Heterotic Kerr-Schild Double Field Theory and its double Yang-Mills
formulation

Eric Lescano1 and Sourav Roychowdhury2

1 Instituto de Astronomía y Física del Espacio (IAFE-UBA-CONICET)
Ciudad Universitaria, Pabellón IAFE, Ciudad de Buenos Aires (1428), Argentina

2 Chennai Mathematical Institute
SIPCOT IT Park, Siruseri 603 103, India

<EMAIL_ADDRESS><EMAIL_ADDRESS>
Abstract

We present a formulation of heterotic Double Field Theory (DFT), where the
fundamental fields are in $O(D,D)$ representations. The theory is obtained
splitting an $O(D,D+K)$ duality invariant DFT. This procedure produces a
Green-Schwarz mechanism for the generalized metric, and a fundamental gauge
field which transforms as a gauge connection only to leading order. After
parametrization, the former induces a non-covariant transformation on the
metric tensor, which can be removed considering field redefinitions, and an
ordinary Green-Schwarz mechanism on the b-field. Within this framework we
explore perturbative properties of heterotic DFT. We use a relaxed version of
the generalized Kerr-Schild ansatz (GKSA), where the generalized background
metric is perturbed up to quadratic order considering a single null vector and
the gauge field is linearly perturbed before parametrization. Finally we
compare the dynamics of the gauge field and the generalized metric in order to
inspect the behavior of the classical double copy correspondence at the DFT
level.

###### Contents

  1. 1 Introduction
  2. 2 Review of heterotic Kerr-Schild Double Field Theory
    1. 2.1 Basics of heterotic Double Field Theory
    2. 2.2 Heterotic Kerr-Schild ansatz in metric formulation
    3. 2.3 Heterotic Kerr-Schild ansatz in frame formulation
  3. 3 Double Yang-Mills formulation
    1. 3.1 Transformation rules
    2. 3.2 Flux formulation
    3. 3.3 Action principle
    4. 3.4 Leading order terms
    5. 3.5 Parametrization
  4. 4 Double Yang-Mills and the relaxed GKSA
    1. 4.1 Generalized metric formulation
    2. 4.2 Generalized frame formulation
    3. 4.3 Field redefinitions
  5. 5 Application
    1. 5.1 Classical double copy at the DFT level
  6. 6 Discussion
  7. 7 Summary
  8. A Relaxed GKSA and the DFT Lagrangian

##  1 Introduction

T-duality is an exact symmetry of string theory [1]. Its low energy limit, or
supergravity limit, can be written in a duality invariant way before
compactification within the framework of Double Field Theory (DFT) [2] [3]
111For reviews we recommend [10] and Section 2 part A of [11]. We follow the
notation of the latter.. Among the different formulations of string theory
here we focus on the low energy limit of the heterotic string and its embedded
into the double geometry. The canonical way to construct heterotic DFT was
given in [4]. Following the proposal of [5], the standard construction consist
of regrouping some of the fundamental fields into a generalized metric. This
object is constructed reorganizing the D-dimensional metric tensor, the
b-field and a non-Abelian gauge field, i.e., ${\cal H}={\cal H}(g,b,A)$. The
generalized metric is a multiplet of the duality group and together with the
generalized dilaton are the fundamental fields of heterotic DFT. Moreover, in
[6] the generalized frame formalism of heterotic DFT was constructed, and the
role of the generalized metric is replaced by the generalized version of the
vielbein, which has the same field dependence, ${\cal E}={\cal E}(g,b,A)$.
Independently of the fundamental fields of the theory, in the standard
formulation of heterotic DFT the gauge field and the gauge symmetry only
appear upon parametrization.

On the other hand, there exists certain tension between the generalized metric
formulation of DFT and its generalized frame formulation since both schemes
are not always equivalent [7]. Similarly to what happens in general
relativity, the generalized frame contains extra degrees of freedom which give
rise to a gauge fixing procedure: some components of this field are fixed and
extra conditions appear if/when, for instance, one desires to work considering
perturbations around a background. One simple and powerful proposal to explore
this tension is the generalized Kerr-Schild ansatz (GKSA). This proposal was
introduced in [8] for ordinary DFT, while the heterotic version of the ansatz
was given in [9]. In both cases the ansatz consists of a linear perturbation
of a generalized metric/frame background using a pair of generalized null
vectors, plus an arbitrary perturbation for the generalized dilaton. When
inspecting the form of the ansatz at the supergravity level, the generalized
metric formalism produces perturbations around a background metric tensor,
$g_{o}$, a background b-field, $b_{o}$ and a background gauge field, $A_{o}$,
i.e.,

$\displaystyle g_{\mu\nu}$ $\displaystyle=$ $\displaystyle
g_{o\mu\nu}-\frac{\kappa}{1+\frac{1}{2}\kappa
l.\bar{l}}l_{(\mu}\bar{l}_{\nu)}$ (1.1) $\displaystyle b_{\mu\nu}$
$\displaystyle=$ $\displaystyle b_{o\mu\nu}-\frac{\kappa}{1+\frac{1}{2}\kappa
l.\bar{l}}l_{[\mu}(\bar{l}_{\nu]}-\frac{1}{\sqrt{2}}j_{i}A_{\nu]}{}^{i})$
(1.2) $\displaystyle A_{\mu i}$ $\displaystyle=$ $\displaystyle A_{o\mu
i}+\frac{1}{\sqrt{2}}\frac{\kappa}{1+\frac{1}{2}\kappa
l.\bar{l}}l_{\mu}j_{i}\,,$ (1.3)

with $\kappa$ an order parameter, $l$ is a null vector and $\bar{l}$ and $j$
satisfy

$\displaystyle\bar{l}^{2}+j^{2}=0\,.$ (1.4)

Both $l$ and $\bar{l}$, as well as $j$ satisfy geodesic equations. These
relations are useful to simplify part of the dynamics, and can be easily
imposed at the DFT level [9]. Equations (1.1)-(1.3), plus a perturbation for
the background dilaton, integrate the family of theories that can be described
using the GKSA. Some of them are the charged black string [12] and the charged
dilaton black hole [13], both studied in detail in [9] using the generalized
metric formalism. As a corollary, the case $j=0$ and $l=\bar{l}$ describes the
subfamily of theories which can be described using the ordinary Kerr-Schild
ansatz [14].

Contrary to expectation, the family of theories that can be studied using the
GKSA in the generalized frame formalism of DFT is slightly smaller than the
theories that can be described in the generalized metric formulation, since
the gauge fixing condition forces $j=0$, and then both $l$ and $\bar{l}$ are
null vectors. The main goal of this work is two-folded:

  * $-$

On the one hand, we reformulate heterotic DFT in a double Yang-Mills forms,
similarly to the proposal [15]. Our proposal is equivalent to the standard
heterotic DFT, and it is constructed writing $O(D,D+K)$ multiplets in terms of
$O(D,D)$ multiplets. This kind of formulation is advantageous because it
includes a generalized gauge field/symmetry at the DFT level, and the
perturbations of this field can be imposed before parametrization,
independently of the fundamental fields of the theory.

  * $-$

On the other hand, we relax the form of the GKSA in order to obtain
perturbations for the gauged field in the generalized frame formulation of
DFT. The relaxed form of the GKSA was firstly used in [16], and here we adapt
the proposal to use it in heterotic DFT.

Furthermore, the previous points allow us to explore the double copy relation
at the DFT level: the equation of motion of the generalized metric can be
contracted with a generalized killing vector and single/zeroth copy can be
explore before parametrization.

This work is organized as follows: in Section 2 we review the basic aspects of
DFT and the GKSA in the generalized metric/frame formulation. In Section 3 we
introduce the double Yang-Mills formulation of heterotic DFT, while we explore
the relaxed form of the GKSA in Section 4. In Section 5 we explore the
classical double copy relation at the DFT level. In Section 6 we discuss the
results of this work and in Section 7 we conclude with a summary.

##  2 Review of heterotic Kerr-Schild Double Field Theory

###  2.1 Basics of heterotic Double Field Theory

Heterotic DFT is defined on a double space with coordinates $X^{\cal M}$ which
transform under the fundamental representation of the symmetry group
$G=O(D,D+K)$, with ${\cal M}=0,\dots,D-1+K$, and $K$ the dimension of the
heterotic gauge group.

The theory is invariant under a global $G$ symmetry which infinitesimally
reads

$\displaystyle\delta_{G}V_{\cal M}=V_{\cal N}h^{\cal N}{}_{\cal M}\,,$ (2.1)

where $V_{\cal M}$ is a generic $G$-multiplet, $h\in o(D,D+K)$ is the
$G$-parameter. The invariant metric of $G$ is $\eta_{\cal MN}\in G$ and
$G$-invariance forces

$\displaystyle h_{\cal MN}=-h_{\cal NM}\,.$ (2.2)

We use $\eta$ and $\eta^{-1}$ in order to lower and raise all the G-indices.

Infinitesimal generalized diffeomorphisms are defined through a generalized
Lie derivative,

$\displaystyle\hat{\cal L}_{\xi}V_{\cal M}=\xi^{\cal N}\partial_{\cal
N}V_{\cal M}+(\partial_{\cal M}\xi^{\cal N}-\partial^{\cal N}\xi_{\cal
M})V_{\cal N}+f_{\cal MNP}\xi^{\cal N}V^{\cal P}+t\partial_{\cal M}\xi^{\cal
M}\,,$ (2.3)

where $V_{\cal M}$ is an arbitrary generalized tensor, $t$ is a weight
constant and $f_{\cal MNP}$ is the generalized version of the structure
constants which satisfy

$\displaystyle f_{{\cal MNP}}=f_{[{\cal MNP}]}\,,\qquad f_{[\cal MN}{}^{\cal
R}f_{{\cal P}]R}{}^{\cal Q}=0\,.$ (2.4)

The strong constraint

$\displaystyle\partial_{\cal M}\blacktriangledown\partial^{\cal
M}\blacktriangle$ $\displaystyle=$ $\displaystyle 0\,,$ (2.5)
$\displaystyle\partial_{\cal M}(\partial^{\cal M}\blacktriangledown)$
$\displaystyle=$ $\displaystyle 0\,,$ (2.6) $\displaystyle f^{\cal
MNP}\partial_{\cal M}\blacktriangledown$ $\displaystyle=$ $\displaystyle 0\,,$
(2.7)

is a sufficient condition to ensure the closure of the generalized
diffeomorphisms. In the previous expression $\blacktriangledown$ and
$\blacktriangle$ are generic fields or parameters in heterotic DFT.

Considering a generic double vector $V_{\cal N}$, the covariant derivative can
be defined as

$\displaystyle\nabla_{\cal M}V_{\cal N}=\partial_{\cal M}V_{\cal
N}-\Gamma_{\cal MN}{}^{\cal P}V_{\cal P}\,,$ (2.8)

where $\Gamma_{\cal MNP}$ is a generalized affine connection. We demand

$\displaystyle\nabla_{\cal M}{\cal H}_{\cal NP}$ $\displaystyle=$
$\displaystyle 0\,,\quad\nabla_{\cal M}{\cal\eta}_{\cal NP}=0\,,$ (2.9)

and a vanishing generalized torsion

$\displaystyle\Gamma_{[\cal MNP]}=0\,,$ (2.10)

in order to determine some projections of the generalized affine connection
[17].

Heterotic DFT is also invariant under a local double Lorentz ${\cal
H}=O(D-1,1)_{L}\times O(1,D-1+K)_{R}$ symmetry generated infinitesimally by a
generalized parameter $\Gamma_{\cal AB}$ where ${\cal A}=(\underline{\cal
A},\overline{\cal A})$ splitting into $O(D-1,1)_{L}$ and $O(1,D-1+K)_{R}$
vector indices, $\underline{\cal A}={\underline{A}}=0,\dots,D-1$ and
$\overline{A}=(\overline{A},\overline{i})=0,\dots,D-1+K$, i.e.,

$\displaystyle\delta_{\cal H}V_{\cal A}=V_{\cal B}\Gamma^{\cal B}{}_{\cal
A}\,,$ (2.11)

for a generic ${\cal H}$-vector. The $\cal H$-invariance of $\eta_{\cal AB}$
forces $\Gamma_{\cal AB}=-\Gamma_{\cal BA}\,$.

The fundamental fields of the theory consist of a generalized frame ${\cal
E}_{\cal M}{}^{\cal A}$ and a generalized dilaton field $d$. The frame-
formulation of DFT demands the existence of two constant, symmetric and
invertible ${\cal H}$-invariant metrics $\eta_{{\cal AB}}$ and ${\cal
H}_{{\cal AB}}$. The former is used to raise and lower the indices that are
rotated by ${\cal H}$ and the latter is constrained according to

$\displaystyle{\cal H}_{\cal A}{}^{\cal C}{\cal H}_{\cal C}{}^{\cal
B}=\delta_{\cal A}^{\cal B}\,.$ (2.12)

The generalized frame is constrained to relate the metrics $\eta_{{\cal AB}}$
and $\eta_{{\cal MN}}$ and defines a generalized metric ${\cal H}_{\cal MN}$
from ${\cal H}_{\cal AB}$

$\eta_{\cal AB}={\cal E}^{\cal M}{}_{\cal A}\eta_{\cal MN}{\cal E}^{\cal
N}{}_{\cal B}\,,\quad{\cal H}_{\cal MN}={\cal E}_{\cal M}{}^{\cal A}{\cal
H}_{\cal AB}{\cal E}_{\cal N}{}^{\cal B}\,.$ (2.13)

${\cal H}_{\cal MN}$ is also an element of $O(D,D+K)$, i.e.

${\cal H}_{\cal MP}\eta^{\cal PQ}{\cal H}_{\cal QN}=\eta_{\cal MN}\,.$ (2.14)

The DFT curved projectors are defined as

$\displaystyle{\cal P}_{\cal MN}=\frac{1}{2}\left(\eta_{\cal MN}-{\cal
H}_{\cal MN}\right)\ \ {\rm and}\ \ \overline{\cal P}_{\cal
MN}=\frac{1}{2}\left(\eta_{\cal MN}+{\cal H}_{\cal MN}\right)\ ,$ (2.15)

and the same can be done with $\eta_{\cal AB}$ and ${\cal H}_{\cal AB}$ to
define ${\cal P}_{AB}$ , $\overline{\cal P}_{{\cal AB}}$. We use the
convention that the projectors and their inverses lower and raise projected
indices.

The flat covariant derivative acting on a generic vector $V_{\cal B}$ is

$\displaystyle{\cal D}_{\cal A}V_{\cal B}={\cal E}_{\cal A}V_{\cal
B}+{\omega}_{\cal AB}{}^{\cal C}V_{\cal C}\,,$ (2.16)

where ${\cal E}_{\cal A}=\sqrt{2}{\cal E}^{\cal M}{}_{\cal A}\partial_{\cal
M}$ and ${\omega}_{\cal AB}{}^{\cal C}$ is the generalized spin connection
which satisfies

${\omega}_{\cal ABC}=-{\omega}_{\cal
ACB}\,\qquad\mathrm{and}\qquad{\omega}_{\cal
A\overline{B}\underline{C}}={\omega}_{\cal A\underline{B}\overline{C}}=0\,.$
(2.17)

Unlike general relativity, DFT consists of a generalized notion of geometry
and there are not enough compatibility conditions to fully determine a
generalized spin connection, $\omega_{\cal ABC}$. Only the totally
antisymmetric and trace parts of $\omega_{\cal ABC}$ can be determined in
terms of ${\cal E}_{\cal M}{}^{\cal A}$ and $d$, i.e.

$\omega_{\cal[ABC]}=-{\cal E}_{\cal[A}{\cal E}^{\cal N}{}_{\cal B}{\cal
E}_{\cal NC]}-\frac{\sqrt{2}}{3}f_{\cal MNP}{\cal E}^{\cal M}{}_{\cal A}{\cal
E}^{\cal N}{}_{\cal B}{\cal E}^{\cal P}{}_{\cal C}\equiv-\frac{1}{3}{\cal
F}_{\cal ABC}\,,$ (2.18) $\omega_{\cal BA}{}^{\cal
B}=-\sqrt{2}e^{2d}\partial_{\cal M}({\cal E}^{\cal M}{}_{\cal
A}e^{-2d})=-{\cal F}_{\cal A}\,.$ (2.19)

The action principle of the $O(D,D+K)$ invariant DFT, in the generalized
metric formulation, is given by

$\displaystyle S=\int d^{2D+K}Xe^{-2d}{\cal L}\,,$ (2.20)

where,

$\displaystyle{\cal L}$ $\displaystyle=$ $\displaystyle\frac{1}{8}{\cal
H}^{\cal MN}\partial_{\cal M}{{\cal H}^{\cal KL}}\partial_{\cal N}{{\cal
H}_{\cal KL}}-\frac{1}{2}{\cal H}^{\cal MN}\partial_{\cal N}{{\cal H}^{\cal
KL}}\partial_{\cal L}{{\cal H}_{\cal MK}}$ (2.21) $\displaystyle+4{\cal
H}^{\cal MN}\partial_{\cal M}{d}\partial_{\cal N}{d}-2\partial_{\cal M}{{\cal
H}^{\cal MN}}\partial_{\cal N}{d}-\frac{1}{2}f^{\cal M}{}_{\cal NK}{\cal
H}^{\cal NP}{\cal H}^{\cal KQ}\partial_{\cal P}{\cal H}_{\cal MQ}$
$\displaystyle-\frac{1}{12}f^{\cal M}{}_{\cal KP}f^{\cal N}{}_{\cal LQ}{\cal
H}_{\cal MN}{\cal H}^{\cal KL}{\cal H}^{\cal PQ}$
$\displaystyle-\frac{1}{4}f^{\cal M}{}_{\cal NK}f^{\cal N}{}_{\cal ML}{\cal
H}^{\cal KL}-\frac{1}{6}f^{\cal MNK}f_{\cal MNK}\,.$

The Lagrangian (2.21) can be written in terms of generalized fluxes [18] as
222We are neglecting a total derivative.

$\displaystyle 2{\cal E}_{\underline{\cal A}}{\cal F}^{\underline{\cal
A}}+{\cal F}_{\underline{\cal A}}{\cal F}^{\underline{\cal
A}}-\frac{1}{6}{\cal F}_{\underline{\cal ABC}}{\cal F}^{\underline{\cal
ABC}}-\frac{1}{2}{\cal F}_{\overline{\cal A}\underline{\cal BC}}{\cal
F}^{\overline{\cal A}\underline{\cal BC}}\,.$ (2.22)

The DFT equations of motion are given by the generalized Ricci scalar,

$\displaystyle{\cal R}=$ $\displaystyle\frac{1}{8}{\cal H}^{\cal
MN}\partial_{\cal M}{\cal H}^{\cal KL}\partial_{\cal N}{\cal H}_{\cal
KL}-\frac{1}{2}{\cal H}^{\cal MN}\partial_{\cal N}{\cal H}^{\cal
KL}\partial_{\cal L}{\cal H}_{\cal MK}+4{\cal H}^{\cal MN}\partial_{\cal
M}\partial_{\cal N}d$ (2.23) $\displaystyle+4\partial_{\cal M}{\cal H}^{\cal
MN}\partial_{\cal N}d-4{\cal H}^{\cal MN}\partial_{\cal M}d\partial_{\cal
N}d-\partial_{\cal M}\partial_{\cal N}{\cal H}^{\cal MN}-\frac{1}{2}f^{\cal
M}{}_{\cal NK}{\cal H}^{\cal NP}{\cal H}^{\cal KQ}\partial_{\cal P}{\cal
H}_{\cal MQ}$ $\displaystyle-\frac{1}{12}f^{\cal M}{}_{\cal KP}f^{\cal
N}{}_{\cal LQ}{\cal H}_{\cal MN}{\cal H}^{\cal KL}{\cal H}^{\cal
PQ}-\frac{1}{4}f^{\cal M}{}_{\cal NK}f^{\cal N}{}_{\cal ML}{\cal H}^{\cal KL}$
$\displaystyle-\frac{1}{6}f^{\cal MNK}f_{\cal MNK}\,=0\,,$

and a generalized Ricci tensor

$\displaystyle{\cal R}_{\cal MN}=P_{\cal M}{}^{\cal P}{\cal K}_{\cal
PQ}{\overline{P}}^{\cal Q}{}_{\cal N}+{\overline{P}}_{\cal M}{}^{\cal P}{\cal
K}_{\cal PQ}P^{\cal Q}{}_{\cal N}=0\,,$ (2.24)

where

$\displaystyle{\cal K}_{\cal MN}$ $\displaystyle=$
$\displaystyle\frac{1}{8}\partial_{\cal M}{\cal H}^{\cal KL}\partial_{\cal
N}{\cal H}_{\cal KL}-\frac{1}{4}\left(\partial_{\cal L}-2\partial_{\cal
L}d\right)\left({\cal H}^{\cal LK}\partial_{\cal K}{\cal H}_{\cal
MN}\right)+2\partial_{\cal M}\partial_{\cal N}d$ (2.25)
$\displaystyle-\frac{1}{2}\partial_{(\cal M}{\cal H}^{\cal KL}\partial_{\cal
L}{\cal H}_{\cal N)K}+\frac{1}{2}\left(\partial_{\cal L}-2\partial_{\cal
L}d\right)\left({\cal H}^{\cal KL}\partial_{\cal(M}{\cal H}_{\cal N)K}+{\cal
H}^{\cal K}{}_{\cal(M}\partial_{\cal K}{\cal H}^{\cal L}{}_{\cal N)}\right)$
$\displaystyle+\frac{1}{2}f_{(\cal ML}{}^{\cal K}{\cal H}^{\cal
LP}\partial_{\cal P}{\cal H}_{\cal N)K}-\frac{1}{2}f_{\cal(ML}{}^{\cal K}{\cal
H}^{\cal LP}\partial_{\cal N)}{\cal H}_{\cal PK}-\frac{1}{4}f_{\cal
ML}{}^{\cal K}f_{\cal NK}{}^{\cal L}$
$\displaystyle+\frac{1}{2}e^{2d}\partial_{\cal P}(e^{-2d}{\cal H}^{\cal
LP}{\cal H}_{\cal Q(M})f_{\cal N)L}{}^{\cal Q}-\frac{1}{4}f_{\cal MK}{}^{\cal
P}f_{\cal NL}{}^{\cal Q}{\cal H}^{\cal KL}{\cal H}_{\cal PQ}\,.$

So far we have reviewed the standard construction of heterotic DFT. As we
mention before in this formulation there are no signs of a gauge field since
it is embedded in the generalized metric/frame, while the gauge symmetry is
embedded in the generalized diffeomorphisms.

###  2.2 Heterotic Kerr-Schild ansatz in metric formulation

The formulation of the GKSA for heterotic DFT consists of an exact and linear
perturbation of a generalized background metric [9]

$\displaystyle{\cal H}_{\cal MN}={\cal H}_{o\cal MN}+\kappa(\overline{\cal
K}_{\cal M}{\cal K}_{\cal N}+{\cal K}_{\cal M}\overline{\cal K}_{\cal N})\,,$
(2.26)

where $\overline{\cal K}_{\cal M}=\overline{\cal P}_{\cal M}{}^{\cal
N}\overline{\cal K}_{\cal N}$ and ${\cal K}_{\cal M}={\cal P}_{\cal M}{}^{\cal
N}{\cal K}_{\cal N}$ are a pair of generalized null vectors

$\displaystyle\eta^{\cal MN}\overline{\cal K}_{\cal M}\overline{\cal K}_{\cal
N}$ $\displaystyle=$ $\displaystyle 0\,,$ $\displaystyle\eta^{\cal MN}{\cal
K}_{\cal M}{\cal K}_{\cal N}$ $\displaystyle=$ $\displaystyle 0\,.$ (2.27)

According to (2.26), the DFT projectors are perturbed as follows,

$\displaystyle{\cal P}_{\cal MN}$ $\displaystyle=$ $\displaystyle{\cal
P}_{o\cal MN}-\frac{1}{2}\kappa(\overline{K}_{\cal M}{\cal K}_{\cal N}+{\cal
K}_{\cal M}\overline{\cal K}_{\cal N})$ $\displaystyle\overline{\cal P}_{\cal
MN}$ $\displaystyle=$ $\displaystyle\overline{\cal P}_{\cal
MN}+\frac{1}{2}\kappa(\overline{\cal K}_{\cal M}{\cal K}_{\cal N}+{\cal
K}_{\cal M}\overline{\cal K}_{\cal N})\,.$ (2.28)

The generalized background dilaton can be perturbed with a generic $\kappa$
expansion,

$d=d_{o}+\kappa f\,,\qquad f=\sum_{n=0}^{\infty}\kappa^{n}f_{n}\,.$ (2.29)

Mimicking the ordinary Kerr-Schild ansatz [14], the generalized vectors ${\cal
K}_{\cal M}$, $\overline{\cal K}_{\cal M}$ and $f$ obey some conditions in
order to produce finite deformations in the DFT action and EOM’s. Following
the original construction of the GKSA [8] we impose,

$\displaystyle{\overline{\cal K}}^{\cal P}\partial_{\cal P}{\cal K}^{\cal
M}+{\cal K}_{\cal P}\partial^{\cal M}{\overline{K}}^{\cal P}-{\cal K}^{\cal
P}\partial_{\cal P}{\overline{\cal K}}^{\cal M}$ $\displaystyle=$
$\displaystyle 0\,,$ $\displaystyle{\cal K}^{\cal P}\partial_{\cal
P}{\overline{\cal K}}^{\cal M}+\overline{\cal K}_{\cal P}\partial^{\cal
M}{\cal K}^{\cal P}-\overline{\cal K}^{\cal P}\partial_{\cal P}{\cal K}^{\cal
M}$ $\displaystyle=$ $\displaystyle 0\,,$ (2.30)

and

$\displaystyle{\cal K}^{\cal M}\partial_{\cal M}f=\overline{\cal K}^{\cal
M}\partial_{\cal M}f=0\,.$ (2.31)

Using (2.10), we can change $\partial\rightarrow\nabla$ in (2.30) obtaining,

$\displaystyle\overline{\cal K}^{\cal P}\nabla_{\cal P}{\cal K}^{\cal M}+{\cal
K}_{\cal P}\nabla^{\cal M}\overline{\cal K}^{\cal P}-{\cal K}^{\cal
P}\nabla_{\cal P}\overline{\cal K}^{\cal M}$ $\displaystyle=$ $\displaystyle
0\,,$ $\displaystyle{\cal K}^{\cal P}\nabla_{\cal P}\overline{\cal K}^{\cal
M}+\overline{\cal K}_{\cal P}\nabla^{\cal M}{\cal K}^{\cal P}-\overline{\cal
K}^{\cal P}\nabla_{\cal P}{\cal K}^{\cal M}$ $\displaystyle=$ $\displaystyle
0\,.$ (2.32)

In order to make contact with the ordinary heterotic supergravity we need to
parametrize our background fields and their perturbations. The parametrization
of the generalized metric is given by

$\displaystyle{\cal H}_{\cal
MN}=\left(\begin{matrix}g_{o}^{\mu\nu}&-g_{o}^{\mu\rho}C_{o\rho\nu}&-g_{o}^{\mu\rho}A_{o\rho
i}\\\
-g_{o}^{\nu\rho}C_{o\rho\mu}&g_{o\mu\nu}+C_{o\rho\mu}C_{o\sigma\nu}g_{o}^{\rho\sigma}+A_{o\mu}{}^{i}\kappa_{ij}A_{o\nu}{}^{j}&C_{o\rho\mu}g_{o}^{\rho\sigma}A_{o\sigma
i}+A_{o\mu}{}^{j}\kappa_{ji}\\\ -g_{o}^{\nu\rho}A_{o\rho
i}&C_{o\rho\nu}g_{o}^{\rho\sigma}A_{o\sigma
i}+A_{o\nu}{}^{j}\kappa_{ij}&\kappa_{ij}+A_{o\rho
i}g_{o}^{\rho\sigma}A_{o\sigma j}\end{matrix}\right)\ ,$ (2.33)

where $C_{o\mu\nu}=b_{o\mu\nu}+\frac{1}{2}A_{o\mu}{}^{i}A_{o\nu i}$ and
$\mu=0,\dots,D-1$, $i=1,\dots,K$, while the parametrization of the generalized
background dilaton is given by

$\displaystyle e^{-2d_{o}}=\sqrt{-g_{o}}e^{-2\phi_{o}}.$ (2.34)

The generalized vectors ${\cal K}_{\cal M}$ and $\overline{\cal K}_{\cal M}$
can be parametrized as

$\displaystyle K_{M}=\,\frac{1}{\sqrt{2}}\left(\begin{matrix}l^{\mu}\\\
-l_{\mu}-C_{o\rho\mu}l^{\rho}\\\ -A_{o\rho
i}{l}^{\rho}\end{matrix}\right)\,,\quad\bar{K}_{M}=\,\frac{1}{\sqrt{2}}\left(\begin{matrix}{\bar{l}}^{\mu}\\\
{\bar{l}}_{\mu}-C_{o\rho\mu}{\bar{l}}^{\rho}-\sqrt{2}A_{o\mu i}j^{i}\\\
-A_{o\rho i}{\bar{l}}^{\rho}+\sqrt{2}j_{i}\end{matrix}\right)\,,$ (2.35)

where $l$ and $\overline{l}$ satisfy,

$\displaystyle l_{\mu}l^{\mu}$ $\displaystyle=$ $\displaystyle 0\,,$ (2.36)
$\displaystyle\overline{l}_{\mu}\overline{l}^{\mu}+j^{i}j_{i}$
$\displaystyle=$ $\displaystyle 0\,.$ (2.37)

These objects also satisfy geodesic conditions, which are inherited from
generalized conditions at the DFT level.

Finally, the heterotic supergravity field content is perturbed in the
following way,

$\displaystyle g_{\mu\nu}$ $\displaystyle=$ $\displaystyle
g_{o\mu\nu}-\frac{\kappa}{1+\frac{1}{2}\kappa
l.\bar{l}}l_{(\mu}\bar{l}_{\nu)}$ (2.38) $\displaystyle b_{\mu\nu}$
$\displaystyle=$ $\displaystyle b_{o\mu\nu}-\frac{\kappa}{1+\frac{1}{2}\kappa
l.\bar{l}}l_{[\mu}(\bar{l}_{\nu]}-\frac{1}{\sqrt{2}}j_{i}A_{\nu]}{}^{i})$
(2.39) $\displaystyle A_{\mu i}$ $\displaystyle=$ $\displaystyle A_{o\mu
i}+\frac{1}{\sqrt{2}}\frac{\kappa}{1+\frac{1}{2}\kappa l.\bar{l}}l_{\mu}j_{i}$
(2.40) $\displaystyle\phi$ $\displaystyle=$ $\displaystyle\phi_{o}+\kappa
f\,,$ (2.41)

where we keep the same notation for the perturbation of the standard dilaton.
While the ordinary Kerr-Schild ansatz is based on linear perturbations on the
metric tensor, the generalized Kerr-Schild ansatz contains a tower of
perturbations due to $l.\overline{l}\neq 0$. Moreover, only $l$ is a null
vector when the gauge sector is taking into account.

In the next part of the work we discuss about an obstruction when the
generalized frame formulation is used. Particularly, we show that the
background gauge field $A_{o\mu i}$ cannot be perturbed when one considers
heterotic DFT written in terms of a fundamental $O(D,D+K)$ covariant frame.

###  2.3 Heterotic Kerr-Schild ansatz in frame formulation

The generalized Kerr-Schild ansatz for the DFT frame is given by,

$\displaystyle{\cal E}_{\cal M}{}^{\overline{\cal A}}={\cal E}_{o\cal
M}{}^{\overline{\cal A}}+\frac{1}{2}\kappa{\cal E}_{o\cal N}{}^{\overline{\cal
A}}{\cal K}_{\cal M}\overline{\cal K}^{\cal N}\,,$ $\displaystyle{\cal
E}_{\cal M}{}^{\underline{\cal A}}={\cal E}_{oM}{}^{\underline{\cal
A}}-\frac{1}{2}\kappa{\cal E}_{o\cal N}{}^{\underline{\cal A}}\overline{\cal
K}_{\cal M}{\cal K}^{\cal N}\,.$ (2.42)

In the frame formulation of DFT, conditions (2.30) and (2.31) become [23]

$\displaystyle{\cal K}^{\underline{\cal A}}{\cal E}_{\underline{\cal
A}}\overline{\cal K}^{\overline{\cal C}}+{\cal K}^{\underline{\cal
A}}\overline{\cal K}^{\overline{\cal B}}{\cal F}_{\underline{\cal
A}\overline{\cal B}}{}^{\overline{\cal C}}$ $\displaystyle=$ $\displaystyle
0\,,$ $\displaystyle\overline{\cal K}^{\overline{\cal A}}{\cal
E}_{\overline{\cal A}}{\cal K}^{\underline{\cal C}}+\overline{\cal
K}^{\overline{\cal A}}{\cal K}^{\underline{\cal B}}{\cal F}_{\overline{\cal
A}\underline{\cal B}}{}^{\underline{\cal C}}$ $\displaystyle=$ $\displaystyle
0\,,$ (2.43)

and

$\displaystyle{\cal K}^{\underline{\cal A}}{\cal E}_{\underline{\cal
A}}f=\overline{\cal K}^{\cal A}{\cal E}_{\overline{\cal A}}f=0\,,$ (2.44)

where ${\cal K}_{\underline{\cal A}}={\cal E}^{\cal M}{}_{\underline{\cal
A}}{\cal K}_{\cal M}$ and $\overline{\cal K}_{\overline{\cal A}}={\cal
E}^{\cal M}{}_{\overline{\cal A}}\overline{\cal K}_{\cal M}$.

Using these identifications, conditions (2.43) and (2.44) can be written as

$\displaystyle{\cal K}_{\underline{\cal A}}{\cal D}^{\underline{\cal
A}}\overline{\cal K}^{\overline{\cal B}}$ $\displaystyle=$
$\displaystyle\overline{\cal K}_{\overline{\cal A}}{\cal D}^{\overline{\cal
A}}{\cal K}^{\underline{\cal B}}=0\,,$ $\displaystyle{\cal K}_{\underline{\cal
A}}{\cal D}^{\underline{\cal A}}f$ $\displaystyle=$ $\displaystyle{\cal
K}_{\overline{\cal A}}{\cal D}^{\overline{\cal A}}f=0\,,$ (2.45)

where ${\cal D}_{\cal A}$ is a background Lorentz covariant derivative.

The parametrization of the generalized background frame is given by

$\displaystyle{\cal E}^{\cal M}{}_{\cal
A}=\frac{1}{\sqrt{2}}\left(\begin{matrix}-{e}_{o\mu
a}-C_{o\rho\mu}{e}_{o}^{\rho}{}_{a}&{e}_{o}^{\mu}{}_{a}&-A_{o\rho}{}^{i}{e}_{o}^{\rho}{}_{{a}}\,,\\\
{\overline{e}}_{o\mu
a}-C_{o\rho\mu}{}{\overline{e}}_{o}^{\rho}{}_{{a}}&{\overline{e}}_{o}^{\mu}{}_{a}&-A_{o\rho}{}^{i}{\overline{e}}_{o}^{\rho}{}_{a}\\\
\sqrt{2}A_{o\mu
i}e^{i}{}_{\overline{i}}&0&\sqrt{2}e^{i}{}_{\overline{i}}\end{matrix}\right)\,,$
(2.46)

and we impose the standard gauge fixing for the double Lorentz group,

$\displaystyle e_{o\mu a}\eta^{ab}e_{o\nu b}={\overline{e}}_{o\mu
a}\eta^{ab}{\overline{e}}_{o\nu b}={g}_{o\mu\nu}\,,$ (2.47)

with $\eta_{ab}$ the ten dimensional flat metric, $a,b=0,\dots,D-1$.

Since ${\cal E}^{\mu}{}_{\bar{i}}$ cannot be perturbed, we are forced to
impose $j_{i}=0$ from the very beginning. Then, the parametrization of the
generalized vector fields is

$\displaystyle{\cal K}_{\cal
M}=\,\frac{1}{\sqrt{2}}\left(\begin{matrix}l^{\mu}\\\
-l_{\mu}-C_{o\rho\mu}l^{\rho}\\\
-A_{oi\rho}{l}^{\rho}\end{matrix}\right)\,,\quad\overline{\cal K}_{\cal
M}=\,\frac{1}{\sqrt{2}}\left(\begin{matrix}{\bar{l}}^{\mu}\\\
{\bar{l}}_{\mu}-C_{o\rho\mu}{\bar{l}}^{\rho}\\\
-A_{oi\rho}{\bar{l}}^{\rho}\end{matrix}\right)\,$ (2.48)

In this framework, both $l$ and $\overline{l}$ are null vectors in order to
preserve the condition

$\displaystyle{\cal E}^{\mu}{}_{\bar{i}}=0\,.$ (2.49)

Then, it is possible to recover perturbations for the supergravity fields but
part of the perturbations of the b-field and the full perturbation for the
gauge field is missed.

In the next section we rewrite heterotic DFT in terms of $O(D,D)$ fundamental
fields in order to present a double Yang-Mills formulation of this theory. The
leading order terms of our construction is closely related to the construction
given in [15].

##  3 Double Yang-Mills formulation

We start by considering a splitting of heterotic DFT. We parametrize the
$O(D,D+K)$ generalized frame in terms of $O(D,D)$ multiplets, i.e.,

${\cal E}_{\cal M}{}^{\cal A}=\left(\begin{matrix}{\cal E}_{M}{}^{A}&{\cal
E}_{M}{}^{\bar{\alpha}}\\\ {\cal E}_{\alpha}{}^{A}&{\cal
E}_{\alpha}{}^{\bar{\alpha}}\end{matrix}\right)=\left(\begin{matrix}({\chi}^{\frac{1}{2}})_{M}{}^{N}E_{N}{}^{A}&-A_{M}{}^{\beta}e_{\beta}{}^{\overline{\alpha}}\\\
{A}^{M}{}_{\alpha}E_{M}{}^{A}&({\Box}^{\frac{1}{2}})_{\alpha}{}^{\beta}e_{\beta}{}^{\overline{\alpha}}\end{matrix}\right)\
,$ (3.1)

where $e_{\alpha}{}^{\overline{\alpha}}$ satisfies

$\displaystyle
e_{\alpha}{}^{\overline{\alpha}}\kappa_{\overline{\alpha}\overline{\beta}}e_{\beta}{}^{\overline{\beta}}$
$\displaystyle=$ $\displaystyle\kappa_{\alpha\beta}\,,$ (3.2) $\displaystyle
e^{\alpha}{}_{\overline{\alpha}}\kappa_{\alpha\beta}e^{\beta}{}_{\overline{\beta}}$
$\displaystyle=$ $\displaystyle\kappa_{\overline{\alpha}\overline{\beta}}\,.$
(3.3)

The matrices $\chi_{MN}$ and $\Box_{\alpha\beta}$ are defined in the following
way,

$\displaystyle{\chi}_{MN}=\eta_{MN}-A_{M}{}^{\alpha}A_{N\alpha}\,,\quad{\Box}_{\alpha\beta}=\kappa_{\alpha\beta}-A_{M\alpha}A^{M}{}_{\beta}\,,$
(3.4)

and we also impose

$\displaystyle E^{M\overline{A}}A_{M\alpha}=0\,$ (3.5)

which requires a gauge fixing. The projectors ${\cal P}_{\cal MN}$,
$\overline{\cal P}_{\cal MN}$ and ${\cal P}_{\cal AB}$, $\overline{\cal
P}_{\cal AB}$ are decomposed as

${\cal P}_{\cal MN}=\left(\begin{matrix}{P}_{MN}&0&\ \ 0\\\ 0&0&\ \ 0\\\ 0&0&\
\ 0\end{matrix}\right)\,,\quad\overline{\cal P}_{\cal
MN}=\left(\begin{matrix}0&0&0\\\ 0&{\overline{P}}_{MN}&0\\\
0&0&{\kappa}_{\alpha\beta}\end{matrix}\right)\,,$ (3.6) ${\cal P}_{\cal
AB}=\left(\begin{matrix}{P}_{AB}&0&\ \ 0\\\ 0&0&\ \ 0\\\ 0&0&\ \
0\end{matrix}\right)\,,\quad\overline{\cal P}_{\cal
AB}=\left(\begin{matrix}0&0&0\\\ 0&{\overline{P}}_{AB}&0\\\
0&0&{\kappa}_{\overline{\alpha}\overline{\beta}}\end{matrix}\right)\,.$ (3.7)

In [19], the ansatz (3.1) was used to obtain $\alpha^{\prime}$-corrections.
There, the authors considered that the $A_{M\alpha}$ is not a fundamental
field and, consequently, they identified it with some projections of the DFT
fluxes. Here we follow a different philosophy. We consider $A_{M\alpha}$ as a
fundamental field and we construct a double Yang-Mills action. Moreover, the
parametrization of this field gives raise to the ordinary Yang-Mills
connection $A_{\mu i}$. As we previously mentioned,
$\Big{\\{}E_{MA},A_{M\alpha},d\Big{\\}}$ are now the fundamental fields of
this alternative formulation of heterotic DFT. Similarly, the symmetry rules
given by $\hat{\xi}_{\cal M}$ and $\Gamma_{\cal A\cal B}$ must be decomposed
in a consistent way.

The symmetry transformations of the $O(D,D+K)$ fields are given by

$\displaystyle\delta{\cal E}_{\cal MA}$ $\displaystyle=$
$\displaystyle\hat{\xi}^{\cal N}\partial_{\cal N}{\cal E}_{\cal
MA}+(\partial_{\cal M}\hat{\xi}^{\cal P}-\partial^{\cal P}\hat{\xi}_{\cal
M}){\cal E}_{\cal PA}+f_{\cal MNP}\hat{\xi}^{\cal N}{\cal E}^{\cal P}{}_{\cal
A}+{\cal E}_{\cal MB}\Gamma^{\cal B}{}_{\cal A}\,,$ (3.8) $\displaystyle\delta
d$ $\displaystyle=$ $\displaystyle\hat{\xi}^{\cal N}\partial_{\cal
N}d-\frac{1}{2}\partial_{\cal M}\hat{\xi}^{\cal M}\,,$ (3.9)

where $f_{\cal MNP}$ only takes values when ${\cal MNP}=\alpha\beta\gamma$.
The $\partial_{\cal M}$ derivative is split according to $\partial_{\cal
M}=(\partial_{M},0)$ and the $O(D,D)$ strong constraint is

$\displaystyle\partial_{M}\blacktriangledown\partial^{M}\blacktriangle$
$\displaystyle=$ $\displaystyle 0\,,$ (3.10)
$\displaystyle\partial_{M}(\partial^{M}\blacktriangledown)$ $\displaystyle=$
$\displaystyle 0\,.$ (3.11)

The generalized diffeomorphisms parameter is split as

$\displaystyle\hat{\xi}^{\cal M}$ $\displaystyle=$
$\displaystyle(\hat{\xi}^{M},\lambda^{\alpha})\,,$ (3.12)

with $\lambda_{\alpha}$ a gauge parameter.

Some of the components of the double Lorentz parameter, ${\Gamma_{\cal AB}}$,
require a gauge fixing to ensure $\delta{\cal E}_{\alpha}{}^{\overline{A}}=0$
and $\delta e_{i}{}^{\overline{i}}=0$. From the former we find,

$\displaystyle\Gamma_{\overline{\beta}}{}^{\overline{A}}=(\Box^{-1/2})^{\alpha}{}_{\beta}e^{\beta}{}_{\overline{\beta}}(\partial^{P}\lambda_{\alpha})E_{P}{}^{\overline{A}}\,.$
(3.13)

Similarly, we can now demand $\delta{\cal
E}_{\alpha}{}^{\overline{\alpha}}=\delta(\Box^{\frac{1}{2}})_{\alpha}{}^{\beta}e_{\beta}{}^{\overline{\alpha}}$,
and then $\delta e_{\alpha}{}^{\overline{\alpha}}=0$. The previous condition
is satisfied if

$\displaystyle\Gamma_{\overline{\beta}\overline{\alpha}}=(\Box^{-1/2})^{\alpha}{}_{\beta}e^{\beta}{}_{[\overline{\beta}}\Big{(}-\hat{\xi}_{P}\partial^{P}{\cal
E}_{\alpha\overline{\alpha}]}+\partial^{P}\lambda_{\alpha}{\cal
E}_{P\overline{\alpha}]}-f_{\alpha\delta\gamma}\lambda^{\delta}{\cal
E}^{\gamma}{}_{\overline{\alpha}]}\Big{)}\,.$ (3.14)

Equations (3.13) and (3.14) are the gauge fixing conditions that we need to
write heterotic DFT in terms of fields which are in representations of
$O(D,D)$. In the next part of the work we explore the transformation rules for
these fields.

###  3.1 Transformation rules

We start by defining the following field

$\displaystyle
C_{M\alpha}=-A_{M}{}^{\beta}(\Box^{-\frac{1}{2}})_{\beta\alpha}$ (3.15)

which is constrained by $E_{M}{}^{\overline{A}}C_{M\alpha}=0$. We also define

$\displaystyle\Delta_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\kappa_{\alpha\beta}+C_{M\alpha}C^{M}{}_{\beta}$ (3.16)
$\displaystyle\Theta_{MN}$ $\displaystyle=$
$\displaystyle\eta_{MN}+C_{M}{}^{\alpha}C_{N\alpha}\,.$ (3.17)

The previous objects satisfy

$\displaystyle\Delta_{\alpha\beta}=(\Box^{-1}){}_{\alpha\beta}\ ,\ \ \ \
\Theta_{MN}=(\chi^{-1}){}_{MN}\ .$ (3.18)

There relations are useful to write the action principle in terms of
$A_{M\alpha}$ or in terms of $C_{M\alpha}$.

Now we inspect the transformation of $\delta{\cal
E}_{M\overline{A}}=\delta{E}_{M\overline{A}}$. In terms of the $C_{M\alpha}$
field we have,

$\displaystyle\delta{E}_{M\overline{A}}={\cal
L}_{\hat{\xi}}E_{M\overline{A}}+E_{M}{}^{\overline{B}}\Lambda_{\overline{B}\overline{A}}+C_{M}{}^{\gamma}\partial^{P}\lambda_{\gamma}E_{P\overline{A}}\,,$
(3.19)

where we have identified

$\displaystyle\Gamma_{\overline{A}\overline{B}}=\Lambda_{\overline{A}\overline{B}}\,.$
(3.20)

At this point we need to inspect the other generalized frame projection
$\delta E_{M\underline{A}}$,

$\displaystyle\delta{\cal
E}_{M\underline{A}}=\delta(\chi^{\frac{1}{2}})_{M}{}^{N}E_{N}{}^{\underline{A}}+(\chi^{\frac{1}{2}})_{M}{}^{N}\delta
E_{N}{}^{\underline{A}}\,.$ (3.21)

From the previous expression we find,

$\displaystyle\delta E_{N\underline{A}}={\cal
L}_{\hat{\xi}}E_{N\underline{A}}+E_{N}{}^{\underline{B}}\Gamma_{\underline{B}\underline{A}}-(\Theta^{\frac{1}{2}})^{M}{}_{N}\Big{(}\partial_{M}\lambda^{\alpha}C^{R}{}_{\beta}(\Delta^{\frac{1}{2}})^{\beta}{}_{\alpha}+\delta_{\Lambda}(\Theta^{-\frac{1}{2}})_{M}{}^{R}\Big{)}E_{R\underline{A}}\,,$
(3.22)

which need a parameter redefinition since we want (3.22) to have a generalized
Green-Schwarz form. Then, we must define

$\displaystyle
Q_{N}{}^{R}=(\Theta^{\frac{1}{2}})^{M}{}_{N}\Big{(}\partial_{M}\lambda^{\alpha}C^{R}{}_{\beta}(\Delta^{\frac{1}{2}})^{\beta}{}_{\alpha}+\delta_{\Lambda}(\Theta^{-\frac{1}{2}})_{M}{}^{R}\Big{)}E_{R\underline{A}}\,,$
(3.23)

and
$S_{MN}=\overline{P}_{M}{}^{P}\partial_{P}\lambda^{\alpha}C_{N\alpha}-Q_{MN}$,
in order to identify

$\displaystyle\Gamma_{\underline{A}\underline{B}}=\Lambda_{\underline{A}\underline{B}}-E^{M}{}_{\underline{A}}S_{MN}E^{N}{}_{\underline{B}}\,.$
(3.24)

In terms of the previous parameter, the transformation of this component takes
its expected form,

$\displaystyle\delta E_{N\underline{A}}={\cal
L}_{\hat{\xi}}E_{N\underline{A}}+E_{N}{}^{\underline{B}}\Lambda_{\underline{B}\underline{A}}-\partial_{\overline{N}}\lambda^{\alpha}C^{R}{}_{\alpha}E_{R\underline{A}}\,.$
(3.25)

Finally, the transformation of $C_{M\alpha}$ reads

$\displaystyle\delta C_{M}{}^{\gamma}={\cal
L}_{\hat{\xi}}C_{M}{}^{\gamma}+\partial_{M}\lambda^{\gamma}-(\Box^{-1})^{\gamma\alpha}\partial_{\overline{M}}\lambda_{\alpha}+C_{M}{}^{\alpha}\partial^{P}\lambda_{\alpha}C_{P}{}^{\gamma}-C_{M}{}^{\alpha}f_{\alpha\beta}{}^{\gamma}\lambda^{\beta}\,.$
(3.26)

So far we have computed the transformation rules for all the fundamental
fields of this double Yang-Mills formulation of heterotic DFT.

###  3.2 Flux formulation

The $O(D,D+K)$ invariant fluxes [18] can be easily decomposed in terms of
$O(D,D)$ multiplets using (3.1),

$\displaystyle{\cal F}_{\underline{A}\underline{B}\underline{C}}$
$\displaystyle=$ $\displaystyle
3\sqrt{2}(\chi^{\frac{1}{2}})^{M}{}_{P}E^{P}{}_{[\underline{A}}\Big{(}\partial_{M}((\chi^{\frac{1}{2}})^{NQ}E_{Q\underline{B}})(\chi^{\frac{1}{2}})_{NR}E^{R}{}_{\underline{C}]}+\partial_{M}A_{\underline{B}}{}^{\alpha}A_{\underline{C}]\alpha}\Big{)}$
(3.27)
$\displaystyle+\sqrt{2}f^{\alpha\beta\gamma}A_{\underline{A}\alpha}A_{\underline{B}\beta}A_{\underline{C}\gamma}\,,$
$\displaystyle{\cal F}_{\overline{A}\underline{B}\underline{C}}$
$\displaystyle=$
$\displaystyle\sqrt{2}E^{M}{}_{\overline{A}}\partial_{M}((\chi^{\frac{1}{2}})^{NQ}E_{Q\underline{B}})(\chi^{\frac{1}{2}})_{NR}E^{R}{}_{\underline{C}}+2\sqrt{2}(\chi^{\frac{1}{2}})^{M}{}_{P}E^{P}{}_{[\underline{C}}\partial_{M}(E^{N}{}_{\overline{A}})(\chi^{\frac{1}{2}})_{NR}E^{R}{}_{\underline{B}]}$
(3.28)
$\displaystyle+2\sqrt{2}E^{M}{}_{\overline{A}}\partial_{M}(A_{[\underline{B}}{}^{\alpha})A_{\underline{C}]\alpha}\,,$
$\displaystyle{\cal F}_{\overline{\alpha}\underline{B}\underline{C}}$
$\displaystyle=$
$\displaystyle-\sqrt{2}A^{M}{}_{[\alpha}e^{\alpha}{}_{\overline{\alpha}}\partial_{M}((\chi^{\frac{1}{2}})^{NQ}E_{Q\underline{B}})(\chi^{\frac{1}{2}})_{NR}E^{R}{}_{\underline{C}}+2\sqrt{2}(\chi^{\frac{1}{2}})^{M}{}_{P}E^{P}{}_{[\underline{C}}\partial_{M}((\Box^{\frac{1}{2}})_{\alpha\beta}e^{\beta}{}_{\overline{\alpha}})A_{\underline{B}]}{}^{\alpha}$
(3.29)
$\displaystyle+\sqrt{2}A^{M}{}_{\beta}e^{\beta}{}_{\overline{\alpha}}\partial_{M}A_{\underline{B}}{}^{\alpha}A_{\underline{C}\alpha}+\sqrt{2}f^{\alpha\beta\gamma}(\Box^{\frac{1}{2}})^{\alpha\delta}e_{\delta\overline{\alpha}}A_{\underline{B}\beta}A_{\underline{C}\gamma}\,,$
$\displaystyle{\cal F}_{\underline{A}}$ $\displaystyle=$
$\displaystyle\sqrt{2}\partial_{M}((\chi^{\frac{1}{2}})^{MP}E_{P\underline{A}})-2\sqrt{2}(\chi^{\frac{1}{2}})^{MP}E_{P\underline{A}}\partial_{M}d\,.$
(3.30)

where $A_{\underline{B}\alpha}=A_{M\alpha}E^{M}{}_{\underline{B}}$. Other
projections can be decomposed in the same way. Here we prioritize the ones
which appear in the heterotic DFT action.

###  3.3 Action principle

The components of the $O(D,D+K)$ generalized metric can be written in terms of
the following $O(D,D)$ multiples $\left\\{H_{MN},A_{M\alpha}\right\\}$
similarly to the decomposition given in [20]. According to our conventions,

$\displaystyle{\cal H}_{MN}$ $\displaystyle=$ $\displaystyle
H_{MN}+2A_{M\alpha}A_{N}{}^{\alpha}=H_{MN}+2\eta_{MN}-2(\Theta^{-1})_{MN}\,,$
(3.31) $\displaystyle{\cal H}_{M\beta}$ $\displaystyle=$
$\displaystyle-2(\chi^{\frac{1}{2}})_{M}{}^{P}A_{P\beta}=2C_{M\beta}\,,$
(3.32) $\displaystyle{\cal H}_{\alpha N}$ $\displaystyle=$
$\displaystyle-2(\chi^{\frac{1}{2}})_{N}{}^{Q}A_{Q\alpha}=2C_{N\alpha}\,,$
(3.33) $\displaystyle{\cal H}_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\kappa_{\alpha\beta}-2A^{M}{}_{\alpha}A_{M\beta}=-\kappa_{\alpha\beta}+2(\Delta^{-1})_{\alpha\beta}\,.$
(3.34)

The decomposition of the DFT Lagrangian is given by

$\displaystyle\mathcal{L}=$
$\displaystyle\frac{1}{8}\mathcal{H}^{MN}\partial_{M}\mathcal{H}^{KL}\partial_{N}\mathcal{H}_{KL}+\frac{1}{4}\mathcal{H}^{MN}\partial_{M}\mathcal{H}^{K\alpha}\partial_{N}\mathcal{H}_{K\alpha}+\frac{1}{8}\mathcal{H}^{MN}\partial_{M}\mathcal{H}^{\alpha\beta}\partial_{N}\mathcal{H}_{\alpha\beta}$
$\displaystyle-\frac{1}{2}\mathcal{H}^{MN}\partial_{N}\mathcal{H}^{KL}\partial_{L}\mathcal{H}_{MK}-\frac{1}{2}\mathcal{H}^{MN}\partial_{N}\mathcal{H}^{\beta
K}\partial_{K}\mathcal{H}_{M\beta}-\frac{1}{2}\mathcal{H}^{\alpha
K}\partial_{K}\mathcal{H}^{MN}\partial_{N}\mathcal{H}_{\alpha M}$
$\displaystyle-\frac{1}{2}\mathcal{H}^{\alpha K}\partial_{K}\mathcal{H}^{\beta
N}\partial_{N}\mathcal{H}_{\alpha\beta}+4\mathcal{H}^{MN}\partial_{M}d\partial_{N}d-2\partial_{M}\mathcal{H}^{MN}\partial_{N}d-\frac{1}{2}f_{\beta\gamma}^{\alpha}\mathcal{H}^{\beta
M}\mathcal{H}^{\gamma N}\partial_{M}\mathcal{H}_{\alpha N}$
$\displaystyle-\frac{1}{2}f_{\beta\gamma}^{\alpha}\mathcal{H}^{\beta
M}\mathcal{H}^{\gamma\lambda}\partial_{M}\mathcal{H}_{\alpha\lambda}-\frac{1}{12}f_{\beta\gamma}^{\alpha}f_{\mu\rho}^{\lambda}\mathcal{H}_{\alpha\lambda}\mathcal{H}^{\beta\mu}\mathcal{H}^{\gamma\rho}-\frac{1}{4}f_{\beta\gamma}^{\alpha}f_{\alpha\lambda}^{\beta}\mathcal{H}^{\gamma\lambda}-\frac{1}{6}f^{\alpha\beta\gamma}f_{\alpha\beta\gamma}\
,$

and the double Yang-Mills action principle takes the following form,

$\displaystyle S$ $\displaystyle=$ $\displaystyle\int
d^{2D}Xe^{-2d}\Big{(}\frac{1}{8}(H^{MN}-2(\Theta^{-1})^{MN})\partial_{M}(H^{KL}-2(\Theta^{-1})^{KL})\partial_{N}(H_{KL}-2(\Theta^{-1})_{KL})$
(3.36)
$\displaystyle+(H^{MN}-2(\Theta^{-1})^{MN})\partial_{M}C^{K\alpha}\partial_{N}C_{K\alpha}+\frac{1}{2}(H^{MN}-2(\Theta^{-1})^{MN})\partial_{M}(\Delta^{-1})^{\alpha\beta}\partial_{N}(\Delta^{-1})_{\alpha\beta}$
$\displaystyle-\frac{1}{2}(H^{MN}+2\eta^{MN}-2(\Theta^{-1})^{MN})\partial_{N}(H^{KL}-2(\Theta^{-1})^{KL})\partial_{L}(H_{MK}-2(\Theta^{-1})_{MK})$
$\displaystyle-2(H^{MN}+2\eta^{MN}-2(\Theta^{-1})^{MN})\partial_{N}C^{K\beta}\partial_{K}C_{M\beta}-2C^{K\alpha}\partial_{K}(H^{MN}-2(\Theta^{-1})^{MN})\partial_{N}C_{M\alpha}$
$\displaystyle-4C^{K\alpha}\partial_{K}C^{N\beta}\partial_{N}(\Delta^{-1})_{\alpha\beta}+4(H^{MN}-2(\Theta^{-1})^{MN})\partial_{M}d\partial_{N}d-2\partial_{M}(H^{MN}-2(\Theta^{-1})^{MN})\partial_{N}d$
$\displaystyle-4f_{\beta\gamma}^{\alpha}C^{M\beta}C^{N\gamma}\partial_{M}C_{N\alpha}-2f_{\beta\gamma}^{\alpha}C^{M\beta}(-\kappa^{\gamma\lambda}+2(\Delta^{-1})^{\gamma\lambda})\partial_{M}(\Delta^{-1})_{\alpha\lambda}$
$\displaystyle-\frac{1}{12}f_{\beta\gamma}^{\alpha}f_{\mu\rho}^{\lambda}(-\kappa_{\alpha\lambda}+2(\Delta^{-1})_{\alpha\lambda})(-\kappa^{\beta\mu}+2(\Delta^{-1})^{\beta\mu})(-\kappa^{\gamma\rho}+2(\Delta^{-1})^{\gamma\rho})$
$\displaystyle-\frac{1}{4}f_{\beta\gamma}^{\alpha}f_{\alpha\lambda}^{\beta}(-\kappa^{\gamma\lambda}+2(\Delta^{-1})^{\gamma\lambda})-\frac{1}{6}f^{\alpha\beta\gamma}f_{\alpha\beta\gamma}\Big{)}\
.$

The dynamics can be computed considering variations on the previous Lagrangian
or splitting the equations of motion from the $O(D,D+K)$ perspective. Using
the latter method the equation of motion for the $O(D,D)$ dilaton reads

$\displaystyle{\cal R}$ $\displaystyle=$ $\displaystyle 0\,,$ (3.37)

or, equivalently, ${\cal L}=0$. The equation of motion for the $O(D,D)$ metric
reads

$\displaystyle{\cal
K}_{MN}-(H_{M}{}^{P}+2\delta_{M}{}^{P}-2(\Theta^{-1})_{M}{}^{P}){\cal
K}_{QP}(H^{Q}{}_{N}+2\delta^{Q}{}_{N}-2(\Theta^{-1})^{Q}{}_{N})$
$\displaystyle-4C_{M}{}^{\alpha}{\cal K}_{\alpha
Q}(H^{Q}{}_{N}+2\delta^{Q}{}_{N}-2(\Theta^{-1})^{Q}{}_{N})-4C_{M}{}^{\alpha}{\cal
K}_{\alpha\beta}C_{N}{}^{\beta}=0\,,$ (3.38)

and the equation of motion for the gauge field is given by

$\displaystyle{\cal
K}_{M\alpha}-2(H_{M}{}^{P}+2\delta_{M}{}^{P}-2(\Theta^{-1})_{M}{}^{P}){\cal
K}_{PQ}C^{Q}{}_{\alpha}$
$\displaystyle-(H_{M}{}^{P}+2\delta_{M}{}^{P}-2(\Theta^{-1})_{M}{}^{P}){\cal
K}_{P\beta}(-\kappa^{\beta}{}_{\alpha}+2(\Delta^{-1})^{\beta}{}_{\alpha})$
$\displaystyle-2C_{M}{}^{\gamma}{\cal
K}_{\gamma\beta}(-\kappa^{\beta}{}_{\alpha}+2(\Delta^{-1})^{\beta}{}_{\alpha})-4C_{M}{}^{\gamma}{\cal
K}_{\gamma Q}C^{Q}{}_{\alpha}=0\,,$ (3.39)

where

$\displaystyle\mathcal{K}_{MN}=$
$\displaystyle\frac{1}{8}\partial_{M}({H}^{KL}-2(\Theta^{-1})^{KL})\partial_{N}({H}_{KL}-2(\Theta^{-1})_{KL})+\partial_{M}C^{K\alpha}\partial_{N}C_{K\alpha}$
(3.40)
$\displaystyle-\frac{1}{4}\big{(}\partial_{L}-2\partial_{L}d\big{)}\big{(}({H}^{KL}-2(\Theta^{-1})^{KL})\partial_{K}({H}_{MN}-2(\Theta^{-1})_{MN})\big{)}+2\partial_{M}\partial_{N}d$
$\displaystyle-\frac{1}{2}\partial_{(M}({H}^{KL}-2(\Theta^{-1})^{KL})\partial_{L}({H}_{N)K}-2(\Theta^{-1})_{N)K})-2\partial_{(M}C^{L\alpha}\partial_{L}C_{N)\alpha}$
$\displaystyle+\frac{1}{2}\big{(}\partial_{L}-2\partial_{L}d\big{)}\big{(}({H}^{KL}+2\eta^{KL}-2(\Theta^{-1})^{KL})\partial_{(M}({H}_{N)K}-2(\Theta^{-1})_{N)K})$
$\displaystyle+4C^{L\alpha}\partial_{(M}C_{N)\alpha}+({H}_{(MK}+2\eta_{(MK}-2(\Theta^{-1})_{(MK})\partial^{K}({H}^{L}{}_{N)}-2(\Theta^{-1})^{L}{}_{N)})\big{)}$
$\displaystyle+\frac{1}{2}\partial_{M}(\Delta^{-1})^{\alpha\beta}\partial_{N}(\Delta^{-1})_{\alpha\beta}\,,$
$\displaystyle\mathcal{K}_{M\alpha}=$
$\displaystyle-\frac{1}{2}\big{(}\partial_{L}-2\partial_{L}d\big{)}\big{(}({H}^{LK}-2(\Theta^{-1})^{LK})\partial_{K}C_{M\alpha}\big{)}-\frac{1}{2}\partial_{M}({H}^{LK}-2(\Theta^{-1})^{LK})\partial_{L}C_{K\alpha}$
(3.41)
$\displaystyle-\partial_{M}C^{L\beta}\partial_{L}(\Delta^{-1})_{\alpha\beta}+\frac{1}{2}\big{(}\partial_{L}-2\partial_{L}d\big{)}\big{(}({H}^{LK}+2\eta^{LK}-2(\Theta^{-1})^{LK})\partial_{M}C_{K\alpha}$
$\displaystyle+2C^{L\beta}\partial_{M}(\Delta^{-1})_{\alpha\beta}+2({H}_{M}{}^{K}+2\delta_{M}{}^{K}-2(\Theta^{-1})_{M}{}^{K})\partial_{K}C^{L}{}_{\alpha)}\big{)}$
$\displaystyle+e^{2d}\partial_{P}\Big{(}e^{-2d}C^{P\beta}C_{M\gamma}\Big{)}f_{\alpha\beta}^{\gamma}\,,$

and

$\displaystyle\mathcal{K}_{\alpha\beta}=$
$\displaystyle-\frac{1}{4}\big{(}\partial_{L}-2\partial_{L}d\big{)}\big{(}({H}^{LK}-2(\Theta^{-1})^{LK})\partial_{K}(\Delta^{-1})_{\alpha\beta}\big{)}+2\big{(}\partial_{L}-2\partial_{L}d\big{)}\big{(}C^{K}{}_{(\alpha}\partial_{K}C^{L}{}_{\beta)}\big{)}$
(3.42)
$\displaystyle+f_{(\alpha\gamma}^{\lambda}C^{P\gamma}\partial_{P}(\Delta^{-1})_{\beta)\lambda}-\frac{1}{4}f_{\alpha\gamma}^{\lambda}f_{\beta\lambda}^{\gamma}+e^{2d}\partial_{P}\Big{(}e^{-2d}C^{P\gamma}(\Delta^{-1})_{\lambda(\alpha}\Big{)}f_{\beta)\gamma}^{\lambda}$
$\displaystyle-\frac{1}{4}f_{\alpha\gamma}^{\lambda}f_{\beta\rho}^{\sigma}(-\kappa^{\gamma\rho}+2(\Delta^{-1})^{\gamma\rho})(-\kappa_{\lambda\sigma}+2(\Delta^{-1})_{\lambda\sigma})\,.$

In the next subsection we explore the leading order terms of this double Yang-
Mills formulation of heterotic DFT.

###  3.4 Leading order terms

This formulation of heterotic DFT contains an infinite expansion of gauge
fields in the action principle. Let us focus in the leading order terms. In
this limit the non-covariant transformation of the gauged vector is given by
[15],

$\displaystyle\delta_{\textrm{non-
cov}}A_{M}{}^{\gamma}=-\partial_{\underline{M}}\lambda^{\gamma}+\mathcal{O}(A^{2})\,.$
(3.43)

A generic DFT gauged vector transforms as

$\displaystyle\delta_{\lambda}V^{\gamma}=-A_{M}{}^{\alpha}f_{\alpha\beta}{}^{\gamma}\lambda^{\beta}\,,$
(3.44)

and hence a projected covariant derivative can be defined in the following
form,

$\displaystyle\nabla_{\underline{M}}V^{\gamma}=\partial_{\underline{M}}V^{\gamma}-V^{\alpha}f_{\alpha\beta}{}^{\gamma}A_{\underline{M}}{}^{\beta}\,.$
(3.45)

Interestingly enough, the combination $\partial_{\overline{M}}V^{\gamma}$
cannot be put in a covariant form using the gauge connection. Moreover, when
deriving a more general object one needs to consider the full DFT covariant
derivative,

$\displaystyle\nabla_{\underline{M}}V_{N}{}^{\gamma}=\partial_{\underline{M}}V_{N}{}^{\gamma}-\Gamma_{\underline{M}N}{}^{P}V_{P}{}^{\gamma}-V_{N}{}^{\alpha}f_{\alpha\beta}{}^{\gamma}A_{\underline{M}}{}^{\beta}\,,$
(3.46)

where $\Gamma_{MNP}$ is the generalized affine connection of DFT, which is not
fully-determined [17].

The curvature for the generalized gauge connection is given by

$\displaystyle
F_{MN}{}^{\gamma}=2\partial_{[M}A_{N]}{}^{\gamma}+f^{\gamma}{}_{\alpha\beta}A_{M}{}^{\alpha}A_{N}{}^{\beta}\,.$
(3.47)

The previous object transforms covariantly with respect to (3.44) upon
considering generalized Jacobi identities for the structure constant, in
agreement with the constraints of the heterotic DFT formulation [4].

The transformation rule of the generalized metric with respect to gauge
transformations is given by

$\displaystyle\delta_{\lambda}H_{MN}=4C_{(\underline{M}}{}^{\alpha}\partial_{\overline{N})}\lambda_{\alpha}\,=-4A_{(\underline{M}}{}^{\alpha}\partial_{\overline{N})}\lambda_{\alpha}+\mathcal{O}(A^{3}).$
(3.48)

Since the previous transformation is non-covariant, only $F^{2}$ terms
considering $\eta$ contractions are fully covariant. The leading order term of
(3.48) is a generalized Green-Schwarz transformation when one considers
$A_{M\alpha}$ as a fundamental field. Upon parametrization, the b-field
inherits this kind of transformation, as well as the metric tensor, which must
be redefined. We will return to this point when we parametrize the theory.

In order to construct the leading order Lagrangian, we focus on Abelian terms
such that their variation are linear in the gauge field. The leading order
terms of the double Yang-Mills action are given by

$\displaystyle S$ $\displaystyle=$ $\displaystyle\int
d^{2D}Xe^{-2d}(\frac{1}{8}{H}^{MN}\partial_{M}{H}^{KL}\partial_{N}{H}_{KL}-\frac{1}{2}{H}^{MN}\partial_{N}{H}^{KL}\partial_{L}{H}_{MK}$
(3.49)
$\displaystyle+4{H}^{MN}\partial_{M}d\partial_{N}d-2\partial_{M}{H}^{MN}\partial_{N}d+H^{MN}\eta^{KL}F_{LM}^{\alpha}F_{KN\alpha}$
$\displaystyle-2A^{N\alpha}(\partial_{N}H^{KL})(\partial_{L}A_{K\alpha})-2H^{MN}\partial_{N}H^{KL}(\partial_{L}A_{(K}{}^{\alpha})A_{M)\alpha}$
$\displaystyle+\frac{1}{4}A_{M\alpha}A_{N}{}^{\alpha}\partial^{M}H^{KL}\partial^{N}H_{KL}-A^{M}{}_{\alpha}A^{N\alpha}\partial_{N}H^{KL}\partial_{L}H_{MK}$
$\displaystyle+H_{MN}\partial^{M}A^{K}{}_{\alpha}A^{L\alpha}\partial^{N}H_{KL}-2H^{MN}\partial_{N}A^{(K\alpha}A^{L)}{}_{\alpha}\partial_{L}H_{MK}$
$\displaystyle+8A^{M\alpha}A^{N}{}_{\alpha}(\partial_{M}d)(\partial_{N}d)-8(\partial_{M}A^{(M\alpha})A^{N)}{}_{\alpha}(\partial_{N}d)\,+\dots)\,.$

In this limit $A_{M\alpha}=-C_{M\alpha}$. Since the combination
$H^{MN}\eta^{KL}F_{LM}^{\alpha}F_{KN\alpha}$ is not fully covariant, extra
pieces are required to construct the invariant action. Following the same
logic, the ungauged part of the Lagrangian is not gauge invariant because of
the generalized Green-Schwarz mechanism. The double gauge field, $A_{M\alpha}$
transforms as a gauge connection to leading order in gauge fields. Then, part
of the leading order Lagrangian can be written in terms of its curvature but
also non-covariant contributions appear.

So far we have shown that heterotic DFT can be formulated as a deformed double
Yang-Mills theory. In the next section we parametrize this theory and show the
required field redefinition in order to obtain the standard heterotic
supergravity.

###  3.5 Parametrization

We parametrize the fundamental fields as follows,

$\displaystyle{H}_{MN}$ $\displaystyle=$
$\displaystyle\left(\begin{matrix}\bar{g}^{\mu\nu}&-\bar{g}^{\mu\rho}b_{\rho\nu}\\\
b_{\mu\rho}\bar{g}^{\rho\nu}&\bar{g}_{\mu\nu}-b_{\mu\rho}\bar{g}^{\rho\sigma}b_{\sigma\nu}\end{matrix}\right)\
,\ \ \ \ \ \ \ \ $ (3.50) $\displaystyle{C}_{M}{}^{\alpha}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\begin{matrix}-\bar{g}^{\mu\rho}A_{\rho}{}^{i}\\\
-b_{\mu\rho}\bar{g}^{\rho\sigma}A_{\sigma}{}^{i}+A_{\mu}{}^{i}\end{matrix}\right)\
.$ (3.51)

The metric tensor $\bar{g}_{\mu\nu}$ inherits an anomalous gauge
transformation from (3.48) and the following field redefinition is required,

$\displaystyle\bar{g}_{\mu\nu}=g_{\mu\nu}+\frac{1}{2}A_{\mu}{}^{i}A_{\nu i}\
.$ (3.52)

This procedure is analogous to the gravitational Green-Schwarz mechanism where
the metric tensor is redefined considering terms proportional to the spin
connection [21].

The generalized dilaton is parametrized in the following way,

$\displaystyle e^{-2d}$ $\displaystyle=$
$\displaystyle\sqrt{-\tilde{g}}e^{-2\tilde{\phi}}=\sqrt{-g}e^{-2\phi}\ ,$
(3.53)

since the metric redefinition (3.52) implies a dilaton redefinition to obtain
the standard integral measure. The transformation rules for the supergravity
fields take their standard forms,

$\displaystyle\delta g_{\mu\nu}$ $\displaystyle=$ $\displaystyle
L_{\xi}g_{\mu\nu}\ ,$ (3.54) $\displaystyle\delta b_{\mu\nu}$ $\displaystyle=$
$\displaystyle
L_{\xi}b_{\mu\nu}+2\partial_{[\mu}\zeta_{\nu]}-\partial_{[\mu}\lambda^{i}A_{\nu]i}\
,$ (3.55) $\displaystyle\delta A_{\mu}{}^{i}$ $\displaystyle=$ $\displaystyle
L_{\xi}A_{\mu}{}^{i}+\partial_{\mu}\lambda^{i}-f_{jk}{}^{i}\lambda^{j}A_{\mu}{}^{k}\
,$ (3.56) $\displaystyle\delta\phi$ $\displaystyle=$ $\displaystyle
L_{\xi}\phi\ ,$ (3.57)

where we use $L_{\xi}$ for the ordinary Lie derivative and
$\hat{\xi}_{\mu}=\zeta_{\mu}$ for the abelian gauge transformation of the
b-field. Finally, the action is given by

$\displaystyle S=\int
d^{D}x\sqrt{-g}e^{-2\phi}\Big{(}R-4\partial_{\mu}\phi\partial^{\mu}\phi-\frac{1}{12}\hat{H}_{\mu\nu\rho}\hat{H}^{\mu\nu\rho}-\frac{1}{4}F_{\mu\nu
i}F^{\mu\nu i}\Big{)}\,,$ (3.58)

with

$\displaystyle\hat{H}_{\mu\nu\rho}$ $\displaystyle=$ $\displaystyle
3\left[\partial_{[\mu}b_{\nu\rho]}-\left(A_{[\mu}^{i}\partial_{\nu}A_{\rho]i}-\frac{1}{3}f_{ijk}A_{\mu}^{i}A_{\nu}^{j}A_{\rho}^{k}\right)\right]$
(3.59) $\displaystyle F_{\mu\nu}^{i}$ $\displaystyle=$ $\displaystyle
2\partial_{[\mu}A^{i}_{\nu]}-f^{i}{}_{jk}A_{\mu}^{j}A_{\nu}^{k}\,.$ (3.60)

In this section we show how the double Yang-Mills formulation of DFT can
reproduce the standard transformation rules and action of the heterotic
supergravity. In the next one we inspect perturbative aspects of this theory
using the relaxed form of the GKSA.

##  4 Double Yang-Mills and the relaxed GKSA

The GKSA was originally formulated in [8] and it has been explored in
heterotic DFT [9] [22] [23], Kaluza-Klein DFT [24], Exceptional Field Theory
[25], among others [16] [26]. In this part of the work we extend this ansatz
to the double Yang-Mills formulation of heterotic field theory. We use a
relaxed form of the GKSA, firstly introduced in [16].

###  4.1 Generalized metric formulation

We start by considering a generalized metric ${H}_{MN}$, a generalized dilaton
$d$ and a generalized gauged field $A_{M\alpha}$. The generalized metric
satisfies a relaxed Kerr-Schild ansatz [16] while the gauge field is linearly
perturbed and the generalized dilaton has an arbitrary perturbation, i.e.,

$\displaystyle{H}_{MN}$ $\displaystyle=$
$\displaystyle{H}_{oMN}+\kappa(\overline{K}_{M}K_{N}+K_{M}\overline{K}_{N})+\frac{\kappa^{2}}{2}\overline{K}^{2}K_{M}K_{N}\,,$
(4.1) $\displaystyle{C}_{M\alpha}$ $\displaystyle=$
$\displaystyle{C}_{oM\alpha}-\kappa K_{M}J_{\alpha}\,,$ (4.2) $\displaystyle
d$ $\displaystyle=$ $\displaystyle d_{o}+\kappa f\,,$ (4.3)

where ${H}_{oMN}$, ${C}_{oMN}$ and $d_{o}$ are generalized
backgrounds,$f=\sum_{n=0}^{\infty}f^{(n)}\kappa^{n}$, $K_{M}$ and
$\bar{K}_{M}$ are projected vectors,

$\displaystyle\bar{K}_{M}$ $\displaystyle=$
$\displaystyle\frac{1}{2}({\eta}_{MN}+{H}_{MN})\bar{K}^{N}=\bar{P}_{MN}\bar{K}^{N}\,,$
(4.4) $\displaystyle K_{M}$ $\displaystyle=$
$\displaystyle\frac{1}{2}({\eta}_{MN}-{H}_{MN})K^{N}={P}_{MN}{K}^{N}\,,$ (4.5)

and $\kappa$ is an order parameter. The null condition is given only on
$K_{M}$,

$\eta^{MN}K_{M}K_{N}=0\,,$ (4.6)

since $\overline{K}_{M}$ is related to $J_{\alpha}$ through

$\eta^{MN}\overline{K}_{M}\overline{K}_{N}=-J_{\alpha}J_{\beta}\kappa^{\alpha\beta}\,.$
(4.7)

The parametrization of the ansatz is the same as (2.38)-(2.41) but the inverse
metric receives a second order perturbation,

$\displaystyle g^{\mu\nu}$ $\displaystyle=$
$\displaystyle\tilde{g}_{o}^{\mu\nu}+\kappa
l^{(\mu}\bar{l}^{\nu)}+\frac{\kappa^{2}}{4}\bar{l}^{2}l^{\mu}l^{\nu}\,.$ (4.8)

Let us now analyze the pure Abelian Yang-Mills theory at the DFT level. We
start with

$\displaystyle S$ $\displaystyle=$ $\displaystyle\int
d^{2D}Xe^{-2d}H^{MN}F^{K}{}_{M\alpha}F_{KN}{}^{\alpha}$ (4.9) $\displaystyle=$
$\displaystyle\int
d^{2D}Xe^{-2d}(H^{MN}\partial_{M}A^{K\alpha}\partial_{N}A_{K\alpha}-2H^{MN}\partial_{M}A^{K\alpha}\partial_{K}A_{N\alpha})\,,$
(4.10)

where $d$ and $H^{MN}$ are constants and the only dynamical field is
$A_{M\alpha}$. The equation of motion of the later is given by

$\displaystyle H^{MN}\partial_{M}F_{KN}+2H^{M}{}_{K}\partial^{N}F_{MN}=0\,,$
(4.11)

which can be understood as a duality invariant generalization of the Maxwell
equation. Now we use the generalized Kerr-Schild ansatz on $A_{M\alpha}$ as

$\displaystyle A_{M\alpha}=A_{oM\alpha}+\kappa K_{M}J_{\alpha}\,,$ (4.12)

in agreement with (4.3). The dynamics turns into

$\displaystyle H^{MN}\partial_{M}(\partial_{K}(A_{oN}{}^{\alpha}+\kappa
K_{N}J^{\alpha})-\partial_{N}(A_{oK}{}^{\alpha}+\kappa K_{K}J^{\alpha}))$
$\displaystyle+H^{M}{}_{K}\partial_{M}\partial_{N}(A_{o}^{N\alpha}+\kappa
K^{N}J^{\alpha})=0\,.$ (4.13)

To make contact with supergravity we parametrize the background fields as

$\displaystyle{H}_{oMN}$ $\displaystyle=$
$\displaystyle\left(\begin{matrix}\eta^{\mu\nu}&0\\\
0&\eta_{\mu\nu}\end{matrix}\right)\ ,\ \ \ \ \ \ \ \ $ (4.14)
$\displaystyle{C}_{oM}{}^{\alpha}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\begin{matrix}-\eta^{\mu\rho}A_{o\rho}{}^{i}\\\
A_{o\mu}{}^{i}\end{matrix}\right)\ ,$ (4.15)

and the perturbations as

$\displaystyle{K}_{M}=\,\frac{1}{\sqrt{2}}\left(\begin{matrix}l^{\mu}\\\
-l_{\mu}\end{matrix}\right)\,,\quad\overline{K}_{M}=\,\frac{1}{\sqrt{2}}\left(\begin{matrix}{\bar{l}}^{\mu}\\\
{\bar{l}}_{\mu}\end{matrix}\right)\,,\quad
J_{\alpha}=\frac{1}{\sqrt{2}}j_{i}\,.$ (4.16)

The parametrization of the pure Abelian Yang-mills dynamics gives rise to the
following first order contributions

$\displaystyle-\frac{\kappa}{2}\eta^{\mu\nu}\partial_{\mu\rho}(l_{\nu}j^{i})+\frac{\kappa}{2}\eta^{\mu\nu}\partial_{\mu\nu}(l_{\rho}j^{i})=0\,.$
(4.17)

In the next section we inspect the form of the relaxed generalized Kerr-Schild
ansatz in the generalized frame formulation of DFT. As previously described in
[7], we discussed about the equivalences of both formulations.

###  4.2 Generalized frame formulation

The relaxed generalized Kerr-Schild ansatz for the double Yang-Mills fields in
the generalized frame formulation is given by,

$\displaystyle{E}_{M}{}^{\overline{A}}$ $\displaystyle=$
$\displaystyle{E}_{oM}{}^{\overline{A}}+\frac{1}{2}\kappa{E}_{oN}{}^{\overline{A}}{K}_{M}\overline{K}^{N}\,,$
(4.18) $\displaystyle{E}_{M}{}^{\underline{A}}$ $\displaystyle=$
$\displaystyle{E}_{oM}{}^{\underline{A}}-\frac{1}{2}\kappa{E}_{oN}{}^{\underline{A}}\overline{K}_{M}{K}^{N}\,-\frac{1}{8}\kappa^{2}\overline{K}^{2}K_{M}K_{N}E_{o}^{N}{}_{\underline{A}}\,,$
(4.19)

while the generalized gauge field and the generalized dilaton are perturbed as
in the generalized metric formalism. The second order perturbation of
$E_{M}{}^{\underline{A}}$ has its coefficient fixed through

$\displaystyle
E_{M\underline{A}}E^{M}{}_{\underline{B}}+E_{M\overline{A}}E^{M}{}_{\overline{B}}=\eta_{AB}\,,$
(4.20) $\displaystyle-
E_{M\underline{A}}E^{M}{}_{\underline{B}}+E_{M\overline{A}}E^{M}{}_{\overline{B}}=H_{AB}$
(4.21)

since the RHS of the previous expressions must remain invariant. In this
formulation, the generalized metric is an element of $O(D,D)$ only if
$\overline{K}^{2}=0$. In other words, the perturbation (4.1) is not compatible
with (4.19), which explains why the gauge field cannot be perturbed in the
ordinary $O(D,D+K)$ framework. This formulation still contains an obstruction,
but it appears before parametrization. This is very reasonable since the
double Yang-Mills formulation of heterotic DFT written in terms of $O(D,D)$
multiplets is equivalent to the standard formulation written in terms of
$O(D,D+K)$ multiplets.

###  4.3 Field redefinitions

One interesting aspect of the different formulations of DFT is the need of
field redefinitions to match with standard supergravity scenarios. This
situation usually appears when the generalized metric has a non-trivial
symmetry invariance because it encodes the b-field. The most canonical example
is the Green-Schwarz mechanism. If we analyze the equation (3.52), this
redefinition is mandatory since the metric $\tilde{g}$ transforms non-
covariantly under gauge transformations according to (3.48). When one works at
the perturbative level, background objects could encode perturbations.

Let us focus in the ordinary Kerr-Schild ansatz [14] for a generic
$\tilde{g}_{\mu\nu}$,

$\displaystyle\tilde{g}_{\mu\nu}=\tilde{g}_{o\mu\nu}+\kappa\tilde{l}_{\mu}\tilde{l}_{\nu}\,.$
(4.22)

In order to have agreement at the perturbative level we are forced to impose a
vector redefinition

$\displaystyle\tilde{l}_{\mu}=l_{\mu}+A_{o\mu}{}^{i}j_{i}\,,$ (4.23)

and a field redefinition for the backgrounds

$\displaystyle\tilde{g}_{o\mu\nu}=g_{o\mu\nu}+\frac{\kappa^{2}}{2}l_{\mu}l_{\nu}j^{2}\,.$
(4.24)

Now it is very clear that the background $\tilde{g}$ encodes a (second order)
perturbation in terms of the ordinary $l$ vector. Moreover, the null and
geodesic equations for the tilde fields impose new conditions. These relations
were discussed in [23] for Lorentz Green-Schwarz contributions in the
generalized Kerr-Schild ansatz, and they are equivalent to the gauge Green-
Schwarz contributions when the Lorentz connection is replaced by the gauge
connection.

##  5 Application

###  5.1 Classical double copy at the DFT level

The main results of this work allow us the inspection of the single and zeroth
copy at the DFT level. The foundations of this relation and DFT can be
consulted in [27].

Let us focus on the ungauged part of the equation of motion of the generalized
metric with vanishing dilaton,

$\displaystyle{\cal R}_{MN}=P_{M}{}^{P}{\cal
K}_{PQ}{\bar{P}}^{Q}{}_{N}+{\bar{P}}_{M}{}^{P}{\cal K}_{PQ}P^{Q}{}_{N}=0\,,$
(5.1)

where

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

Now we can impose the ordinary generalized Kerr-Schild ansatz considering a
flat ${H}_{oMN}$ background,

$\displaystyle{H}_{MN}$ $\displaystyle=$
$\displaystyle{H}_{oMN}+\kappa(\overline{K}_{M}K_{N}+K_{M}\overline{K}_{N})\,.$
(5.2)

In this case it is useful to invoke the generalized version of the geodesic
condition [8]

$\displaystyle{\overline{K}}^{P}\partial_{P}{K}^{M}={K}^{P}\partial_{P}{\overline{K}}^{M}$
$\displaystyle=$ $\displaystyle 0\,.$ (5.3)

After imposing the previous conditions, the linear contributions reads

$\displaystyle-\frac{1}{2}{H}_{o}^{KL}\partial_{K}(\partial_{L}K_{M}\overline{K}_{N}+K_{M}\partial_{L}\overline{K}_{N})$
$\displaystyle+\overline{P}_{oM}{}^{K}\partial_{K}(\partial_{L}K^{L}\overline{K}_{N})-{P}_{oN}{}^{K}\partial_{K}(K_{M}\partial_{L}\overline{K}^{L})=0\,,$
(5.4)

while the quadratic contributions are null since they depend on the
perturbation of the generalized dilaton. Now we define a generalized killing
vector, $\xi_{M}$, in the following way

$\displaystyle{\cal L}_{\xi}{H}_{MN}=0\,,$ (5.5)

where ${\cal L}_{\xi}$ is the generalized Lie derivative. Furthermore, we
choose a set of double coordinates such as $\xi=const.$ and

$\displaystyle\xi^{M}K_{M}=\xi^{M}\overline{K}_{M}=1\,.$ (5.6)

If we contract $\xi^{M}R_{MN}$ we obtain,

$\displaystyle-\frac{1}{2}{H}_{o}^{KL}\partial_{K}\partial_{L}\overline{K}_{N}+\xi^{M}\overline{P}_{oM}{}^{K}\partial_{K}(\partial_{L}K^{L}\overline{K}_{N})-{P}_{oN}{}^{K}\partial_{K}\partial_{L}\overline{K}^{L}=0\,,$
(5.7)

while contracting $\xi^{N}R_{MN}$ gives

$\displaystyle-\frac{1}{2}{H}_{o}^{KL}\partial_{K}\partial_{L}{K}_{M}-\xi^{N}{P}_{oN}{}^{K}\partial_{K}(\partial_{L}\overline{K}^{L}{K}_{M})+\overline{P}_{oM}{}^{K}\partial_{K}\partial_{L}{K}^{L}=0\,.$
(5.8)

Now we identify each null vector with a generalization of a $U(1)$ vector
field, $A_{M}=A_{\underline{M}}$ and
$\overline{A}_{M}=\overline{A}_{\overline{M}}$ and we obtain the
generalization of the single copy expression at the DFT level,

$\displaystyle-\frac{1}{2}{H}_{o}^{KL}\partial_{K}\partial_{L}\overline{A}_{N}+\xi^{M}\overline{P}_{oM}{}^{K}\partial_{K}(\partial_{L}A^{L}\overline{A}_{N})-{P}_{oN}{}^{K}\partial_{K}\partial_{L}\overline{A}^{L}$
$\displaystyle=$ $\displaystyle 0\,,$
$\displaystyle-\frac{1}{2}{H}_{o}^{KL}\partial_{K}\partial_{L}{A}_{M}-\xi^{N}{P}_{oN}{}^{K}\partial_{K}(\partial_{L}\overline{A}^{L}{A}_{M})+\overline{P}_{oM}{}^{K}\partial_{K}\partial_{L}{A}^{L}$
$\displaystyle=$ $\displaystyle 0\,.$ (5.9)

Even when the previous expressions do not match exactly with the
generalization of the Maxwell equation at the DFT level, it is straightforward
to show that they embedded a pair of Maxwell equations at the supergravity
level. We start by considering a null B-field at the supergravity level and
then $\xi^{M}=(0,\xi^{\mu})$. The generalized metric is parametrized as

$\displaystyle{H}_{MN}$ $\displaystyle=$
$\displaystyle\left(\begin{matrix}\eta^{\mu\nu}&0\\\
0&\eta_{\mu\nu}\end{matrix}\right)\ ,$ (5.10)

while the pair of generalized gauge vectors reproduces the ordinary Abelian
fields $A_{M}\rightarrow A_{\mu}$ and
$\overline{A}_{M}\rightarrow\overline{A}_{\mu}$. The geodesic condition turns
into

$\displaystyle A_{\mu}\partial^{\mu}\overline{A}_{\nu}$ $\displaystyle=$
$\displaystyle 0\,,$ $\displaystyle\overline{A}_{\mu}\partial^{\mu}A_{\nu}$
$\displaystyle=$ $\displaystyle 0\,,$

while the ordinary killing vector satisfies

$\displaystyle\xi^{\mu}\partial_{\mu}(A_{\nu}\overline{A}_{\rho})=0\,.$ (5.11)

The parametrization of (5.9) is given by

$\displaystyle\Box A_{\nu}-\partial_{\nu}(\partial^{\rho}A_{\rho})$
$\displaystyle=$ $\displaystyle 0\leftrightarrow\partial_{\mu}F^{\mu\nu}=0$
(5.12) $\displaystyle\Box
A_{\nu}-\partial_{\nu}(\partial^{\rho}\overline{A}_{\rho})$ $\displaystyle=$
$\displaystyle 0\leftrightarrow\partial^{\mu}\overline{F}_{\mu\nu}=0\,,$
(5.13)

where we have defined the curvatures of the Abelian gauge fields in the usual
way

$\displaystyle F_{\mu\nu}$ $\displaystyle=$ $\displaystyle
2\partial_{[\mu}A_{\nu]}\,,$ $\displaystyle\overline{F}_{\mu\nu}$
$\displaystyle=$ $\displaystyle 2\partial_{[\mu}\overline{A}_{\nu]}\,.$

On the The zeroth copy relation can be easily obtained contracting an extra
generalized Killing vector $\xi$ in (5.7) or (5.8), after introducing the
scalar function $\varphi$ using the redefinition
$\kappa_{\textrm{DFT}}\rightarrow\kappa\varphi$. The resulting equation is
given by

$\displaystyle-\frac{1}{2}{H}_{o}^{KL}\partial_{K}\partial_{L}\varphi+\xi^{M}\overline{P}_{oM}{}^{K}\partial_{K}\partial_{L}(\varphi
K^{L})-\xi^{N}{P}_{oN}{}^{K}\partial_{KL}(\varphi\overline{K}^{L})=0\,,$
(5.14)

which embeds the standard supergravity zeroth copy relation,

$\displaystyle\Box\varphi=0\,.$ (5.15)

##  6 Discussion

The double Yang-Mills formulation of heterotic DFT forces the apparition of a
generalized gauge field/symmetry in the double geometry. Moreover, the
$O(D,D)$ generalized metric transforms according to a generalized Green-
Schwarz mechanism, which is inherited to the b-field and also to the metric
tensor. For this reason, a field redefinition is mandatory to remove this last
transformation as we show in (3.52). This formulation explains the agreement
between [4] and the proposal given in [15], where a pure double Yang-Mills
formulation was proposed.

The extra terms (beyond $F^{2}$ contributions) which we find here are directly
related to Chern-Simons contributions at the supergravity level. This kind of
terms cannot be written in the double geometry 333We thank K. Lee for this
observation. since the notion of the exterior derivative is not fully
understood in the present formulation of DFT. In heterotic supergravity, the
need of Chern-Simons terms is related to the lack of invariance of the square
of the curvature of the b-field under gauge transformations. Following this
logic, a more general notion of Chern-Simons terms in the double geometry
could involve particular combinations of the $O(D,D)$ fundamental fields, i.e.
$CS_{MNP}$, which might be used in order to rewrite the double Yang-Mills
action in a compact form, curing the lack of invariance of the pure double
Yang-Mills term.

On the other hand, here we revisit the tension between the generalized
metric/frame formalism of DFT when one considers the GKSA. Particularly, using
the standard form of the GKSA we show that the D-dimensional gauge field,
$A_{\mu i}$, cannot be perturbed in the generalized frame formulation.
Furthermore, we relax the conditions of the GKSA mimicking [16] in order to
perturb this field at the DFT level. As happens in the ordinary case, the
obstruction is still present since the generalized metric is not an $O(D,D)$
element unless $J_{\alpha}=0$.

It would be natural to propose an ordinary GKSA over the $O(D,D+K)$ invariant
DFT and then impose a suitable splitting as

$\displaystyle\overline{\cal K}_{\cal
M}=(\overline{K}_{M},J_{\alpha})\,,\quad\quad{\cal K}_{\cal M}=(K_{M},0)\,.$
(6.1)

Unfortunately, this proposal also leads to $J_{\alpha}=0$ according to the
gauge fixing of the $O(D,D+K)$ generalized frame.

Finally, using the main results of this paper we explore the classical double
copy relation at the DFT level. A very important observation of equation (5.9)
is the fact that after the identification of the generalized null vectors with
Abelian gauge fields in the EOM of the generalized metric contracted with a
generalized Killing vector, the form of the equation is not exactly a Maxwell-
like equation as (4.11) as happens at the supergravity level. However, the
agreement occurs upon parametrization, as expected.

##  7 Summary

We present a double Yang-Mills formulation of heterotic DFT, where the
fundamental fields are in representations of $O(D,D)$ and a generalized gauged
field/symmetry explicitly appears in the double geometry. The formalism is
equivalent to the standard formulation of heterotic DFT and contains a pure
double Yang-Mills theory in its Lagrangian formulation as proposed in [15]. We
use this framework to explore a relaxed version of the GKSA. The generalized
background metric contains up to quadratic perturbations generated by a single
null vector, while the generalized gauge field is linearly perturbed before
parametrization. As an application we explore the classical double copy
correspondence at the DFT level and we find that after obtaining the
generalized version of the single copy, the form of the equation does not
match with the generalization of the Maxwell equation and the agreement occurs
upon parametrization.

## Acknowledgements

We are very grateful to K. Lee for e-mail correspondence and enlighten
comments. We also thank to the organisers of the school “Integrability,
Dualities and Deformations” which ran from 23 to 27 August, 2021, in Santiago
de Compostela and virtually, since they provided discussion sessions where
this project began. Finally, E.L would like to thank CONICET for supporting
his work. The work of SRC is partially supported by grants from the Infosys
Foundation to CMI.

##  Appendix A Relaxed GKSA and the DFT Lagrangian

If we focus on the ungauged part of the double Yang-Mills Lagrangian assuming
constant backgrounds and with vanishing dilaton, i.e.,

$\displaystyle\frac{1}{8}{H}^{MN}\partial_{M}{H}^{KL}\partial_{N}{H}_{KL}-\frac{1}{2}{H}^{MN}\partial_{N}{H}^{KL}\partial_{L}{H}_{MK}\,,$
(A.1)

it is straightforward to verify that the sixth and fifth order contributions
vanish since $K_{M}$ is a null vector. The $\kappa^{4}$ contributions are

$\displaystyle-\frac{1}{8}K^{M}K^{N}{\overline{K}}^{4}\partial_{M}{K_{Q}}\partial_{N}{K^{Q}}-\frac{1}{8}K^{M}K^{N}{\overline{K}}_{P}{\overline{K}}^{Q}{\overline{K}}^{2}\partial_{M}{K_{Q}}\partial_{N}{K^{P}}$
$\displaystyle+\frac{1}{8}K^{M}K^{N}{\overline{K}}^{P}{\overline{K}}^{Q}{\overline{K}}^{2}\partial_{M}{K_{P}}\partial_{N}{K_{Q}}-\frac{1}{8}H_{o}^{MN}K_{M}K^{P}{\overline{K}}^{4}\partial_{P}{K_{Q}}\partial_{N}{K^{Q}}\,,$
(A.2)

the $\kappa^{3}$ contributions are

$\displaystyle-\frac{1}{4}K^{M}{\overline{K}}^{2}{\overline{K}}^{P}\partial_{P}{K_{Q}}\partial_{M}{K^{Q}}-\frac{1}{4}K^{M}{\overline{K}}_{N}{\overline{K}}^{P}{\overline{K}}^{Q}\partial_{P}{K_{Q}}\partial_{M}{K^{N}}$
$\displaystyle+\frac{1}{4}K^{M}{\overline{K}}^{2}{\overline{K}}^{P}\partial_{M}{K_{Q}}\partial_{P}{K^{Q}}+\frac{1}{4}K^{M}{\overline{K}}_{N}{\overline{K}}^{P}{\overline{K}}^{Q}\partial_{M}{K_{P}}\partial_{Q}{K^{N}}$
$\displaystyle-\frac{1}{4}H_{o}^{MN}K_{M}{\overline{K}}^{P}{\overline{K}}^{2}\partial_{P}{K_{Q}}\partial_{N}{K^{Q}}+\frac{1}{4}H_{o}^{MN}K_{M}K^{P}{\overline{K}}^{Q}\partial_{N}{K_{Q}}\partial_{P}{{\overline{K}}^{2}}$
$\displaystyle+\frac{1}{4}H_{o}^{MN}K^{P}{\overline{K}}^{Q}{\overline{K}}^{2}\partial_{M}{K_{Q}}\partial_{P}{K_{N}}-\frac{1}{4}H_{o}^{MN}K_{M}K^{P}{\overline{K}}^{2}\partial_{P}{K_{Q}}\partial_{N}{{\overline{K}}^{Q}}$
$\displaystyle+\frac{1}{4}H_{o}^{MN}K_{M}K^{P}{\overline{K}}^{Q}\partial_{P}{K_{Q}}\partial_{N}{{\overline{K}}^{2}}-\frac{1}{4}H_{o}^{MN}K^{P}{\overline{K}}_{Q}{\overline{K}}^{2}\partial_{P}{K_{M}}\partial_{N}{K^{Q}}$
$\displaystyle-\frac{1}{4}H_{o}^{MN}K_{M}K^{P}{\overline{K}}^{2}\partial_{N}{K^{Q}}\partial_{P}{{\overline{K}}_{Q}}-\frac{1}{4}H_{o}^{MN}K^{P}{\overline{K}}_{M}{\overline{K}}^{2}\partial_{P}{K_{Q}}\partial_{N}{K^{Q}}$
$\displaystyle-K^{M}{\overline{K}}^{2}{\overline{K}}^{P}\partial_{Q}{K_{P}}\partial_{M}{K^{Q}}-\frac{1}{2}K^{M}K^{N}{\overline{K}}^{2}\partial_{M}{K_{Q}}\partial_{N}{{\overline{K}}^{Q}}\,,$
(A.3)

and, finally, the $\kappa^{2}$ contributions are

$\displaystyle\frac{1}{4}H_{o}^{MN}{\overline{K}}^{2}\partial_{M}{K_{Q}}\partial_{N}{K^{Q}}+\frac{1}{4}H_{o}^{MN}{\overline{K}}_{P}{\overline{K}}^{Q}\partial_{M}{K_{Q}}\partial_{N}{K^{P}}$
$\displaystyle+\frac{1}{4}H_{o}^{MN}{\overline{K}}^{P}{\overline{K}}^{Q}\partial_{M}{K_{P}}\partial_{N}{K_{Q}}-\frac{1}{2}H_{o}^{MN}{\overline{K}}_{P}{\overline{K}}^{Q}\partial_{Q}{K_{M}}\partial_{N}{K^{P}}$
$\displaystyle-\frac{1}{2}H_{o}^{MN}K_{M}{\overline{K}}^{P}\partial_{N}{K^{Q}}\partial_{P}{{\overline{K}}_{Q}}+\frac{1}{2}H_{o}^{MN}K_{M}{\overline{K}}^{P}\partial_{Q}{K_{P}}\partial_{N}{{\overline{K}}^{Q}}$
$\displaystyle-\frac{1}{2}H_{o}^{MN}{\overline{K}}_{M}{\overline{K}}^{P}\partial_{P}{K_{Q}}\partial_{N}{K^{Q}}-\frac{1}{2}H_{o}^{MN}K^{P}{\overline{K}}^{Q}\partial_{P}{K_{M}}\partial_{N}{{\overline{K}}_{Q}}$
$\displaystyle-\frac{1}{2}H_{o}^{MN}K_{M}K^{P}\partial_{P}{{\overline{K}}_{Q}}\partial_{N}{{\overline{K}}^{Q}}-\frac{1}{2}H_{o}^{MN}{\overline{K}}^{2}\partial_{Q}{K_{M}}\partial_{N}{K^{Q}}$
$\displaystyle-\frac{1}{2}H_{o}^{MN}K_{M}{\overline{K}}^{P}\partial_{N}{K^{Q}}\partial_{Q}{{\overline{K}}_{P}}+\frac{1}{2}H_{o}^{MN}K^{P}{\overline{K}}^{Q}\partial_{M}{K_{Q}}\partial_{P}{{\overline{K}}_{N}}$
$\displaystyle-\frac{1}{2}H_{o}^{MN}K^{P}{\overline{K}}_{M}\partial_{P}{K_{Q}}\partial_{N}{{\overline{K}}^{Q}}-\frac{1}{2}H_{o}^{MN}{\overline{K}}_{M}{\overline{K}}^{P}\partial_{Q}{K_{P}}\partial_{N}{K^{Q}}\,.$
(A.4)

On the other hand, the generalized dilaton contributions are given by

$\displaystyle\mathcal{L}_{d}$ $\displaystyle=$ $\displaystyle
4H^{MN}\partial_{M}d\partial_{N}d-2\partial_{M}H^{MN}\partial_{N}d$ (A.5)
$\displaystyle=$ $\displaystyle
4\kappa^{2}\Bigg{[}H_{o}^{MN}+\kappa\Big{(}\bar{K}^{M}K^{N}+K^{M}\bar{K}^{N}\Big{)}\Bigg{]}\partial_{M}f\partial_{N}f$
$\displaystyle-2\kappa^{2}\partial_{M}\Bigg{[}\Big{(}\bar{K}^{M}K^{N}+K^{M}\bar{K}^{N}\Big{)}\Bigg{]}\partial_{N}f\
.$

The gauged part can be easily computed using the linear ansatz for the
C-field,

$\displaystyle\Delta_{\alpha\beta}-\Delta_{o\alpha\beta}$ $\displaystyle=$
$\displaystyle-2\kappa K^{M}J_{(\alpha}C_{oM\beta)}$ (A.6)
$\displaystyle\Theta_{MN}-\Theta_{oMN}$ $\displaystyle=$
$\displaystyle-2\kappa
K_{(M}J^{\alpha}C_{oN)\alpha}+\kappa^{2}K_{M}K_{N}J^{2}\,.$ (A.7)

The first order contributions can be easily computed using (3.36) since
$\Delta^{-1}=\Delta^{-1}_{o}+\mathcal{O}(\kappa^{2})$. Then, the $\kappa$
contribution for the gauge sector is,

$\displaystyle
4f_{\beta\gamma}^{\alpha}C^{M\beta}C^{N\gamma}\partial_{M}(K_{N}J_{\alpha})$
(A.8)

Higher-order contributions can be straightforwardly computed following the
same logic to any desired order.

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