# 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.

# Doubled diffeomorphisms and the generalised Ricci curvature

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]].

# Start of new semester, thinking about double field theory cosmology

I haven’t added much to my blog in the past weeks. With university kicking off again, and with Tony and I having our first work sessions of the semester, it has been quite busy. I’ve also been adjusting to being back at university after summer holiday, and with being back on campus for the first time since lock down due to the pandemic. So I’ve been finding my feet again with new daily structure and routine.

I’ve also been working on a number of projects, some short-term and some long-term, which have kept me quite occupied. It is the battle of constantly balancing enticing questions and ideas that define the day. It’s what makes life exciting and keeps me coming back to physics, I suppose.

In the last week or so we’ve been talking more about double field theory cosmology, mainly from the perspective of how matter couples. As a developing area of research there are many interesting questions one can ask. It’s quite interesting stuff, to be honest, and I’m looking forward to potentially pursuing a few side projects in this area. As it relates, I’m interested in higher ${\alpha^{\prime}}$ corrections, non-perturbative solutions, and ${\alpha^{\prime}}$ deformed geometric structures.

To share a bit more, one thing that is quite neat about DFT cosmology is how, under a cosmological ansatz [1,2], the equations coupled to matter take the form

$\displaystyle 4d^{\prime \prime} - 4(d^{\prime})^2 - (D-1)\tilde{H}^2 + 4\ddot{d} - 4 \dot{d}^2 - (D - 1)H^2 = 0$

$\displaystyle (D - 1)\tilde{H}^2 - 2 d^{\prime \prime} - (D - 1)H^2 + 2\ddot{d} = \frac{1}{2}e^{2d} E$

$\displaystyle \tilde{H}^{\prime} - 2\tilde{H}d^{\prime} + \dot{H} - 2h\dot{d} = \frac{1}{2} e^{2d}P. \\ (1)$

Here ${E}$ and ${P}$ denote energy density and pressure, respectively. These equations are duality invariant provided ${E \leftrightarrow -E}$ and ${P \leftrightarrow -P}$. The approaches that make use of these equations are typically restricted to dilaton gravity. That is to say, the B-field is switched off. From what I presently understand the reason for this is because it is generally unknown how proceed with the full massless string sector explicit in the theory.

For a homogeneous and isotropic cosmology the metric takes the form

$\displaystyle dS^2 = -dt^2 + \mathcal{H}_{MN} dx^M dx^N$

$\displaystyle = -dt^2 + a^2(t) dx^2 + a^{-2}(t) d\tilde{x}, \ \ (2)$

where ${t}$ is physical time, ${a(t)}$ is the cosmological scale factor, ${x}$ denote are co-moving spatial coordinates. In general, the basic fields reduce to the cosmological scale factor ${a(t, \tilde{t})}$ and the dilaton ${\phi(t, \tilde{t})}$.

Most pertinently, as we are dealing with a manifestly T-duality invariant theory, what one finds is that T-duality results in scale factor duality. In some ways, this is expected. With the B-field off, the background fields transform

$\displaystyle a(t, \tilde{t}) \rightarrow \frac{1}{a(\tilde{t},t)},$

$\displaystyle \phi(t, \tilde{t}) \rightarrow \phi(\tilde{t}, t). \ \ (3)$

The T-duality invariant combination of the scale factor and the dilaton is

$\displaystyle \phi \equiv \phi - d\ln a, \ \ (4)$

where ${d = D-1}$ is the number of spatial dimensions with D space-time dimensions.

It will be interesting to read more about the work that has so far been done in this area. One thing that is very clear, the approaches to DFT cosmology that I have so far looked at ultimately go back to Tseytlin and Vafa [3], and, also, of course, to efforts in string gas cosmology.

The main thing about these types of approaches behind (1) is that, rather than using T-duality variables, they leverage T-duality frames. The assumption, again, is the use of the section condition (conventional in DFT), which states the fields only depend on a D-dimensional subset of the space-time variables. We’ve talked about this in the past on this blog. There are different, often arbitrary choices, of this condition – what we call frames – and these different frames are related by T-duality.

The most basic example is the supergravity frame with standard coordinates transformed to the winding frame with dual coordinates. And so, what one can do, is calculate supergravity and winding frame solutions of the cosmological equations (1), with these solutions being T-dual to each other [4].

In review of ongoing efforts, it will be interesting to see what ideas might arise in the coming weeks.

References

[1] H. Wu and H. Yang, Double Field Theory Inspired Cosmology. JCAP 1407, 024 (2014) doi:10.1088/1475- 7516/2014/07/024 [arXiv:1307.0159 [hep-th]].

[2] R. Brandenberger, R. Costa, G. Franzmann and A. Welt- man, T-dual cosmological solutions in double field theory. [arXiv:1809.03482 [hep-th]].

[3] A. A. Tseytlin and C. Vafa, Elements of string cosmol- ogy. Nucl. Phys. B 372, 443 (1992) doi:10.1016/0550- 3213(92)90327-8 [hep-th/9109048].

[4] H. Bernardo, R. Brandenberger, G. Franzmann, T-Dual Cosmological Solutions of Double Field Theory II. [ arXiv:1901.01209v1 [hep-th]].

# (n-1)-thoughts, n=4: A return to the North Sea, new string papers, and Strings 2021

Beth and I frequently talk about how we miss the North Sea. We lived on the coast and I think it is our nature that we both prefer its unique countryside. But now that we’re living in East Midlands, landlocked and busy at university, we haven’t been back for a couple of years. So for our summer holiday we’ve travelled to North Norfolk, a place that for many reasons became an adopted home for both of us, to smell the sea again and enjoy the beautiful sights.

It may perhaps sound a bit mawkish, but in many ways North Norfolk provided an opening for discovery, not just in a literal sense but also in a philosophical sense as a space to reflect on the world of ideas. There is a line by John Berger that speaks of a place from which the world can be discovered – that is, a foundation from which one can venture forward but always return if needed. I think this is what the North Norfolk countryside came to represent for us: at the time a much needed place of peace and calm, but also a place of thought and reflection, where time runs slow and where we could gather ourselves, find our footing in life, and sometimes spend entire afternoons contemplating life. If for Plato and Socrates our true home is the eternal world of ideas, the North Sea, with the calm and quiet nature of the hills, valleys, woodlands, and beaches unfolding alongside it, is the continuum from which philosophy and mathematical realism can be pursued. Its the total landscape, the geography, and the open horizon that is grounding.

The little cottage where we used to live, just off the main road which the Romans would have similarly ventured when travelling to the coastal towns, was our first proper home. It was maybe the first place where we both found lasting comfort, coming from difficult situations and experiences. It was modest accommodation – a flintstone cottage, with an old fireplace, pinewood shelves, and steep narrow stairs sharply turning from the kitchen at the back up to the bedroom facing the main road. It was a tiny dwelling, perhaps a bit too cramped at times. But with our library of books, regular philosophical discussion, and no shortage of slow days of reflection, maths, and hobby, it was the perfect place at the right time. On the most difficult days, we could just listen to the birds in the back garden, find peace in our thoughts, or write to our contentment.

The main town, Holt, the origin derived from the Anglo-Saxon word for ‘wood’, is a place defined very much by its surrounding line of trees and small forest areas. It is situated on a hill, and, although a couple miles from the coast, a fresh sweeping wind from the North Sea can often be felt. Out here, the air is fresh. Space is wide and open.

One of my favourite places was, and remains to be, the woodland and heath not far from where we lived. It’s not marked on the map, so it is truly a hidden place to be discovered. Beth and I would take regular walks and spend many afternoons sitting among the heather – purple flowers of Calluna vulgaris – and yellow flowers of Ulex gallii. One year, I remember wild ponies were introduced to help maintain the land, and often we would see them hanging out by the brook or underneath a tree on a hot summer day. Upon our return this week, it was a joy to visit this place again.

I’m not very good at creative writing, although I’ve tried to learn and experiment with it. One thing that we used to do is, during our walks in the heath or through the many wonderful nature reserves, we would treat them like field expeditions. And, like a keen biologist or natural scientist, we would take our time for the finer inspection of all species of plant and animal. I would practice writing in my journal in an observational and documentative way. Being back here, I was inspired to dig up one such piece of writing. At the time I was experimenting with phenomenological-style notetaking, trying to intricately describe my experience of the countryside. I think I also took inspiration from the structure of a number of writings by various authors that I was reading back then.

To the left of me dense thicket carries into the distance, a rolling plain of healthy evergreen and intricate pathways of rotting needle, brown and dank, align in close order row upon row upon row. To the right, at the nearest edge of the forest, golden rays of sunlight spray across an open valley. Its radiating warmth and all-consuming light illuminate the young grasses breaking into the soil, filling the land with a rich, unfolding spectacle of colour. Looking beyond these trees, it is readily noticeable that there is an abundance of wildflower and bracken, a diverse quality of dazzling tones and subjects, which harmonize in a single unified phenomenal pallet like one massive, entanglement of earth. Ramshackle and unexpected, diverse and revealing, from endless rills and rivulets, from ditches to dells, from hedgerows to underbrush, with each new experience pressing deeper into this landscape and, ever from the background of this vast horizon of rolling hills before me, entire swells of breathing life continuing to reveal themselves. From the rabbits and wood pigeons who rather be hidden; the swaying fields are a thoroughfare for creatures of various kinds, from field mice to deer and the odd passing fox; song thrush, jays, long-tailed tits, and spotted woodpeckers - what grows and lives in this place is truly possessed of a beauty all its own.

As for the very belly of the forest, off the heath, there is a rich vision of evergreen, each swirling pine looks entirely similar to the other. But upon closer inspection we see that each particular pine tree is distinguishable and, indeed, of unique character. The pine itself is of course home to many things. Birds, insects, a peering squirrel; all find comfort in these dense woodlands. But row upon row, with its dwarf shoots that spiral from off the axils of scaly bracts, such a dense growth of pine, whose intricate branches are like a massive conic arrangement of narrow needles bundled together by both bark and sap, is a marvel in itself.  Occasionally stepping on fallen seed or the coned fruit, my senses are overwhelmed by the spatially sweet and particular fragrance that lingers throughout the air.


The countryside here is in many ways a place of Tolkein description. It has been nice sitting again by the cliffs and walking through the overgrown footpaths. As I tried to capture, it is Shire-like in its beauty. With its salt marshes; winding roads lined by hedges, wild flowers, sedges, and rushes; and rolling hills demarcated by broad leaved woodland, towering at times with veteran oaks, birch, and, my favourite, Scotch pines – there is so much to be admired in this part of the country. Down by the sea, fishermen sit with lines cast, birds circling overhead. Again, it is perhaps more than a bit mawkish, but it is for me one of the places in our beautiful country that speaks a bit to old Romanticism, with every brook and winding turn outlined by hedgerows evoking a scene from a classic Keats poem.

It is my nature to be reclusive. No doubt, there are many other reasons why I find home by the sea and in the countryside. But as I write from the cottage where we’re staying, I remember why North Norfolk represents more than a place of stillness and beauty. With a cup of tea and some maths by the window, in the quiet thoughtfulness, the appearance and seeming order of the world of phenomena, mental idealisations or not, rushes forth some profound reality.

New string papers

As I was preparing to leave for holiday, three papers appeared of significant interest. I haven’t had a chance to work through them all yet, between being strict with my holiday time and with String 2021 ongoing, but I felt motivated over a cup of tea to take note:

Heterotic duels of M-theory

A nice paper by Bobby Samir Acharya,  Alex Kinsella, and David R. Morrison on the non-perturbative heterotic duels of M-theory was released. This is of particular interest to me as it relates to the wider study of the non-perturbative aspects of M/heterotic duality.

This duality was discovered in the mid 90’s in which one can take M-theory compactified on a $K3$ and find it relates to the $E8 \times E8$ heterotic theory compactified on a three-torus. When you look at the 4D picture, we may instead compactify M-theory on a $G2$ manifold (equipped with a K3 fibration), which is a seven-dimensional Riemannian manifold that is special because it comes with the holonomy group in the exceptional simple Lie group $G_2$. For the $E8 \times E8$, it gets compactified on a Calabi-Yau threefold equipped with a three-torus. I haven’t had a chance to read through and consider the paper in any great detail, but it is noticeable that it starts with a similar set-up, taking low-energy M-theory with $G2$ orbifolds as the choice of compactification, with choice of equipped K3-fiberation to enable comparison with the dual heterotic string spectrum. A key observation, I take it, is that for the heterotic background there is a subtlty with the gauge bundle on $T^3$ such that, when it comes to the non-perturbative physics, there are point-like instantons on orbifold points of the geometry. This is where things get both interesting and complicated, and I’m not sure in what way these instanton effects in the spectrum relate to M-branes. I am keen to read the second half of the study.

Higgs mass in string theory

Another paper that appeared looks at calculating the Higgs mass. It’s by Steven Abel and Keith R. Dienes. This paper is quite the joy, and I’m sure anyone with interest in string theory will enjoy it over a cup of tea. Abel and Dienes harnesses the powers of the world-sheet theory to perform some proper stringy calculations, developing a framework that presents a relationship between the Higgs mass and the cosmological constant. What is neat about the computation is that this connection is generic for all closed string theories and provides a bit of a platform for future studies on gauge hierarchy problems.

Double sigma models and geometric quantisation

With a rush of papers leading up to my holiday, this one immediately caught my attention and got me excited. Luigi Alfonsi and David Berman study geometric quantisation in double field theory and double sigma models. From what I have seen, it is grand.

I was actively thinking about quantisation of double sigma models, as this is one area in which I have been working. In fact, I recall a few discussions a year or more ago about a project looking into the quantisation of the doubled string. In parts, from working in the area, what we see in this paper is kind of what one would expect in that, to start, the zero-mode sector for the closed string is intrinsically non-commutative. This alone is an interesting fact with some deep implications. Commonly, in the set-up where the target-space is treated as a phase-space, one will also equip a symplectic form $\omega$, and one will can construct a theory with an action following Tseytin (we talked about this in a past post). What is found with the inclusion of $\omega$ is an interesting connection with Born geometry (maybe I’ll write about this in a future post) and, furthermore, one will often find discussion on symplectic structures as it relates to Poisson geometry which has some deep relation with T-duality.

In short, in the quantisation procedure there is a choice of polarisation, and the authors want to make a choice of polarisation in conjunction with the strategy for geometric quantisation. What happens, in any case, is that T-duality will give polarisations. And then what one wants to study is the noncommutative algebra associated to the doubled phase space. What the paper shows is that there are, in essence, two types of quantisations going on, because there is one coming from the usual phase space and then another from the duality frame (i.e., what in the formalism is understood in terms of the Lagrangian submanifold).

A deeper idea here has to do with the doubled phase space and para-Hermitean geometry, which I think I’ve mentioned a wee bit in the past. On that note, it is also interesting to think about the findings in this paper as it relates to the idea of metastring theory and quantisation.

As an aside, I’ve been working on a draft essay about a series of papers by Luigi. I wanted to write a bit about double sigma models and double field theory before finishing this essay, with a mind toward giving the reader some reference. They are fantastic papers on the global double space of double field theory, among other things. I also have Luigi’s PhD thesis on hand, which I think is great. There is a lot to discussed here in the context of the doubled geometry of double sigma models and higher structures.

Strings 2021

The annual string conference, Strings 2021, is ongoing (21 June – 2 July). It’s always an event that I look forward to, as it brings together the entire string theory community. Among a large list of great and usual names, my eye immediate caught an anomalous speaker amongst the expected and anticipated: namely, Roger Penrose. I will be most eager to hear what he has to say during his presentation on Friday 2, July. The topic is on gravitational singularities. There are of course a number of talks that I am looking forward to – too many to list! For now, here is the schedule with list of speakers, including links to notes and recordings. If I find the time and motivation, I’ll write a summary of my favourite talks next week.

# O(D,D) and Double Field Theory

1. Introduction

In continuation of a past entry, this week I was intending to write more about double sigma models. I wanted to offer several further remarks on the intrinsic aspects of the doubled world-sheet formalism, and also give the reader a sense of direction when it comes to interesting questions about the geometry of the doubled string.

However, I realised that I have yet to share on this blog many of my notes on Double Field Theory (DFT). We’ve talked a bit about the Courant Bracket and the strong constraint and, in a recent post, we covered a review of Tseytlin’s formulation of the duality symmetric string for interacting chiral bosons that relates to the formulation of DFT. But, as a whole, it would be useful to discuss more about the latter before we continue with the study of double sigma models. There is a wonderfully deep connection between two, with a lot of the notation and concepts employed in the former utilised in the latter, and eventually a lot of concepts become quite interrelated.

We’ll start with some basics about DFT, focusing particularly on the T-duality group ${O(d,d)}$ and the generalised metric formulation. In a later entry, we’ll deepen the discussion with gauge transformations of the generalised metric; generalised Lie derivatives; Courant brackets, generalised Lie brackets, and Dorfman brackets; among other things. The endgame for my notes primarily focuses on the generalised Ricci and the question of DFT’s geometric constitution, which we will also discuss another time.

For the engaged reader interested in working through the seminal papers of Zwiebach, Hull, and Hohm, see [1,2,3,4].

2. What is ${O(d,d)}$?

As we’ve discussed in other places, DFT was formulated with the purpose of incorporating target space duality (T-duality) in way that is manifest on the level of the action. One will recall that, in our review of the duality symmetric string, the same motivation was present from the outset. I won’t discuss T-duality in much depth here, instead see past posts or review Chapter 8 in Polchinski [5]. The main thing to remember, or take note of, is how T-duality is encoded in the transformations $R \leftrightarrow \frac{l_s}{R}$, $p \leftrightarrow w$, which describe an equivalence between radius and inverse radius, with the exchange of momentum modes ${p}$ and the intrinsically stringy winding modes ${w}$ in closed string theory, or in the case of the open string an exchange of Dirichlet and Neumann boundary conditions. More technically, we have an automorphism of conformal field theory. In the case of compactifying on $S^1$ for example, as momentum and winding are exchanged, the coordinates ${x}$ on ${S^1}$ are exchanged with the dual ${S^1}$ coordinates $\tilde{x}$.

When T-duality is explicit we have for the mass operator,

$\displaystyle M^2 = (N + \tilde{N} - 2) + p^2 \frac{l_s^2}{R^2} + \tilde{w}^2 \frac{R^2}{l_s^2}, \ (1)$

where the dual radius is ${\frac{R^2}{l_s} \leftrightarrow \frac{\tilde{R}^2}{l_s} = \frac{l_s}{R^2}}$ with ${p \leftrightarrow \tilde{w}}$. Here ${l_s}$ is the string scale. One may recognise the first terms as the number operators of left and right moving oscillator excitations. The last two terms are proportional to the quantised momentum and winding. Compactified on a circle, the spectrum is invariant under ${\mathbb{Z}_2}$, but for a d-dimensional torus the duality group is the indefinite orthogonal group ${O(d,d; \mathbb{Z})}$, with ${d}$ the number of compact dimensions.

And, actually, since we’re here one can motivate the idea another way [6]. A generic aspect of string compactifications is that there exist subspaces of the moduli space which feature enhanced gauge symmetry. The story goes back to Kaluza-Klein. Take an ${S^1}$ compactification and set ${R = \sqrt{2}}$, one finds four additional massless gauge bosons that correspond to ${pw = \pm 1}$, ${N + \tilde{N} = 1}$. One can combine these states with the two ${U(1)}$ gauge fields to enlarge the ${U(1)^2}$ gauge symmetry in the form

$\displaystyle U(1) \times U(1) \rightarrow SU(2) \times SU(2). \ (2)$

If we want to generalise from the example of an ${S^1}$ compactification to higher-dimensional toroidal compactifications, we can do so such that the massless states at a generic point in the moduli space include Kaluza-Klein gauge bosons of the group ${G = U(1)^{2n}}$ and the toroidal moduli ${g_{ij}, b_{ij}}$, parameterising a moduli space of inequivalent vacua. This moduli space is ${n^2}$-dimensional coset space

$\displaystyle \mathcal{M}^{n} = \frac{O(n,n)}{O(n) \times O(n)} / \Gamma_T, \ (3)$

where ${\Gamma_T = O(n,n; \mathbb{Z})}$. In other words, it is the T-duality group relating equivalent string vacua. (In my proceeding notes I sometimes use $O(d,d)$ and $O(n,n)$ interchangably).

But the example I really want to get to comes from the classical bosonic string sigma model and its Hamiltonian formulation [7]. It is fairly straightforward to work through. Along with the equations of motion, constraints in the conformal gauge are found to be of the form

$\displaystyle G_{ab} (\partial_{\tau} X^{a} \partial_{\tau} X^b + \partial_{\sigma} X^a \partial_{\sigma} X^b) = 0$

and

$\displaystyle G_{ab}\partial_{\tau}X^a \partial_{\sigma} X^b = 0, \ (4)$

which determine the dynamics of the theory. Then in the Hamiltonian description, one can calculate the Hamiltonian density from the standard Lagrangian density. After some calculation, which includes obtaining the canonical momentum and winding, the Hamiltonian density is found to take the form

$\displaystyle H(X; G,B) = -\frac{1}{4 \pi \alpha^{\prime}} \begin{pmatrix}\partial_{\sigma} X \\ 2 \pi \alpha^{\prime} P \end{pmatrix}^T \mathcal{H}(G, B) \begin{pmatrix} \partial_{\sigma} X \\ 2\pi \alpha^{\prime} P \end{pmatrix}$

$\displaystyle = -\frac{1}{4\pi \alpha^{\prime}} \begin{pmatrix} \partial_{\tau} X \\ -2\pi \alpha^{\prime} W \end{pmatrix}^T \mathcal{H}(G, B) \begin{pmatrix} \partial_{\tau}X \\ -2\pi \alpha^{\prime} P \end{pmatrix} \ (5).$

This ${\mathcal{H}(G,B)}$ is what we will eventually come to define as the generalised metric. Keeping to the Hamiltonian formulation of the standard string, the appearance of ${O(d,d)}$follows. We first may define generalised vectors given some generalised geometry ${TM \oplus T \star M}$, in which the tangent bundle ${TM}$ of a manifold ${M}$is doubled in the sum of the tangent and co-tangent bundle. The vectors read:

$\displaystyle A_{P}(X) = \partial_{\sigma} X^a \frac{\partial}{\partial x^a} + 2\pi \alpha^{\prime}P_a dx^a$

and

$\displaystyle A_W(X) = \partial_{\tau} X^a \frac{\partial}{\partial x^a} - 2\pi \alpha^{\prime}W_a dx^a. \ (6)$

Now, in this set-up, ${O(d, d)}$ naturally appears in the classical theory ; because we take the generalised vector (6) with the constraint (4) and, in short, find that the energy-momentum tensor can be written as

$\displaystyle A^T_{P} \mathcal{H} A_P = 0 \ \ \text{and} \ \ A^T_P L A_P = 0. \ (7)$

The two constraints in (7) tell quite a bit: we have the Hamiltonian density set to zero with the second constraint being quite key. It will become all the more clear as we advance in our discussion that this ${L}$ defines the group ${O(d,d)}$. Moreover, a ${d \times d}$ matrix ${Z}$ is an element of ${O(d,d)}$ if and only if

$\displaystyle Z^T L Z = L \ (8),$

where

$\displaystyle L = \begin{pmatrix} 0 & \mathbb{I} \\ \mathbb{I} & 0 \\ \end{pmatrix}. \ (9)$

The moral of the story here is that the generalised vectors solving the constraint in (7) are related by an ${O(d,d)}$ transformation. This transformation is, in fact, T-duality. But to formalise this last example, let us do so finally in the study of DFT and its construction.

3. Target Space Duality, Double Field Theory, and ${O(D,D,\mathbb{Z})}$

From a field theory perspective, there is a lot to unearth about the presence of ${O(d,d)}$, especially given the motivating idea to make T-duality manifest. What we want to do is write everything in terms of T-duality representations. So all objects in our theory should have well-defined transformations.

We can then ask the interesting question about the field content. What one will find is that for the NS-NS sector of closed strings – i.e., gravitational fields ${g_{IJ}}$ with Riemann curvature ${R(g)}$, the Kalb-Ramond field ${b_{IJ}}$ with the conventional definition for the field strength ${H=db}$, and a dilaton scalar field ${\phi}$ – these form a multiplet of T-duality. From a geometric viewpoint, this suggests some sort of unifying geometric description, which, as discussed elsewhere on this blog, may be formalised under the concept of generalised geometry (i.e., geometry generalised beyond the Riemannian formalism).

Earlier, in arriving at (1), we talked about compactification on ${S^1}$. Generalising to a d-dimensional compactification, we of course have ${O(d,d)}$ and for the double internal space we may write the coordinates ${X^i = (x^i, \tilde{x}_i)}$, where ${i = 1,...,d}$. But what we really want to do is to double the entire space such that ${D = d + n}$, with ${I = 1,..., 2D}$, and then see what happens. Consider the standard formulation of DFT known as the generalised metric formulation (for a review of the fundamentals see [8]). The effort begins with the NS-NS supergravity action

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

In the case of toroidal compactification defined by ${D}$-dimensional non-compact coordinates and ${d}$-dimensional compact directions, the target space manifold can be defined as a product between ${d}$-dimensional Minkowski space-time and an ${n}$-torus, such that ${\mathbb{R}^{d-1,1} \times T^{n}}$ where, as mentioned a moment ago, ${D = n + d}$. We have for the full undoubled coordinates ${X^{I} = (X^{a}, X^{\mu})}$ with ${X^{a} = X^{a} + 2\pi}$ being the internal coordinates on the torus. The background fields are ${d \times d}$ matrices taken conventionally to be constant with the properties:

$\displaystyle G_{IJ} = \begin{pmatrix} \hat{G}_{ab} & 0 \\ 0 & \eta_{\mu \nu} \\ \end{pmatrix}, \ \ B_{IJ} = \begin{pmatrix} \hat{B}_{ab} & 0 \\ 0 & 0 \\ \end{pmatrix}, \ \ \text{and} \ \ G^{IJ}G_{JK} = \delta^{I}_K. \ (11)$

We define ${\hat{G}_{ab}}$ as a flat metric on the torus and ${\eta_{\mu \nu}}$ is simply the Minkowski metric on the ${d}$-dimensional spacetime. As usual, the inverse metric is defined with upper indices. In (11) we also have the antisymmetric Kalb-Ramond field. Finally, for purposes of simplicity, we have dropped the dilaton. Of course one must include the dilaton at some point so as to obtain the correct form of the NS-NS supergravity action, but for now it may be dropped because the motivation here is primarily to study the way in which ${G_{IJ}}$ and ${B_{IJ}}$ come together in a single generalised geometric entity, which we begin to construct with the internal metric denoted as

$\displaystyle E_{IJ} = G_{IJ} + B_{IJ} = \begin{pmatrix} \hat{E}_{ab} & 0 \\ 0 & \eta_{\mu \nu} \\ \end{pmatrix} \ (12)$

for the closed string background fields, with ${\hat{E}_{ab} = \hat{G}_{ab} + \hat{B}_{ab}}$ as first formulated by Narain et al [9]. It is important to note that the canonical momentum of the theory is ${2\pi P_{I} = G_{IJ}\dot{X}^{J} + B_{IJ} X^{\prime J}}$, where, in the standard way, ${\dot{X}}$ denotes a ${\tau}$ derivative and ${X^{\prime}}$ denotes a ${\sigma}$ derivative. Famously, the Hamiltonian of the theory may then also be constructed from the expansion of the string modes for coordinate ${X^{I}}$, the canonical momentum, and from the Hamiltonian density to take the following form

$\displaystyle H = \frac{1}{2} Z^{T} \mathcal{H}(E) Z + (N + \bar{N} - 2). \ (13)$

Or, to write it in terms of the mass operator,

$\displaystyle M^{2} = Z^{T} \mathcal{H}(E) Z + (N + \bar{N} - 2). \ (14)$

The structure of the first terms in (14) should look familiar. In summary, in an ${n}$-dimensional toroidal compactification, the momentum ${p^{I}}$ and winding modes ${w_{I}}$ become ${n}$-dimensional objects. So the momentum and the winding are combined in a single object known as the generalised momentum $Z = \begin{pmatrix} w_{I} \\ p^{I} \\ \end{pmatrix}$. This generalised momentum $Z$ is defined as a $2D$-dimensional column vector, and we will return to a discussion of its transformation symmetry in a moment. Meanwhile, in (13) and (14) $N$ and $\bar{N}$ are the usual number operators counting the excitations familiar in the standard bosonic string theory. One typically derives these when obtaining the Virasoro operators. We also see the first appearance of the generalised metric $\mathcal{H}(E)$, which is a $2D \times 2D$ symmetric matrix constructed from $G_{IJ}$ and $B_{IJ}$ with $E = E_{IJ} = G_{IJ} + B_{IJ}$. We will discuss the generalised metric in just a few moments.

As is fundamental to closed string theory there is the Virasoro constraint ${L_{0} - \bar{L}_{0} = 0}$, where ${L_{0}}$ and ${\bar{L}_{0}}$are the Virasoro operators. This fundamental constraint remains true in the case of DFT. Except in DFT this condition on the spectrum gives ${N - \bar{N} = p_{I}w^{I}}$ or, equivalently,

$\displaystyle N - \bar{N} = \frac{1}{2} Z^{T} L Z, \ (15)$

where

$\displaystyle L = \begin{pmatrix} 0 & \mathbb{I} \\ \mathbb{I} & 0 \\ \end{pmatrix}. \ (16)$

This is, indeed, the same ${L}$ we defined before. Given some state and some oscillators, the fundamental constraint (15) must be satisfied, with the energy of such states computed using (13). For the time being, we treat ${L}$ somewhat vaguely and simply consider it as a constant matrix. We denote ${\mathbb{I}}$ as a ${D \times D}$ identity matrix.

Continuing with basic definitions, the generalised metric that appears in (13) and (14) is similar to what one finds using the Buscher rules [10] for T-duality transformations with the standard sigma model [11,12]. That is to say, ${\mathcal{H}}$ takes a form in which there is clear mixing of the background fields. It is defined as follows,

$\displaystyle \mathcal{H}(E) = \begin{pmatrix} G - BG^{-1}B & BG^{-1} \\ -G^{-1}B & G^{-1} \\ \end{pmatrix}. \ (17)$

One inuitive motivation for the appearance of the generalised metric is simply based on the fact that, if we decompose the supergravity fields into the metric ${G_{ij}}$ and the Kalb-Ramond field ${B_{ij}}$, in DFT these then must assume the form of an ${O(d,d)}$tensor. The generalised metric, constructed from the standard spacetime metric and the antisymmetric two-form serves this purpose. On the other hand, the appearance of the generalised metric can be approached from a more general perspective that offers a deeper view on toroidal compactifications. In (13) what we have is in fact an expression that serves to illustrate the underlying moduli space structure of toroidal compactifications [9,13], which, as we have discussed, for a general manifold ${\mathcal{M}}$ may be similarly written as (3).

The overall dimension of the moduli space is ${n^2}$ which follows from the parameters of the background matrix ${E_{ij}}$, with ${n(n+1)/2}$ for ${G_{ij}}$ plus ${n(n-1)/2}$ for ${B_{ij}}$. The zero mode momenta of the theory define the Narain lattice ${\Gamma_{n,n} \subset \mathbb{R}^{2n}}$, and it can be proven that ${\Gamma_{n,n}}$ is even and also self-dual. These properties ensure that, in the study of 1-loop partition functions, the theory is modular invariant with the description enabling a complete classification of all possible toroidal compactifications (for free world-sheet theories). The feature of self-duality contributes ${O(n, \mathbb{R}) \times O(n, \mathbb{R})}$. The Hamiltonian (13) remains invariant from separate ${O(n, \mathbb{R})}$ rotations of the left and right-moving modes that then gives the quotient terms. As for the generalised metric, we may in fact define it as the ${O(n,n) / O(n) \times O(n)}$ coset form of the ${n^2}$ moduli fields.

4. ${O(n,n,\mathbb{Z})}$

In a lightning review of certain particulars of DFT, we may deepen our discussion of the T-duality group by returning first to the generalised momentum ${Z}$ as it appears in (14). If we shuffle the quantum numbers ${w,p}$, which means we exchange ${w}$for ${p}$ and vice versa, the transformation symmetry of ${Z}$ is well known to be

$\displaystyle Z \rightarrow Z = h^{T}Z^{\prime}. \ (18)$

For now, ${h}$\$ is considered generally as a ${2D \times 2D}$invertible transformation matrix with integer entries, which mixes ${p^{I}}$ and ${w_{I}}$ after operating on the generalized momentum. It follows that ${h^{-1}}$ should also have invertible entries, this will be shown to be true later on. Importantly, if we have a symmetry for the theory, this means a transformation in which we may take a set of states and, upon reshuffling the labels, we should obtain the same physics. Famously, it is indeed found that the level-matching condition and the Hamiltonian are preserved. If we take ${Z \rightarrow Z^{\prime}}$ as a one-to-one correspondence, the level-matching condition (15) with the above symmetry transformation (18) gives

$N - \bar{N} = \frac{1}{2} Z^{T}LZ = \frac{1}{2} Z^{T \prime}L Z^{\prime}$

$\displaystyle = \frac{1}{2} Z^{T \prime} h L h^{T} Z^{\prime}. \ (19)$

For this result to be true, it is necessary as a logical consequence that the transformation matrix ${h}$ must preserve the constant matrix ${L}$. This means it is required that

$\displaystyle h L h^{T} = L, \ (20)$

which also implies

$\displaystyle h^{T} L h = L. \ (21)$

These last two statements can be proven, producing several equations that give conditions on the elements of ${h}$. The full derivation will not be provided due to limited space (complete review of all items can again be found in [1,2,3,4,8]); however, to illustrate the logic, let ${a, b, c, d}$ be ${D \times D}$matrices, such that ${h}$ may be represented in terms of these matrices

$\displaystyle h = \begin{pmatrix} a & b \\ c & d \\ \end{pmatrix}. \ (22)$

The condition in which ${h}$ preserves ${L}$demands that the elements ${a, b, c, d}$satisfy in the case of (20)

$\displaystyle a^{T}c + c^{T}a = 0, \ b^{T}d + d^{T}b = 0,$

and

$\displaystyle a^{T}d + c^{T}b = 1. \ (23)$

Likewise, similar conditions are found for the case (21), for which altogether it is proven that ${h^{-1}}$ has invertible entries. What this ultimately means is that although we previously considered ${h}$ vaguely as some transformation matrix, it is in fact an element of ${O(D,D, \mathbb{R})}$ and ${L}$is an ${O(D,D, \mathbb{R})}$invariant metric. Formally, an element ${h \in O(D,D, \mathbb{R})}$ is a ${2D \times 2D}$ matrix that preserves, by its nature, the ${O(D,D, \mathbb{R})}$ invariant metric ${L}$(16) such that

$\displaystyle O(D,D,\mathbb{R}) = \bigg \{h \in GL(2D, \mathbb{R}) \ : \ h^{T}Lh = L \bigg \}. \ (24)$

Finally, if the aim of DFT at this point is to completely fulfil the demand for the invariance of the massless string spectrum, it is required from (13) for the energy that, if the first term is invariant under ${O(D,D)}$ then we must have the following transformation property in the case ${Z^{T} \mathcal{H}(E) Z \rightarrow Z^{\prime T} \mathcal{H}(E^{\prime}) Z^{\prime}}$:

$\displaystyle Z^{\prime T}\mathcal{H}(E^{\prime}) Z^{\prime} = Z^{T}\mathcal{H}(E)Z$

$\displaystyle = Z^{\prime T} h \mathcal{H}(E)h^{T} Z^{\prime}. \ (25)$

By definition, given the principle requirement of (25) it is therefore also required that the generalised metric transforms as

$\displaystyle \mathcal{H}(E^{\prime}) = h\mathcal{H}(E)h^{T}. \ (26)$

The primary claim here is that for the transformation of ${E}$ we find

$\displaystyle (E^{\prime}) = h(E) = \begin{pmatrix} a & b \\ c & d \\ \end{pmatrix}(E) \equiv (aE + b)(cE + d)^{-1}. \ (27)$

One should note that this is not matrix multiplication, and ${h(E)}$ is not a linear map. What we find in (27) is actually a well known transformation in string theory that appears often in different contexts, typically taking on the appearance of a modular transformation. Given the notational convention that ${\mathcal{H}}$is acting on the background ${E}$, what we end up with is the following

$\displaystyle (E^{\prime T}) = \begin{pmatrix} a & -b \\ -c & d \\ \end{pmatrix}(E^{T}) \equiv (aE^{T} - b)(d - cE^{T})^{-1}, \ (28)$

where in the full derivation of this definition it is shown $(E^{\prime T}) = \begin{pmatrix} a & -b \\ -c & d \\ \end{pmatrix} E^T.$

Proof: To work out the full proposition with a proof of (26), we may also demonstrate the rather deep relation between (26) and (28). The basic idea is as follows: imagine creating ${E}$ from the identity background ${E^{\prime} = \mathbb{I}}$, where conventionally ${E = G + B}$ and ${G = AA^{T}}$. Recall, also, the definition for the generalised metric metric (17). Then for ${E = h_{E}(\mathbb{I})}$, what is ${h_{E} \in O(D,D, \mathbb{R})}$? To answer this, suppose we know some ${A}$ such that

$\displaystyle h_{E} = \begin{pmatrix} A & B(A^{T})^{-1} \\ 0 & (A^{T})^{-1} \\ \end{pmatrix}. \ (29)$

It then follows

$\displaystyle h_{E}(I) = (A \cdot \mathbb{I} + B(A^{T})^{-1})(0 \cdot \mathbb{I} + (A^{T})^{-1})^{-1}$

$\displaystyle = (A + B(A^{T})^{-1}) A^{T} = AA^{T} + B = E = G + B. \ (30)$

This means that the ${O(D,D)}$ transformation creates a ${G + B}$ background from the identity. Additionally, the transformation ${h_E}$ is ambiguous because it is always possible to substitute ${h_E}$with ${h_E \cdot g}$, where we define ${g(\mathbb{I}) = \mathbb{I}}$ for ${g \in O(D,D, \mathbb{R})}$. In fact, it is known that ${g}$ defines a ${O(D) \times O(D)}$subgroup of ${O(D,D)}$ ${g^{T}g = gg^{T} = I}$.

In conclusion, one can show that ${\mathcal{H}}$ transforms appropriately, given that up to this point ${h_{E}}$ was constructed in such a way that the metric ${G}$ is split into the product ${A}$ and ${A^{T}}$, with the outcome that only ${A}$ is entered into ${h_{E}}$. To find ${G}$ we simply now consider the product ${h_{E}h_{E}^{T}}$,

$\displaystyle h_{E}h_{E}^{T} = \begin{pmatrix} A & B(A^{T})^{-1} \\ 0 & (A^{T})^{-1} \\ \end{pmatrix} \begin{pmatrix} A^{T} & 0 \\ -A^{-1}B & A^{-1} \\ \end{pmatrix}$

$\displaystyle = \begin{pmatrix} G - BG^{-1}B & BG^{-1} \\ -G^{-1}B & G^{-1} \\ \end{pmatrix} = \mathcal{H}(E). \ (31)$

If we now suppose naturally ${E^{\prime}}$ is a transformation of ${E}$ by ${h}$, such that ${E^{\prime} = h(E) = hh_{E}(\mathbb{I})}$, we also have ${E^{\prime} = h_{E^{\prime}}(\mathbb{I})}$. Notice that this implies ${h_{E^{\prime}} = hh_{Eg}}$ up to some ambiguous and so far undefined ${O(D,D,\mathbb{R})}$ subgroup defined by ${g}$. Putting everything together, we obtain the rather beautiful result

$\displaystyle \mathcal{H}(E^{\prime}) = h_{E^{\prime}}h^{T}_{E^{\prime}} = hh_{Eg}(hh_{Eg})^{T} = hh_{E}h^{T}_{E}h^{T} = h\mathcal{H}(E)h^{T}. \ (31)$

$\Box$

Thus ends the proof of (26). A number of other useful results can be obtained and proven in the formalism, including the fact that the number operators are invariant which gives complete proof of the invariance of the full spectrum under ${O(D,D,\mathbb{R})}$.

In conclusion, and to summarise, in DFT there is an explicit restriction on the winding modes ${w_{I}}$ and the momenta ${p^{I}}$ to take only discrete values and hence their reference up to this point as quantum numbers. The reason has to do with the boundary conditions of ${n}$-dimensional toroidal space, so that in the quantum theory the symmetry group is restricted to ${O(n,n,\mathbb{Z})}$ subgroup to ${O(D,D,\mathbb{R})}$. The group ${O(n,n,\mathbb{Z})}$ is as a matter of fact the T-duality symmetry group in string theory. It is conventional to represent the transformation matrix ${h \in O(n,n,\mathbb{Z})}$ in terms of ${O(D,D,\mathbb{R})}$ such that

$\displaystyle h = \begin{pmatrix} a & b \\ c & d \\ \end{pmatrix}$

with,

$\displaystyle a = \begin{pmatrix} \tilde{a} & 0 \\ 0 & 1 \\ \end{pmatrix},$

$\displaystyle b = \begin{pmatrix} \tilde{b} & 0 \\ 0 & 0 \\ \end{pmatrix},$

$\displaystyle c = \begin{pmatrix} \tilde{c} & 0 \\ 0 & 0 \\ \end{pmatrix}$

and

$\displaystyle d = \begin{pmatrix} \tilde{d} & 0 \\ 0 & 1 \\ \end{pmatrix}. \ (32)$

Each of ${\tilde{a}, \tilde{b}, \tilde{c}, \tilde{d}}$ are ${n \times n}$ matrices. They can be arranged in terms of ${\tilde{h} \in O(n,n,\mathbb{Z})}$ as

$\displaystyle \tilde{h} = \begin{pmatrix} \tilde{a} & \tilde{b} \\ \tilde{c} & \tilde{d} \\ \end{pmatrix}. \ (33)$

Invariance under the ${O(D,D,\mathbb{Z})}$ group of transformations is generated by the following transformations. To simplify matters, let us define generally the action of an ${O(D,D)}$ element as

$\displaystyle \mathcal{O} = \begin{pmatrix} a & b \\ c & d \\ \end{pmatrix} = \mathcal{O}^{T}L\mathcal{O}. \ (34)$

Residual diffeomorphisms: If ${A \in GL(D, \mathbb{Z})}$, then one can change the basis for the compactification lattice ${\Gamma}$ by ${A \Gamma A^{T}}$. The action on the generalised metric is

$\displaystyle \mathcal{O}_{A} = \begin{pmatrix} A^{T} & 0 \\ 0 & A^{-1} \\ \end{pmatrix}, \ \ A \in GL(D, \mathbb{Z}), \ \ \det A = \pm 1. \ (35)$

B-field shifts: If we define ${\Theta}$to be an antisymmetric matrix with integer entries, one can use ${\Theta}$to shift the B-field producing no change in the path integral. For compact d-dimensions, this amounts to ${B_{IJ} \rightarrow B_{IJ} + \Omega_{IJ}}$. It follows that the ${O(D,D)}$ transformation acts on the generalised metric,

$\displaystyle \mathcal{O}_{\Omega} = \begin{pmatrix} 1 & \Omega \\ 0 & 1 \\ \end{pmatrix}, \ \ \Omega_{IJ} = - \Omega_{JI} \in \mathbb{Z}. \ (36)$

Factorised dualities: We define a factorised duality as a ${\mathbb{Z}_2}$ duality corresponding to the ${R \rightarrow \frac{1}{R}}$ transformation for a single circular direction (i.e., radial inversion). It acts on the generalised metric as follows

$\displaystyle \mathcal{O}_{T} = \begin{pmatrix} 1-e_{i} & e_{i} \\ e_i & 1-e_{i} \\ \end{pmatrix}, \ (37)$

where ${e}$ is a ${D \times D}$ matrix with 1 in the ${(i, i)}$-th entry, and zeroes elsewhere ${(e_{i})_{jk} = \delta_{ij}\delta_{ik}}$. Altogether, these three essential transformations define the T-duality group ${O(D,D,\mathbb{Z})}$, as first established in [14,15]. To calculate a T-dual geometry one simply performs the action (26) or (28) using an ${O(D,D,\mathbb{R})}$ transformation and, in general, one may view the formalism with the complete T-duality group as a canonical transformation on the phase space of a given system.

References

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