**1. Introduction **

**2. Yang-Mills / DFT – Quadratic theory **

**Proposition 1**

*The projector defined in (10) satisfies the identities*

*Proof:*The second identity is trivial, while the first identity can be found substituting (10) in (11) and recalling we’ve scaled out . The first identity in (11) implies gauge invariance under the transformation where the gauge parameter 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 by a second set of spacetime indices . This second set of spacetime indices then corresponds to a second set of spacetime momenta . For the fields in momentum space, we define a new doubled field Next, following the double copy formalism, a substitution rule for the Cartan-Killing metric needs to be defined. In [DHP], the authors propose that we replace this metric with a projector carrying barred indices such that 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 , where are kinematic factors, are colour factors, and denote inverse propagators. Then, in the double copy, replace by with . This means that 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 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 , and, just as the duality symmetric string, treat on equal footing. It now seems arbitrary whether there is or 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 , the asymmetry of (15) is resolved by imposing 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 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 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 fields, we find the action (15) to take the following form: 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 (i.e., the dilaton). Doing these steps means we can first rewrite (18) as follows By using the field equations for or, alternatively, using the redefinition 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 .*

Now Fourier transforming (19) to doubled position space, we define in the standard way and . We also of course obtain the usual duality invariant measure. The resulting action takes the form

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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