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

Start with a gauge theory of non-abelian vector fields in D-dimensions

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

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.

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