# Diagrammatic Rules for Triad Census

Enrico Borriello

## Abstract

In network theory, a triad census is a method designed to categorize and
enumerate the various types of subgraphs with three nodes and their connecting
edges within a network. Triads serve as fundamental building blocks for
comprehending the structure and dynamics of networks, and the triad census
offers a systematic approach to their classification. Typically, triad counts
are obtained numerically, but lesser-known methods have been developed to
precisely evaluate them without the need for sampling. In our study, we build
upon Moody’s matrix approach, presenting general diagrammatic rules that
systematically and intuitively generate closed formulas for the occurrence
numbers of triads in a network.

##  1 Introduction

One of the most prolific strategies in the study of a complex system involves
the analysis of its network topology –the graph encoding information about the
connectivity among its interacting components.

Network theory has experienced an enormous increase in adoption by the
scientific community over the past decades. Various complementary strategies
have emerged and have been successfully employed to understand the topology of
real-world networks [1]. On the one hand, global studies have unveiled the
connection between the evolutionary nature of real-world networks and the
imprint this leaves on their topology; Most notably, the ubiquitous presence
of scale-free networks [2, 3, 4]. On the other hand, exploring the local
topology of complex networks has proven equally fruitful. In this case, a
common and systematic strategy involves comparing the frequency of recurrent
motifs in complex networks with their occurrence in random counterparts
–networks sharing similar global characteristics but with erased peculiarities
of the real-world networks they represent. This motif census has demonstrated
particular success when applied to triads [5, 6, 7, 8, 9], specifically the
smallest non-trivial network subsets consisting of just three nodes.

Notably, the census of fully connected, directed triads has uncovered a
limited number of recurring patterns capable of categorizing extensive
ensembles of real-world networks into significant superfamilies distinguished
by distinctive and frequently occurring profiles [10, 11]. Arising from its
roots in social science [12], the triad census has now become a widely
utilized and prolific tool for analyzing biological networks, particularly
gene regulatory networks [10, 13, 14, 15, 16], as well as technology networks
such as the World Wide Web (WWW) [17]. Additionally, it has found applications
in linguistics [18, 19].

Historically, the triad census has primarily been conducted numerically,
likely due to an initial misconception about the feasibility of deriving
closed formulas to count triads [20, 21]. These formulas were eventually
provided by Moody [22]. Nonetheless, to this date, they are still criminally
underutilized.

The advantage of employing closed formulas, instead of an exhausting scan
searching for all triads in a network, not only resides in higher speed across
a wide range of network sizes where both algorithms can operate effectively
but also in the potential to reveal conservation rules, akin to those
identified in [11]. Discovering such conservation rules would be challenging,
if not impossible, through network-specific numerical scans.

In this work, our goal is to revisit Moody’s approach and illustrate how his
rules can be consolidated into a general matrix formula. The matrices in this
formula can be linked to the three edges of a triad in a manner that enables
the provision of just a few diagrammatic rules. These rules correctly allow
the reproduction of expressions for all thirteen fully-connected triads.

In the first part of the next section, we will re-derive matrix expressions
for the triad census that are equivalent to Moody’s expression, but by
following a more consistent strategy across various triads. This will lead, in
the second half of the section, to the generalization of our matrix
expressions through the consistent application of just a few diagrammatic
rules. We will then conclude by explicitly demonstrating how triad-specific
counts can be deduced directly through the applications of our rules.

$i$$k$$j$ |  |  $i$$k$$j$ |  |  $i$$k$$j$
---|---|---|---|---
triad 1 |  | triad 2 |  | triad 3
$i$$k$$j$ |  |  $i$$k$$j$ |  |  $i$$k$$j$
triad 4 |  | triad 5 |  | triad 6
$i$$k$$j$ |  |  $i$$k$$j$ |  |  $i$$k$$j$
triad 7 |  | triad 8 |  | triad 9
$i$$k$$j$ |  |  $i$$k$$j$ |  |  $i$$k$$j$
triad 10 |  | triad 11 |  | triad 12
$i$$k$$j$ |  |  |  |
triad 13 |  |  |  |
Figure 1: There are 16 possible types of triads, each representing a unique
arrangement of relationships between three nodes in a network. In this work we
focus on the thirteen fully connected ones.

##  2 Methods and Results

### Matrix Formulas

As in [22], we focus on unweighted networks with no self-loops. Our
consideration is limited to fully connected triads, numbered from 1 to 13 as
shown in Figure 1. For each triad $i$ (where $i=1,\dots,13$), we aim to
evaluate its occurrence number $t_{i}$ in a closed form that avoids explicit
references to iterations over combinations of nodes. Instead, all our results
are expressed in terms of matrix operations on $A$, the adjacency matrix of
the network. Presented as straightforward transformations of $A$, these
expressions can be efficiently computed using various optimized packages for
vector calculus. (As an example, we complement these notes with a Python
implementation that uses NumPy.)

By standardizing the derivative process of our final matrix expression, we aim
to highlight the generalization discussed in the next section. In that
section, all formulas will be reinterpreted as outcomes derived from the
application of common diagrammatic rules.

$i$$k$$j$ |  |  $i$$k$$j$ |  |  $i$$k$$j$
---|---|---|---|---
triad 1 |  | triad 2 |  | triad 3

Triads 1, 2, and 3: Given the convention on the vertex labels $i,j$ and $k$ as
in figure, triad 1 occurs whenever the elements $A_{pq}$ of the adjacency
matrix $A$ of the network satisfy the conditions

$\displaystyle A_{ki}=A_{kj}=1\ ,$ $\displaystyle
A_{jk}=A_{ik}=A_{ij}=A_{ji}=0\ ,$ $\displaystyle i\neq j\ .$

We do not need to impose $k\neq i$ and $k\neq j$ because we will not account
for the presence of self-loops in this manuscript. Therefore, we can just
assume $A_{ii}=A_{jj}=0$.

In a network with $n$ nodes, and for given nodes $i$ and $j$, we can calculate
the number of nodes like node $k$ in figure:

$t_{1,ij}=(1-A_{ij})(1-A_{ji})\sum_{k=1}^{n}(1-A_{jk})(1-A_{ik})A_{ki}A_{kj}\
.$ (1)

We can then calculate $t_{1}$, the occurrence number of triad 1 in the
network, by summing over all the elements $t_{1,ij}$ with $i\neq j$, and
divide the result by 2 to prevent the double-counting induced by the symmetry
of this triad with respect to $i$ and $j$:

$t_{1}=\frac{1}{2}\sum_{i,j\neq i}t_{1,ij}\ .$ (2)

By defining

$X=({\bf 1}-A)\circ({\bf 1}-A^{T})$ (3)

and

$Y=A\circ A^{T}\ ,$ (4)

where ${\bf 1}_{ij}=1\ ,\forall i,j$, and

$D_{1}=A^{T}A-YA-A^{T}Y+Y^{2}\ ,$ (5)

we can recast $t_{1,ij}$ as $X_{ij}D_{1,ij}$, and

$\boxed{t_{1}=\frac{1}{2}[\textrm{sum}(X\circ D_{1})-\textrm{tr}(X\circ
D_{1})]}\ .$ (6)

In the previous formulae: $\circ$ is the the element-wise product between
matrices (Hadamard product); $\textrm{sum}(\cdot)$ returns the sum of all the
element of a matrix; $\textrm{tr}(\cdot)$ is the trace. We have also made use
of the fact that $Y=A\circ A^{T}$ is a symmetric matrix.

We can calculate $t_{2}$, the occurrence number of triad 2, by a similar
argument. As triad 1 becomes triad 2 once $A$ is transposed, we just need to
replace $A$ with $A^{T}$ in (6): $t_{2}=\left.t_{1}\right|_{A\rightarrow
A^{T}}$

$\boxed{t_{1}=\frac{1}{2}[\textrm{sum}(X\circ D_{2})-\textrm{tr}(X\circ
D_{2})]}\ ,$ (7)

with

$D_{2}=AA^{T}-YA^{T}-AY+Y^{2}\ ,$ (8)

We can count the occurrence number of triad 3 analogously to what we have done
with triads 1 and 2. The new conditions being

$\displaystyle A_{ik}=A_{kj}=1\ ,$ $\displaystyle
A_{jk}=A_{ki}=A_{ij}=A_{ji}=0\ ,$ $\displaystyle i\neq j\ .$

We first define

$D_{3}=A^{2}-YA-AY+Y^{2}\ ,$ (9)

and then cast the count in the form

$\boxed{t_{3}=\textrm{sum}(X\circ D_{3})-\textrm{tr}(X\circ D_{3})}\quad.$
(10)

Notice the lack of the factor $1/2$ in this count, as this triad is not
symmetric with respect to an exchange of nodes $i$ and $j$. Also notice the
common functional form of Equations 6, 7, and 10.

Triads 9, 10, and 11: Let us evaluate the counts for triads 9, 10, and 11
first, because of their similarities with the expressions we have already
obtained for triads 1, 2, and 3.

$i$$k$$j$ |  |  $i$$k$$j$ |  |  $i$$k$$j$
---|---|---|---|---
triad 9 |  | triad 10 |  | triad 11

The difference this time is the presence of a symmetric connection between
nodes $i$ and $j$, translating into factors $A_{ij}=A_{ji}=1$ as opposed to
zero. The result for triad 9 then becomes

$\boxed{t_{9}=\frac{1}{2}\textrm{sum}(Y\circ D_{9})}\ ,$ (11)

Where it is simply $D_{9}=D_{1}$ (Equation 5), and we are omitting the term
$-\textrm{tr}(Y\circ B)$ because $Y_{ii}=0,\forall i$ as we are assuming the
absence of self-loops in the networks. (Notice that, as long as the presence
of self-loops is not accounted for, we can always replace the diagonal
elements $A_{ii}$ of the adjacency matrix with null entries before performing
this census.)

Just like we derived $t_{2}$ from $t_{1}$, we can now get $t_{10}$ by just
replacing $A$ with $A^{T}$ in Equation (11),
$t_{10}=\left.t_{9}\right|_{A\rightarrow A^{T}}$, and obtain

$\boxed{t_{10}=\frac{1}{2}\textrm{sum}(Y\circ D_{10})}\ ,$ (12)

where this time it is $D_{10}=D_{2}$ (Equation 8).

It should be clear at thiss point, and it will look more systematica in the
next section, that we can easily get $t_{11}$ from our expression for $t_{3}$
after replacing $X$ with $Y$, and omitting to subract the trace:

$\boxed{t_{11}=\textrm{sum}(Y\circ D_{11})}\ ,$ (13)

where $D_{11}=D_{3}$ (Equation 9).

Not surprisingly, this time, $t_{9}$, $t_{10}$, and $t_{11}$ once again
exhibit the same functional form as the counts we obtained earlier. The only
(apparent) distinction is the absence of the trace term, as it is evaluated
over matrices with null diagonal entries. Inherited from our previous
expression, the numeric coefficient continues to convey information about the
symmetry of the triad concerning a swap between indices $i$ and $j$.

$i$$k$$j$ |  |  $i$$k$$j$
---|---|---
triad 4 |  | triad 5

Triads 4 and 5: There is not much novelty involved in the deduction of the
occurrence count of triads 4 and 5. For triad 4, we can directly express the
number of nodes such as $k$ in our graph as:

$t_{4,ij}=X_{ij}\sum_{k}Y_{ij}A_{jk}({\bf 1}-A_{kj})$

and then derive

$t_{4}=\sum_{i,j\neq i}t_{4,ij}\ .$

After defining

$D_{4}=YA^{T}-Y^{2}\ ,$

we can express our result in a closed form as

$\boxed{t_{4}=\textrm{sum}(X\circ D_{4})-\textrm{tr}(X\circ D_{4})}\ .$ (14)

Triads of type 5 can be counted in a similar way, with

$D_{5}=YA-Y^{2}$

and

$\boxed{t_{5}=\textrm{sum}(X\circ D_{5})-\textrm{tr}(X\circ D_{5})}\ .$ (15)

Triad 6: We will determine the occurrence number of Triad 6 using a different
approach. Our primary objective is to demonstrate that, even with this
alternative method, we will arrive at a triad count that shares the same
functional form as the expressions obtained thus far.

|  $i$$k$$j$
---|---
| triad 6

Whenever feasible, we will leverage the established property of the powers of
the adjacency matrix of an unweighted network to determine the number of paths
of a given length between pairs of nodes. For instance, the element $(i,j)$ of
$A^{l}$ equals the number of paths of length $l$ from node $i$ to node $j$.

With this in mind, we can determine the occurrence number of triads of type 6
by focusing solely on bidirected paths of length 2 between $i$ and $j$ with
$i\neq j$. It is straightforward to observe that we can identify the
components of matrix $A$ with a non-zero symmetric component by taking the
element-wise product of $A$ and its transpose $A^{T}$, what we previously
denoted as matrix $Y$ (4). Consequently, we can reinterpret $Y$ as the
adjacency matrix of the subgraph of $A$ that exclusively includes the
bidirected edges. Thus, the element $(i,j)$ of $Y^{2}$ represents the count of
bidirected paths of length 2 between nodes $i$ and $j$. Accounting for the
symmetric nature of triad 6 concerning $i$ and $j$, we can express our result
as:

$\boxed{t_{6}=\frac{1}{2}[\textrm{sum}(X\circ Y^{2})-\textrm{tr}(X\circ
Y^{2})]}\ .$ (16)

The Hadamard product with $X$ ensures the absence of connections between nodes
$i$ and $j$. Additionally, the trace, as customary at this point, reinforces
the condition that $i\neq j$. It is important to note that the diagonal
elements of $Y^{2}$ are not automatically zero, hence necessitating the
retention of the trace term.

It is a simple exercise to illustrate that this result can be obtained again
by starting from more fundamental assumptions about the elements $A_{ij}$,
employing some algebra, akin to the approach taken in this section so far. The
more crucial aspect is deducing the functional form for the count that we have
become accustomed to. (Considering the simplicity of the second factor in the
Hadamard product, there did not seem to be a need to define a matrix
$D_{6}=Y^{2}$, but one could do so to emphasize the formal analogy even
further.)

$i$$k$$j$ |  |  |  $i$$k$$j$
---|---|---|---
triad 7 |  |  | triad 12

Triads 7 and 12:

At this point, it should not come as a surprise that triad 7 can be counted as
triads 3, after we replacement of $X$ with

$Z=A\circ(1-A^{T})=A-Y$

in the final expression for $t_{3}$ Therefore,

$\boxed{t_{7}=\textrm{sum}(Z\circ D_{7})}\ ,$ (17)

where it is simply $D_{7}=D_{3}$.

The same is true about the connection between triads 12 and 6:

$\boxed{t_{12}=\textrm{sum}(Z\circ Y^{2})}\ .$ (18)

Again, we have ignored the subtraction of traces in (17) and (18) under the
assumption that $Z_{ii}=0$ for all $i$, provided we disregard self-loops.

Triads 8 and 13:

$i$$k$$j$ |  |  $i$$k$$j$
---|---|---
triad 8 |  | triad 13

Similar to triad 6, the counts for triads 8 and 13 possess a geometric
significance that is too evident to overlook. Consequently, we will assess
them differently.

Given $A_{ii}=0\ ,\forall i$, these triads represent the exclusive ways to
create a closed loop of length 3. In the context of triad 13, we additionally
necessitate that this loop be bidirectional:

$\boxed{t_{8}=\frac{1}{3}\textrm{tr}(Z^{3})}\quad$ (19)

$\boxed{t_{13}=\frac{1}{6}\textrm{tr}(Y^{3})}\quad$ (20)

Interestingly, these expressions exhibit the highest dissimilarity among all
we have derived. If they genuinely share the same functional form for all
remaining triad counts, this expression implies that the sum terms are null.

This could be demonstrated explicitly, commencing with expressions such as

$t_{8,ijk}=A_{ij}A_{jk}A_{ki}({\bf 1}-A_{ji})({\bf 1}-A_{kj})({\bf 1}-A_{ik})$

and proceeding with the necessary algebraic steps. It will be important to
establish this proof to reconcile these counts with the expression derived
from the straightforward application of the diagrammatic rules we are about to
present.

### Diagrammatic Rules

We can now establish diagrammatic rules for deducing the matrix expression of
the triad count directly from its graph. We have seen that, for a given
ordered pair of generic nodes $p$ and $q$, the absence of a connection between
them corresponds to the presence in the result of a matrix $X$. An outgoing
edge (from $p$ to $q$) is associated with a matrix $Z$, an incoming edge ($q$
to $p$) with $Z^{T}$, and a symmetric edge between the two with a matrix $Y$.
The required matrix operations involving the matrices associated to the three
edges in a triad can be derived from the two general steps applied in
calculating the terms $t_{\alpha}$ $(\alpha=1,\dots,13)$, like in Equations 1
and 2. Initially, we sum over the intermediate nodes that, along with $p$ and
$q$, satisfy the graph topology of the triad being counted. Subsequently, an
additional sum is conducted over nodes $p$ and $q$, taking into account the
graph’s symmetry.

The list below outlines these characteristic matrices along with their
explicit expressions in terms of $A$ for any arbitrary ordered pair of nodes
$p$ and $q$:

$p$ $q$ | $C^{[p,q]}=X$ | $\equiv\quad(A-Y)\circ(A^{T}-Y)$
---|---|---
$p$ $q$ | $C^{[p,q]}=Z$ | $\equiv\quad A\circ(1-A^{T})=A-Y$
$p$ $q$ | $C^{[p,q]}=Z^{T}$ | $\equiv\quad A^{T}\circ(1-A)=A^{T}-Y$
$p$ $q$ | $C^{[p,q]}=Y$ | $\equiv\quad A\circ A^{T}$

With the conventions and labeling established in Figure 1, We will refer to
the first two nodes we selected, $i$ and $j$ as the basis of the triad, and to
the intermediate node $k$ as the vertex.

We first build the matrix

$\boxed{B_{\alpha}=C^{[i,j]}\circ\left(C^{[k,i]\,T}\cdot C^{[k,j]}\right)}\ ,$
(21)

where the dot product corresponds to the sum over $k$ (e.g. Equation 2), and
the Hadamard product instead corresponds to the sum over $i$ and $j$ (e.g.
Equation 1). The final result can then be expressed as

$\boxed{t_{\alpha}=\frac{1}{s_{\alpha}}\left[\textrm{sum}(B_{\alpha})-\textrm{tr}(B_{\alpha})\right]}\
,$ (22)

where $s_{\alpha}$ is a symmetry factor, defined as follows.

We will define a triad as “rotationally-symmetric” if it remains unchanged
under a cyclic permutation of all three indices. A triad that remains
unchanged after a label exchange of the nodes in its basis will be said to be
“parity-symmetric.”

According to these definitions:

  * •

Triads 3, 4, 5, 7, 11, and 12 do not possess any of these symmetries.

  * •

Triads 1, 2, 6, 9, and 10 are parity-symmetric.

  * •

Triad 8 is rotationally-symmetric.

  * •

Triad 13 is both parity- and rotationally-symmetric.

For asymmetric triads, $s_{\alpha}=1$. For parity-symmetric triads,
$s_{\alpha}=2$. For parity- and rotationally-symmetric triads,
$s_{\alpha}=2\times 3=6$.

Now, we need to demonstrate that equation 22 indeed provides the general
functional form for all triad counts. The proofs for triads 1 to 5, 7, and 9
to 12 are already provided explicitly in the previous section by the method we
adopted to count their occurrence numbers, leaving us with the task of
confirming the correctness of our diagrammatic rules for triads 6, 8, and 13.

In the case of triad 6, the basis of the triad contributes the matrix
$C^{[i,j]}=X$, while the vertices correspond to matrices
$C^{[k,i]}=C^{[k,j]}=Y$. With $s_{6}=2$, Equations 22 and 16 actually yield
the same result.

The remaining proofs for triads 8 and 13 are similar. We will only present the
quicker proof for triad 13, but the interested reader can readily perform the
necessary algebra to verify the other one.

According to the diagrammatic rules provided here,
$t_{13}=\frac{1}{6}\textrm{sum}(Y\circ Y^{2})$, to be compared to
$t_{13}=\frac{1}{6}\textrm{tr}(Y^{3})$ as obtained from counting closed and
bidirectional paths of length 3 in the network. Therefore, we just need to
proof that $\textrm{tr}(Y^{3})=\textrm{sum}(Y\circ Y^{2})$. The element $ij$
of $Y^{3}$ can be written as

$(Y^{3})_{ij}=\sum_{q,p}Y_{ip}Y_{pq}Y_{qj}\ .$

Therefore

$\textrm{tr}(Y^{3})=\sum_{i}(Y^{3})_{ii}=\sum_{i,p,q}Y_{ip}Y_{pq}Y_{qj}\ .$

On the other hand,

$(Y\circ Y^{2})_{ij}=Y_{ij}\sum_{p}Y_{ip}Y_{pj}$

and

$\textrm{sum}(Y\circ Y^{2})=\sum_{i,j,p}Y_{ij}Y_{ip}Y_{pj}\ .$

By virtue of the symmetry of Y:

$\textrm{sum}(Y\circ Y^{2})=\sum_{i,j,p}Y_{ij}Y_{jp}Y_{pi}=\textrm{tr}(Y^{3})\
,$

as we wanted to prove.

We can therefore establish the equivalence between the general expression in
Equation 22 and the individual triad counts from the previous subsection.

##  3 Conclusions

In network theory, a triad census is a method aimed at categorizing and
enumerating different types of subgraphs with three nodes and their connecting
edges within a network. Triads, as fundamental building blocks, play a crucial
role in understanding the structure and dynamics of networks. The triad census
provides a systematic approach to classify these structures, allowing
researchers to analyze the prevalence of various triad types in a given
network. This method is particularly valuable in social science for studying
local structures and clustering properties of social networks.

Moody is credited with the initial derivation of closed formulas for counting
the occurrence number of triads. Despite its significance, the scientific
community has been hesitant to fully embrace the potential of deducing triad
census through efficient matrix operations. Traditionally, numerical methods
have been utilized to count occurrences of each triad type through exhaustive
network traversal algorithms. These algorithms, however, only suboptimally
leverage the peculiar structure of sparse connectedness present in most real-
world networks.

Our study introduces diagrammatic rules that systematically and intuitively
generate closed formulas for the occurrence number of triads in a network.
This unifies Moody’s matrix expressions into a common generating rule. The
simplification presented in this work, characterized by an extremely concise
mathematical representation, has the potential to enhance the understanding of
underlying patterns and dynamics within a network. This is achieved by
directly relating the triad counts to common matrix operations performed over
the network’s adjacency matrix. Additionally, we envision this work as an
initial step towards generalizing diagrammatic rules for the systematic
counting of larger and more complex motifs.

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# Aspects of Conformal Gravity and Double Field Theory from a Double Copy Map

Eric Lescano<EMAIL_ADDRESS>Division of Theoretical Physics, Rudjer Bošković
Institute, Bijenička 54, 10000 Zagreb, Croatia  Gabriel Menezes
<EMAIL_ADDRESS>Departamento de Física, Universidade Federal Rural do
Rio de Janeiro, 23897-000, Seropédica, RJ, Brazil  Jesús A. Rodríguez
<EMAIL_ADDRESS>Universidad de Buenos Aires, FCEyN, Departamento de
Física, Ciudad Universitaria, 1428 Buenos Aires, Argentina

###### Abstract

Double Field Theory (DFT) can be constructed as the double copy of a Yang-
Mills theory. In this work we extend this statement by including higher-
derivative terms. Starting from a four-derivative extension of Yang-Mills
whose double copy is known to correspond to a conformal-gravity theory, we
obtain a four-derivative theory formulated in double space, which in the pure
gravity limit reduces to conformal gravity at quadratic order. This result
reveals important aspects for the study of conformal symmetry in the context
of DFT through double copy maps.

##  I Introduction

Recently research activities have witnessed a surge of interest in exploring
possible relationships between gauge and gravity theories. Such a connection
can be instrumental in determining fundamental relations in order to
understand more deeply the quantum formulation of gravitational interactions.
One successful approach in this direction is the so-called double copy
prescription Dcopy , which is a map between scattering amplitudes in a broad
variety of gravitational and gauge theories (see Snow  and references therein
for an introduction to the subject and its applications). In its original
manifestation revealed by Kawai, Lewellen and Tye, closed string tree-level
amplitudes can be written in terms of open-string amplitudes KLT . The double-
copy formulation also presents correspondences of classical solutions of Yang-
Mills theory and gravity class  and it has been of paramount importance for
scattering-amplitudes methods in the investigation of classical gravitational
physics class2 .

A very interesting result was recently obtained in HohmDC . In this paper the
authors demonstrated that the gravitational framework which results from the
double copy map of Yang-Mills theory contains a chiral structure, such as the
one that happens in the geometry of Double Field Theory (DFT) Siegel ; DFT .
In particular, after applying the double copy procedure to the Yang-Mills
theory, the resulting gravitational theory is indeed described by a weak-
constrained perturbative DFT, both at quadratic and cubic order, which
requires the integration of the generalized dilaton as well as a particular
gauge fixing (Siegel’s gauge) for describing the cubic interactions, while the
quadratic interactions can be described without fixing any gauge.

In DFT a doubled set of coordinates is included. This approach allows for a
consistent treatment of both momentum and winding modes in a way that is
invariant under T-duality transformations. DFT provides a natural setting for
studying the physics of non-geometric backgrounds and has applications in
various areas of theoretical physics, including black hole physics, cosmology,
and the study of non-commutative geometries 111For reviews see reviewDFT  and
the second lecture of Electure .. Much of the work on DFT has been done on the
so-called strongly constrained DFT: The inclusion of a T-duality invariant
metric as well as a dynamical generalized metric generates a notion of
chirality which can be explicitly manifest in terms of projectors. There is an
extra fundamental field, the generalized dilaton, which can be used to
construct a measure in the double geometry. These fields encode the
information related with the massless bosonic fields of the NS-NS sector of
string theory, once the antisymmetric field is identified with the Kalb-Ramond
field. Pure Yang-Mills theory can be coupled at the DFT level as in PK ; EyS ,
but in this work we will consider only gravitational degrees of freedom in the
double geometry.

The double structure of DFT suggest a natural relation with the double copy
approach. This was initially explored in KL  where DFT was used to derive the
double copy relationship by showing that the equations of motion of DFT can be
mapped onto the equations of motion for Yang-Mills theory. This was done from
a perturbative method related with a suitable generalization of the Kerr-
Schild ansatz KS . This framework can be extended for heterotic DFT hetKS ;
AE-HD:21 , KK-DFT KK  and ExFT EFT . The connection between the double copy
and the geometry of DFT can also explain properties of the $L_{\infty}$
structure of the latter. It is well known that the DFT Jacobiator is not
trivial (but it is given by a trivial parameter) and therefore the algebraic
structure of DFT is given by an $L_{\infty}$ algebra with a non-trivial
$l_{3}$ product HZ , which measures the failure of the Jacobi identity in the
double geometry. The $L_{\infty}$ structure of DFT when the generalized Kerr-
Schild ansatz is imposed was studied in EM , while the relation between the
double copy prescription given in HohmDC  was recently studied in HLinfty
generalizing the results of the former.

Based on these previous researches, in this work we go a step further and
explore the double copy of higher-derivative theories. In particular we are
interested in a result coming from the computation of scattering amplitudes –
double copy of certain higher-derivative Yang-Mills theories corresponds to
conformal (super)gravity DFJ ; DFF . As it was noted in HohmDC , double copy
requires replacing color factors with a second set of kinematic factors, which
come with their own momenta. This substitution leads to a theory in double
momentum space or, in position space, a doubled set of coordinates. This
suggest a close relation between conformal gravity and a higher-derivative
deformation of DFT via double-copy prescription. We start by reviewing the
double copy prescription on the Yang-Mills Lagrangian as was done in HohmDC ,
which serves us to present the procedure for the leading order theory (two-
derivative case). In the following section we extend the analysis to a
Lagrangian containing higher-derivative terms. We find a consistent action on
the double space to quadratic order by imposing gauge fixing conditions which
reduces to Weyl gravity. Then we compute the full cubic order of the theory,
which in the pure gravity case contains a larger structure of terms beyond
Weyl gravity. Finally we discuss different alternatives to address the
inclusion of conformal symmetry in non-perturbative DFT. We work with units
such as $\hbar=c=G=1$.

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

As we mentioned previously, it was shown in HohmDC  that DFT may arise quite
naturally from the color-kinematics double copy of Yang-Mills theory. In that
work the authors showed that after implementing a double-copy prescription on
a Yang-Mills theory the resulting gravitational theory is, at least to cubic
order in fields, a week-constrained perturbative DFT which requires the
integration of the generalized dilaton. More precisely, at quadratic order
they obtained a gauge invariant DFT, whereas the cubic order requires a
particular gauge-fixing procedure, the use of the Siegel gauge.

In this section we briefly review the analysis made in Ref. HohmDC . This will
help us establish the notation, but mainly the point here is to introduce the
double-copy prescription before applying it to a higher-derivative gauge
theory related to conformal gravity in the next section. The starting point is
a $D$-dimensional Yang-Mills action,

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

where the Yang-Mills field strength is defined in the standard way

$F_{\mu\nu}{}^{a}=2\partial_{[\mu}A_{\nu]}{}^{a}+g_{\rm{YM}}f^{a}{}_{bc}A_{\mu}{}^{b}A_{\nu}{}^{c}\,,$
(2)

and space-time indices are contracted with a Minkowski metric
$\eta_{\mu\nu}={\rm{diag}}(-,+,+,+)$. Passing over to momentum space, the
quadratic terms of the gauge action reads (up to a total derivative)

$S_{\rm{YM}}^{(2)}=-\frac{1}{2}\int_{k}\,\kappa_{ab}\,k^{2}\,\Pi^{\mu\nu}(k)A_{\mu}{}^{a}(-k)A_{\nu}{}^{b}(k)\;,$
(3)

where $\int_{k}\equiv\int d^{D}k$. The projector $\Pi^{\mu\nu}(k)$ is defined
as

$\Pi^{\mu\nu}(k)=\eta^{\mu\nu}-\frac{k^{\mu}k^{\nu}}{k^{2}}\,,$ (4)

and obeys the identities

$\Pi^{\mu\nu}(k)k_{\nu}=0\;,\quad\quad\Pi^{\mu\nu}\Pi_{\nu\rho}=\Pi^{\mu}{}_{\rho}\;.$
(5)

The next step is to take the double copy prescription to construct a
gravitational theory. It consists on replacing the color indices by a second
set of space-time indices ($a\rightarrow\bar{\mu}$) corresponding to a second
set of space-time momenta $\bar{k}^{\bar{\mu}}$. This implies

$A_{\mu}{}^{a}(k)\;\longrightarrow\;e_{\mu\bar{\mu}}(k,\bar{k})\;.$ (6)

The Cartan-Killing metric $\kappa_{ab}$ must also be substituted following the
relation HohmDC

$\kappa_{ab}\;\longrightarrow\;{\textstyle\frac{1}{2}}\,\bar{\Pi}^{\bar{\mu}\bar{\nu}}(\bar{k})\,,$
(7)

where the projector $\bar{\Pi}^{\bar{\mu}\bar{\nu}}$ is defined in the same
way as the projector $\Pi^{\mu\nu}$ but for barred momenta and indices instead
of the original ones. Using these rules the quadratic action (3) becomes,

$S_{\rm
DC}^{(2)}=-\frac{1}{4}\int_{k,\bar{k}}k^{2}\,\Pi^{\mu\nu}(k)\bar{\Pi}^{\bar{\mu}\bar{\nu}}(\bar{k})e_{\mu\bar{\mu}}(-k,-\bar{k})e_{\nu\bar{\nu}}(k,\bar{k})\,.$
(8)

It is obvious that the action is symmetric in $k$ and $\bar{k}$ except for the
factor $k^{2}$. This asymmetry is solved in DFT due to the so-called level-
matching constraint, which states that $\bar{k}^{2}=k^{2}$.

To complete the construction of the gravitational theory related to (3) we
have to Fourier transform the previous action to position space. After
expanding the projectors and using the level-matching constraint we find the
presence of a non-local term which forces the introduction of an auxiliary
scalar field $\phi(k,\bar{k})$ such that, when integrated out, the action (8)
can be explicitly recovered. It is straightforward to Fourier transform to a
local action in doubled position space:

$\displaystyle S_{\rm DC}^{(2)}=\frac{1}{4}$ $\displaystyle\int
d^{D}x\,d^{D}\bar{x}\Big{(}e^{\mu\bar{\nu}}\square
e_{\mu\bar{\nu}}+\partial^{\mu}e_{\mu\bar{\nu}}\,\partial^{\rho}e_{\rho}{}^{\bar{\nu}}$
$\displaystyle+\bar{\partial}^{\bar{\nu}}e_{\mu\bar{\nu}}\,\bar{\partial}^{\bar{\sigma}}e^{\mu}{}_{\bar{\sigma}}-\phi\square\phi+2\phi\partial^{\mu}\bar{\partial}^{\bar{\nu}}e_{\mu\bar{\nu}}\Big{)}\,,$
(9)

which reproduces the standard quadratic DFT action without the assumption of
any gauge choice.

We have reviewed how to obtain quadratic DFT from the ordinary Yang-Mills
action. Next we are going to discuss some relevant aspects of the cubic
construction. After Fourier transforming to momentum space the cubic
contributions from (1), the three-point vertex function arises naturally
($k_{ij}=k_{i}-k_{j}$)

$\pi^{\mu\nu\rho}(k_{1},k_{2},k_{3})=\eta^{\mu\nu}k_{12}^{\rho}+\eta^{\nu\rho}k_{23}^{\mu}+\eta^{\rho\mu}k_{31}^{\nu},$
(10)

which satisfies the antisymmetric properties of the structure constant. The
action at this point can be written as ($A_{i}\equiv A(k_{i})$)

$\displaystyle S^{(3)}_{\rm{YM}}$ $\displaystyle=$
$\displaystyle-{\textstyle\frac{ig_{\rm{YM}}}{6\left(2\pi\right)^{D/2}}}\int_{k_{i}}\delta\left(k_{1}+k_{2}+k_{3}\right)$
(11) $\displaystyle\times$ $\displaystyle
f_{abc}\pi^{\mu\nu\rho}A_{1\mu}^{a}A_{2\nu}^{b}A_{3\rho}^{c}\,,$

which shows that an extension of the double copy prescription (6),(7) must be
considered in order to include the structure constant. The proper substitution
rule is

$f_{abc}\longrightarrow\frac{i}{4}\bar{\pi}^{\bar{\mu}\bar{\nu}\bar{\rho}}\,,$
(12)

defined in the same way as (10) but for barred momentum. One obtains

$\displaystyle S_{\rm{DC}}^{(3)}$ $\displaystyle=$
$\displaystyle\frac{1}{48(2\pi)^{D/2}}\int
dK_{1}dK_{2}dK_{3}\delta(K_{1}+K_{2}+K_{3})$ (13) $\displaystyle\times$
$\displaystyle\bar{\pi}^{\bar{\mu}\bar{\nu}\bar{\rho}}\pi^{\rho\mu\nu}e_{1\mu\bar{\mu}}e_{2\nu\bar{\nu}}e_{3\rho\bar{\rho}}$

where $K=(k,\bar{k})$, $dK=d^{2D}K$ and
$e_{i\mu\bar{\mu}}=e_{\mu\bar{\mu}}(K_{i})$. After some manipulations, Fourier
transformation to position space and integration by parts, the authors obtain
the following cubic action for the double copy of Yang-Mills theory

$\displaystyle S^{(3)}_{\rm DC}$ $\displaystyle=$
$\displaystyle\frac{1}{8}\int d^{D}x\,d^{D}\bar{x}\
e_{\mu\bar{\mu}}\Big{[}\,2\partial^{\mu}e_{\rho\overline{\rho}}\,\overline{\partial}^{\overline{\mu}}e^{\rho\overline{\rho}}-2\partial^{\mu}e_{\nu\overline{\rho}}\,\overline{\partial}^{\overline{\rho}}e^{\nu\overline{\mu}}\,$
(14) $\displaystyle-$ $\displaystyle
2\partial^{\rho}e^{\mu\overline{\rho}}\,\overline{\partial}^{\overline{\mu}}e_{\rho\overline{\rho}}+\partial^{\rho}e_{\rho\overline{\rho}}\,\overline{\partial}^{\overline{\rho}}e^{\mu\overline{\mu}}+\overline{\partial}_{\overline{\rho}}e^{\mu\overline{\rho}}\,\partial_{\rho}e^{\rho\overline{\mu}}\,\Big{]}.$

It was proven that this action agrees with the cubic DFT action by a gauge
fixing condition, integrating out the dilaton. The imposition of a gauge
fixing condition is expected considering amplitude computations.

In the next section we are going to apply the double copy map (6), (7) and
(12) on the minimal $\left(DF\right)^{2}$ theory DFJ  in order to explore the
relation between higher-derivative Double Field Theory and conformal gravity.

##  III Higher-derivative double field theory from the minimal
$\left(DF\right)^{2}$ theory

Our main interest in this work lies in exploring the existence of a relation
between conformal gravity and DFT. It was demonstrated that, at the level of
amplitudes, the double copy of the following higher-derivative extension of
the usual Yang-Mills theory, given by the Lagrangian

${\cal L}=\frac{1}{2}\kappa_{ab}D_{\mu}F^{\mu\nu
a}D_{\rho}F^{\rho}{}_{\nu}{}^{b}\,,$ (15)

corresponds to conformal (super)gravity DFJ ; DFF . Here, we take this
deformation and follow the procedure presented in HohmDC . The idea is to
explore if this resulting double copy presents a structure that can be
interpreted from the double geometry framework.

Considering the gauge covariant derivative defined as

$D_{\rho}F_{\mu\nu}{}^{a}=\partial_{\rho}F_{\mu\nu}{}^{a}+g_{\rm{YM}}f^{a}{}_{bc}A_{\rho}{}^{b}F_{\mu\nu}{}^{c}.$
(16)

The expansion of the action (15) up to quadratic terms and its subsequent
integration by parts becomes

$S^{(2)}_{\rm{HD}}=\frac{1}{2}\int d^{D}x\kappa_{ab}\square A^{\mu
a}\left(\square
A_{\mu}{}^{b}-\partial_{\mu}\partial^{\nu}A_{\nu}{}^{b}\right)\,.$ (17)

This expression is interesting because it contains the quadratic expansion of
the pure Yang-Mills action (1). Going to momentum space it becomes

$S_{\rm{HD}}^{(2)}=-\frac{1}{2}\int_{k}\,\kappa_{ab}\,k^{4}\,\Pi^{\mu\nu}(k)A_{\mu}{}^{a}(-k)A_{\nu}{}^{b}(k)\;,$
(18)

After imposing the double copy relations (6) and (7) on the previous action,
we obtain

$S_{\rm{HD/DC}}^{(2)}=-\frac{1}{4}\int_{k,\bar{k}}k^{4}\,\Pi^{\mu\nu}(k)\bar{\Pi}^{\bar{\mu}\bar{\nu}}(\bar{k})e_{\mu\bar{\mu}}(-k,-\bar{k})e_{\nu\bar{\nu}}(k,\bar{k}),$
(19)

which, not surprisingly, takes the same form as the Yang-Mills case except for
the additional $k^{2}$ contribution. However, this additional contribution
avoids the emergence of non-local terms in the action as we can observe after
expanding the projectors

$\displaystyle S_{\rm{HD/DC}}^{(2)}=$
$\displaystyle-\frac{1}{4}\int_{k,\bar{k}}k^{2}\Big{(}k^{2}e^{\mu\bar{\nu}}e_{\mu\bar{\nu}}-k^{\mu}k^{\rho}e_{\mu\bar{\nu}}e_{\rho}{}^{\bar{\nu}}$
$\displaystyle-\bar{k}^{\bar{\nu}}\bar{k}^{\bar{\sigma}}e_{\mu\bar{\nu}}e^{\mu}{}_{\bar{\sigma}}+\frac{1}{k^{2}}k^{\mu}\bar{k}^{\bar{\nu}}k^{\rho}\bar{k}^{\bar{\sigma}}e_{\mu\bar{\nu}}e_{\rho\bar{\sigma}}\Big{)}\;,$
(20)

and hence the introduction of auxiliary fields is not necessary. Transforming
the last expression to doubled position space we find the following higher-
derivative contributions

$\displaystyle S_{\rm{HD/DC}}^{(2)}=$ $\displaystyle-\frac{1}{4}\int
d^{D}xd^{D}\bar{x}\Big{[}\Box e^{\mu\bar{\nu}}\Box e_{\mu\bar{\nu}}-\Box
e^{\mu\bar{\nu}}\partial_{\mu}\partial^{\rho}e_{\rho\bar{\nu}}\,$
$\displaystyle-\Box
e^{\mu\bar{\nu}}\bar{\partial}_{\bar{\nu}}\bar{\partial}^{\bar{\sigma}}e_{\mu\bar{\sigma}}+\partial^{\mu}\bar{\partial}^{\bar{\nu}}e_{\mu\bar{\nu}}\partial^{\rho}\bar{\partial}^{\bar{\sigma}}e_{\rho\bar{\sigma}}\Big{]}.$
(21)

This action can be understood as a higher-derivative extension of DFT with
conformal symmetry in the double space (in a classical sense). To clarify this
point, we are going to explore the conformal gravity action

$S_{\rm{CG}}=\int d^{D}x\ \sqrt{-g}\
C_{\mu\nu\rho\lambda}C^{\mu\nu\rho\lambda}\,,$ (22)

with the Weyl tensor given by

$\displaystyle C_{\mu\nu\rho\lambda}$ $\displaystyle=$ $\displaystyle
R_{\mu\nu\rho\lambda}-\frac{2}{D-2}\left(g_{\mu[\rho}R_{\lambda]\nu}-g_{\nu[\rho}R_{\lambda]\mu}\right)\,$
(23)
$\displaystyle+\frac{2}{\left(D-1\right)\left(D-2\right)}Rg_{\mu[\rho}g_{\lambda]\nu}.$

Considering the expansion of $S_{\rm{CG}}$ up to quadratic order for
$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$; we obtain

$\displaystyle S_{\rm{CG}}^{(2)}$ $\displaystyle=$
$\displaystyle\frac{D-3}{D-2}\int d^{D}x\left[\left(\Box h^{\mu\nu}\Box
h_{\mu\nu}-2\Box
h^{\mu\nu}\partial_{\mu}\partial^{\rho}h_{\rho\nu}\right.\right.$
$\displaystyle+$
$\displaystyle\left.\left.\partial^{\mu}\partial^{\nu}h_{\mu\nu}\partial^{\rho}\partial^{\lambda}h_{\rho\lambda}\right)-\frac{1}{D-1}\left(\Box
h-\partial_{\mu}\partial_{\nu}h^{\mu\nu}\right)^{2}\right],$

where $h=h_{\mu}{}^{\mu}$.

The second term in (III) can be removed by imposing the gauge fixing condition
$\Box h=\partial_{\mu}\partial_{\nu}h^{\mu\nu}$, related to dilatation
symmetry $\delta h_{\mu\nu}=-2\lambda_{D}\eta_{\mu\nu}$. It is straightforward
to prove that, after setting $x=\bar{x}$ and properly rescaling the metric
(and/or considering a particular volume for the double space when we
integrate), in the pure gravity case ($e_{\mu\bar{\nu}}\sim h_{\mu\nu}$) the
action (III) reduces to (III).

We will now address the double-copy prescription for the cubic terms in (15),
which can be written as

$\displaystyle S^{(3)}_{\rm{HD}}$ $\displaystyle=$ $\displaystyle
g_{\rm{YM}}\int
d^{D}xf_{abc}\left[\triangle_{\nu}{}^{a}\left(A_{\rho}{}^{b}\partial^{\rho}A^{\nu
c}-A_{\rho}{}^{b}\partial^{\nu}A^{\rho c}\right)\right.\,$ (25)
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\left.-\partial_{\rho}\triangle_{\nu}{}^{a}A^{\rho b}A^{\nu c}\right]\,,$

with $\triangle_{\nu}{}^{a}=\Box
A_{\nu}{}^{a}-\partial_{\nu}\partial^{\mu}A_{\mu}{}^{a}$. Following the same
logic as in the quadratic case we transform the action to momentum space
obtaining

$\displaystyle S_{\rm{HD}}^{(3)}$ $\displaystyle=$
$\displaystyle\frac{ig_{\rm{YM}}}{(2\pi)^{D/2}}\int
dk_{1}dk_{2}dk_{3}\delta(k_{1}+k_{2}+k_{3})\,$ (26) $\displaystyle\times$
$\displaystyle
f_{abc}k_{1}^{2}\Pi^{\rho\mu}\left(k_{3}^{\nu}-k_{1}^{\nu}\right)A_{1\mu}{}^{a}A_{2\nu}{}^{b}A_{3\rho}{}^{c},$

where $\Pi^{\rho\mu}$ was defined in (4). Before applying the double copy
prescription we consider a deformation to the three-point vertex function,
given by

$\Pi^{\mu\nu\rho}(k_{1},k_{2},k_{3})=\Pi^{\mu\nu}k_{12}^{\rho}+\Pi^{\nu\rho}k_{23}^{\mu}+\Pi^{\rho\mu}k_{31}^{\nu}.$
(27)

Notice how similar this is to the standard three-point vertex function defined
in Eq. (10) – indeed, $\Pi^{\mu\nu\rho}$ can be obtained from
$\pi^{\mu\nu\rho}$ by replacing the metric $\eta^{\mu\nu}$ with
$\Pi^{\mu\nu}$. This object retains all the properties of the leading-order
definition and allows us to write the cubic action as

$S_{\rm{HD}}^{(3)}=\frac{ig_{\rm{YM}}}{3(2\pi)^{D/2}}\int_{k_{i}}\delta(k_{i})f_{abc}k_{1}^{2}\Pi^{\mu\nu\rho}A_{1\mu}{}^{a}A_{2\nu}{}^{b}A_{3\rho}{}^{c}\,.$
(28)

Analyzing the structure of the action one can realize that the only way to
apply the double copy prescription (6) and (7) preserving the symmetry between
the coordinates $\mu$ and $\bar{\mu}$ is to consider $\Pi^{\mu\nu\rho}$
instead of $\pi^{\mu\nu\rho}$ in the substitution rule for the structure
constant (12), this is

$f_{abc}\longrightarrow\frac{i}{4}\bar{\Pi}^{\bar{\mu}\bar{\nu}\bar{\rho}}\,.$
(29)

Then the higher-derivative cubic contributions read,

$\displaystyle S_{\rm{HD}/\rm{DC}}^{(3)}$ $\displaystyle=$
$\displaystyle-\frac{1}{24(2\pi)^{D/2}}\int
dK_{1}dK_{2}dK_{3}\delta(K_{1}+K_{2}+K_{3})\,$ (30) $\displaystyle\times$
$\displaystyle
k_{1}^{2}\bar{\Pi}^{\bar{\mu}\bar{\nu}\bar{\rho}}\Pi^{\rho\mu\nu}e_{1\mu\bar{\mu}}e_{2\nu\bar{\nu}}e_{3\rho\bar{\rho}}.$

This expression resembles Eq. (13) but encodes the higher-derivative terms of
the action (15). Expanding this last equation we obtain

$\displaystyle S_{\rm{HD}/\rm{DC}}^{(3)}$ $\displaystyle=$
$\displaystyle-\frac{1}{4(2\pi)^{D/2}}\int
dK_{1}dK_{2}dK_{3}\delta(K_{1}+K_{2}+K_{3})\left[\frac{}{}k_{1}^{2}k_{1}^{\rho}\bar{k}_{1}^{\bar{\sigma}}e_{1\mu\bar{\nu}}e_{2}^{\mu\bar{\nu}}e_{3\rho\bar{\sigma}}-k_{1}^{2}k_{2}^{\rho}\bar{k}_{1}^{\bar{\sigma}}e_{1\mu\bar{\nu}}e_{2}^{\mu\bar{\nu}}e_{3\rho\bar{\sigma}}\right.\,$
(31) $\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
+k_{1}^{2}k_{2}^{\mu}\bar{k}_{1}^{\bar{\sigma}}e_{1\mu\bar{\nu}}e_{2}^{\rho\bar{\nu}}e_{3\rho\bar{\sigma}}-k_{1}^{2}k_{3}^{\mu}\bar{k}_{1}^{\bar{\sigma}}e_{1\mu\bar{\nu}}e_{2}^{\rho\bar{\nu}}e_{3\rho\bar{\sigma}}-k_{1}^{2}k_{2}^{\mu}\bar{k}_{2}^{\bar{\sigma}}e_{1\mu\bar{\nu}}e_{2}^{\rho\bar{\nu}}e_{3\rho\bar{\sigma}}+k_{1}^{2}k_{3}^{\mu}\bar{k}_{2}^{\bar{\sigma}}e_{1\mu\bar{\nu}}e_{2}^{\rho\bar{\nu}}e_{3\rho\bar{\sigma}}\,$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
-k_{1}^{\mu}k_{1}^{\rho}k_{1}^{\lambda}\bar{k}_{1}^{\bar{\sigma}}e_{1\mu\bar{\nu}}e_{2\rho}{}^{\bar{\nu}}e_{3\lambda\bar{\sigma}}+k_{1}^{\mu}k_{1}^{\rho}k_{2}^{\lambda}\bar{k}_{1}^{\bar{\sigma}}e_{1\mu\bar{\nu}}e_{2\rho}{}^{\bar{\nu}}e_{3\lambda\bar{\sigma}}-k_{2}^{\mu}k_{1}^{\rho}k_{1}^{\lambda}\bar{k}_{1}^{\bar{\sigma}}e_{1\mu\bar{\nu}}e_{2\rho}{}^{\bar{\nu}}e_{3\lambda\bar{\sigma}}\,$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
+k_{3}^{\mu}k_{1}^{\rho}k_{1}^{\lambda}\bar{k}_{1}^{\bar{\sigma}}e_{1\mu\bar{\nu}}e_{2\rho}{}^{\bar{\nu}}e_{3\lambda\bar{\sigma}}+k_{2}^{\mu}k_{1}^{\rho}k_{1}^{\lambda}\bar{k}_{2}^{\bar{\sigma}}e_{1\mu\bar{\nu}}e_{2\rho}{}^{\bar{\nu}}e_{3\lambda\bar{\sigma}}-k_{3}^{\mu}k_{1}^{\rho}k_{1}^{\lambda}\bar{k}_{2}^{\bar{\sigma}}e_{1\mu\bar{\nu}}e_{2\rho}{}^{\bar{\nu}}e_{3\lambda\bar{\sigma}}\,$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
-\bar{k}_{1}^{\bar{\nu}}\bar{k}_{1}^{\bar{\sigma}}\bar{k}_{1}^{\bar{\kappa}}k_{1}^{\rho}e_{1\mu\bar{\nu}}e_{2}^{\mu}{}_{\bar{\sigma}}e_{3\rho\bar{\kappa}}+\bar{k}_{1}^{\bar{\nu}}\bar{k}_{1}^{\bar{\sigma}}\bar{k}_{1}^{\bar{\kappa}}k_{2}^{\rho}e_{1\mu\bar{\nu}}e_{2}^{\mu}{}_{\bar{\sigma}}e_{3\rho\bar{\kappa}}+\bar{k}_{1}^{\bar{\nu}}\bar{k}_{1}^{\bar{\sigma}}\bar{k}_{2}^{\bar{\kappa}}k_{2}^{\mu}e_{1\mu\bar{\nu}}e_{2}^{\rho}{}_{\bar{\sigma}}e_{3\rho\bar{\kappa}}\,$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
-\bar{k}_{1}^{\bar{\nu}}\bar{k}_{1}^{\bar{\sigma}}\bar{k}_{2}^{\bar{\kappa}}k_{3}^{\mu}e_{1\mu\bar{\nu}}e_{2}^{\rho}{}_{\bar{\sigma}}e_{3\rho\bar{\kappa}}+\frac{1}{k_{1}^{2}}\bar{k}_{1}^{\bar{\mu}}\bar{k}_{1}^{\bar{\nu}}k_{1}^{\mu}k_{1}^{\nu}\bar{k}_{1}^{\bar{\rho}}k_{1}^{\rho}e_{1\mu\bar{\mu}}e_{2\nu\bar{\nu}}e_{3\rho\bar{\rho}}-\frac{1}{k_{1}^{2}}\bar{k}_{1}^{\bar{\mu}}\bar{k}_{1}^{\bar{\nu}}k_{1}^{\mu}k_{1}^{\nu}\bar{k}_{1}^{\bar{\rho}}k_{2}^{\rho}e_{1\mu\bar{\mu}}e_{2\nu\bar{\nu}}e_{3\rho\bar{\rho}}\,$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\left.+\frac{1}{k_{1}^{2}}\bar{k}_{1}^{\bar{\mu}}\bar{k}_{1}^{\bar{\nu}}k_{1}^{\nu}k_{1}^{\rho}\bar{k}_{2}^{\bar{\rho}}k_{2}^{\mu}e_{1\mu\bar{\mu}}e_{2\nu\bar{\nu}}e_{3\rho\bar{\rho}}-\frac{1}{k_{1}^{2}}\bar{k}_{1}^{\bar{\mu}}\bar{k}_{1}^{\bar{\nu}}k_{1}^{\nu}k_{1}^{\rho}\bar{k}_{2}^{\bar{\rho}}k_{3}^{\mu}e_{1\mu\bar{\mu}}e_{2\nu\bar{\nu}}e_{3\rho\bar{\rho}}\right]\,.$

Unlike the quadratic case, the cubic action contains terms that would give
rise to non-local terms once we transform to coordinate space, as in the Yang-
Mills theory. As we learned from that case, the introduction of an auxiliary
scalar field solves this problem. Considering this, the action becomes

$\displaystyle S_{\rm{HD}/\rm{DC}}^{(3)}$ $\displaystyle=$
$\displaystyle-\frac{1}{4(2\pi)^{D/2}}\int
dK_{1}dK_{2}dK_{3}\delta(K_{1}+K_{2}+K_{3})\left[k_{1}^{2}k_{1}^{\rho}\bar{k}_{1}^{\bar{\sigma}}e_{1\mu\bar{\nu}}e_{2}^{\mu\bar{\nu}}e_{3\rho\bar{\sigma}}-k_{1}^{2}k_{2}^{\rho}\bar{k}_{1}^{\bar{\sigma}}e_{1\mu\bar{\nu}}e_{2}^{\mu\bar{\nu}}e_{3\rho\bar{\sigma}}\right.\,$
(32) $\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
+k_{1}^{2}k_{2}^{\mu}\bar{k}_{1}^{\bar{\sigma}}e_{1\mu\bar{\nu}}e_{2}^{\rho\bar{\nu}}e_{3\rho\bar{\sigma}}-k_{1}^{2}k_{3}^{\mu}\bar{k}_{1}^{\bar{\sigma}}e_{1\mu\bar{\nu}}e_{2}^{\rho\bar{\nu}}e_{3\rho\bar{\sigma}}-k_{1}^{2}k_{2}^{\mu}\bar{k}_{2}^{\bar{\sigma}}e_{1\mu\bar{\nu}}e_{2}^{\rho\bar{\nu}}e_{3\rho\bar{\sigma}}+k_{1}^{2}k_{3}^{\mu}\bar{k}_{2}^{\bar{\sigma}}e_{1\mu\bar{\nu}}e_{2}^{\rho\bar{\nu}}e_{3\rho\bar{\sigma}}\,$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
-k_{1}^{\mu}k_{1}^{\rho}k_{1}^{\lambda}\bar{k}_{1}^{\bar{\sigma}}e_{1\mu\bar{\nu}}e_{2\rho}{}^{\bar{\nu}}e_{3\lambda\bar{\sigma}}+k_{1}^{\mu}k_{1}^{\rho}k_{2}^{\lambda}\bar{k}_{1}^{\bar{\sigma}}e_{1\mu\bar{\nu}}e_{2\rho}{}^{\bar{\nu}}e_{3\lambda\bar{\sigma}}-k_{2}^{\mu}k_{1}^{\rho}k_{1}^{\lambda}\bar{k}_{1}^{\bar{\sigma}}e_{1\mu\bar{\nu}}e_{2\rho}{}^{\bar{\nu}}e_{3\lambda\bar{\sigma}}\,$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
+k_{3}^{\mu}k_{1}^{\rho}k_{1}^{\lambda}\bar{k}_{1}^{\bar{\sigma}}e_{1\mu\bar{\nu}}e_{2\rho}{}^{\bar{\nu}}e_{3\lambda\bar{\sigma}}+k_{2}^{\mu}k_{1}^{\rho}k_{1}^{\lambda}\bar{k}_{2}^{\bar{\sigma}}e_{1\mu\bar{\nu}}e_{2\rho}{}^{\bar{\nu}}e_{3\lambda\bar{\sigma}}-k_{3}^{\mu}k_{1}^{\rho}k_{1}^{\lambda}\bar{k}_{2}^{\bar{\sigma}}e_{1\mu\bar{\nu}}e_{2\rho}{}^{\bar{\nu}}e_{3\lambda\bar{\sigma}}\,$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
-\bar{k}_{1}^{\bar{\nu}}\bar{k}_{1}^{\bar{\sigma}}\bar{k}_{1}^{\bar{\kappa}}k_{1}^{\rho}e_{1\mu\bar{\nu}}e_{2}^{\mu}{}_{\bar{\sigma}}e_{3\rho\bar{\kappa}}+\bar{k}_{1}^{\bar{\nu}}\bar{k}_{1}^{\bar{\sigma}}\bar{k}_{1}^{\bar{\kappa}}k_{2}^{\rho}e_{1\mu\bar{\nu}}e_{2}^{\mu}{}_{\bar{\sigma}}e_{3\rho\bar{\kappa}}+\bar{k}_{1}^{\bar{\nu}}\bar{k}_{1}^{\bar{\sigma}}\bar{k}_{2}^{\bar{\kappa}}k_{2}^{\mu}e_{1\mu\bar{\nu}}e_{2}^{\rho}{}_{\bar{\sigma}}e_{3\rho\bar{\kappa}}\,$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
-\bar{k}_{1}^{\bar{\nu}}\bar{k}_{1}^{\bar{\sigma}}\bar{k}_{2}^{\bar{\kappa}}k_{3}^{\mu}e_{1\mu\bar{\nu}}e_{2}^{\rho}{}_{\bar{\sigma}}e_{3\rho\bar{\kappa}}+k_{1}^{\mu}k_{1}^{\rho}\bar{k}_{1}^{\bar{\nu}}\bar{k}_{1}^{\bar{\sigma}}\phi_{1}e_{2\mu\bar{\nu}}e_{3\rho\bar{\sigma}}-k_{1}^{\mu}k_{2}^{\rho}\bar{k}_{1}^{\bar{\nu}}\bar{k}_{1}^{\bar{\sigma}}\phi_{1}e_{2\mu\bar{\nu}}e_{3\rho\bar{\sigma}}\,$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\left.+k_{2}^{\mu}k_{2}^{\rho}\bar{k}_{1}^{\bar{\nu}}\bar{k}_{1}^{\bar{\sigma}}\phi_{2}e_{1\mu\bar{\nu}}e_{3\rho\bar{\sigma}}-k_{3}^{\mu}k_{2}^{\rho}\bar{k}_{1}^{\bar{\nu}}\bar{k}_{1}^{\bar{\sigma}}\phi_{2}e_{1\mu\bar{\nu}}e_{3\rho\bar{\sigma}}\right]\,.$

The final step in the procedure consists in transforming back to coordinate
space. We finally obtain

$\displaystyle S_{\rm{HD}/\rm{DC}}^{(3)}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}\int d^{D}x\
d^{D}\bar{x}\left[\Box\partial^{\rho}\bar{\partial}^{\bar{\sigma}}e_{\mu\bar{\nu}}e^{\mu\bar{\nu}}e_{\rho\bar{\sigma}}-\Box\bar{\partial}^{\bar{\sigma}}e_{\mu\bar{\nu}}\partial^{\rho}e^{\mu\bar{\nu}}e_{\rho\bar{\sigma}}+\Box\bar{\partial}^{\bar{\sigma}}e_{\mu\bar{\nu}}\partial^{\mu}e^{\rho\bar{\nu}}e_{\rho\bar{\sigma}}-\Box\bar{\partial}^{\bar{\sigma}}e_{\mu\bar{\nu}}e^{\rho\bar{\nu}}\partial^{\mu}e_{\rho\bar{\sigma}}\right.\,$
(33) $\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -\Box
e_{\mu\bar{\nu}}\partial^{\mu}\bar{\partial}^{\bar{\sigma}}e^{\rho\bar{\nu}}e_{\rho\bar{\sigma}}+\Box
e_{\mu\bar{\nu}}\bar{\partial}^{\bar{\sigma}}e^{\rho\bar{\nu}}\partial^{\mu}e_{\rho\bar{\sigma}}-\partial^{\mu}\partial^{\rho}\partial^{\lambda}\bar{\partial}^{\bar{\sigma}}e_{\mu\bar{\nu}}e_{\rho}{}^{\bar{\nu}}e_{\lambda\bar{\sigma}}+\partial^{\mu}\partial^{\rho}\bar{\partial}^{\bar{\sigma}}e_{\mu\bar{\nu}}\partial^{\lambda}e_{\rho}{}^{\bar{\nu}}e_{\lambda\bar{\sigma}}\,$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
-\partial^{\rho}\partial^{\lambda}\bar{\partial}^{\bar{\sigma}}e_{\mu\bar{\nu}}\partial^{\mu}e_{\rho}{}^{\bar{\nu}}e_{\lambda\bar{\sigma}}+\partial^{\rho}\partial^{\lambda}\bar{\partial}^{\bar{\sigma}}e_{\mu\bar{\nu}}e_{\rho}{}^{\bar{\nu}}\partial^{\mu}e_{\lambda\bar{\sigma}}+\partial^{\rho}\partial^{\lambda}e_{\mu\bar{\nu}}\partial^{\mu}\bar{\partial}^{\bar{\sigma}}e_{\rho}{}^{\bar{\nu}}e_{\lambda\bar{\sigma}}-\partial^{\rho}\partial^{\lambda}e_{\mu\bar{\nu}}\bar{\partial}^{\bar{\sigma}}e_{\rho}{}^{\bar{\nu}}\partial^{\mu}e_{\lambda\bar{\sigma}}\,$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
-\bar{\partial}^{\bar{\nu}}\bar{\partial}^{\bar{\sigma}}\bar{\partial}^{\bar{\kappa}}\partial^{\rho}e_{\mu\bar{\nu}}e^{\mu}{}_{\bar{\sigma}}e_{\rho\bar{\kappa}}+\bar{\partial}^{\bar{\nu}}\bar{\partial}^{\bar{\sigma}}\bar{\partial}^{\bar{\kappa}}e_{\mu\bar{\nu}}\partial^{\rho}e^{\mu}{}_{\bar{\sigma}}e_{\rho\bar{\kappa}}+\bar{\partial}^{\bar{\nu}}\bar{\partial}^{\bar{\sigma}}e_{\mu\bar{\nu}}\bar{\partial}^{\bar{\kappa}}\partial^{\mu}e^{\rho}{}_{\bar{\sigma}}e_{\rho\bar{\kappa}}-\bar{\partial}^{\bar{\nu}}\bar{\partial}^{\bar{\sigma}}e_{\mu\bar{\nu}}\bar{\partial}^{\bar{\kappa}}e^{\rho}{}_{\bar{\sigma}}\partial^{\mu}e_{\rho\bar{\kappa}}\,$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\left.+\partial^{\mu}\partial^{\rho}\bar{\partial}^{\bar{\nu}}\bar{\partial}^{\bar{\sigma}}\phi
e_{\mu\bar{\nu}}e_{\rho\bar{\sigma}}-\partial^{\mu}\bar{\partial}^{\bar{\nu}}\bar{\partial}^{\bar{\sigma}}\phi\partial^{\rho}e_{\mu\bar{\nu}}e_{\rho\bar{\sigma}}+\partial^{\mu}\partial^{\rho}\phi\bar{\partial}^{\bar{\nu}}\bar{\partial}^{\bar{\sigma}}e_{\mu\bar{\nu}}e_{\rho\bar{\sigma}}-\partial^{\rho}\phi\bar{\partial}^{\bar{\nu}}\bar{\partial}^{\bar{\sigma}}e_{\mu\bar{\nu}}\partial^{\mu}e_{\rho\bar{\sigma}}\right]\,.$

Just as in the quadratic case, we will set $x=\bar{x}$ for the pure gravity
case ($e_{\mu\bar{\nu}}\sim h_{\mu\nu}$, $\phi=0$) in order to compare with
the cubic contributions coming from (22). As a first observation we notice
that the cubic action obtained from the double copy of the
$\left(DF\right)^{2}$ theory contains further contributions apart from the
Weyl square action. Some of these contributions vanish after imposing a
particular gauge, e.g. the harmonic gauge condition, as well as by choosing a
particular space-time dimension. It would be interesting to fully understand
the physical interpretation of the action (33) in terms of symmetry arguments.
In the ideal case, one expects a four derivative gauge theory whose double
copy map matches with the higher-derivative DFT introduced in MN . It is
important to take into account that, once we are interested in making contact
with a theory coming from a double formalism, it is possible that there are
remnants of $O(D,D)$ symmetry in addition to conformal symmetry. Therefore,
one possibility is to study this formalism under a generic $D$-dimensional
toroidal compactification and to inspect how the $O(D,D)$ multiplets are
constructed.

##  IV Discussion

As well known, DFT is defined on a $2D$-dimensional space with coordinates
$X_{M}=(\tilde{x}_{\mu},x^{\mu})$ where $M$ is in the fundamental
representation of the $O(D,D)$ group. The field content is the generalized
metric ${\cal H}_{MN}$, a symmetric $O(D,D)$ tensor encoding the standard
metric $g_{\mu\nu}$ and the Kalb-Ramond field $b_{\mu\nu}$, and the
generalized dilaton $d$ which is related with the standard supergravity
dilaton $\phi$.

Besides $O(D,D)$ symmetry, the theory is invariant under a generalized notion
of diffeomorphism transformations which accounts for usual diffeomorphisms as
well as abelian gauge transformations of the $b$-field. Additional symmetries
can be incorporated, for instance, the tangent space is enhanced with an
extended Lorentz symmetry. Supersymmetry can also be naturally considered.
Closure of the generalized diffeomorphisms require the imposition of the so-
called strong constraint

$\partial_{M}\star\partial^{M}\star=0\,,\ \qquad\
\partial_{M}\partial^{M}\star=0\,,$ (34)

where $\star$ denotes $O(D,D)$ fields or combinations of them.

The theory is equipped with an action principle from which one can derive
equations of motion

$S_{DFT}=\int d^{D}xd^{D}\tilde{x}\ {\cal R}\left({\cal H},d\right)\,.$ (35)

As we can see, the Lagrangian of DFT is given by the generalized Ricci scalar
${\cal R}$, which depends on both the generalized metric and the generalized
dilaton,

$\displaystyle{\cal R}$ $\displaystyle=$ $\displaystyle\frac{1}{4}{\cal
H}^{MN}\partial_{M}{\cal H}^{KL}\partial_{N}{\cal H}_{KL}-{\cal
H}^{MN}\partial_{N}{\cal H}^{KL}\partial_{L}{\cal H}_{MK}$ (36)
$\displaystyle+8{\cal H}^{MN}\partial_{M}\partial_{N}d+8\partial_{M}{\cal
H}^{MN}\partial_{N}d$ $\displaystyle-8{\cal
H}^{MN}\partial_{M}d\partial_{N}d-2\partial_{M}\partial_{N}{\cal H}^{MN}\,.$

For a particular solution of the strong constraint, this action reproduces
exactly the NS-NS sector of string theory at leading order in
$\alpha^{\prime}$.

At the moment we do not have a conformal formulation of DFT in the sense that
it is not clear how to implement conformal symmetry in the double space of the
theory. So it is difficult to compare our cubic action (33) with a given
result derived directly from DFT. We speculate this action could be the cubic
contribution from an appropriate conformal DFT defined in a suitable
generalized Siegel gauge, a sort of higher-derivative realization of the
interpretation given in Ref. HohmDC . Indeed, our results of the previous
section clearly suggest the existence of a relation between conformal gravity
and some higher-derivative extension of the usual DFT. This scenario is
possible due to the symmetric structure between $k$ and $\bar{k}$ that appears
in equations (19) and (30), similar to the ones that emerge in the pure Yang-
Mills case. This gives us the opportunity to venture into some intriguing
proposals that could account for this relation, which we quickly discuss
below.

  * •

Generalized Kerr-Schild ansatz – In Ref. KL  a duality invariant analogous of
the widely known Kerr-Schild (KS) ansatz was introduced. There, the background
generalized metric is linearly and exactly perturbed by a pair of generalized
null vectors

${\cal H}_{MN}=\widetilde{{\cal
H}}_{MN}+\kappa\left(K_{M}\bar{K}_{N}+K_{N}\bar{K}_{M}\right)\,,$ (37)

with $\kappa$ an arbitrary parameter which quantifies the order of the
perturbations and tildes denotes background quantities. Among other
conditions, this generalization of the KS ansatz leads to a linearization of
the equations of motion of DFT. The parameterization of (37) leads to

$\displaystyle g_{\mu\nu}$ $\displaystyle=$
$\displaystyle\widetilde{g}_{\mu\nu}+\frac{\kappa}{1+{\textstyle\frac{1}{2}}\kappa\left(k\cdot\bar{k}\right)}k_{(\mu}\bar{k}_{\nu)}\,$
$\displaystyle b_{\mu\nu}$ $\displaystyle=$
$\displaystyle\widetilde{b}_{\mu\nu}-\frac{\kappa}{1+{\textstyle\frac{1}{2}}\kappa\left(k\cdot\bar{k}\right)}k_{[\mu}\bar{k}_{\nu]}\,.$
(38)

Contrary to the ordinary KS ansatz, now the $b$-field can be perturbed but as
a consequence, the perturbation of the metric is no longer linear. However, it
was proven in the same work that, with this perturbations, the classical
double copy structure at the level of the equations of motion can be extended
to the entire string NS-NS sector.

In Ref. AE-HD:21  the authors extended the analysis to heterotic DFT at order
$\alpha^{\prime}$ and using the classical double copy, they found higher-
derivative corrections to the Maxwell equations at order $\kappa$ in the
perturbations. Based on this result one of our proposals is to continue this
path by extending the order of the perturbations to $\kappa^{2}$ which match
with the order of gauge fields expected for the equations of motion coming
from (15).

  * •

Conformal symmetry in DFT – A different path consists in introducing a new
symmetry in the framework of DFT such that Weyl transformations at the
supergravity level could be obtained. This seems to be possible with the
introduction of an $O(D,D)$ invariant $\Phi$ parameterizing double Weyl
transformations.

Consideration of this kind of transformation leads to a particular
parameterization of the generalized metric as well as a deformation of the
$O(D,D)$ metric. As a consequence, the theory requires a coupling with an
extra scalar field to have a conformal invariant action at leading order.

  * •

Generalized Weyl tensor – It is well known that the generalized Riemann tensor
of DFT is not fully determined RiemannDFT  since there are not enough
compatibility conditions in the double geometry to fully determine the
generalized connection $\Gamma_{MNP}$ in terms of the degrees of freedom of
the theory. This makes the direct construction of higher order terms with the
structure

${\cal R}_{MNPQ}{\cal R}^{MNPQ}\,$

a difficult task.

This, however, has not been a sufficiently great obstacle to calculate higher-
derivative extensions of DFT. A very interesting construction in this sense
was put forward in Ref. MN , based on a generalization of the Green-Schwarz
mechanism of anomaly cancellation. This theory depends on two parameters whose
values are related to different theories: bosonic and heterotic strings and
also HSZ theory HSZ . Therefore it would be interesting to explore if it is
possible to obtain a combination of the parameters such that conformal gravity
could be obtained after supergravity reduction.

A different but related approach is to look for the existence of a two-
derivative combination of the DFT fields ${\cal C}_{MNPQ}$, mimicking the
usual Weyl tensor, such that an action

${\cal C}_{MNPQ}{\cal C}^{MNPQ}\,$

could be constructed and reduced to conformal gravity.

##  V Outlook

We used the double copy prescription introduced in HohmDC  as a guiding
principle to study the relation between higher-derivative gauge theories and
higher-derivative extensions of double field theory (DFT). Exploring the
double copy structure of the action (15) we obtained promising results for
addressing the generalization of conformal symmetry to the double space in
which DFT is formulated.

It is known that the double copy of (15) is related to a gravity theory with
conformal symmetry. As it is possible to express this double copy in a double
space formalism, we suggest a relation between conformal gravity and a higher-
derivative deformation of DFT. Furthermore, at quadratic order our
prescription reduces to Weyl gravity upon enforcing a gauge fixing condition.

Here we prove that in the pure gravity case, the quadratic action (III)
precisely agrees with the quadratic graviton terms of conformal gravity. This
is expected since these contributions are common to all theories of conformal
gravity. The cubic action has additional contributions beyond the Weyl square
action. Some of these, but not all, vanishes with specific conditions. The
physical interpretation of the action (32) in terms of symmetry arguments is
of interest, and different ways of approaching it were proposed.

Finally, based on the relation of the action (15) and conformal gravity and
our construction in double space, we discuss different alternatives to
consider conformal symmetry in the context of Double Field Theory. We believe
that the proposal discussed in this paper is an important step further in the
construction of a non-perturbative conformal double geometry with applications
to scattering amplitudes.

### Acknowledgements

We would like to kindly thank to Felipe Diaz-Jaramillo, Olaf Hohm and Silvia
Nagy for very interesting comments in the first manuscript of this work. EL is
supported by the Croatian Science Foundation project IP-2019-04-4168. GM
acknowledges the hospitality of the Mani L. Bhaumik Institute for Theoretical
Physics at UCLA, where part of this research was carried out. The work of GM
has been partially supported by Conselho Nacional de Desenvolvimento
Científico e Tecnológico - CNPq under grant 317548/2021-2 and Fundacão Carlos
Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro - FAPERJ under
grants E-26/202.725/2018 and E-26/201.142/2022. The work of JAR is supported
by CONICET, and would also like to thank the Hanrieder Foundation for
Excellence and the Max Planck Institut für Physik (MPP) for support and
hospitality during the early stages of this project.

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