. \ \ \ \ \ (30)$

This means we can write the more compact equation for the line element

$\displaystyle -ds^{2} = h_{\alpha \beta} d\xi^{\alpha}d\xi^{\beta}, \ \ \ \ \ (31)$

because, while the induced metric describes distances on the string worldsheet, it also includes the metric of the background spacetime in its definition. But, to ensure clarity of knowledge, let’s think about this induced metric a bit more. In matrix form, it is a ${2 \times 2}$ matrix with components

$\displaystyle h_{\tau \tau} = \eta_{\mu \nu} \frac{\partial X^{\mu}}{\partial \tau} \frac{\partial X^{\nu}}{\partial \tau} = \dot{X}^{2},$

$\displaystyle h_{\sigma \tau} = \eta_{\mu \nu} \frac{\partial X^{\mu}}{\partial \sigma} \frac{\partial X^{\nu}}{\partial \tau} = \dot{X} \cdot X^{\prime} = h_{\tau \sigma},$

$\displaystyle h_{\sigma \sigma} = \eta_{\mu \nu} \frac{\partial X^{\mu}}{\partial \sigma} \frac{\partial X^{\nu}}{\partial \sigma} = X^{\prime 2}. \ \ \ \ \ (32)$

And so, the induced metric may be written in matrix form as

$\displaystyle h_{\alpha \beta} = \begin{pmatrix} \dot{X}^2 & \dot{X} \cdot X^{\prime} \\ \dot{X} \cdot X^{\prime} & X^2 \prime \\ \end{pmatrix}. \ \ \ \ \ (33)$

Therefore, as we showed in the case of the spatial surface, we extend the logic of the previous examples and manifest reparameterisation invariance is seen to be featured with the help of the induced metric

$\displaystyle S_{NG} = -\frac{T_{0}}{c} \int_{\sum} d\tau d\sigma \sqrt{-h}, \ \ \ \ \ (34)$

where ${h = \det h_{\alpha \beta}}$.

The final observation is that, as Polchinski notes (p.11), for the string tension an alternative parameter is ${\alpha^{\prime}}$. This proportionality constant ${\alpha^{\prime}}$ has been used since the early days of string theory; one may recognise it as the Regge slope, which has to do with the relation between the angular momentum, ${J}$, of a rotating string and the square of the energy ${E}$. In that ${\alpha^{\prime}}$ has units of spacetime-length-squared, we therefore observe

$\displaystyle T = \frac{1}{2 \pi \alpha \prime}, \ \ \ \ \ (35)$

where we’ve set ${\hbar = c = 1}$. This is equation (1.2.10) in Polchinski. Hence, we may now rewrite ${S_{NG}}$ in its more conventional form as read in equations (1.2.9a-1.29b):

$\displaystyle S_{NG} = - \frac{1}{2 \pi \alpha^{\prime}} \int_{\Sigma} d\tau d\sigma \ (- \det h_{\alpha \beta})^{1/2}. \ \ \ \ \ (36)$

The answer in Exercise 1.1b reveals more explicitly how the string tension is related to the Regge slope. It just requires that we write things in terms of the transverse velocity. To keep these notes focused and organised, at the conclusion of each chapter we’ll go over the solutions to the exercises and so we’ll return to this question then.

6.1. Equations of motion

Before we get to the symmetries of the action (36), let’s quickly look at its equations of motion. To simplify matters, we can write the Lagrangian as ${\mathcal{L} = \sqrt{-h}}$ with the Euler-Lagrange equations reading as

$\displaystyle \partial_\alpha\left(\frac{\partial\mathcal{L}}{\partial_\alpha X^\mu}\right)=\frac{\partial\mathcal{L}}{\partial_\alpha X^\mu}=0. \ \ \ \ \ (37)$

From the chain rule, it is therefore clear that we need to calculate

$\displaystyle \partial_\alpha\left(\frac{\partial\mathcal{L}}{\partial_\alpha X^\mu}\right)=\partial_\alpha\left(\frac{\partial\mathcal{L}}{\partial h_{\beta\gamma}}\frac{\partial h_{\beta\gamma}}{\partial_\alpha X^\mu}\right). \ \ \ \ \ (38)$

For the first term in brackets, we use the identity for the variation of the determinant ${\delta\sqrt{-g}=-\frac{1}{2}\sqrt{-g}g_{\alpha\beta}\delta g^{\alpha\beta}}$, which can be easily verified. Hence,

$\displaystyle \frac{\partial\mathcal{L}}{\partial h_{\beta\gamma}}=\frac{\partial\sqrt{-h}}{\partial h_{\beta\gamma}}=-\frac{1}{2}\sqrt{-h}\frac{h_{\rho\kappa}\delta h^{\rho\kappa}}{\delta h_{\beta\gamma}}=-\frac{1}{2}\sqrt{-h}h^{\beta\gamma}. \ \ \ \ \ (39)$

For the next term we find

$\displaystyle \frac{\partial h_{\beta\gamma}}{\partial(\partial_\alpha X^\mu)} =\eta^{\mu\nu}\delta^{\alpha}_\beta\partial_\gamma X_\nu +\eta^{\mu\nu}\delta^\alpha_\gamma\partial_\beta X_\nu =\delta^\alpha_\beta\partial_\gamma X^\mu +\delta^\alpha_\gamma\partial_\beta X^\mu. \ \ \ \ \ (40)$

Putting everything together

$\displaystyle \partial_\alpha\left(\frac{\partial\mathcal{L}}{\partial h_{\beta\gamma}}\frac{\partial h_{\beta\gamma}}{\partial_\alpha X^\mu}\right)=\partial_\alpha(-\frac{1}{2}\sqrt{-h}h^{\beta\gamma}(\delta_{\alpha}^\beta\partial_\gamma X^\mu+\delta_\alpha^\gamma\partial_\beta X^\mu))=0 \ \ \ \ \ (41)$

$\displaystyle \frac{1}{2}\partial_\alpha(\sqrt{-h}h^{\alpha\gamma}\partial_\gamma X^\mu+\sqrt{-h}h^{\beta\alpha}\partial_\beta X^\mu)=0 \ \ \ \ \ (42)$

$\displaystyle \partial_\alpha(\sqrt{-h}h^{\alpha\beta}\partial_\beta X^\mu)=0. \ \ \ \ \ (43)$

As the metric ${h}$ contains the embedding ${X^{\mu}}$, these equations are highly non-linear. But this is not unexpected given the fact that the action (35) is non-linear. One way to interrept these equations is that, as a minimal surface area is being demanded by the stationary action, in Zwiebach’s textbook one is motivated to think analogously of the image of static soap film in some Lorentz frame. In this case, we think of the film as a spatial surface in which every point is a saddle point.

6.2. Symmetries

Finally, the last topic covered concerns the symmetries of the action (36).

Poincare group: As the Nambu-Goto action is completely and directly analogous to the action for a relativistic point particle, one might rightly anticipate that the action for a string is invariant under the isometry group of flat spacetime, which is the D-dimensional Poincare group

$\displaystyle X^{\prime \mu}(\tau, \sigma) = \Lambda^{\mu}_{\nu}X^{\nu}(\tau, \sigma) + a^{\mu}. \ \ \ \ \ (44)$

This symmetry group consists of consists of Lorentz transformations ${\Lambda^{\mu}_{\nu}}$ satisfying ${SO(D-1, 1)}$ algebra and ${a^{\mu}}$ transformations. This symmetry is manifest and can be read-off from (36) since the Lorentz indices are contracted in the correct way to obtain a Lorentz scalar. But to see it explicitly just note that ${X^{\mu}}$ are flat spacetime vectors. Under the transformation ${X^{\mu} \rightarrow X^{\prime \mu} = \Lambda^{\mu}_{\nu}X^{\nu}(\tau, \sigma) + a^{\mu}}$ we see that ${\partial_{\alpha}X^{\prime \mu} = \Lambda^{\mu}_{\nu}\partial_{\beta}X^{\nu}}$. Hence

$\displaystyle \eta_{\mu}\partial_{\alpha} X^{\prime \mu}\partial_{\beta}X^{\prime \nu} = = {\Lambda^{\mu}}_{\gamma} \eta_{\mu \nu} {\Lambda^{\nu}}_{\sigma}\partial_{\alpha} X^{\gamma} \partial_{\beta} X^{\sigma} = \eta_{\gamma \sigma} \partial_{\alpha} X^{\gamma} \partial_{\beta} X^{\sigma}, \ \ \ \ \ (45)$

where we used ${\Lambda^{\mu}_{\gamma} \eta_{\mu \nu} \Lambda^{\nu}_{\sigma} = \eta_{\gamma \sigma}}$.

From the perspective of the worldsheet theory, the Nambu-Goto action is a 2-dimensional field theory of scalar fields ${X^{\mu}(\tau, \sigma)}$, and Poincare invariance is in fact an internal symmetry.

Diffeomorphism invariance The Nambu-Goto action is also invariant under diffeomorphism transformations, or reparamterisation of the coordinates, which we’ve already observed by the very nature of how we construct (36) such that ${X^{\prime \mu} (\tau^{\prime}, \sigma^{\prime}) = X^{\mu}(\tau, \sigma)}$.

6.3. Concluding remarks

The main issue with the action (36) is the presence of the square root, which complicates matters when we attempt to quantise the theory or take the massless limit ${m \rightarrow 0}$. That is why, analogous to the case of the relativistic point particle, we’ll want to get rid of this square root and construct a classically equivalent action. This is known as the Polyakov action and, following the progression in Polchinski, it is the topic of the next note.

In the meantime, I want to point out that there is still much more to be learned about the Nambu-Goto action and its dynamics. There are some quite famous and important results, which are not covered in Polchinski’s textbook. It is notable, for instance, that from an analysis of the worldsheet momentum densities

$\displaystyle P^{\alpha}_{\mu} = \frac{\partial \mathcal{L}}{\partial \partial_{\alpha}X^{\mu}} \ \ \ \ \ (46)$

we can evaluate the components of the canonical momenta explicitly

$\displaystyle \Pi = P_{\mu}^{\sigma} = \frac{\partial L}{\partial X^{\prime \mu}} = \frac{\partial}{\partial X^{\prime \mu}} (-T \sqrt{(\dot{X} \cdot X^{\prime})^{2} - (\dot{X}^2)(X^{\prime})^2})$

$\displaystyle = -\frac{T}{2}[(\dot{X} \cdot X^{\prime})^{2} - (\dot{X}^2)(X^{\prime})^2]^{-1/2} [2(\dot{X} \cdot X^{\prime})\dot{X}_{\mu} - 2 \dot{X}^{2}X_{\mu}^{\prime}]$

$\displaystyle = \frac{(\dot{X} \cdot X^{\prime})\dot{X}_{\mu} - \dot{X}^{2}X_{\mu}^{\prime}}{\sqrt{(\dot{X} \cdot X^{\prime})^{2} - (\dot{X}^2)(X^{\prime})^2}}, \ \ \ \ \ (47)$

and,

$\displaystyle P_{\mu}^{\tau} = \frac{\partial L}{\partial \dot{X}^{\mu}} = \frac{\partial}{\partial \dot{X}^{\mu}}(-T \sqrt{(\dot{X} \cdot X^{\prime})^{2} - (\dot{X}^2)(X^{\prime})^2})$

$\displaystyle = -\frac{T}{2}[(\dot{X} \cdot X^{\prime})^{2} - (\dot{X}^2)(X^{\prime})^2]^{-1/2} [2(\dot{X} \cdot X^{\prime})X_{\mu}^{\prime} - 2 X^{\prime} \dot{X}_{\mu}^{2}]$

$\displaystyle = \frac{(\dot{X} \cdot X^{\prime})X_{\mu}^{\prime} - X^{\prime 2} \dot{X}_{\mu}}{\sqrt{(\dot{X} \cdot X^{\prime})^{2} - (\dot{X}^2)(X^{\prime})^2}}. \ \ \ \ \ (48)$

From this analysis, we can obtain an equation that we can interpret as the generalised momentum flow of the particle worldline. This helps give a bit more insight and intuition into the direct analogue we’ve established between point particle theory and the theory of strings. Furthermore, by imposing the appropriate boundary conditions we can show for the equations of motion that

$\displaystyle \partial_{\alpha}P^{\alpha}_{\mu} = 0, \ \ \ \ \ (49)$

which, given the equations for the worldsheet momentum, we find the 2-dimensional wave equation given the choice of coordinates ${\dot{X} \cdot X^{\prime} = 0}$, ${\dot{X}^2 = -1}$, and ${X^{\prime} = 1}$. In the same analysis, we can find very important conditions such as the Virasaro constraints that govern the dynamics of the string.

There is also much more that can be studied: boundary conditions and motion of the string endpoints, which provides a first introduction to D-branes; tension and energy of the stretched string; transverse velocity; among other interesting topics. All of this of course comes up also in our study of the Polyakov action; but for the interested reader, Zwiebach’s textbook referenced at the outset covers all of these topics in detail in the context of the Nambu-Goto action.

# Cosmological constant, the duality symmetric string, and Atkin-Lehner symmetry

I was going through one of my notebooks and I came across a page with several comments on old papers by Arkady Tseytlin [1] and Gregory Moore [3], respectively. The notes must have been written last autumn at the start of the academic year, because it was around this time my supervisor and I were talking about the cosmological constant problem. In the referenced papers, two interesting approaches to the CC in string theory are presented.

Let’s start with Tseytlin. We’ve discussed in the past Tseytlin’s formulation of the duality symmetric string for interacting chiral bosons, so I direct the reader to that entry for a background introduction. Jumping straight to the point, what we find in the final sections of [2] is that, upon computing the 3-graviton amplitudes, the following 3-graviton interaction is obtained

$\displaystyle S_3 = \int d^D x_{+} d^D x_{-} [h_{\alpha \beta} (h_{\lambda \rho}\partial_{+ \alpha} \partial_{- \beta} h_{\lambda \rho} + 2\partial_{+ \alpha} h_{\lambda \rho}\partial_{- \rho}h_{\beta \lambda})], \ \ (1)$

where $\partial_{\pm \mu} \equiv \partial / \partial x^{\mu}_{\pm}$ and $h_{\mu \nu} \equiv H_{(\mu \nu)} (x_{+}, x_{-})$. When (1) is written in terms of doubled coordintes $(x, \tilde{x})$ the low-energy effective theory takes the form

$\displaystyle S_3 = \int d^D x d^D \tilde{x} [R_3 (\partial) - R_3 (\tilde{\partial})], \ \ (2)$

where $\partial_{\mu} = 1/\sqrt{2} (\partial_{+ \mu} + \partial_{-\mu}) = \partial / \partial x^{\mu}$ and $\tilde{\partial}_{\mu} = 1 / \sqrt{2} (\partial_{+ \mu} - \partial_{-\mu} = \partial / \partial \tilde{x}_{\mu}$. The 3-graviton term $R_3 (\partial)(R_3(\tilde{\partial}))$ in the expansion of the scalar curvature for the metric $G_{\mu \nu} = \delta_{\mu \nu} + h_{\mu \nu}$ with $h_{\mu \nu}(x, \tilde{x})$ can be written

$\displaystyle R_3 (\partial) = 1/4 h_{\mu \nu} \partial^2 h_{\mu \nu} - 1/4 h_{\alpha \beta}(h_{\lambda \rho} \partial_{\alpha} \partial_{\beta} h_{\lambda \rho} + 2\partial_{\alpha} h_{\lambda \rho}\partial_{\rho}h_{\beta \lambda}) + ..., \ \ (3)$

$\displaystyle \equiv R_2 + R_3 + ..., \ \ (4)$

with $\partial_{\mu} h_{\mu \nu}= 0$ and $h^{\mu}_{\mu} = 0$.

As we then see in [2], in the case $\tilde(\partial)_{\lambda} h_{\mu \nu} = 0$ it follows (2) reduces to the standard Einstein vertex. But as Tseytlin also notes, there is a contradiction in the structure of (2) owed to the presence of the minus sign. What happens is that, if $R_3(\partial)$ and $R_3 (\tilde{\partial})$ are replaced for the full Einstein scalars, the corresponding linearised equations for $h_{\mu \nu}$ contains the difference of $\partial^2$ and $\tilde{\partial}^2$ which does not match the mass-shell condition $(\partial^2 + \tilde{\partial}^2)H_{\mu \nu} = 0$. To remedy this, the full off-shell generalisation of (2) is considered

$\displaystyle S_{Eff} = \int d^D x d^D \tilde{x} \sqrt{g(x,\tilde{x})} \sqrt{\tilde{g}(x, \tilde{x})} [R(g, \partial) + R(\tilde{g}, \tilde{\partial}) + ...], \ \ (5)$

which I think is fair to say is a quite famous result. Take particular notice of the structure of this effective action. For me, I could stare at it for lengths of time; it is one of my current favourite results in the context of duality symmetric string theory and I have several thoughts about it. In fact, some of my ongoing research is focused on thinking more broadly about the geometric structure of the full 2D-dimensional space, and I think there is still quite a bit left to be said about potential insight offered in (5).

But for the interests of the present post, we want to focus on an altogether different matter: the cosmological consant. To share something else that is interesting, in [1] perhaps a lesser known about ansatz is presented for the large distance effective gravitational action based on the effective theory (5). It takes the form

$\displaystyle \bar{S} = \frac{S}{V} = \frac{\int d^D x \sqrt{g} (R + L_M)}{\int d^D x \sqrt{g}}. \ \ (6)$

What we have here is a gravity plus matter system $\bar{S}$ that is given by the standard action $S$ divided by the volume $V$ of spacetime. How to make sense of it? Much of [1] is spent arriving at (6), and so I’ll spare the details as they are quite clear in that paper. The main idea, in summary, is that from (5) in which the coordinates are doubled at the Planck scale, one can essentially integrate out the dual coordinates $\tilde{x}$ (really, the dual coordinates are treated in Kaluza-Klein fashion and as such one sees that the integral over the dual coordinates decouples) so that, as a step to arriving at (6), an action is obtained for the standard curvature scalar $R$ that includes the dual volume $\tilde{V}$ that is the inverse of the usual volume. It looks like this

$\displaystyle \hat{S} \simeq \tilde{V} \int d^D x \sqrt{g} R + ..., \ \ (7)$

with

$\displaystyle \tilde{V} = \int d^D \tilde{x} \sqrt{\tilde{g}(\tilde{x})}. \ \ (8)$

What was really clever by Tseytlin resides in how, motivated by an earlier proposal by Linde, he saw that although some mechanism to solve the CC problem at the level of the Planck scale looked unlikely, one might be able to explain why the CC looked small through some modification of the low-energy effective gravitational action using a sort of nonlocality. He saw, quite rightly, such a possibility naturally emerges within the structure of duality symmetric string theory. However, as it stands, there are issues with radiative stability in this set-up, despite some claims in the literature. This was most recently explored in relation to vacuum energy sequestering. But despite these issues, among a number of other questions, I think there could still be something in the general line of thought; hence my interests in the target space of this theory.

***

The other paper [3] I started taking notes on was by another legend, Gregory Moore. One of the issues with the CC in string theory is the contribution to it by the massless sector. One can easily see this from an analysis of the standard string. But what Moore observes is how this contribution may be cancelled by a tower of massive states, such as by using the Atkin-Lehner symmetry for instance.

Atkin-Lehner (AL) symmetry is really quite neat. It originates from number theory and the study of modular forms, but there is some suggestion and deep hints that AL symmetry is present in string theory. Admittedly, I am not deeply familiar with this topic and have merely flagged this paper as interesting for when I have some time to go back and think about the CC. But from my understanding is that, given the fact that the string path integral can be viewed as an inner product of modular forms over some moduli space, then in the case of certain backgrounds the moduli space can be seen to exhibit AL symmetry.

In short, the motivation for Moore is to look for any kind of enhanced albeit hidden symmetry (for instance, in parameter space). In the expansion of the trace for a complete set of stringy states, the one-loop path integral can be interpreted as an inner product of left and right-moving wave-functions $Z = \langle \Psi_R \vert \Psi_L \rangle$. From a stringy point of view, it is argued that the vanishing of the cosmological constant in our universe could then be interpreted from understanding why $\Psi_R$ and $\Psi_L$ are orthogonal. Naturally, Moore turns to heterotic theory. He finds that the one-loop string cosmological constant vanishes in non-trivial non-supersymmetric backgrounds when viewing the path integral as an inner product of orthogonal wave-functions.

But from what I understand, there are issues with the construction in [3], for example when applied in the case of four-dimensional spacetime. There is also another paper that I am aware of on twisted modular forms, but I have not read it. That said, I would like to understand AL better and also the issues faced in [3]. It is a very interesting paper. Given time with a return to thinking about the CC, it would be a fun to properly work through. For that reason I share it here.

References

[1] A. A. Tseytlin. Duality-Symmetric String Theory and the Cosmological-Constant Problem. Phys. Rev. Lett. 66 (1991), 545-548. doi:10.1103/PhysRevLett.66.545. url: https://inspirehep.net/literature/299973

[2] A. A. Tseytlin. Duality symmetric closed string theory and interacting chiral scalars. Nucl. Phys. B 350 (1991), 395-440. doi:10.1016/0550-3213(91)90266-Z.

[3] G. Moore. Atkin-Lehner Symmetry. Nucl. Phys. B293 (1987) 139. url: https://lib-extopc.kek.jp/preprints/PDF/1987/8705/8705249.pdf.

# 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

# Back home by the North Sea

As we usually do at the end of term, Beth and I have returned home to North Norfolk. Throughout the academic year, I greatly miss the North Sea from coast to countryside. In the past I’ve written about the coastal footpaths we’ve come to know intimately; the secret gardens and ponds off the path, with many hidden nooks etched into the landscape for a quiet picnic or some outdoor reading; the little cottage that we lived in for many years, where the clock seemed to tick slower; the sand dunes and the cliffs towering over the beaches; the rolling heathland where Shetland ponies roam free; and the poetry of tall Scotch pines seamlessly blending with farmland and rich marshes. It’s a romantic vision, I admit. It’s a land we know deeply, such that as we return each year from university we notice many small and intricate changes to the tangled ditches that line familiar country lanes; the assorted growth of wildflowers and the poppies that border the rolling fields; not to mention the forest paths we frequent where new saplings seem to be constantly breaking the soil. There are so many changes and developments in a year as the land takes new shape.

***

Aside from studying birds and other wildlife, and when not engaged in spontaneous philosophical and scientific enquiry, a reoccurring topic of discussion this week has been sociological. I haven’t taken too many pictures of the towns and villages, nor of the people, but for every eloquent description of the countryside it is interesting to also think about the sociological dimension of life here.

Beth and I spent important formative years growing up in North Norfolk, and we know first-hand how young locals might eventually yearn for more opportunities and possibilities than what can be offered by the local towns and villages. In our visit to Sheringham we spoke to a young lad at one of our favourite chip shops. He was quite sweet, working in what was probably his new summer job. When we mentioned in passing that we used to live in the area he commented in agreement that it is difficult because there aren’t many opportunities. We call this lack opportunity in rural England and the eventual forced migration ‘the brain drain’. It gives description to how young people feel they must leave rural areas to find economic opportunity, perhaps most prominently university graduates. This brain drain relates to the common, if not universal paradox in many western countries as far as I can tell: urbanisation versus rural life.

To my mind, there hasn’t been a single period of British government (left or right on the spectrum) that has successfully confronted this paradox, although some more than others have certainly exacerbated the issue over the years. In North Norfolk I think it is fair to say that for a place that is so tranquil its sociological dimension can feel restrictive and absent of future. The reality is in the juxtaposition. The beauty of the landscape and the richness of rural life on the one hand, and then the creaking public infrastructure, rural poverty, lack of opportunity and social mobility, along with lower wages combined with higher living costs on the other. Then there is the ongoing issue of the loss of small farms, the struggle of small business, and the erosion of important civic assets like access to land and well-connected transportation networks. (I imagine these issues stretch across the entire country).

In the right situation with a well-paid job nearby or with one in the nearest city; established money; the special circumstance of successfully establishing a small business; or economically viable farming (not at all an easy thing to do nowadays with many farmers selling off their businesses); one could live here in great peace and enjoy a genuine slice of heaven. Otherwise, it can easily be a desperate situation with few channels for basic work and with fewer opportunities in which one might secure a decent living. (I remember reading a sociological study where the youth of old industrial cities and mining towns offered similar accounts).

I would say East Anglia shares similar sociological features as most counties ranging north from the Midlands to the Scottish boarder. For every coastal town or rural village with a healthy economy, there is another that is quite obviously deteriorating. In the heart of North Norfolk, with its glorious scenes, quaint flint-stone cottages, and well-off neighbourhoods – of which I often speak poetically – there are also obvious signs of poverty. Towns like Holt are a good example. From the view of an outside observer Holt has the appearance of an affluent neighbourhood; but that general appearance of affluence can be misleading when it comes to the particular realities of people that call it home. Generally, a lot of the wealth does not seem to be sourced, or kept, locally. Many people have holiday homes in this area and therefore reside here only part-time or less, bringing their wealth from London and elsewhere. I think this partly contributes to the affluent appearance. Its impact is noticeable in many little details of life, like the quality of the roads, perhaps because a lot of the money is not actually taxed here; or in how the buildings on the high street are owned by only one or two companies.

I am quite sensitive to people. I like watching and observing. After a certain amount of time, it becomes easy to distinguish a local from a part-time resident or a tourist. The distinguishing features between the locals, part-time residents, and temporary holiday-goers can be much more obvious in towns like Cromer, North Walsham, and parts of King’s Lynn. When passing through Cromer it is possible to see local families wait in line at the foodbank as tourists pass with ice creams after enjoying a day on the beach. In many cases, poverty seems to be entrenched; such that it is not surprising when the Office for National Statistics releases a report indicating massive income disparities in this area. Extrapolate across the whole of England, and one finds a fairly straightforward explanation for Brexit and other recent political trends, with people in rural areas feeling abandoned or perhaps even completely forgotten. All it takes is appropriately directed political rhetoric to exploit the underlying currents of resentment and frustration.

But that’s enough observation and reflection for now. My main purpose was to share some photographs of the countryside. Perhaps next time I will focus more on writing about the many different towns and villages, with their unique histories and characteristics, and about the people that live in this area. I can then also focus on taking more pictures from that perspective.

***

The most popular castle in Norfolk is Norwich Castle. Outside of Norwich in rural Norfolk there were over 20 other castles built during the Middle Ages. The other day we returned to Baconsthorpe Castle, which has a very interesting history. I hadn’t been here for almost 10 years, so it was lovely to see it again.

The main court yard would have been full with residences and stables. You can see a bit of it behind me, although much of the structure has been lost over centuries.

The castle is situated next to a pond. There were lots of baby ducks and baby swans swimming in it, but we didn’t manage to get a good picture. Here is one photograph of the pond facing away from the castle’s courtyard.

The lane to get to the castle cuts through a number of fields. It makes for a painting in-itself. Quintessential Norfolk.

This picture reminds me a lot of the family dairy farm. It was nice to visit. There wasn’t time to see the cows on this occasion, but I did take some photos of the fields that Beth and I used to walk through.

We had lunch in the orchard under one of the apple trees.

Weybourne beach is a place we like to have picnic dinners. Watching the seabirds and the fishermen is one of my favourite pastimes. It’s a place where many locals will spend the evening with a comfortable chair, a good book, and a bottle of wine.

The sea was especially calm this day. We managed to capture a few lovely photos.

Beth and I will sometimes search the shingle beaches for cool pebbles and cobbles. I’ve been experimenting with taking more focused photographs, and this is one of the cobbles that I think looks good.

One of our favourite forests in North Norfolk is captured below. I’ve written about this forest a few times in the past. It is one of the first places we visit whenever we return to Holt. We used to go for walks here a few times a week, and it is always interesting to compare the growth over the years.

Here I am walking down one of my favourite footpaths that runs under a canopy of trees. On the other side of the gate is an entrance to the heath.

Below are some pictures of the heath, which has changed dramatically in the last 10 years. We used to be able to see very far down into the valley, but the goss has grown so much that this is no longer possible. In recent visits we’ve also spotted more garden snakes than usual (in fact, we never used to see any at all!), suggesting perhaps that their population has grown. I am curious as to why. Maybe there has been a decline in the population of birds of prey?

Here we are having a sit down.

Beth did take one image of Sheringham high street on our way to the beach beneath the cliffs.

Here is a photo of the beach and the cliffs.

Sheringham Golf Club is situated just a top these cliffs. It’s a beautiful links course that I was looking forward to play if not for an injury.

One of the most beautiful places on planet Earth, as far as I am concerned, is Holkham beach.

Here is a tile gallery of a few more of my favourite pictures.

It was especially hot one day we were there, so we receded to the forest. The tall Scotch pines provided lots of shade. The treeline also made for a nice photo.

Finally, a picture of Beth and I having a stop at a countrypark. We’ve never been here before, and I was excited to learn that it had a fresh water lake.

# Double Field Theory as the double copy of Yang-Mills

1. Introduction

A few weeks ago I came across this paper [DHP] on Double Field Theory and the double copy of Yang-Mills. Its result is most curious.

As a matter of introduction, recall how fundamental interactions in nature are governed by two kinds of theories: On the one hand, Einstein’s theory of relativity. On the other hand we have Yang-Mills theory, which provides a description of the gauge bosons of the standard model of particle physics. Yang-Mills is one example of gauge theory; however, not all gauge theories must necessarily be of Yang-Mills form. In a very broad picture view, gravity is also a gauge theory. This can be most easily seen in the diffeomorphism group symmetry.

Of course, Yang-Mills is the best quantum field theory that we have; it yields remarkable simplicity and is at the heart of the unification of the electromagnetic force and weak forces as well as the theory of the strong force, i.e., quantum chromodynamics. Similarly one might think that, given gravity is an incredibly symmetric theory, it should also yield a beautiful QFT. It doesn’t. When doing perturbation theory, even at quadratic order things already start to get hairy; but then at cubic and quartic order the theory is so complicated that attempting to do calculations with the interaction vertices becomes nightmarish. So instead of a beautiful QFT, what we actually find is incredibly complicated.

In this precise sense, on a quantum level there is quite an old juxtaposition between gauge theory in the sense of Yang-Mills (nice and simple) versus gravity (a hot mess). In other parts, the two can be seen to be quite close (at least we have have a lot of hints that they are close). Indeed, putting aside gauge formulations of gravity, even simply under the gauge theory of Lorentz symmetries we can start to draw a comparison between gravity and Yang-Mills, and this has been the case since at least the 1970s. Around a similar time gauge theory of super Poincare symmetries produced another collection of hints. And, one of the most important examples without question is the holographic principle and the AdS/CFT correspondence.

Yet another highly fruitful way to drill down into gauge-gravity, especially over the last decade, has followed the important work of Bern-Carrasco-Johansson in [BCJ1] and [BCJ2]. Here, a remarkable observation is made: gravity scattering amplitudes can be seen as the exact double copy of Yang-Mills amplitudes, suggesting even further a deeply formal and profoundly intimate relationship between gauge theory and gravity.

Schematically put, following the double copy technique it is observed that gravity = gauge x gauge. This leads to the somewhat misleading statement that gravity is gauge theory squared.

A lot goes back to the KLT relations of string theory. The general idea of the double copy method is that, from within perturbation theory, Yang-Mills (and gauge theories in general) can be appropriately constructed so that their building blocks obey a property known as color-kinematics duality. (This is, in itself, a fascinating property worthy of more discussion in the future. To somewhat foreshadow what is to come, there were already suspicions in the early 1990s that it may relate to T-duality, which one will recall is a fundamental symmetry of the string). Simply put, this is a duality between color and kinematics for gauge theories leaving the amplitudes unaltered.

For instance, to understand the relation between gravity and gauge theory amplitudes at tree-level, we can consider a gauge theory amplitude where all particles are in the adjoint color representation. So if we take pure Yang-Mills

$\displaystyle S_{YM} = \frac{1}{g^2} \int \text{Tr} F \wedge \star F \ \ (1)$

there is an organisation of the n-point L-loop gluon amplitude in terms of only cubic diagrams

$\displaystyle \mathcal{A}_{YM}^{n,L} = \sum \limits_i \frac{c_i n_i}{S_i d_i}, \ \ (2)$

where ${c_i}$ are the colour factors, ${n_i}$ the kinetic numerical factors, and ${d_i}$ the propagator. Then the color-kinematic duality states that, given some choice of numerators, such that if those numerators are known, it is required there exists a transformation from any valid representation to one where the numerators satisfy equations in one-to-one correspondence with the Jacobi identity of the color factors,

$\displaystyle c_i + c_j + c_k = 0 \Rightarrow n_i + n_j + n_k = 0$

$\displaystyle c_i \rightarrow -c_i \Rightarrow n_i \rightarrow -n_i. \ \ (3)$

So, as the kinematic numerators satisfy the same Jacobi identities as the structure constants do, for some choice of numerators (from what I understand the choice is not unique), we can obtain the gravity amplitude. For example, given double copy ${c_i \rightarrow n_i}$ it is possible to obtain an amplitude of ${\mathcal{N} = 0}$ supergravity

$\displaystyle \mathcal{A}_{\mathcal{N}=0}^{n,L} = \sum \limits_i \frac{n_1 n_i}{S_i d_i}, \ \ (4)$

where one will notice in the numerator that we’ve striped off the colour and replaced with kinematics, and where the supergravity action is

$\displaystyle S_{\mathcal{N}=0} = \frac{1}{2\kappa^2} \int \star R - \frac{1}{d-2} d\psi \wedge \star d\psi - \frac{1}{2} \exp(- \frac{4}{d-2}\psi) dB \wedge \star dB. \ \ (5)$

In summary, the colour factors that contribute in the gauge theory appear on equal footing as the purely kinematical numerator factors (functions of momenta and polarizations), and all the while the Jacobi identities are satisfied. When all is said and done, the hot mess of a QFT in the gravity theory can be related to the nicest QFT in terms of Yang-Mills.

But notice that none of what has been said has anything to do with a description of physics at the level of the Lagrangian. For a long time, some attempts were made but there was no reason to think the double copy method should work at the level of an action. As stated in [Nico]: ‘no amount of fiddling with the Einstein-Hilbert action will reduce it to a square of a Yang-Mills action.’ Although many attempts have been made, with some notable results, this question of applying the double copy method on the level of the action takes us to [DHP].

In this paper, the authors use the double copy techniques to replace colour factors with a second set of kinematic factors, which come with their own momenta, and it ultimately leads to a double field theory (see past posts for discussion on DFT) with doubled momenta or, in position space, a doubled set of coordinates. In other words, the double copy of Yang-Mills theory (at the level of the action) yields at quadratic and cubic order double field theory upon integrating out the duality invariant dilaton.

When I first read this paper, the result of obtaining the background independent DFT action was astounding to me. In what follows, I want to quickly review the calculation (we’ll only consider the quadratic action, where the Lagrangian remains gauge invariant).

2. Yang-Mills / DFT – Quadratic theory

$\displaystyle S_{YM} = -\frac{1}{4} \int \ d^Dx \ \kappa_{ab} F^{\mu \nu a} F_{\mu \nu}^{b}, \ \ (6)$

with the field strength for the gauge bosons ${A_{\mu}^{a}}$ defined as

$\displaystyle F_{\mu \nu}^{a} = \partial_{\mu} A^{a}_{\nu} - \partial_{\nu}A^{a}_{\mu} + g_{YM} f^{a}_{bc}A_{\mu}^{b} A_{\nu}^{c}. \ \ (7)$

Here ${g_{YM}}$ is the usual gauge coupling. The ${f^{a}_{bc}}$ term denotes the structure constants of a compact Lie group (i.e., in this case a non-Abelian gauge group). This group represents the color gauge group, and we define ${a,b,...}$ as adjoint indices. The invariant Cartan-Killing form ${\kappa_{ab}}$ lowers the adjoint indices such that ${f_{abc} \equiv \kappa_{ad}f^d_{bc}}$ is antisymmetric.

Expanding the action (3) to quadratic order in ${A^{\mu}}$ and then integrating by parts we find

$\displaystyle -\frac{1}{4} \int d^{D}x \ \kappa_{ab} \ (-2 \Box A^{\mu a} A_{\mu}^{b} + \partial_{\mu}\partial^{\nu} A^{\mu a}A_{\nu}^b). \ \ (8)$

Pulling out ${A^{\mu a}}$ and the factor of 2, we obtain the second-order action as given in [DHP]

$\displaystyle S_{YM}^{(2)} = \frac{1}{2} \int d^{D}x \ \kappa_{ab} \ A^{\mu a}(\Box A^{b}_{\mu} - \partial_{\mu} \partial^{\nu} A^b_{\nu}). \ \ (9)$

To make contact with the double copy formalism, we next move to momentum space with momenta ${k}$. Define ${A^{a}_{\mu}(k) = 1/(2\pi)^{D/2} \int d^D x \ A_{\mu}^{a}(x) \exp(ikx)}$. In these notes we use the shorthand ${\int_k := \int d^{D} k}$. In [DHP], the convention is used where ${k^2}$ is scaled out, which then allows us to define the following projector

$\displaystyle \Pi^{\mu \nu}(k) \equiv \eta^{\mu \nu} - \frac{k^{\mu} k^{\nu}}{k^2}, \ \ (10)$

where we have the Minkowski metric ${\eta_{\mu \nu} = (-,+,+,+)}$.

Proposition 1 The projector defined in (10) satisfies the identities

$\displaystyle \Pi^{\mu \nu}(k)k_{\nu} \equiv 0, \ \text{and} \ \Pi^{\mu \nu}\Pi_{\nu \rho} = \Pi^{\mu}_{\rho}. \ \ (11)$

Proof: The second identity is trivial, while the first identity can be found substituting (10) in (11) and recalling we’ve scaled out ${k^2}$. $\Box$

The first identity in (11) implies gauge invariance under the transformation

$\displaystyle \delta A^{a}_{\mu}(k) = k_{\mu}\lambda^a(k), \ \ (12)$

where the gauge parameter ${\lambda^a(k)}$ is defined as an arbitrary function.

3. Double copy of gravity theory

Proposition 2 The double copy prescription of gravity theory leads to double field theory.

Proof: Begin by replacing the color indices ${a}$ by a second set of spacetime indices ${a \rightarrow \bar{\mu}}$. This second set of spacetime indices then corresponds to a second set of spacetime momenta ${\bar{k}^{\bar{\mu}}}$. For the fields ${A^a_{\mu}(k)}$ in momentum space, we define a new doubled field

$\displaystyle A^a_{\mu}(k) \rightarrow e_{\mu \bar{\mu}}(k, \bar{k}). \ \ (13)$

Next, following the double copy formalism, a substitution rule for the Cartan-Killing metric ${\kappa_{ab}}$ needs to be defined. In [DHP], the authors propose that we replace this metric with a projector carrying barred indices such that

$\displaystyle \kappa_{ab} \rightarrow \frac{1}{2} \bar{\Pi}^{\bar{\mu} \bar{\nu}}(\bar{k}). \ \ (14)$

Notice, this expression exists entirely in the barred space.

Remark 1 (Argument for why (14) is correct) It is argued that the replacement (14) is derived from the double copy rule at the level of amplitudes. Schematically, one can consider a gauge theory amplitude of the form ${\mathcal{A} = \Sigma_i n_i c_i / D_i}$, where ${n_i}$ are kinematic factors, ${c_i}$ are colour factors, and ${D_i}$ denote inverse propagators. Then, in the double copy, replace ${c_i}$ by ${n_i}$ with ${D_i \sim k^2}$. This means that ${k^2}$ may be scaled out as before, leaving only the propagator to be doubled.

Making the appropriate substitutions, we obtain a double copy action for gravity of the form

$\displaystyle S_{grav}^{(2)} = - \frac{1}{4} \int_{k, \bar{k}} \ k^2 \ \Pi^{\mu \nu}(k) \bar{\Pi}^{\bar{\mu}\bar{\nu}}(\bar{k}) \ e_{\mu \bar{\mu}}(-k, -\bar{k})e_{\nu \bar{\nu}}(k, \bar{k}). \ \ (15)$

The structure of this action is really quite nice; in some ways, it is what one might expect as it is very reminiscent of the structure of the duality symmetric string.

To make the doubled nature of the action (15) more explicit, define doubled momenta ${K = (k, \bar{k})}$, and, just as the duality symmetric string, treat ${k, \bar{k}}$ on equal footing. It now seems arbitrary whether there is ${k^2}$ or ${\bar{k}^2}$ at the front of the integrand. In any case, unlike the measure factor for the duality symmetric string which, in momentum space, takes the form ${k, \tilde{k}}$, the asymmetry of (15) is resolved by imposing

$\displaystyle k^2 = \bar{k}^2, \ \ (16)$

which one might notice is just the level-matching condition. To obtain DFT, the imposition of this constraint is necessary (indeed, just like it is in pure DFT).

Remark 2 (More general solutions) The solution ${k = \bar{k}}$ should be familiar from studying the linearised theory. However, here exists more general solutions and it might be interesting to think more about this matter.

It is fairly straightforward to see that under

$\displaystyle \delta e_{\mu \bar{\nu}} = k_{\mu}\bar{\lambda}_{\bar{\nu}} + \bar{k}_{}\bar{\nu}\lambda_{\mu} \ \ (17)$

the action (15) is invariant. Now we have two gauge parameters dependent on doubled momenta.

Upon writing out the projectors (11) and then imposing the level-matching condition (16), we can use the metric to lower indices. Then taking the product with the ${e}$ fields, we find the action (15) to take the following form:

$\displaystyle S_{grav}^{(2)} = -\frac{1}{4} \int \ \int_{k, \bar{k}} (k^{2}e^{\mu \bar{\nu}}e_{\mu \bar{\nu}} - k^{\mu}k^{\rho}e_{\mu \bar{\nu}}e^{\bar{\nu}}_{\rho} - \bar{k}^{\bar{\nu}}\bar{k}^{\bar{\sigma}}e_{\mu \bar{\nu}}e^{\mu}_{\bar{\sigma}} + \frac{1}{k^2}k^{\mu}k^{\rho}\bar{k}^{\bar{\nu}}\bar{k}^{\bar{\sigma}}e_{\mu \bar{\nu}}e_{\rho \bar{\sigma}}). \ \ (18)$

Already one can see this looks very similar to the background independent quadratic action of DFT. To get a better comparison, we can Fourier transform to doubled position space. In doing so, it is observed that every term transforms without a problem except the last term which results in a non-local piece. The trick, as noted in [DHP], is to introduce an auxiliary scalar field ${\phi(k, \bar{k})}$ (i.e., the dilaton).

Doing these steps means we can first rewrite (18) as follows

$\displaystyle S_{grav}^{(2)} = -\frac{1}{4} \int \ \int_{k, \bar{k}} (k^{2}e^{\mu \bar{\nu}}e_{\mu \bar{\nu}} - k^{\mu}k^{\rho}e_{\mu \bar{\nu}}e^{\bar{\nu}}_{\rho} - \bar{k}^{\bar{\nu}}\bar{k}^{\bar{\sigma}}e_{\mu \bar{\nu}}e^{\mu}_{\bar{\sigma}} - k^2 \phi^2 + 2\phi k^{\mu}\bar{k}^{\bar{\nu}}e_{\mu \bar{\nu}}). \ \ (19)$

By using the field equations for ${\phi}$

$\displaystyle \phi = \frac{1}{k^2} k^{\mu}\bar{k}^{\bar{\nu}}e_{\mu \bar{\nu}} \ \ (20)$

or, alternatively, using the redefinition

$\displaystyle \phi \rightarrow \phi^{\prime} = \phi - \frac{1}{k^2} k^{\mu}\bar{k}^{\bar{\nu}}e_{\mu \bar{\nu}} \ \ (21)$

we then get back the non-local action (18).

Remark 3 (Maintaining gauge invariance) What’s nice is that (19) is still gauge invariant, which can be checked using also the gauge transformation for the dilaton ${\delta \phi = k_{\mu}\lambda^{\mu} + \bar{k}_{\bar{\mu}}\bar{\lambda}^{\bar{\mu}}}$.

Now Fourier transforming (19) to doubled position space, we define in the standard way ${\partial_{\mu} / \partial x^{\mu}}$ and ${\bar{\partial}_{\bar{\mu}} = \partial / \partial \bar{x}^{\bar{\mu}}}$. We also of course obtain the usual duality invariant measure. The resulting action takes the form

$\displaystyle S_{grav}^{(2)} = \frac{1}{4} \int d^D x \ d^D \bar{x} \ (e^{\mu \bar{\nu}}\Box e_{\mu \bar{\nu}} + \partial^{\mu}e_{\mu \bar{\nu}}\partial^{\rho}e_{\rho}^{\bar{\nu}}$

$\displaystyle + \bar{\partial}^{\bar{\nu}}e_{\mu \bar{\nu}}\bar{\partial}^{\bar{\sigma}}e^{\mu}_{\bar{\sigma}} - \phi \Box \phi + 2\phi \partial^{\mu}\bar{\partial}^{\bar{\nu}}e_{\mu \bar{\nu}}. \ \ (22)$

$\Box$

This is the standard quadratic double field theory action. As such, it maintains gauge invariance – notice, we haven’t had to impose a gauge condition and the only extra field introduced was the dilaton.

Very cool.

References

[BCJ1] Z. Bern, J.J. M. Carrasco, and H. Johansson, New Relations for Gauge-Theory Amplitudes. [0805.3993 [hep-ph]].

[BCJ2] Z. Bern, J.J. M. Carrasco, and H. Johansson, Perturbative Quantum Gravity as a Double Copy of Gauge Theory. [1004.0476 [hep-th]].

[BJH] R. Bonezzi, F. Diaz-Jaramillo, O. Hohm, The Gauge Structure of Double Field Theory follows from Yang-Mills Theory. [2203.07397 [hep-th]]

[DHP] F. Dıaz-Jaramillo, O. Hohm, and J. Plefka, Double Field Theory as the Double Copy of Yang-Mills. [2109.01153 [hep-th]].

[Nico] H. Nicolai, “From Grassmann to maximal (N=8) supergravity,” Annalen Phys. 19, 150–160 (2010).

*Cover image: Z. Bern lecture notes, Gravity as a Double Copy of Gauge Theory.