July 2022

HU-EP-22/17-RTG

Cosmological Perturbations in Double Field Theory

Olaf Hohm and Allison F. Pinto

Institute for Physics, Humboldt University Berlin,
Zum Großen Windkanal 6, D-12489 Berlin, Germany

<EMAIL_ADDRESS><EMAIL_ADDRESS>

Abstract

> We explore perturbative double field theory about time-dependent
> (cosmological) backgrounds to cubic order. To this order the theory is
> consistent in a weakly constrained sense, so that for a toroidal geometry it
> encodes both momentum and genuine winding modes. We give a self-contained
> discussion of the consistency problems and their resolution, including the
> role of cocycle factors and the $O(d,d,\mathbb{Z})$ duality. Finally, as a
> first step toward the computation of cosmological correlation functions, we
> propose a generalized scalar-vector-tensor decomposition and use it to
> construct gauge invariant generalized Bardeen variables. Compared to
> standard cosmology there are fewer tensor modes but more vector and scalar
> modes.

###### Contents

  1. 1 Introduction
  2. 2 Double field theory in time-dependent backgrounds
    1. 2.1 Generalities
    2. 2.2 Quadratic theory
    3. 2.3 Canonical formulation of quadratic theory
    4. 2.4 Cubic theory
  3. 3 Dualities and cocycles
    1. 3.1 Dualities in time-dependent backgrounds
    2. 3.2 Friedmann-Robertson-Walker backgrounds
    3. 3.3 Weak constraint and cocycles
  4. 4 Decompositions
    1. 4.1 Scalar-vector-tensor decomposition
    2. 4.2 Gauge fixing
    3. 4.3 Gauge invariant variables
  5. 5 Summary and Outlook

##  1 Introduction

Cosmology requires, ultimately, a gravitational theory beyond Einstein’s
general relativity in order to resolve the Big Bang singularity. As a
consistent theory of quantum gravity, string theory is a promising candidate
for such a theory. While it has turned out to be very challenging to obtain
predictions from string theory in the realm of particle physics, in cosmology
the situation may be better, given that here effects of fundamental physics at
microscopic scales may be amplified to very large scales. Since the early days
of string theory there have been striking ideas of how to exploit some of the
unique characteristics of string theory, such as dualities, in cosmological
scenarios [1, 2, 3, 4, 5] (see [6, 7, 8, 9, 10, 11] for some recent
developments).

The simplest duality emerges directly in string cosmologies with vanishing
spatial curvature (that luckily appears to be a feature of the universe we
live in). In this case, the dynamics of the Friedmann-Robertson-Walker (FRW)
backgrounds are encoded in the dimensional reduction of the string target
space actions to one dimension (cosmic time), for which classical string
theory exhibits the enhanced symmetry group $O(d,d,\mathbb{R})$ to all orders
in the inverse string tension $\alpha^{\prime}$ [12, 13, 14], with $d$ being
the number of spatial dimensions. This group contains the duality
transformation $a(t)\rightarrow\frac{1}{a(t)}$ of the cosmological scale
factor (‘scale factor duality’) that is at the heart of some of the string
cosmology proposals [1, 2, 3].

In modern cosmology the small deviations or fluctuations around FRW spacetimes
are equally important, as these relate to the temperature fluctuations of the
cosmic microwave background (CMB), see [15] for a review. In order to include
such fluctuations it is no longer sufficient to employ the dimensionally
reduced theory; rather, in cosmological perturbation theory the fundamental
fields are allowed to depend on all spatial coordinates, in which case the
$O(d,d,\mathbb{R})$ symmetry is generically broken to $GL(d,\mathbb{R})$.
However, if the spatial geometry of the FRW spacetime contains a torus (as
assumed in the string-gas cosmology proposal of [1]) then in string theory
there is still an $O(d,d,\mathbb{Z})$ duality invariance. This is due to the
appearance of new states or fields that are not present in standard gravity,
the so-called winding modes of the string, which under $O(d,d,\mathbb{Z})$ are
mapped to the more familiar Kaluza-Klein modes. It is of great interest to
explore whether these modes, or more general massive string modes, assuming
they were excited in the very early universe, can leave an imprint on the CMB.

In this paper we explore some general aspects of the corresponding field
theory, using double field theory [16, 17, 18, 19]. Here the fields depend in
addition to cosmic time $t$ on doubled spatial coordinates ${\bf
X}=(x,\tilde{x})$, where upon Fourier expansion the dependence on the dual
coordinates $\tilde{x}$ leads to the infinite tower of winding modes. More
precisely, in string theory the fields are subject to the level-matching
constraint, which requires that for a given Fourier mode the momenta ${\bf k}$
and the winding vector ${\bf w}$ are orthogonal, i.e., ${\bf k}\cdot{\bf
w}=0$. The implementation of this constraint in a consistent theory (other
than the full string theory) is a difficult and so far unsolved
problem,111However, on general grounds this theory has to exist [20, 21]. but
on flat backgrounds to cubic order such a double field theory was derived by
Hull and Zwiebach in [17] and is entirely consistent to this order. This
theory is usually referred to as ‘weakly constrained’. Most of the subsequent
work on double field theory was done on the ‘strongly constrained’ subsector
for which a stronger version of the level-matching constraint is assumed,
which in turn implies that the doubling of coordinates is only formal.
(However, this theory still makes the $O(d,d,\mathbb{R})$ symmetry arising
upon dimensional reduction manifest before reduction and thus has found
numerous applications, see [22] for a review.)

Since in cosmological perturbation theory much information is already
contained in the quadratic theory around FRW backgrounds, which encodes the
two-point functions, or the cubic theory, which would be required in order to
compute the so far unobserved three-point functions possibly encoding non-
Gaussianity [23], we will here revisit true or weakly constrained double field
theory to cubic order, but allowing the background to be a time-dependent FRW
spacetime whose spatial geometry is a torus. It is a notoriously difficult
problem to treat time-dependent backgrounds in string theory, and therefore we
will not attempt to derive this double field theory directly from closed
string field theory. Instead we use the short cut of starting from the
strongly constrained double field theory which is known in a background
independent form [16, 18], so that we may expand about an arbitrary background
solution, including time-dependent ones. Following Hull and Zwiebach we may
then reinterpret the resulting theory to cubic order (and so far to cubic
order only) as a weakly constrained theory. More precisely, this holds up to
so-called cocycle factors, which are claimed to be necessary for consistency
to quartic and higher orders [24, 25, 26], and which have not been included in
[17]. For this issue the time dependence of the background does not seem
relevant, and we use the opportunity to give a complete discussion of the
consistency of the cubic theory including the cocycles, following the
treatment by Kugo and Zwiebach [26]. While we thus do not provide a first-
principle derivation of this theory, the latter is consistent to cubic order
and has all features expected of string theory for such backgrounds.

Although the above program seems a straightforward extension of double field
theory on flat space, it turns out that the structure of the theory is
significantly more involved for time-dependent backgrounds. Ultimately these
complications trace back to the fact that in string theory on FRW backgrounds
the symmetry group is reduced compared to flat space. More precisely, there is
the duality group $O(d,d,\mathbb{Z})$ that maps different backgrounds into
each other in such a way that string theories on such backgrounds are
equivalent. The subgroup that leaves the background genuinely invariant is
then a symmetry group of string theory. For a flat metric this group is
$O(d)\times O(d)$, but for non-trivial FRW backgrounds it is reduced to the
diagonal subgroup hereof,

$(O(d)\times O(d))_{\rm diag}\ \subset\ O(d)\times O(d)\ \subset\ O(d,d)\;.$
(1.1)

Concretely, the complication is that while perturbative double field theory on
any background features a ‘double copy’ type factorization into
unbarred/barred indices [27], only on flat space does this imply two
independent rotational symmetries. In contrast, for FRW backgrounds a tensor
$L_{a}{}^{\bar{b}}$ enters the theory that is proportional to the Hubble
parameter $H=\frac{\dot{a}}{a}$ and which connects unbarred and barred
indices. Consequently, the double field theory action allows for contractions
of the derivative operators $D_{a}$ and $D_{\bar{a}}$ that are linearly
independent combinations of momentum and winding operators. Concretely, for a
flat FRW background with a single scale factor $a(t)$, these operators are
given in terms of the dual momentum derivatives $\partial$ and winding
derivatives $\tilde{\partial}$ by

$D\ \propto\ a^{-1}(t)\partial-a(t)\tilde{\partial}\;,\qquad\bar{D}\ \propto\
a^{-1}(t)\partial+a(t)\tilde{\partial}\;.$ (1.2)

In contrast to flat space, where $a\equiv 1$, there is no doubled spatial
rotation invariance, which in turn complicates the decomposition of fields
into irreducible vector and tensor components subject to divergence-free and
other constraints. Here we propose a scalar-vector-tensor (SVT) decomposition
that has two desirable features: First, it allows for the definition of gauge
invariant field variables, generalizing those introduced by Bardeen in
conventional cosmology [28]. Second, in terms of these variables the quadratic
action completely decouples between tensor, vector and scalar modes. However,
this SVT decomposition is significantly more involved than in standard
cosmology since, owing to the presence of twice as many derivatives, one may
impose more constraints, say on the divergence of a tensor. Consequently, a
decomposition into fully irreducible objects (that cannot be subjected to
further constraints) leads to more vector and scalar modes than in standard
cosmology. It is tempting to speculate about the possible observability of
these extra modes.

The rest of this paper is organized as follows. In sec. 2 we give a brief
review of the background independent double field theory, and we expand it
about an arbitrary time-dependent background to cubic order. In particular, we
display the gauge symmetries to this order and verify invariance. In sec. 3 we
discuss the duality properties of time-dependent backgrounds in double field
theory and then specialize to a particular class of FRW-type backgrounds and
discuss their invariance group. In the final subsection we explain that the
theory to cubic order is consistent in a weakly constrained sense, and we
introduce the cocyle factors that appear to be necessary in string theory. In
sec. 4 we introduce a scalar-vector-tensor decomposition and use it to define
gauge invariant variables. These technical results should be the basis, for
instance, for the computation of cosmological correlation functions, which we
leave for future work. We briefly discuss this and other aspects in a
conclusion section.

##  2 Double field theory in time-dependent backgrounds

In this section we give a brief review of double field theory and then expand
its action around a general time-dependent background. We display the action
quadratic in fluctuations and discuss its gauge invariance together with its
canonical formulation. Finally, we give the complete cubic action.

###  2.1 Generalities

The double field theory action can be written in terms of the generalized
metric and the dilaton on a doubled space. The canonical formulation was
derived in [29], based on earlier results in [30], via splitting the doubled
coordinates and indices into temporal and spatial components and imposing the
condition that all fields are independent of dual time. This results in a
theory with fields:

${\cal H}_{MN}\,,\quad\Phi\,,\quad n\,,\;\quad{\cal N}^{M}\;,$ (2.1)

where ${\cal H}_{MN}$ denotes the spatial generalized metric, satisfying
$\mathcal{H}_{MK}\mathcal{H}^{KN}=\delta_{M}{}^{N}$, $\Phi$ is the duality
invariant dilaton, $n$ denotes the lapse function ensuring time
reparametrization invariance, and ${\cal N}^{M}$ denotes the doubled shift
vector. The action is formulated on a $(1+2d)$-dimensional space and reads

$S=\int dt\int d^{2d}{\bf
X}\,n\,e^{-2\Phi}\bigg{(}-4(D_{t}\Phi)^{2}-\frac{1}{8}D_{t}\mathcal{H}_{MN}D_{t}\mathcal{H}^{MN}+\mathcal{R}(\Phi,\mathcal{H}_{MN})\bigg{)}\,.$
(2.2)

Here doubled spatial coordinates are denoted by ${\bf
X}^{M}=(\tilde{x}_{i},x^{i})$, with $i=1,...,d$ and $M=1,...,2d$, and all
fields depend on $(t,{\bf X}^{M})$. The covariant derivatives are defined as

$\displaystyle
D_{t}\equiv\frac{1}{n}\left(\partial_{t}-\mathcal{L}_{\mathcal{N}}\right)\;,$
(2.3)

where $\mathcal{L}_{\mathcal{N}}$ is the generalized Lie derivative with
respect to the generalized shift vector $\mathcal{N}^{M}$:

$\begin{split}{\cal L}_{{\cal N}}\Phi\ &=\ {\cal
N}^{M}\partial_{M}\Phi-\frac{1}{2}\partial_{M}{\cal N}^{M}\;,\\\ {\cal
L}_{\cal N}{\cal H}_{MN}\ &=\ {\cal N}^{K}\partial_{K}{\cal
H}_{MN}+K_{M}{}^{K}({\cal N}){\cal H}_{KN}+K_{N}{}^{K}({\cal N}){\cal
H}_{KN}\;.\end{split}$ (2.4)

Here we defined $K_{MN}(\xi)=2\partial_{[M}\xi_{N]}$, where $O(d,d)$ indices
are contracted by the $O(d,d)$ metric

$\eta_{MN}\ =\ \begin{pmatrix}0&\delta^{i}{}_{j}\\\
\delta_{i}{}^{j}&0\end{pmatrix}\,.$ (2.5)

The last term in the action (2.2) uses the generalized curvature scalar

$\displaystyle\begin{split}\mathcal{R}(\Phi,\mathcal{H}_{MN})\ &\equiv\
4\mathcal{H}^{MN}\partial_{M}\partial_{N}\Phi-\partial_{M}\partial_{N}\mathcal{H}^{MN}-4\mathcal{H}^{MN}\partial_{M}\Phi\partial_{N}\Phi+4\partial_{M}\mathcal{H}^{MN}\partial_{N}\Phi\\\
&\;\;+\frac{1}{8}\mathcal{H}^{MN}\partial_{M}\mathcal{H}^{KL}\partial_{N}\mathcal{H}_{KL}-\frac{1}{2}\mathcal{H}^{MN}\partial_{M}\mathcal{H}^{KL}\partial_{K}\mathcal{H}_{NL}\;.\end{split}$
(2.6)

For now we assume all fields to be subject to the strong version of the
constraint

$\eta^{MN}\partial_{M}\partial_{N}f=2\tilde{\partial}^{i}\partial_{i}f=0\,,$
(2.7)

meaning that also terms of the form $\partial^{M}f\,\partial_{M}g$ are set to
zero.

In the following we will employ a frame formalism for the internal generalized
metric [16, 31]. We introduce a generalized frame or vielbein $E_{A}{}^{M}$,
with inverse $E_{M}{}^{A}$, satisfying
$E_{A}{}^{M}E_{M}{}^{B}=\delta_{A}{}^{B}$, from which the generalized metric
can be constructed via

$\displaystyle\mathcal{H}_{MN}\ =\ E_{M}{}^{A}E_{N}{}^{B}S_{AB}\;,$ (2.8)

where $S_{AB}$ denotes a positive-definite (tangent space) metric to be given
momentarily. The flat indices split as $A=(a,\bar{a})$. The frame field is
subject to the constraint that the “flattened” version of the $O(d,d)$ metric
(2.5) is block-diagonal:

${\cal G}_{AB}\ \equiv\ E_{A}{}^{M}E_{B}{}^{N}\eta_{MN}\ =\
\begin{pmatrix}{\cal G}_{ab}&0\\\ 0&{\cal G}_{\bar{a}\bar{b}}\end{pmatrix}\;,$
(2.9)

with no further constraints on the (generally spacetime dependent) metrics
${\cal G}_{ab}$ and ${\cal G}_{\bar{a}\bar{b}}$. Thus, the local frame
transformations comprise the group $GL(d,\mathbb{R})\times GL(d,\mathbb{R})$.
The metric ${\cal G}_{AB}$, which is used to raise and lower flat indices, has
signature $(d,d)$, and so we can assume without loss of generality that ${\cal
G}_{ab}$ is negative-definite and that ${\cal G}_{\bar{a}\bar{b}}$ is
positive-definite. Then, the metric $S_{AB}$ is defined as

$S_{AB}\ =\ \begin{pmatrix}-{\cal G}_{ab}&0\\\ 0&{\cal
G}_{\bar{a}\bar{b}}\end{pmatrix}\;.$ (2.10)

Finally, we give the symmetries of the action, which is invariant under
$O(d,d)$ transformations and generalized diffeomorphisms. The transformation
rules can be inferred from [30]. The gauge parameters are ${\xi}^{0}$,
$\tilde{\xi}_{0}$, $\xi^{M}$, and $\Lambda_{A}{}^{B}$, all depending on
coordinates $(t,{\bf X}^{M})$, and act infinitesimally via

$\displaystyle\begin{split}\delta
E_{A}{}^{M}&=\mathcal{L}_{\xi}E_{A}{}^{M}+n\xi^{0}D_{t}E_{A}{}^{M}+\Lambda_{A}{}^{B}E_{B}{}^{M}\,,\\\
\delta n&=n\xi^{0}D_{t}n+n^{2}D_{t}\xi^{0}+\xi^{M}\partial_{M}n\,,\\\
\delta\mathcal{N}^{M}&=\partial^{M}\tilde{\xi}_{0}+\partial_{t}\xi^{M}-n^{2}\mathcal{H}^{MN}\partial_{N}\xi^{0}+\mathcal{L}_{\xi}\mathcal{N}^{M}\,,\\\
\delta\Phi&=n\xi^{0}D_{t}\Phi+\mathcal{L}_{\xi}\Phi\,.\end{split}$ (2.11)

###  2.2 Quadratic theory

Now let us expand around a background that is purely time-dependent as
follows:

$\displaystyle E_{A}{}^{M}(t,{\bf X})\ $ $\displaystyle=\
\bar{E}_{A}{}^{M}(t)-h_{A}{}^{B}(t,{\bf X})\bar{E}_{B}{}^{M}(t)\,,$ (2.12)
$\displaystyle\Phi(t,{\bf X})\ $ $\displaystyle=\ \bar{\Phi}(t)+\varphi(t,{\bf
X})\,,$ (2.13) $\displaystyle n(t,{\bf X})\ $ $\displaystyle=\
\bar{n}(t)(1+\phi(t,{\bf X}))\,,$ (2.14) $\displaystyle\mathcal{N}^{M}(t,{\bf
X})\ $ $\displaystyle=\ \bar{n}(t)\mathcal{A}^{M}(t,{\bf X})\,.$ (2.15)

In the following we will omit the bar on $E_{A}{}^{M}(t)$, $\Phi$ and $n$ as
these will exclusively refer to the background quantities. The background
tangent space metric $\bar{\cal G}_{AB}$ constructed as in (2.9) will be used
to raise and lower flat indices, while the background ${E}_{A}{}^{M}$ and its
inverse will be used to flatten and unflatten indices. Moreover, we assume
that a gauge has been chosen for the (background) frame transformations for
which $\bar{\mathcal{G}}_{AB}$ is constant and hence does not depend on time.
The corresponding background version of the metric (2.10) is then also
constant. This has the important advantage that we can freely raise and lower
flat indices under time derivatives.

Finally, we note that the first-order frame transformations allow one to fix a
gauge with $h_{ab}=h_{\bar{a}\bar{b}}=0$, so that the independent fluctuation
is given by

$h_{a\bar{b}}\ =\ -h_{\bar{b}a}\;,$ (2.16)

where the last equality follows from the constraint that (2.9) is block-
diagonal, i.e., ${\cal G}_{a\bar{b}}=0$. We have thus completely fixed the
first-order frame transformations.

#### Background equations

Inserting the above expansion into the action, one obtains to leading order an
action for the purely time-dependent background fields encoding their
dynamics:

$S_{0}=\int dt\int d^{2d}{\bf
X}\,n^{-1}\,e^{-2\Phi}\bigg{(}-4\dot{\Phi}^{2}-\frac{1}{8}{\rm
tr}\big{(}\dot{S}^{2}\big{)}\bigg{)}\,,$ (2.17)

where we employ matrix notation, with $S^{M}{}_{N}$ the background generalized
metric with one index raised, and the dot denotes the time derivative. The
field equations read

$\begin{split}\ddot{S}+S\dot{S}^{2}-2\bigg{(}\dot{\Phi}+\frac{1}{2}\partial_{t}\log{n}\bigg{)}\dot{S}&=0\,,\\\
-4\ddot{\Phi}+4\dot{\Phi}^{2}+4\dot{\Phi}\partial_{t}\log{n}-\frac{1}{8}{\rm
tr}\big{(}\dot{S}^{2}\big{)}&=0\,,\\\ 4\dot{\Phi}^{2}+\frac{1}{8}{\rm
tr}\big{(}\dot{S}^{2}\big{)}&=0\,.\end{split}$ (2.18)

It will be convenient to express these equations explicitly in terms of the
background frame field. We define

$L_{A}{}^{B}\ \equiv\ \frac{1}{n}\,\partial_{t}{E}_{A}{}^{M}{E}_{M}{}^{B}\;,$
(2.19)

in terms of which the equations of motion are

$\displaystyle\frac{1}{n}\dot{L}_{a}{}^{\bar{b}}-L_{a}{}^{c}L_{c}{}^{\bar{b}}+L_{a}{}^{\bar{c}}L_{\bar{c}}{}^{\bar{b}}-\frac{2}{n}\dot{\Phi}L_{a}{}^{\bar{b}}\
$ $\displaystyle=\ 0\,,$ (2.20)
$\displaystyle-4\ddot{\Phi}+4\dot{\Phi}^{2}+4\dot{\Phi}\partial_{t}\log{n}+n^{2}\,L_{a}{}^{\bar{b}}L_{\bar{b}}{}^{a}\
$ $\displaystyle=\ 0\,,$ (2.21) $\displaystyle
4\dot{\Phi}^{2}-n^{2}\,L_{a}{}^{\bar{b}}L_{\bar{b}}{}^{a}\ $ $\displaystyle=\
0\,.$ (2.22)

It is instructive to rewrite these equations in terms of derivatives that are
covariant under background frame transformations with parameter
$\bar{\Lambda}_{A}{}^{B}$ and time reparametrization with parameter
$\bar{\xi}^{0}$. These transformations act on $L_{A}{}^{B}$, the lapse
function $n$ and generic vectors ${\cal V}_{a}$ and ${\cal V}_{\bar{a}}$ as

$\displaystyle\begin{split}\bar{\delta}L_{A}{}^{B}&=\bar{\xi}^{0}\partial_{t}L_{A}{}^{B}+\frac{1}{n}\partial_{t}\bar{\Lambda}_{A}{}^{B}+\bar{\Lambda}_{A}{}^{C}L_{C}{}^{B}-\bar{\Lambda}_{C}{}^{B}L_{A}{}^{C}\;,\\\
\bar{\delta}n&=\partial_{t}(\bar{\xi}^{0}n)\,,\\\
\bar{\delta}\mathcal{V}_{a}&=\bar{\xi}^{0}\partial_{t}\mathcal{V}_{a}+\bar{\Lambda}_{a}{}^{b}\mathcal{V}_{b}\,,\\\
\bar{\delta}\mathcal{V}_{\bar{a}}&=\bar{\xi}^{0}\partial_{t}\mathcal{V}_{\bar{a}}+\bar{\Lambda}_{\bar{a}}{}^{\bar{b}}\mathcal{V}_{\bar{b}}\,.\end{split}$
(2.23)

All other fields transform as scalars under $\bar{\xi}^{0}$ and as tensors
under $\bar{\Lambda}_{A}{}^{B}$. The first line implies that $L_{a}{}^{b}$ and
$L_{\bar{a}}{}^{\bar{b}}$ transform as connections under background frame
transformations, while $L_{a}{}^{\bar{b}}$ transforms as a tensor. We then
have the covariant derivatives

$\displaystyle\begin{split}{\nabla}_{t}\mathcal{V}_{a}\ &=\
\frac{1}{n}\partial_{t}\mathcal{V}_{a}-L_{a}{}^{b}\mathcal{V}_{b}\,,\\\
{\nabla}_{t}\mathcal{V}_{\bar{a}}\ &=\
\frac{1}{n}\partial_{t}\mathcal{V}_{\bar{a}}-L_{\bar{a}}{}^{\bar{b}}\mathcal{V}_{\bar{b}}\,,\end{split}$
(2.24)

which transform covariantly in that

$\displaystyle\begin{split}\bar{\delta}\left(\nabla_{t}\mathcal{V}_{a}\right)\
&=\
\bar{\xi}^{0}\partial_{t}\big{(}\nabla_{t}\mathcal{V}_{a}\big{)}+\bar{\Lambda}_{a}{}^{b}\nabla_{t}\mathcal{V}_{b}\,,\\\
\bar{\delta}\left(\nabla_{t}\mathcal{V}_{\bar{a}}\right)\ &=\
\bar{\xi}^{0}\partial_{t}\big{(}\nabla_{t}\mathcal{V}_{\bar{a}}\big{)}+\bar{\Lambda}_{\bar{a}}{}^{\bar{b}}\nabla_{t}\mathcal{V}_{\bar{b}}\,.\end{split}$
(2.25)

(2.20) can now be written more compactly as

${\nabla}_{t}L_{a}{}^{\bar{b}}-\frac{2}{n}\dot{\Phi}L_{a}{}^{\bar{b}}\ =\
0\,,$ (2.26)

or, equivalently,

$\nabla_{t}\left(e^{-2\Phi}L_{a\bar{b}}\right)\ =\ 0\;.$ (2.27)

In addition, upon adding (2.21) and (2.22) one obtains the useful equation

$\ddot{\Phi}-2\dot{\Phi}^{2}-\dot{\Phi}\,\partial_{t}\log{n}\ =\ 0\,,$ (2.28)

or, equivalently,

$\partial_{t}\left(e^{-2\Phi}n^{-1}\dot{\Phi}\right)\ =\ 0\;.$ (2.29)

#### Quadratic fluctuations

Next, we assume that the background equations are satisfied, so that the terms
linear in fluctuations drop out. The action for the quadratic fluctuations is
given by

$\displaystyle\begin{split}S^{(2)}=&\int dt\int d^{2d}{\bf
X}\,\,\mathcal{L}\,,\end{split}$ (2.30)

where

$\begin{split}\mathcal{L}=n\,e^{-2\Phi}\bigg{\\{}&-4\Gamma^{2}+2\phi\bigg{(}4\frac{\dot{\Phi}}{n}\Gamma-L^{a\bar{b}}\omega_{a\bar{b}}\bigg{)}-\omega_{a\bar{b}}\omega^{a\bar{b}}+\frac{1}{2}(\omega_{ab}\omega^{ab}+\omega_{\bar{a}\bar{b}}\omega^{\bar{a}\bar{b}})\\\
&-\frac{1}{2}\big{(}K_{ab}K^{ab}+K_{\bar{a}\bar{b}}K^{\bar{a}\bar{b}}\big{)}+4L^{a\bar{b}}(\varphi
K_{a\bar{b}}+\Gamma
h_{a\bar{b}})+\mathcal{V}^{(2)}(\phi,\varphi,h)\bigg{\\}}\,,\end{split}$
(2.31)

and we defined

$\displaystyle\begin{split}\Gamma&\equiv\nabla_{t}\varphi+\frac{1}{2}\partial_{M}\mathcal{A}^{M}\,,\\\
\ \omega_{a\bar{b}}&\equiv\nabla_{t}h_{a\bar{b}}-K_{a\bar{b}}\,,\\\
\omega^{ab}&\equiv 2h^{[a}{}_{\bar{c}}L^{b]\bar{c}}+K^{ab}\,,\\\
\omega^{\bar{a}\bar{b}}&\equiv
2h^{c[\bar{a}}L_{c}{}^{\bar{b}]}+K^{\bar{a}\bar{b}}\,,\end{split}$ (2.32)

with

$K_{AB}\equiv K_{AB}({\cal A})=D_{A}{\cal A}_{B}-D_{B}{\cal A}_{A}\;,$ (2.33)

and

$D_{A}\ \equiv\ \bar{E}_{A}{}^{M}\partial_{M}\;.$ (2.34)

$K_{AB}$ satisfies the identities:

$D_{A}K_{BC}+D_{B}K_{CA}+D_{C}K_{AB}=0\,,$ (2.35)
$D^{B}K_{AB}-D_{A}D^{B}\mathcal{A}_{B}=0\,,$ (2.36)

the latter being a consequence of the strong constraint (2.7), which in terms
of this flattened derivative takes the form

$\Delta\equiv-2D^{a}D_{a}=2D^{\bar{a}}D_{\bar{a}}\,.$ (2.37)

Also note that the flattened spatial derivatives and the covariant (time)
derivative satisfy the commutation relations:

$\displaystyle\big{[}D_{a},\nabla_{t}\big{]}\mathcal{V}_{B}$
$\displaystyle=-L_{a}{}^{\bar{c}}D_{\bar{c}}\mathcal{V}_{B}\,,$ (2.38)
$\displaystyle\big{[}D_{\bar{a}},\nabla_{t}\big{]}\mathcal{V}_{B}$
$\displaystyle=-L_{\bar{a}}{}^{c}D_{c}\mathcal{V}_{B}\,.$ (2.39)

In the action we collected the terms with only spatial derivatives into ${\cal
V}^{(2)}$, defined as

$\displaystyle\begin{split}\mathcal{V}^{(2)}(\phi,\varphi,h_{a\bar{b}})\ =\
\;&8D^{a}\phi D_{a}\varphi-8D^{a}\varphi D_{a}\varphi-8D_{a}\varphi
D_{\bar{b}}h^{a\bar{b}}+4D_{a}\phi
D_{\bar{b}}h^{a\bar{b}}-2D^{a}h^{b\bar{c}}D_{a}h_{b\bar{c}}\\\
&\,+2D^{c}h^{a\bar{b}}D_{a}h_{c\bar{b}}-2D^{\bar{c}}h^{a\bar{b}}D_{\bar{b}}h_{a\bar{c}}\,.\end{split}$
(2.40)

This can be rewritten in terms of

$\varphi_{\pm}\ \equiv\ \varphi\pm\frac{1}{2}\phi\;,$ (2.41)

which yields

$\displaystyle\begin{split}\mathcal{V}^{(2)}(\phi,\varphi_{-},h_{a\bar{b}})\
=\ \;&2D^{a}\phi
D_{a}\phi-8D^{a}\varphi_{-}D_{a}\varphi_{-}-8D_{a}\varphi_{-}D_{\bar{b}}h^{a\bar{b}}-2D^{a}h^{b\bar{c}}D_{a}h_{b\bar{c}}\\\
&\,+2D^{c}h^{a\bar{b}}D_{a}h_{c\bar{b}}-2D^{\bar{c}}h^{a\bar{b}}D_{\bar{b}}h_{a\bar{c}}\,.\end{split}$
(2.42)

We next turn to the gauge invariance of the quadratic action, which for time-
dependent backgrounds is quite subtle even for the free theory. Under
linearized gauge transformations one finds

$\displaystyle\begin{split}\delta^{(0)}h_{a\bar{b}}\ &=\
D_{a}\xi_{\bar{b}}-D_{\bar{b}}\xi_{a}-\xi^{0}L_{a\bar{b}}\,,\\\
\delta^{(0)}\phi\ &=\ \frac{1}{n}\partial_{t}\xi^{0}\,,\\\
\delta^{(0)}\varphi\ &=\
\xi^{0}\frac{\dot{\Phi}}{n}-\frac{1}{2}D_{a}\xi^{a}-\frac{1}{2}D_{\bar{a}}\xi^{\bar{a}}\,,\\\
\delta^{(0)}\mathcal{A}^{a}\ &=\
\nabla_{t}\xi^{a}+L_{\bar{b}}{}^{a}\xi^{\bar{b}}+D^{a}(\tilde{\xi}_{0}+\xi^{0})\,,\\\
\delta^{(0)}\mathcal{A}^{\bar{a}}\ &=\
\nabla_{t}\xi^{\bar{a}}+L_{b}{}^{\bar{a}}\xi^{b}+D^{\bar{a}}(\tilde{\xi}_{0}-\xi^{0})\,,\end{split}$
(2.43)

where we performed the following rescaling of gauge parameters,

$\displaystyle\begin{split}\xi^{0}\ \rightarrow\
\frac{1}{n}\xi^{0}\,,\qquad\tilde{\xi}_{0}\ \rightarrow\
n\tilde{\xi}_{0}\,.\end{split}$ (2.44)

It is also convenient to note that with respect to spatial generalized
diffeomorphisms with parameters $\xi^{M}=\bar{E}_{A}{}^{M}\xi^{A}$ the gauge
transformations for $\varphi$ and ${\cal A}^{M}=\bar{E}_{A}{}^{M}{\cal A}^{A}$
simplify as follows:

$\begin{split}\delta^{(0)}\varphi\ &=\ -\frac{1}{2}\partial_{M}\xi^{M}\;,\\\
\delta^{(0)}{\cal A}^{M}\ &=\ \frac{1}{n}\partial_{t}{\xi}^{M}\;.\end{split}$
(2.45)

The gauge transformations in (2.43) act trivially for the special case:

$\xi^{a}\ =\ D^{a}\chi\,,\quad\xi^{\bar{a}}\ =\
D^{\bar{a}}\chi\,,\quad\tilde{\xi}_{0}=-\nabla_{t}\chi\,,$ (2.46)

where $\chi$ is an arbitrary function. For $\delta^{(0)}h_{a\bar{b}}$ this can
be seen by inspection, for $\delta^{(0)}\varphi$ by the constraint (2.7), and
for $\delta^{(0)}\mathcal{A}^{a}$ and $\delta^{(0)}\mathcal{A}^{\bar{a}}$ by
using the commutators (2.38) and (2.39).

###  2.3 Canonical formulation of quadratic theory

In order to elucidate the gauge structure of the quadratic double field theory
on a time-dependent backgrounds we find it convenient to introduce a canonical
formulation. We begin by computing the canonical momenta of the fields
$\varphi$ and $h_{a\bar{b}}$:

$\begin{split}p_{\varphi}&\equiv\frac{\delta\mathcal{L}}{\delta\partial_{t}\varphi}=e^{-2\Phi}\big{(}-8\Gamma+8n^{-1}\dot{\Phi}\phi+4L^{a\bar{b}}h_{a\bar{b}}\big{)}\,,\\\
p^{a\bar{b}}&\equiv\frac{\delta\mathcal{L}}{\delta\partial_{t}h_{a\bar{b}}}=e^{-2\Phi}\big{(}-2\phi
L^{a\bar{b}}-2\omega^{a\bar{b}}\big{)}\,.\end{split}$ (2.47)

In the following we rescale the dot by a factor of $n^{-1}$: $\dot{f}\equiv
n^{-1}\partial_{t}f$. It is also convenient to rescale the canonical momenta
by multiplying by the background quantity $e^{2\Phi}$:

$\begin{split}P_{\varphi}&\equiv e^{2\Phi}p_{\varphi}\ =\
-8\Gamma+8\dot{\Phi}\phi+4L^{a\bar{b}}h_{a\bar{b}}\;,\\\ P_{a\bar{b}}&\equiv
e^{2\Phi}p^{a\bar{b}}=-2\phi L_{a\bar{b}}-2\omega_{a\bar{b}}\,.\end{split}$
(2.48)

Under the gauge transformations in (2.43), the canonical momenta transform as:

$\begin{split}\delta
P_{\varphi}&=4\Delta\xi^{0}+4L^{a\bar{b}}(D_{a}\xi_{\bar{b}}-D_{\bar{b}}\xi_{a})\,,\\\
\delta
P_{a\bar{b}}&=4\dot{\Phi}\xi^{0}L_{a\bar{b}}-4D_{a}D_{\bar{b}}\xi^{0}-2L_{a}{}^{\bar{c}}(D_{\bar{c}}\xi_{\bar{b}}-D_{\bar{b}}\xi_{\bar{c}})+2L_{\bar{b}}{}^{c}(D_{c}\xi_{a}-D_{a}\xi_{c})\,.\end{split}$
(2.49)

The Hamiltonian density is the Legendre transform of $\mathcal{L}$, defined as
follows:

$\mathcal{H}=n\,e^{-2\Phi}(P_{\varphi}\dot{\varphi}+P^{a\bar{b}}\dot{h}_{a\bar{b}})-\mathcal{L}\,,$
(2.50)

which yields explicitly for the above fields

$\begin{split}\mathcal{H}=n\,e^{-2\Phi}\bigg{\\{}&-\frac{1}{4}P_{a\bar{b}}P^{a\bar{b}}-\frac{1}{16}P_{\varphi}^{2}-\phi
L_{a\bar{b}}P^{a\bar{b}}+P^{a\bar{b}}(K_{a\bar{b}}+L_{a}{}^{c}h_{c\bar{b}}+L_{\bar{b}}{}^{\bar{c}}h_{a\bar{c}})\\\
&-\frac{1}{2}\partial_{M}\mathcal{A}^{M}P_{\varphi}+\dot{\Phi}\phi
P_{\varphi}+\frac{1}{2}(L^{a\bar{b}}h_{a\bar{b}})P_{\varphi}\\\
&-4\dot{\Phi}(L^{a\bar{b}}h_{a\bar{b}})\phi+L_{b}{}^{\bar{d}}L_{\bar{c}}{}^{b}h_{a}{}^{\bar{c}}h^{a}{}_{\bar{d}}+L_{c}{}^{\bar{b}}L_{\bar{b}}{}^{d}h^{c}{}_{\bar{a}}h_{d}{}^{\bar{a}}+2L_{a}{}^{\bar{b}}L_{c}{}^{\bar{d}}h^{a}{}_{\bar{d}}h^{c}{}_{\bar{b}}\\\
&-2L_{c}{}^{\bar{b}}h^{c\bar{a}}K_{\bar{a}\bar{b}}-2L^{b\bar{c}}h^{a}{}_{\bar{c}}K_{ab}\\\
&-4L^{a\bar{b}}K_{a\bar{b}}\varphi-(L^{a\bar{b}}h_{a\bar{b}})^{2}-\mathcal{V}^{(2)}\bigg{\\}}\,.\end{split}$
(2.51)

In terms of this Hamiltonian density the complete quadratic action can be
written with the original fields and the canonical momenta in the following
first-order form:

$S=\int dt\int d^{2d}{\bf
X}\Big{[}n\,e^{-2\Phi}\big{(}P^{a\bar{b}}\dot{h}_{a\bar{b}}+P_{\varphi}\dot{\varphi}\big{)}-\mathcal{H}\Big{]}\,.$
(2.52)

Naturally, upon integrating out the canonical momenta one recovers the
original second-order action. Specifically, the equations of motion $E=0$
following from this action are encoded in the following components:

$\displaystyle E_{P_{\varphi}}\ $ $\displaystyle=\
\frac{1}{8}P_{\varphi}+\dot{\varphi}+\frac{1}{2}\partial_{M}\mathcal{A}^{M}-\dot{\Phi}\phi-\frac{1}{2}L^{a\bar{b}}h_{a\bar{b}}\,,$
(2.53) $\displaystyle E_{Pa\bar{b}}\ $ $\displaystyle=\
\frac{1}{2}P_{a\bar{b}}+\nabla_{t}h_{a\bar{b}}-K_{a\bar{b}}+\phi
L_{a\bar{b}}\,,$ (2.54) $\displaystyle E_{\varphi}\ $ $\displaystyle=\
-\nabla_{t}P_{\varphi}+2\dot{\Phi}P_{\varphi}+4L^{a\bar{b}}K_{a\bar{b}}+4\Delta\phi-8\Delta\varphi+8D^{a}D^{\bar{b}}h_{a\bar{b}}\,,$
(2.55) $\displaystyle E_{a\bar{b}}\ $ $\displaystyle=\
-\nabla_{t}P_{a\bar{b}}+2\dot{\Phi}P_{a\bar{b}}-\frac{1}{2}L_{a\bar{b}}P_{\varphi}+2(L^{c\bar{d}}h_{c\bar{d}})L_{a\bar{b}}+4\dot{\Phi}\phi
L_{a\bar{b}}$
$\displaystyle\qquad-2L_{\bar{d}}{}^{c}L_{c\bar{b}}h_{a}{}^{\bar{d}}-2L_{a}{}^{\bar{d}}L_{\bar{d}}{}^{c}h_{c\bar{b}}+4L_{a}{}^{\bar{d}}L_{\bar{b}}{}^{c}h_{c\bar{d}}+2L_{a}{}^{\bar{d}}K_{\bar{b}\bar{d}}-2L_{\bar{b}}{}^{c}K_{ac}$
$\displaystyle\qquad+8D_{a}D_{\bar{b}}\varphi-4D_{a}D_{\bar{b}}\phi-2\Delta
h_{a\bar{b}}-4D_{a}D^{c}h_{c\bar{b}}+4D_{\bar{b}}D^{\bar{c}}h_{a\bar{c}}\,,$
(2.56) $\displaystyle E_{a}\ $ $\displaystyle=\
-D^{\bar{b}}P_{a\bar{b}}-\frac{1}{2}D_{a}P_{\varphi}+4L_{a\bar{b}}D^{\bar{b}}\varphi+2L_{b}{}^{\bar{c}}D^{b}h_{a\bar{c}}-2L_{a}{}^{\bar{c}}D_{b}h^{b}{}_{\bar{c}}\,,$
(2.57) $\displaystyle E_{\bar{a}}\ $ $\displaystyle=\
D^{b}P_{b\bar{a}}-\frac{1}{2}D_{\bar{a}}P_{\varphi}-4L_{b\bar{a}}D^{b}\varphi-2L_{c\bar{a}}D_{\bar{b}}h^{c\bar{b}}+2L_{c}{}^{\bar{b}}D_{\bar{b}}h^{c}{}_{\bar{a}}\,,$
(2.58) $\displaystyle E_{\phi}\ $ $\displaystyle=\
L_{a\bar{b}}P^{a\bar{b}}-\dot{\Phi}P_{\varphi}+4\dot{\Phi}L^{a\bar{b}}h_{a\bar{b}}+4\Delta\varphi-4D^{a}D^{\bar{b}}h_{a\bar{b}}\,.$
(2.59)

If one solves $E_{P_{\varphi}}=0$ and $E_{Pa\bar{b}}=0$ for the canonical
momenta and substitutes the expressions into the action, one recovers the
original Lagrangian (2.31). The tensors defining the equations of motion above
satisfy Bianchi identities $G=0$ with the following components:

$\displaystyle G^{0}\equiv D_{a}E^{a}+D_{\bar{a}}E^{\bar{a}}\,,$ (2.60)
$\displaystyle G_{0}\equiv
D^{\bar{a}}E_{\bar{a}}-D^{a}E_{a}-\dot{E}_{\phi}+2\dot{\Phi}E_{\phi}+\dot{\Phi}E_{\varphi}-L^{a\bar{b}}E_{a\bar{b}}+4\dot{\Phi}L^{a\bar{b}}E_{Pa\bar{b}}+4\Delta
E_{P\varphi}-4D^{a}D^{\bar{b}}E_{Pa\bar{b}}\,,$ (2.61) $\displaystyle
G_{a}\equiv
D^{\bar{b}}E_{a\bar{b}}-\nabla_{t}E_{a}+2\dot{\Phi}E_{a}+L_{a}{}^{\bar{b}}E_{\bar{b}}+\frac{1}{2}D_{a}E_{\varphi}+2L_{c}{}^{\bar{b}}D^{c}E_{Pa\bar{b}}-2L_{a}{}^{\bar{b}}D^{c}E_{Pc\bar{b}}+4L_{a}{}^{\bar{b}}D_{\bar{b}}E_{P\varphi}\,,$
(2.62) $\displaystyle
G_{\bar{a}}\equiv-D^{b}E_{b\bar{a}}-\nabla_{t}E_{\bar{a}}+2\dot{\Phi}E_{\bar{a}}+L_{\bar{a}}{}^{b}E_{b}+\frac{1}{2}D_{\bar{a}}E_{\varphi}-2L_{\bar{b}}{}^{c}D^{\bar{b}}E_{Pc\bar{a}}+2L_{\bar{a}}{}^{c}D^{\bar{b}}E_{Pc\bar{b}}+4L_{\bar{a}}{}^{b}D_{b}E_{P\varphi}.$
(2.63)

For the explicit expressions for the $E$ tensors in (2.53)–(2.59) the $G$
tensors vanish identically. This fact expresses the Bianchi identities.
Furthermore, the Bianchi identities provide a straightforward proof of gauge
invariance of the action, since one may verify with (2.43) and (2.49) that the
gauge variation of the action can be written as the sum over the gauge
parameter times the corresponding Bianchi identity:

$\delta^{(0)}S=\int dt\int d^{2d}{\bf
X}\,n\,e^{-2\Phi}\big{(}\tilde{\xi}_{0}G^{0}+\xi^{0}G_{0}+\xi^{a}G_{a}+\xi^{\bar{a}}G_{\bar{a}}\big{)}=0\,.$
(2.64)

###  2.4 Cubic theory

We now turn to the cubic terms in the action obtained by expanding about a
generic time-dependent background to third order in fluctuations. The
corresponding Lagrangian reads

$\begin{split}{\cal
L}^{(3)}=n\,e^{-2\Phi}\bigg{\\{}&2\omega^{a\bar{b}}\omega_{a}{}^{c}h_{c\bar{b}}+2\omega^{a\bar{b}}\omega_{\bar{b}}{}^{\bar{c}}h_{a\bar{c}}-2L^{a\bar{b}}\omega^{c\bar{d}}h_{a\bar{d}}h_{c\bar{b}}-2L^{a\bar{b}}K^{c\bar{d}}h_{c\bar{b}}h_{a\bar{d}}+4L_{a\bar{b}}\omega^{a\bar{b}}\varphi^{2}\\\
&+2\omega^{a\bar{b}}\mathcal{A}^{M}\partial_{M}h_{a\bar{b}}+8(\Gamma-\dot{\Phi}\phi)\mathcal{A}^{M}\partial_{M}\varphi+4L^{a\bar{b}}\varphi_{+}\mathcal{A}^{M}\partial_{M}h_{a\bar{b}}\\\
&-2\varphi_{+}\mathcal{W}(\phi,\mathcal{A},\varphi,h)+\mathcal{V}^{(3)}(\phi,\varphi_{-},h)\bigg{\\}}\,,\\\
\end{split}$ (2.65)

where we have grouped objects familiar from the quadratic action into

$\displaystyle\begin{split}\mathcal{W}(\phi,\mathcal{A},\varphi,h)\ =\
&-4\Gamma^{2}-\omega_{a\bar{b}}\omega^{a\bar{b}}+\frac{1}{2}\big{(}\omega_{ab}\omega^{ab}+\omega_{\bar{a}\bar{b}}\omega^{\bar{a}\bar{b}}\big{)}-\frac{1}{2}\big{(}K_{ab}K^{ab}+K_{\bar{a}\bar{b}}K^{\bar{a}\bar{b}}\big{)}\\\
&+2\phi(4\dot{\Phi}\Gamma-L_{a\bar{b}}\omega^{a\bar{b}})\,.\end{split}$ (2.66)

Moreover, we defined

$\mathcal{V}^{(3)}(\phi,\varphi_{-},h)=\mathcal{T}+\mathcal{U}\,,$ (2.67)

where

$\displaystyle\begin{split}\mathcal{T}\ =\
&4h^{a\bar{b}}(D_{a}h^{c\bar{d}}D_{\bar{b}}h_{c\bar{d}}-D_{a}h^{c\bar{d}}D_{\bar{d}}h_{c\bar{b}}-D_{\bar{b}}h^{c\bar{d}}D_{c}h_{a\bar{d}})\\\
&+4\varphi_{-}(D^{a}h^{c\bar{d}}D_{a}h_{c\bar{d}}+D^{a}h_{a\bar{b}}D_{c}h^{c\bar{b}}-D^{\bar{b}}h_{a\bar{b}}D_{\bar{c}}h^{a\bar{c}}+2h_{a\bar{b}}D^{a}D_{c}h^{c\bar{b}}-2h_{a\bar{b}}D^{\bar{b}}D_{\bar{c}}h^{a\bar{c}})\\\
&-16h_{a\bar{b}}\varphi_{-}D^{a}D^{\bar{b}}\varphi_{-}-8\varphi_{-}^{2}D^{a}D_{a}\varphi_{-}\,,\\\
\mathcal{U}\ =\ &4\varphi_{-}D^{a}\phi D_{a}\phi+8\varphi_{-}\phi
D^{a}D_{a}\phi-4\phi D^{a}\phi D_{a}\phi+4h^{a\bar{b}}\phi
D_{a}D_{\bar{b}}\phi\,.\\\ \end{split}$ (2.68)

We note that $\mathcal{T}$ matches the form of the cubic expansion of the
double field theory action around a constant background without space-time
split in [27], with $\varphi_{-}$ playing the role of the dilaton.

Finally, expanding the full gauge transformations (2.11) to first order in
fields (i.e. including all terms quadratic in fields and gauge parameters) one
finds:

$\displaystyle\begin{split}\delta^{(1)}h_{a\bar{b}}&\ =\
\xi^{0}\omega_{a\bar{b}}+\xi^{N}\partial_{N}h_{a\bar{b}}+(D_{\bar{b}}\xi^{\bar{c}}-D^{\bar{c}}\xi_{\bar{b}})h_{a\bar{c}}+(D_{a}\xi^{c}-D^{c}\xi_{a})h_{c\bar{b}}\,,\\\
\delta^{(1)}\phi&\ =\
\frac{1}{n}\partial_{t}(\xi^{0}\phi)-\mathcal{A}^{M}\partial_{M}\xi^{0}+\xi^{M}\partial_{M}\phi\,,\\\
\delta^{(1)}\varphi&\ =\ \xi^{0}\Gamma+\xi^{N}\partial_{N}\varphi\,,\\\
\delta^{(1)}\mathcal{A}^{a}&\ =\ -2h^{a\bar{b}}D_{\bar{b}}\xi^{0}+2\phi
D^{a}\xi^{0}+\xi^{B}D_{B}\mathcal{A}^{a}+(D^{a}\xi_{B}-D_{B}\xi^{a})\mathcal{A}^{B}\,,\\\
\delta^{(1)}\mathcal{A}^{\bar{a}}&\ =\ -2h^{b\bar{a}}D_{b}\xi^{0}-2\phi
D^{\bar{a}}\xi^{0}+\xi^{B}D_{B}\mathcal{A}^{\bar{a}}+(D^{\bar{a}}\xi_{B}-D_{B}\xi^{\bar{a}})\mathcal{A}^{B}\,.\end{split}$
(2.69)

Gauge invariance of the action to cubic order in fluctuations requires

$\delta^{(1)}S^{(2)}+\delta^{(0)}S^{(3)}=0\,,$ (2.70)

as the reader may verify with the above formulas by a straightforward but
tedious computation.

##  3 Dualities and cocycles

In this section we discuss the duality properties of double field theory in
time-dependent backgrounds and the role of cocycle factors in the weakly
constrained theory. After discussing in the first subsection some general
features of duality transformations of time-dependent backgrounds, in the
second subsection we turn to a special class of Friedmann-Robertson-Walker
(FRW) backgrounds and discuss their invariance group. In the final subsection
we point out that the theory to cubic order is well-defined in a weakly
constrained sense, for which string theory suggests the presence of cocycle
factors, which we discuss.

###  3.1 Dualities in time-dependent backgrounds

Double field theory is manifestly $O(d,d)$ invariant and so are the quadratic
and cubic actions about time-dependent backgrounds that we determined above.
More precisely, this $O(d,d)$ invariance implies, in general, the following
duality property: changing the background $E_{A}{}^{M}(t)$ by an $O(d,d)$
transformation leads a physically equivalent theory. This follows immediately
from the $O(d,d)$ invariance of the action since the $O(d,d)$ transformation
of $E_{A}{}^{M}(t)$ can then be absorbed by an inverse transformation of the
fields, which in turn can be viewed as a field redefinition. Thus, the actions
for two backgrounds related by an $O(d,d)$ transformation are equal up to
field redefinitions and hence physically equivalent.

While this result is guaranteed by the general double field theory formalism,
it is instructive to give a parametrization of the most general time-dependent
background and to inspect the duality properties explicitly. A general time-
dependent metric $G_{ij}(t)$ and Kalb-Ramond B-field can be packaged into

${\cal E}_{ij}(t)=G_{ij}(t)+B_{ij}(t)\;.$ (3.1)

In order to write a general frame $E_{A}{}^{M}(t)$ without gauge fixing one
needs to introduce on top of this two independent $d$-dimensional frame fields
$e_{a}{}^{i}(t)$ and $\bar{e}_{\bar{a}}{}^{i}(t)$, for then we can write

${E}_{A}{}^{M}=\begin{pmatrix}{E}_{ai}&{E}_{a}{}^{i}\\\
{E}_{\bar{a}i}&{E}_{\bar{a}}{}^{i}\end{pmatrix}=\frac{1}{\sqrt{2}}\begin{pmatrix}{\cal
E}_{ji}e_{a}{}^{j}&-e_{a}{}^{i}\\\ {\cal
E}_{ij}\bar{e}_{\bar{a}}{}^{j}&\bar{e}_{\bar{a}}{}^{i}\end{pmatrix}\;,$ (3.2)

or, in matrix notation with $E=(E_{A}{}^{M})$, $e=(e_{a}{}^{i})$ and
$\bar{e}=(\bar{e}_{\bar{a}}{}^{i})$,

$E=\frac{1}{\sqrt{2}}\begin{pmatrix}e{\cal E}&-e\\\ \bar{e}{\cal
E}^{t}&\bar{e}\end{pmatrix}\;.$ (3.3)

This background satisfies the constraint (2.9) in that

${\mathcal{G}}_{AB}=\begin{pmatrix}-e_{a}{}^{i}e_{b}{}^{j}G_{ij}&0\\\
0&\bar{e}_{\bar{a}}{}^{i}\bar{e}_{\bar{b}}{}^{j}G_{ij}\end{pmatrix}\;.$ (3.4)

This parametrization of the frame preserves the full $O(d,d)$ and $GL(d)\times
GL(d)$ covariance, since we have $3d^{2}$ degrees of freedom ($e$, $\bar{e}$
and ${\cal E}$), as it should be for a frame with $(2d)^{2}=4d^{2}$ components
satisfying the $d^{2}$ constraints (2.9) (i.e., the $GL(d)\times GL(d)$
covariant constraints that the off-diagonal blocks of (3.4) vanish).

Under global $O(d,d)$ and local (time-dependent) $GL(d)\times GL(d)$
transformations this frame transforms as

$E^{\prime}_{A}{}^{M}(t)=\Lambda_{A}{}^{B}(t)E_{B}{}^{N}(t)h^{M}{}_{N}\;,\qquad
h=\begin{pmatrix}a&b\\\
c&d\end{pmatrix},\quad\Lambda_{A}{}^{B}=\begin{pmatrix}\Lambda_{a}{}^{b}&0\\\
0&\bar{\Lambda}_{\bar{a}}{}^{\bar{b}}\end{pmatrix}\;,$ (3.5)

or, in matrix notation, as $E^{\prime}=\Lambda Eh^{t}$. Note that the
transformation does not act on any coordinate argument since the background
depends only on time on which $O(d,d)$ does not act. In the matrix notation
(3.3) this yields

$e^{\prime}=\Lambda\,e\,M\;,\qquad\bar{e}^{\prime}=\bar{\Lambda}\,\bar{e}\,\bar{M}\;,\qquad{\cal
E}^{\prime}=(a{\cal E}+b)(c{\cal E}+d)^{-1}\,,$ (3.6)

where

$M(t):=d^{t}-{\cal E}(t)c^{t}\;,\qquad\bar{M}(t):=d^{t}+{\cal
E}(t)^{t}c^{t}\;.$ (3.7)

The matrices $M$ and $\bar{M}$ are familiar from the double field theory on
flat space [17], with the important difference that here they are time-
dependent. We could now completely fix the background $GL(d)\times GL(d)$
frame transformations by setting $e=\bar{e}={\bf 1}$, which in turn would
require compensating frame transformations given by $\Lambda=M^{-1}$ and
$\bar{\Lambda}=\bar{M}^{-1}$, as follows from (3.6). However, due to (3.4)
this gauge choice leads to a time-dependent tangent space metric, so that the
operations of raising and lowering indices do not commute with time
derivatives. It is instead more convenient to choose a gauge for which the
tangent space metric is constant, as done in the previous sections. We will
next consider particular frames with this property.

###  3.2 Friedmann-Robertson-Walker backgrounds

We now turn to the class of time-dependent backgrounds with the largest degree
of symmetry: the Friedmann-Robertson-Walker (FRW) spaces with vanishing
spatial curvature. These are characterized by a single time-dependent
function, the scale factor $a(t)$, while the B-field vanishes. Specifically,
(3.1) then reduces to ${\cal E}_{ij}(t)=a^{2}(t)\delta_{ij}$. It will
furthermore be convenient to introduce two constant but otherwise arbitrary
bases or frames $e_{a}{}^{i}$ and $\bar{e}_{\bar{a}}{}^{i}$ for the doubled
spatial geometry. Since these bases are unconstrained, the ‘tangent space’
metrics defined in terms of the flat spatial metric by

$g_{ab}=e_{a}{}^{i}e_{b}{}^{j}\delta_{ij}\,,\qquad
g_{\bar{a}\bar{b}}=\bar{e}_{\bar{a}}{}^{i}\bar{e}_{\bar{b}}{}^{j}\delta_{ij}\,,$
(3.8)

are then independent constant metrics (of Euclidean signature). As usual, we
use $\delta_{ij}$ and $\delta^{ij}$ to lower and raise indices $i,j,\ldots$,
while $g_{ab}$ and $g^{ab}$, respectively $g_{\bar{a}\bar{b}}$ and
$g^{\bar{a}\bar{b}}$, are used to lower and raise indices $a,b,\ldots$ and
$\bar{a},\bar{b},\ldots$. Using the frames $e_{a}{}^{i}$ and
$\bar{e}_{\bar{a}}{}^{i}$, together with their inverses denoted by
$e_{i}{}^{a}$ and $\bar{e}_{i}{}^{\bar{a}}$, to convert indices $a,b,\ldots$
and $\bar{a},\bar{b},\ldots$ to $i,j,\ldots$, one may verify that the
different operations of raising and lowering indices are mutually compatible.
For instance, in

$e^{ia}=\delta^{ij}e_{j}{}^{a}=g^{ab}e_{b}{}^{i}\;,$ (3.9)

the second equation follows by contraction with $e_{a}{}^{k}$ and using the
inverse of (3.8). Notably, there are tensors

$g_{a}{}^{\bar{b}}:=e_{a}{}^{i}\,\bar{e}_{i}{}^{\bar{b}}\,,\qquad
g_{\bar{a}}{}^{b}:=\bar{e}_{\bar{a}}{}^{i}\,e_{i}{}^{b}\,$ (3.10)

that connect unbarred and barred indices and that satisfy

$\begin{split}g_{a}{}^{\bar{b}}\,g_{\bar{b}}{}^{c}=\delta_{a}{}^{c}\,,\quad{\rm
etc.}\end{split}$ (3.11)

With these objects we can write the full background frame as

${E}_{A}{}^{M}(t)=\frac{1}{\sqrt{2}}\begin{pmatrix}a(t)\delta_{ij}e_{a}{}^{j}&-a^{-1}(t)e_{a}{}^{i}\\\
a(t)\delta_{ij}\bar{e}_{\bar{a}}{}^{j}&a^{-1}(t)\bar{e}_{\bar{a}}{}^{i}\end{pmatrix}\,.$
(3.12)

This is of the general form (3.2), except that the frames $e_{a}{}^{i}$ and
$\bar{e}_{\bar{a}}{}^{i}$ have been rescaled by $a(t)$ to make them constant.
The corresponding tangent space metric then satisfies the constraint (2.9)
with the constant metric on the doubled tangent space

$\mathcal{G}_{AB}=\begin{pmatrix}-g_{ab}&0\\\
0&g_{\bar{a}\bar{b}}\end{pmatrix}\,.$ (3.13)

Note the relative sign in the upper-left block, which will lead to a change of
convention in raising and lowering indices in sec. 4 below. The differential
operators $D_{A}=(D_{a},D_{\bar{a}})\equiv E_{A}{}^{M}\partial_{M}$ are given
by

$\begin{split}D_{a}&=-\tfrac{1}{\sqrt{2}}\left(a^{-1}(t)\partial_{a}-a(t)\tilde{\partial}_{a}\right)\;,\\\
D_{\bar{a}}&=\tfrac{1}{\sqrt{2}}\left(a^{-1}(t)\partial_{\bar{a}}\ +\
a(t)\tilde{\partial}_{\bar{a}}\right)\;,\end{split}$ (3.14)

using the notation $\partial_{a}=e_{a}{}^{i}\partial_{i}$,
$\tilde{\partial}_{a}=e_{a}{}^{i}\tilde{\partial}_{i}$, and similarly for
barred indices. Note that despite the notation there is only one kind of
momentum and one kind of winding derivative: with (3.10) we have
$\partial_{\bar{a}}=g_{\bar{a}}{}^{b}\partial_{b}$ and
$\tilde{\partial}_{\bar{a}}=g_{\bar{a}}{}^{b}\tilde{\partial}_{b}$.

We can next give the explicit form of the tensor

$L_{A}{}^{B}\equiv\dot{E}_{A}{}^{M}E_{M}{}^{B}\,,$ (3.15)

used in the main text, where we recall the notation $\dot{X}\equiv
n^{-1}\partial_{t}X$. Using (3.12) one finds for its components

$\begin{split}L_{a}{}^{\bar{b}}&=Hg_{a}{}^{\bar{b}}\,,\qquad
L_{\bar{a}}{}^{b}=Hg_{\bar{a}}{}^{b}\,,\qquad
L_{a}{}^{b}=L_{\bar{a}}{}^{\bar{b}}=0\,,\end{split}$ (3.16)

with Hubble parameter

$H\equiv\frac{\dot{a}}{a}\;.$ (3.17)

With this choice of backgrounds, the background equations of motion in
(2.20)-(2.22) become:

$\displaystyle(\dot{H}-2\dot{\Phi}H)g_{a}{}^{\bar{b}}=0\,,$ (3.18)
$\displaystyle-4\ddot{\Phi}+4\dot{\Phi}^{2}+dH^{2}=0\,,$ (3.19) $\displaystyle
4\dot{\Phi}^{2}-dH^{2}=0\,.$ (3.20)

As done before in sec. 2, we can add (3.19) and (3.20) to obtain the equation

$-\ddot{\Phi}+2\dot{\Phi}^{2}=0\,.$ (3.21)

The equations of motion imply that the following quantity is conserved,

$\beta\equiv\frac{\dot{\Phi}}{H}\,,$ (3.22)

since by taking its time derivative we have

$\dot{\beta}=\frac{(\ddot{\Phi}H-\dot{\Phi}\dot{H})}{H^{2}}=\frac{2\dot{\Phi}^{2}H-\dot{\Phi}(2\dot{\Phi}H)}{H^{2}}=0\,.$
(3.23)

Let us next analyze the invariance groups of these FRW backgrounds. We first
note that since we are now considering a rather special class of backgrounds,
under a general $O(d,d)$ or $GL(d)\times GL(d)$ transformation the background
frame will of course not stay in the same class. However, under constant or
time independent $GL(d)\times GL(d)$ transformations the above backgrounds
transform into themselves, just with the frames $e_{a}{}^{i}$ and
$\bar{e}_{\bar{a}}{}^{i}$ rotated, which follows as in (3.6). The genuine
duality transformation left in $O(d,d)$ is given by

$h=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}=\begin{pmatrix}0&1\\\
1&0\end{pmatrix}:\qquad{\cal E}^{\prime}(t)={\cal
E}^{-1}(t)\qquad\Leftrightarrow\qquad a^{\prime}(t)=\frac{1}{a(t)}\;,$ (3.24)

as follows with the last relation in (3.6).222Depending on the form of $e$ and
$\bar{e}$ this transformation requires compensating $GL(d)\times GL(d)$
transformations. For instance, if $e=\bar{e}={\bf 1}$ then the compensating
transformations are $\Lambda=-{\cal E}^{-1}$, $\bar{\Lambda}=({\cal
E}^{t})^{-1}$, as can be verified with (3.6). This is the expected T-duality
or scale-factor duality property of string cosmology.

We can then turn to the problem of determining the invariance group, i.e., the
subgroup of $O(d,d)$ (possibly accompanied by $GL(d)\times GL(d)$
transformations) for which $E^{\prime}=E$. For flat space backgrounds, which
are contained in the above class of backgrounds for $a\equiv 1$, the
invariance group is $O(d)\times O(d)$. Thus, the invariance group of genuine
FRW backgrounds must be a subgroup of $O(d)\times O(d)$, which in turn is
embedded into $O(d,d)$ as

$h\ =\ \frac{1}{2}\begin{pmatrix}\ell+\bar{\ell}&\bar{\ell}-\ell\\\
\bar{\ell}-\ell&\ell+\bar{\ell}\end{pmatrix},\qquad\ell\in
O(d)\;,\quad\bar{\ell}\in O(d)\,.$ (3.25)

Therefore, for ${\cal E}_{ij}=a^{2}(t)\delta_{ij}$ one finds for the factors
in the transformation formula (3.6)

$a{\cal
E}+b=\frac{1}{2}\left((\ell+\bar{\ell})a^{2}(t)+\bar{\ell}-\ell\right)\,,\qquad
c{\cal
E}+d=\frac{1}{2}\left((\bar{\ell}-\ell)a^{2}(t)+\ell+\bar{\ell}\right)\;.$
(3.26)

For flat backgrounds with $a\equiv 1$ this yields $a{\cal E}+b=\bar{\ell}$ and
$c{\cal E}+d=\bar{\ell}$ and hence ${\cal
E}^{\prime}=\bar{\ell}\bar{\ell}^{-1}={\bf 1}={\cal E}$. This confirms that
$O(d)\times O(d)$ is an invariance group of ${\cal E}={\bf 1}$. This argument
fails if $a(t)$ is a genuine function of time, but in this case we note that
upon identifying $\ell=\bar{\ell}$ we have $a{\cal E}+b=a^{2}(t)\ell$ and
$c{\cal E}+d=\ell$, so that the background is invariant:

${\cal E}^{\prime}(t)=a^{2}(t)\ell\ell^{-1}=a^{2}(t){\bf 1}={\cal E}(t)\;.$
(3.27)

Thus, the invariance group of generic FRW backgrounds is given by the diagonal
subgroup ${\rm diag}(O(d)\times O(d))$, as mentioned in the introduction.

###  3.3 Weak constraint and cocycles

We now discuss in more detail the weak constraint originating from the level-
matching constraint of string theory on toroidal backgrounds and explain how
the cubic theory is consistent as a weakly constrained theory. We denote a
generic field by $\phi$ (that need not be the lapse fluctuation above) and
expand in Fourier modes of the torus as

$\phi(t,{\bf X})=\sum_{{\bf K}\in\mathbb{Z}^{2d}}\phi_{\bf K}(t)\,e^{i{\bf
K}^{t}{\bf X}}\;.$ (3.28)

We find it convenient to take the sum over ${\bf K}\in\mathbb{Z}^{2d}$ to be
unconstrained and to implement the weak constraint
$\partial^{M}\partial_{M}\phi=0$ by demanding

${\bf K}^{2}\equiv\eta^{MN}{\bf K}_{M}{\bf K}_{N}\neq
0\quad\Rightarrow\quad\phi_{\bf K}=0\;.$ (3.29)

Put differently, we take the Fourier modes $\phi_{\bf K}$ to be non-zero only
if the corresponding Fourier label ${\bf K}$ is null with respect to the
$O(d,d)$ metric. The weak constraint is then obeyed for (3.28).

The consistency problem for a weakly constrained theory arises because the
usual point-wise product of functions satisfying the weak constraint in
general does not satisfy the weak constraint. In string field theory this
consistency problem is resolved by having a modified product, which simply
projects out all Fourier modes that do not obey (3.29). Specifically, we
define the product

$\big{(}\phi^{1}\bullet\phi^{2}\big{)}(t,{\bf X})\equiv\sum_{{\bf K}_{1},{\bf
K}_{2}\in\mathbb{Z}^{2d}}\delta_{{\bf K}_{1}\cdot{\bf K_{2}},0}\,C({\bf
K}_{1},{\bf K}_{2})\,\phi_{{\bf K}_{1}}^{1}(t)\phi_{{\bf
K}_{2}}^{2}(t)e^{i({\bf K}_{1}+{\bf K}_{2})^{t}{\bf X}}\;,$ (3.30)

where a Kronecker delta was introduced to set to zero any term that does not
obey ${\bf K}_{1}\cdot{\bf K}_{2}=0$. Since by (3.29) any term in the above
sum vanishes unless ${\bf K}_{1}^{2}={\bf K}_{2}^{2}=0$ it then follows that
only terms with $({\bf K}_{1}+{\bf K}_{2})^{2}=0$ have non-vanishing
coefficients. Therefore, the $\bullet$ product of two functions obeying the
weak constraint also obeys the weak constraint:
$\partial^{M}\partial_{M}(\phi^{1}\bullet\phi^{2}\big{)}=0$. The so-called
cocycle function $C({\bf K}_{1},{\bf K}_{2})$ appearing in the definition is a
sign factor that is symmetric in its two arguments and defined in terms of the
components of ${\bf K}=(w,k)$ by

$C({\bf K}_{1},{\bf K}_{2})\equiv(-1)^{k_{1}w_{2}}\equiv(-1)^{{\bf
K}_{1}^{t}Q{\bf K}_{2}}\;,\qquad Q^{MN}\equiv\begin{pmatrix}0&0\\\
1&0\end{pmatrix}\;,$ (3.31)

where $kw$ denotes the usual (Euclidean) dot product between $d$-dimensional
vectors. More precisely, given the cocycle’s role in the deformed product
(3.30), it only needs to be defined for arguments that are already null and
mutually orthogonal. We then have $0={\bf K}_{1}\cdot{\bf
K}_{2}=k_{1}w_{2}+w_{1}k_{2}$ and hence $(-1)^{k_{1}w_{2}}=(-1)^{k_{2}w_{1}}$,
so that the cocycle is indeed symmetric. Similarly, if ${\bf K}_{1}$ is
proportional to ${\bf K}_{2}$ we have $k_{1}w_{2}\propto k_{1}w_{1}=0$ and
hence

${\bf K}_{1}\propto{\bf K}_{2}\quad\Rightarrow\quad C({\bf K}_{1},{\bf
K}_{2})=1\;.$ (3.32)

Let us also record the formula for the mode of the product with Fourier label
${\bf K}$:

$(\phi^{1}\bullet\phi^{2})_{\bf K}=\sum_{{\bf
P}\in\mathbb{Z}^{2d}}\delta_{{\bf P}\cdot{\bf K},0}\,C({\bf P},{\bf
K})\,\phi_{{\bf P}}^{1}\cdot\phi_{{\bf K}-{\bf P}}^{2}\;,$ (3.33)

where we used $C({\bf P},{\bf K}-{\bf P})=C({\bf P},{\bf K})$, which follows
quickly with (3.31). Since the tensor $Q^{MN}$ in (3.31) is not $O(d,d)$
invariant a theory employing the $\bullet$ product seems to violate the
$O(d,d,\mathbb{Z})$ duality invariance, but we will see below that the duality
is realized through a non-standard action involving a similar sign factor.

We now discuss some properties of the $\bullet$ product. Since the cocycle is
symmetric, the product is still commutative. Furthermore, it is easy to
convince oneself that the modified product (3.30) still obeys the Leibniz
rule:

$\partial_{M}(\phi^{1}\bullet\phi^{2})=(\partial_{M}\phi^{1})\bullet\phi^{2}+\phi^{1}\bullet(\partial_{M}\phi^{2})\;.$
(3.34)

The product is, however, not associative, as can be seen by considering the
following three ‘pure Fourier mode’ functions for $d=1$:333We thank Henning
Samtleben for suggesting this example.

$f=e^{i\tilde{x}}\;,\quad g=e^{ix}\;,\quad h=e^{-ix}\;,$ (3.35)

which all, depending only on $x$ or only on $\tilde{x}$, obey the weak
constraint. Moreover, since these functions thus carry only momentum or only
winding, the cocycle factor between any two of them is trivially unity. Since
the point-wise product $f\cdot g=e^{ix+i\tilde{x}}$ violates the weak
constraint the $\bullet$ product is just zero, $f\bullet g=0$, and thus
$(f\bullet g)\bullet h=0$. On the other hand, $g\cdot h=1$ obeys the weak
constraint so that $g\bullet h=1$ and hence $f\bullet(g\bullet h)=f$. Thus,
associativity does not hold. While associativity does not hold in general it
does hold when integrated over the whole (doubled) torus:

$\int_{T^{2d}}d^{2d}{\bf
X}\Big{[}(\phi^{1}\bullet\phi^{2})\bullet\phi^{3}-\phi^{1}\bullet(\phi^{2}\bullet\phi^{3})\Big{]}=0\;.$
(3.36)

In order to see this we use $\frac{1}{(2\pi)^{2d}}\int d^{2d}{\bf X}\,e^{i{\bf
K}\cdot{\bf X}}=\delta({\bf K})$, where $\delta({\bf K})$ is the Kronecker
delta that is equal to $1$ if all components of ${\bf K}$ are zero and zero
otherwise. This implies for the integral of a generic scalar with Fourier
series (3.28):

$\int dt\int d^{2d}{\bf X}\,\phi(t,{\bf X})=(2\pi)^{2d}\int dt\,\phi_{{\bf
0}}(t)\;,$ (3.37)

i.e. the integral reduces to that over the zero mode with ${\bf K}={\bf 0}$.
(In the following we will omit the time integrals as they play no role.) Using
this we compute

$\int_{T^{2d}}d^{2d}{\bf X}\,(\phi^{1}\bullet\phi^{2})=(2\pi)^{2d}\sum_{{\bf
K}\in\mathbb{Z}^{2d}}\phi_{-{\bf K}}^{1}\,\phi_{{\bf
K}}^{2}=\int_{T^{2d}}d^{2d}{\bf X}\,(\phi^{1}\cdot\phi^{2})\;.$ (3.38)

Here we used that the Kronecker delta enforces ${\bf K}_{2}=-{\bf K}_{1}$ in
(3.30), so that ${\bf K}_{1}\cdot{\bf K}_{2}=0$ automatically, and the cocycle
factor is $1$ due to (3.32). Therefore, under an integral the modified product
equals the ordinary product. Similarly,

$\begin{split}\int_{T^{2d}}d^{2d}{\bf
X}\Big{[}(\phi^{1}\bullet\phi^{2})\bullet\phi^{3}\Big{]}&=\int_{T^{2d}}d^{2d}{\bf
X}\Big{[}(\phi^{1}\bullet\phi^{2})\cdot\phi^{3}\Big{]}=(2\pi)^{2d}\sum_{{\bf
K}\in\mathbb{Z}^{2d}}(\phi^{1}\bullet\phi^{2})_{\bf K}\cdot\phi^{3}_{-{\bf
K}}\\\ &=(2\pi)^{2d}\sum_{{\bf K},{\bf P}\in\mathbb{Z}^{2d}}\delta_{{\bf
P}\cdot{\bf K},0}\,C({\bf P},{\bf K})\,\phi_{{\bf P}}^{1}\cdot\phi_{{\bf
K}-{\bf P}}^{2}\cdot\phi_{-{\bf K}}^{3}\;,\end{split}$ (3.39)

where we used (3.33). Recalling the convention (3.29) we can use ${\bf
P}^{2}=({\bf K}-{\bf P})^{2}={\bf K}^{2}=0$ and hence ${\bf P}\cdot{\bf K}=0$.
It is then easy to verify that the second term in (3.36), i.e., the integral
over $\phi^{1}\bullet(\phi^{2}\bullet\phi^{3})$, equals (3.39). This proves
(3.36).

We can now see that the cubic theory is consistent as a weakly constrained
theory. For the free part this is immediate: in the quadratic action, due to
(3.38), the regular point-wise product equals the modified product. Similarly,
the linear gauge transformations needed to check invariance of the quadratic
action are consistent with the weak constraint since the gauge parameters also
obey the weak constraint. (Note that the spatial integral is always the
ordinary integral over the full doubled torus; only the fields are assumed to
be restricted by the weak constraint.) Turning to the cubic theory, we first
note that due to (3.36) there is no ambiguity in writing the terms of cubic
order in the action with the $\bullet$ product. Furthermore, all quadratic
terms in the non-linear gauge transformations (2.69) must be reinterpreted as
employing the $\bullet$ product in the quadratic terms, but since to this
order all terms in the gauge transformed action are still at most cubic in
fields and parameters, gauge invariance follows as before, using (3.34) and
(3.36). It should be emphasized that this argument to cubic order goes through
whether a cocycle factor is included or not. Thus, as far as the cubic
truncation is concerned there is no need to include a cocycle factor, but the
claim in the string field theory literature is that to quartic and higher
order gauge invariance requires the cocycle factors [24, 25, 26], although it
remains a possibility that for the pure DFT sector a theory without cocycles
may exist.444OH thanks Barton Zwiebach for collaboration on the issues
discussed here.

We close this section by returning to the $O(d,d,\mathbb{Z})$ duality
invariance, whose realization is non-standard in the presence of cocycle
factors [26]. To understand this we first consider the standard case without
cocycles, for which $h\in O(d,d,\mathbb{Z})$ acts as

$\phi(t,{\bf X})\ \rightarrow\ \phi^{\prime}(t,{\bf
X}^{\prime})=\rho(h)\phi(t,{\bf X})\,,\quad\text{where}\quad{\bf
X}^{\prime}=h{\bf X}\;.$ (3.40)

Here $\rho(h)$ denotes the matrix in the (finite-dimensional) $O(d,d)$
representation of $\phi$. Expanding both sides into Fourier modes according to
(3.28) then yields for the Fourier components

$\phi_{{\bf K}^{\prime}}^{\prime}(t)=\rho(h)\phi_{\bf
K}(t)\,,\quad\text{where}\quad{\bf K}^{\prime}=(h^{t})^{-1}{\bf K}\,.$ (3.41)

Note that this transformation rule is the reason that the duality group is the
discrete $O(d,d,\mathbb{Z})$: Since the components of ${\bf K}$ are integers,
the matrix components of $h\in O(d,d)$ must be integers too in order to
preserve this property, hence breaking $O(d,d,\mathbb{R})$ to
$O(d,d,\mathbb{Z})$.

Any conventional contractions of products of $O(d,d)$ tensors are covariant
under this transformation, but since the modified product (3.33) involves the
cocycle factor that is not $O(d,d)$ invariant, the modified product is not
covariant under the standard transformation. We will now show, however, that
the modified product is covariant under a similarly modified
$O(d,d,\mathbb{Z})$ action. Specifically, (3.41) is modified to

$\phi_{{\bf K}^{\prime}}^{\prime}(t)=(-1)^{F({\bf K},h)}\rho(h)\phi_{\bf
K}(t)\,,$ (3.42)

with a sign factor depending on ${\bf K}$ and $h\in O(d,d)$ to be determined.
We begin by computing the anomalous transformation of the cocycle factor
(3.31) under

${\bf K}\rightarrow{\bf K}^{\prime}=(h^{t})^{-1}{\bf K}\;,\qquad
h=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in O(d,d,\mathbb{Z})\;,$ (3.43)

i.e.,

$C({\bf K}_{1}^{\prime},{\bf K}_{2}^{\prime})\ =\ (-1)^{{\bf
K}_{1}^{t}h^{-1}Q(h^{t})^{-1}{\bf K}_{2}}\ =\ (-1)^{{\bf K}_{1}^{t}Q{\bf
K}_{2}}(-1)^{{\bf K}_{1}^{t}(h^{-1}Q(h^{t})^{-1}-Q){\bf K}_{2}}\;.$ (3.44)

The failure of the cocycle to be $O(d,d,\mathbb{Z})$ invariant is encoded in
the exponent of the second factor, which we compute with the help of the group
property $h^{-1}=\eta^{-1}h^{t}\eta$ to be

$\begin{split}{\cal A}(h)\ :=\ h^{-1}Q(h^{t})^{-1}-Q\ =\
\begin{pmatrix}b^{t}d&b^{t}c\\\ a^{t}d-1&a^{t}c\end{pmatrix}\ =\
\begin{pmatrix}b^{t}d&b^{t}c\\\ -c^{t}b&a^{t}c\end{pmatrix}\;.\end{split}$
(3.45)

Here we used in the last step the $O(d,d)$ group properties, which also imply
that $a^{t}c$ and $b^{t}d$ are antisymmetric (see, e.g., sec. 4.1 in [26]).
From this it follows that ${\cal A}(h)$ is antisymmetric, ${\cal A}^{t}=-{\cal
A}$, so that it is determined by its upper-triangular part ${\cal A}_{u}$ via
${\cal A}={\cal A}_{u}-{\cal A}_{u}^{t}$. Let us record the transformation
behavior of the cocycle factor:

$C({\bf K}_{1}^{\prime},{\bf K}_{2}^{\prime})\ =\ C({\bf K}_{1},{\bf
K}_{2})(-1)^{{\bf K}_{1}^{t}{\cal A}(h){\bf K}_{2}}\;.$ (3.46)

Our goal is now to determine the sign factor in (3.42) by requiring that the
$\bullet$ product transforms covariantly. For ease of notation we take the two
factors $\phi^{1}$ and $\phi^{2}$ to be scalars so that $\rho(h)=1$. (This we
can do without loss of generality since these finite-dimensional rotations are
not relevant for the sign issues related to cocycles.) Therefore, the product
(3.33) of two transformed fields reads

$\begin{split}(\phi^{\prime 1}\bullet\phi^{\prime 2})_{{\bf
K}^{\prime}}&=\sum_{{\bf P}^{\prime}\in\mathbb{Z}^{2d}}\delta_{{\bf
P}^{\prime}\cdot{\bf K}^{\prime},0}\,C({\bf P}^{\prime},{\bf
K}^{\prime})\,\phi_{{\bf P}^{\prime}}^{\prime 1}\cdot\phi_{{\bf
K}^{\prime}-{\bf P}^{\prime}}^{\prime 2}\\\ &=\sum_{{\bf
P}\in\mathbb{Z}^{2d}}\delta_{{\bf P}\cdot{\bf K},0}\,C({\bf P},{\bf
K})\,(-1)^{{\bf P}^{t}{\cal A}(h){\bf K}}(-1)^{F({\bf P},h)}(-1)^{F({\bf
K}-{\bf P},h)}\phi_{{\bf P}}^{1}\cdot\phi_{{\bf K}-{\bf P}}^{2}\;,\end{split}$
(3.47)

where we used (3.42) and (3.46). Covariance requires according to (3.42) that
this is equal to

$(\phi^{\prime 1}\bullet\phi^{\prime 2})_{{\bf K}^{\prime}}=(-1)^{F({\bf
K},h)}(\phi^{1}\bullet\phi^{2})_{\bf K}\;,$ (3.48)

which is satisfied if

${\bf P}^{t}\,{\cal A}(h)\,{\bf K}=F({\bf K},h)+F({\bf P},h)+F({\bf K}-{\bf
P},h)\;\;{\rm mod}\;2\;.$ (3.49)

We now prove that this relation holds for

$F({\bf K},h)\equiv{\bf K}^{t}\,{\cal A}_{u}(h)\,{\bf K}\;,$ (3.50)

where ${\cal A}_{u}$ denotes the upper-triangular part of ${\cal A}$. This
follows by a direct computation:

$\begin{split}F({\bf K},h)+F({\bf P},h)+F({\bf K}-{\bf P},h)&={\bf K}^{t}{\cal
A}_{u}(h){\bf K}+{\bf P}^{t}{\cal A}_{u}(h){\bf P}+({\bf K}-{\bf P})^{t}{\cal
A}_{u}(h)({\bf K}-{\bf P})\\\ &={\bf P}^{t}\left({\cal A}_{u}(h)-{\cal
A}_{u}^{t}(h)\right){\bf K}\;\;{\rm mod}\;2\\\ &={\bf P}^{t}\,{\cal
A}_{u}(h)\,{\bf K}\;,\end{split}$ (3.51)

where we used repeatedly that this only needs to hold mod 2. This completes
the proof that under (3.42) with (3.50) the modified $\bullet$ product
transforms covariantly. More generally it follows that an $O(d,d)$ singlet
that is built with any number of $\bullet$ products still transforms
‘covariantly’ in this new sense. As a consequence, any action integral based
on such an $O(d,d)$ singlet is still $O(d,d,\mathbb{Z})$ invariant, because
the integral picks out the zero mode, c.f. (3.37), for which (3.50) vanishes.
The modified $O(d,d,\mathbb{Z})$ transformation then reduces to the standard
transformation (3.41) under which the zero mode of a scalar is invariant.

##  4 Decompositions

In this section we decompose the fundamental fields of double field theory,
specialized to the FRW backgrounds of sec. 3.2, into irreducible components by
performing a scalar-vector-tensor (SVT) decomposition. This allows us to
express the quadratic double field theory action in terms of gauge invariant
variables. This analysis extends what was already done for fluctuations with
respect to a flat background in [32].

###  4.1 Scalar-vector-tensor decomposition

We now take the backgrounds to be of the form (3.12) with scale factor $a(t)$.
According to (3.16) we then have $L_{a}{}^{b}=L_{\bar{a}}{}^{\bar{b}}=0$, so
that the covariant derivatives defined in (2.24) simplify:

$\displaystyle\begin{split}\nabla_{t}\mathcal{V}_{a}=\dot{\mathcal{V}}_{a}\,,\qquad\nabla_{t}\mathcal{V}_{\bar{a}}=\dot{\mathcal{V}}_{\bar{a}}\,.\end{split}$
(4.1)

As for double field theory on flat space we also have the spatial Laplacians
that are related by the level-matching constraint as follows:

$\Delta\equiv 2g^{ab}D_{a}D_{b}=2g^{\bar{a}\bar{b}}D_{\bar{a}}D_{\bar{b}}\,.$
(4.2)

In contrast to flat space there is a new differential operator based on the
invariant tensor (3.10):

$\Diamond\equiv
2g_{a}{}^{\bar{b}}D^{a}D_{\bar{b}}=2g_{\bar{b}}{}^{a}D_{a}D^{\bar{b}}\,,$
(4.3)

which satisfies the commutation relations:

$\begin{split}[\nabla_{t},\Delta]&=2H\Diamond\,,\\\
[\nabla_{t},\Diamond]&=2H\Delta\,.\\\ \end{split}$ (4.4)

Explicitly, these two Laplace-type operators are given by

$\begin{split}\Delta&=a^{-2}(t)\partial^{2}+a^{2}(t)\tilde{\partial}^{2}\;,\\\
\Diamond&=-a^{-2}(t)\partial^{2}+a^{2}(t)\tilde{\partial}^{2}\;,\end{split}$
(4.5)

where

$\partial^{2}\equiv\partial^{i}\partial_{i}\equiv\delta^{ij}\partial_{i}\partial_{j}\;,\qquad\tilde{\partial}^{2}\equiv\tilde{\partial}^{i}\tilde{\partial}_{i}\equiv\delta_{ij}\tilde{\partial}^{i}\tilde{\partial}^{j}\;,$
(4.6)

are the independent spatial (Euclidean) Laplacians of the doubled space.

At this stage a comment is in order regarding our conventions for raising and
lowering indices. In the general frame formulation of double field theory the
tangent space metric ${\cal G}_{AB}$ of signature $(d,d)$ is used to raise and
lower flat indices. For the FRW backgrounds to be used here this metric takes
the form (3.13) in terms of the positive-definite (Euclidean) metrics $g_{ab}$
and ${g}_{\bar{a}\bar{b}}$, respectively, which will be used from now on to
raise and lower indices (as alluded to after (3.8)). Since ${\cal
G}_{ab}=-g_{ab}$ this amounts to a change in convention. To be definite, let
us take the elementary fields of the theory to be given by

$\phi\,,\quad\varphi\;,\quad h_{a\bar{b}}\;,\quad{\cal A}_{a}\;,\quad{\cal
A}_{\bar{a}}\;,$ (4.7)

i.e., these fields are considered to be metric-independent, as are the
differential operators $\partial_{a}$, $\tilde{\partial}_{a}$, etc., with
lower indices. Any expression involving these objects with an upper index is
then interpreted to mean that the index is raised with $g^{ab}$ or
$g^{\bar{a}\bar{b}}$, respectively. This change of convention leads to sign
changes but has desirable internal consistency properties. For instance, the
Laplacians in (4.6) are then also writable as
$\partial^{2}=\partial^{a}\partial_{a}=\partial^{\bar{a}}\partial_{\bar{a}}$
and
$\tilde{\partial}^{2}=\tilde{\partial}^{a}\tilde{\partial}_{a}=\tilde{\partial}^{\bar{a}}\tilde{\partial}_{\bar{a}}$,
as follows quickly by recalling $\partial_{a}=e_{a}{}^{i}\partial_{i}$,
$\tilde{\partial}_{a}=e_{a}{}^{i}\tilde{\partial}_{i}$, and similarly for
barred indices. As a consequence, in all formulas to follow we may freely
raise and lower indices without having to worry about sign factors.

After this digression, we turn to the problem of decomposing the complete list
of fields (4.7) into ‘irreducible’ components by performing a scalar-vector-
tensor (SVT) decomposition. For the ${\cal A}$ fields we write

$\begin{split}{\cal
A}_{a}&=A_{a}+\partial_{a}A+\tilde{\partial}_{a}{\tilde{A}}\;,\\\ {\cal
A}_{\bar{a}}&=A_{\bar{a}}+\partial_{\bar{a}}\bar{A}+\tilde{\partial}_{\bar{a}}\bar{\tilde{A}}\;.\end{split}$
(4.8)

Here we see the first instance of an important novelty of a genuinely doubled
field theory on cosmological backgrounds: the SVT decomposition of a vector
yields, compared to standard gravity, an additional scalar mode, corresponding
to the possibility of subtracting the divergence with respect to winding
derivatives, in addition to ordinary derivatives. Correspondingly, the
remaining vector mode is now divergence-free (transverse) with respect to both
derivatives:

$\partial^{a}A_{a}=\tilde{\partial}^{a}A_{a}=\partial^{\bar{a}}A_{\bar{a}}=\tilde{\partial}^{\bar{a}}A_{\bar{a}}=0\;.$
(4.9)

Thus, each of the transverse vectors has $d-2$ degrees of freedom. The logic
here is that one imposes as many constraints as possible on the remaining
vector or tensor mode. This ultimately guarantees the complete decoupling
among tensor, vector and scalar modes. Turning then to the tensor field
$h_{a\bar{b}}$ we postulate the decomposition

$h_{a\bar{b}}=\widehat{h}_{a\bar{b}}+g_{a\bar{b}}E+\partial_{a}B_{\bar{b}}-\partial_{\bar{b}}B_{a}+\tilde{\partial}_{a}\tilde{B}_{\bar{b}}-\tilde{\partial}_{\bar{b}}\tilde{B}_{a}+\partial_{a}\partial_{\bar{b}}C+\tilde{\partial}_{a}\tilde{\partial}_{\bar{b}}\tilde{C}+\partial_{a}\tilde{\partial}_{\bar{b}}D+\tilde{\partial}_{a}\partial_{\bar{b}}\tilde{D}\;,$
(4.10)

with now five independent scalar modes and four independent vector modes,
which are subject to the constraints analogous to (4.9), i.e., every
divergence vanishes:

$\partial^{a}B_{a}=\tilde{\partial}^{a}B_{a}=\partial^{\bar{a}}B_{\bar{a}}=\tilde{\partial}^{\bar{a}}B_{\bar{a}}=0\,,\qquad\partial^{a}\tilde{B}_{a}=\tilde{\partial}^{a}\tilde{B}_{a}=\partial^{\bar{a}}\tilde{B}_{\bar{a}}=\tilde{\partial}^{\bar{a}}\tilde{B}_{\bar{a}}=0\,.$
(4.11)

Similarly, the irreducible tensor mode obeys

$\begin{split}Q_{a}&\equiv\partial^{\bar{b}}\widehat{h}_{a\bar{b}}=0\,,\qquad
Q_{\bar{b}}\equiv\partial^{a}\widehat{h}_{a\bar{b}}=0\,,\\\
\tilde{Q}_{a}&\equiv\tilde{\partial}^{\bar{b}}\widehat{h}_{a\bar{b}}=0\,,\qquad\tilde{Q}_{\bar{b}}\equiv\tilde{\partial}^{a}\widehat{h}_{a\bar{b}}=0\,,\qquad
Q\equiv g_{\bar{b}}{}^{a}\widehat{h}_{a}{}^{\bar{b}}=0\,.\end{split}$ (4.12)

These constraints are not independent but rather subject to

$\begin{split}\partial^{a}Q_{a}&=\partial^{\bar{b}}Q_{\bar{b}}\,,\qquad\tilde{\partial}^{a}Q_{a}=\partial^{\bar{b}}\tilde{Q}_{\bar{b}}\,,\\\
\partial^{a}\tilde{Q}_{a}&=\tilde{\partial}^{\bar{b}}Q_{\bar{b}}\,,\qquad\tilde{\partial}^{a}\tilde{Q}_{a}=\tilde{\partial}^{\bar{b}}\tilde{Q}_{\bar{b}}\,.\end{split}$
(4.13)

Let us verify that ${h}_{a\bar{b}}$ so written encodes the right number of
(off-shell) degrees of freedom. As for $A$, the vector modes $B_{a}$,
$B_{\bar{a}}$, $\tilde{B}_{a}$, $\tilde{B}_{\bar{a}}$ together encode $4(d-2)$
degrees of freedom. The number of components of $\widehat{h}_{a\bar{b}}$ is
$d^{2}$ minus the number of constraints. Since the constraints (4.12) in turn
are subject to (4.13) we have $4d+1-4=4d-3$ independent constraints, so that
$\widehat{h}_{a\bar{b}}$ carries $d^{2}-4d+3$ degrees of freedom. In total,
together with the five scalar modes $E,C,\tilde{C},D,\tilde{D}$ and the
$4(d-2)$ vector modes, the irreducible components carry
$4(d-2)+d^{2}-4d+3+5=d^{2}$ degrees of freedom, as it should be.

At this stage it is appropriate to briefly discuss the counting of degrees of
freedom done above and to relate it to the familiar counting in, say, four
spacetime dimensions (for which $d=3$). Consider, for instance, the Fourier
expansion of a vector mode in (4.8):

$A_{a}(x,\tilde{x})=\sum_{k,\tilde{k}}A_{a}(k,\tilde{k})e^{i(k\cdot
x+\tilde{k}\cdot\tilde{x})}\;.$ (4.14)

As before we assume that the Fourier modes $A_{a}(k,\tilde{k})$ are only non-
zero provided $k\cdot\tilde{k}=0$, so that the weak constraint (level-matching
constraint) is obeyed. The two constraints in (4.9) yield

$k^{a}A_{a}(k,\tilde{k})=\tilde{k}^{a}A_{a}(k,\tilde{k})=0\;,$ (4.15)

which implies that among the three components of $A_{a}$ (for $d=3$)
generically only one survives. For instance, for $k=(0,0,1)$ and
$\tilde{k}=(0,1,0)$, for which $k\cdot\tilde{k}=0$, the constraints imply
$A_{2}=A_{3}=0$, so that only $A_{1}$ remains as a physical degree of freedom.
So what happened to the familiar two polarizations of a spin-1 vector mode in
four dimensions? These spin-1 modes are still present for the special case
that the individual modes carry only momentum or only winding, say in the form

$A_{a}(x,\tilde{x})=\sum_{k}A_{a}(k)e^{ik\cdot
x}+\sum_{\tilde{k}}\tilde{A}_{a}(\tilde{k})e^{i\tilde{k}\cdot\tilde{x}}\;,$
(4.16)

for which the level-matching constraint is trivially satisfied. Then for each
of the two terms one of the constraints (4.9) trivializes, so that the
corresponding vector mode carries the expected two polarizations. Therefore,
if we consider vector modes that live only in $x$-space, or vector modes that
live only in $\tilde{x}$-space, they do carry the familiar two polarizations
(effectively eliminating the additional scalar modes). It is only for modes
that both carry genuine momentum and winding that the degrees of freedom
organize differently. Similar remarks apply to the tensor modes. Indeed,
naively $\widehat{h}_{a\bar{b}}$ carries zero degrees of freedom in four
dimensions, but if we consider tensor modes that live only in $x$-space, or
only in $\tilde{x}$-space, some of the constraints trivialize, so that they do
carry the two polarizations of a spin-2 mode.

Returning to our discussion of the SVT decomposition, let us verify that these
decompositions exist by proving that the SVT components satisfying the
appropriate constraints can always be defined from the given fields (4.7).
More precisely, this is the case if $\partial^{2}$ and $\tilde{\partial}^{2}$
are invertible operators, as we will assume in the following. For instance,
the scalar modes of $h_{a\bar{b}}$ can be expressed in terms of the original
fields as:

$\begin{split}C&=\frac{d-1}{d-2}\partial^{-4}(\partial^{a}\partial^{\bar{b}}h_{a\bar{b}})-\frac{1}{d-2}\partial^{-2}\big{(}g_{\bar{b}}{}^{a}h_{a}{}^{\bar{b}}-\tilde{\partial}^{-2}(\tilde{\partial}^{a}\tilde{\partial}^{\bar{b}}h_{a\bar{b}})\big{)}\,,\\\
\tilde{C}&=\frac{d-1}{d-2}\tilde{\partial}^{-4}(\tilde{\partial}^{a}\tilde{\partial}^{\bar{b}}h_{a\bar{b}})-\frac{1}{d-2}\tilde{\partial}^{-2}\big{(}g_{\bar{b}}{}^{a}h_{a}{}^{\bar{b}}-\partial^{-2}(\partial^{a}\partial^{\bar{b}}h_{a\bar{b}})\big{)}\,,\\\
D&=\partial^{-2}\tilde{\partial}^{-2}(\partial^{a}\tilde{\partial}^{\bar{b}}h_{a\bar{b}})\,,\qquad\tilde{D}=\tilde{\partial}^{-2}\partial^{-2}(\tilde{\partial}^{a}\partial^{\bar{b}}h_{a\bar{b}})\,,\\\
E&=\frac{1}{d-2}\big{(}g_{\bar{b}}{}^{a}h_{a}{}^{\bar{b}}-\partial^{-2}\partial^{a}\partial^{\bar{b}}h_{a\bar{b}}-\tilde{\partial}^{-2}\tilde{\partial}^{a}\tilde{\partial}^{\bar{b}}h_{a\bar{b}}\big{)}\,.\end{split}$
(4.17)

The vector modes of $h_{a\bar{b}}$ in turn can be defined as

$\begin{split}B_{a}&=-\partial^{-2}\big{(}\partial^{\bar{b}}h_{a\bar{b}}-g_{a\bar{b}}\partial^{\bar{b}}E\big{)}+\partial_{a}C+\tilde{\partial}_{a}\tilde{D}\,,\\\
B_{\bar{b}}&=\partial^{-2}\big{(}\partial^{a}h_{a\bar{b}}-g_{a\bar{b}}\partial^{a}E\big{)}-\partial_{\bar{b}}C-\tilde{\partial}_{\bar{b}}D\,,\\\
\tilde{B}_{a}&=-\tilde{\partial}^{-2}\big{(}\tilde{\partial}^{\bar{b}}h_{a\bar{b}}-g_{a\bar{b}}\tilde{\partial}^{\bar{b}}E\big{)}+\tilde{\partial}_{a}\tilde{C}+\partial_{a}\tilde{D}\,,\\\
\tilde{B}_{\bar{b}}&=\tilde{\partial}^{-2}\big{(}\tilde{\partial}^{a}h_{a\bar{b}}-g_{a\bar{b}}\tilde{\partial}^{a}E\big{)}-\tilde{\partial}_{\bar{b}}\tilde{C}-\partial_{\bar{b}}D\,,\end{split}$
(4.18)

where one should view the scalar modes in here as defined in terms of
$h_{a\bar{b}}$ via (4.17). Finally, $\widehat{h}_{a\bar{b}}$ can then be
defined by inserting these scalar and vector modes (4.17) and (4.18) into
(4.10) and solving for $\widehat{h}_{a\bar{b}}$. Similarly, the scalar and
vector modes of the vector fields $\mathcal{A}_{a}$ and
$\mathcal{A}_{\bar{a}}$ can be defined in terms of these fields as

$\begin{split}A&=\partial^{-2}(\partial^{a}\mathcal{A}_{a})\,,\qquad\tilde{A}=\tilde{\partial}^{2}(\tilde{\partial}^{a}\mathcal{A}_{a})\,,\qquad\bar{A}=\partial^{-2}(\partial^{\bar{a}}\mathcal{A}_{\bar{a}})\,,\qquad\bar{\tilde{A}}=\tilde{\partial}^{-2}(\tilde{\partial}^{\bar{a}}\mathcal{A}_{\bar{a}})\,,\\\
A_{a}&=\mathcal{A}_{a}-\partial_{a}\big{(}\partial^{-2}(\partial^{b}\mathcal{A}_{b})\big{)}-\tilde{\partial}_{a}\big{(}\partial^{-2}(\partial^{\bar{b}}\mathcal{A}_{\bar{b}})\big{)}\,,\\\
A_{\bar{a}}&=\mathcal{A}_{\bar{a}}-\partial_{\bar{a}}\big{(}\tilde{\partial}^{-2}(\tilde{\partial}^{\bar{b}}\mathcal{A}_{\bar{b}})\big{)}-\tilde{\partial}_{\bar{a}}\big{(}\tilde{\partial}^{-2}(\tilde{\partial}^{\bar{b}}\mathcal{A}_{\bar{b}})\big{)}\,.\end{split}$
(4.19)

We will next determine the gauge transformations of the SVT components. To
this end we decompose the gauge parameters $\xi_{a}$ and $\xi_{\bar{a}}$ into
scalar and divergenceless vector components:

$\xi_{a}=\zeta_{a}+\partial_{a}\lambda+\tilde{\partial}_{a}\chi\,,\qquad\xi_{\bar{a}}=\zeta_{\bar{a}}+\partial_{\bar{a}}\bar{\lambda}+\tilde{\partial}_{\bar{a}}\bar{\chi}\,,$
(4.20)

where
$\partial^{a}\zeta_{a}=\tilde{\partial}^{a}\zeta_{a}=\partial^{\bar{a}}\zeta_{\bar{a}}=\tilde{\partial}^{\bar{a}}\zeta_{\bar{a}}=0$.
With this decomposition, the gauge transformations (2.43) become:

$\begin{split}\delta
h_{a\bar{b}}&=\frac{1}{\sqrt{2}}\big{(}a\tilde{\partial}_{a}\zeta_{\bar{b}}-a^{-1}\partial_{a}\zeta_{\bar{b}}-a\tilde{\partial}_{\bar{b}}\zeta_{a}-a^{-1}\partial_{\bar{b}}\zeta_{a}\big{)}\\\
&+\frac{1}{\sqrt{2}}\big{(}\tilde{\partial}_{a}\partial_{\bar{b}}(a\bar{\lambda}-a^{-1}\chi)+a\tilde{\partial}_{a}\tilde{\partial}_{\bar{b}}(\bar{\chi}-\chi)-a^{-1}\partial_{a}\partial_{\bar{b}}(\lambda+\bar{\lambda})-\partial_{a}\tilde{\partial}_{\bar{b}}(a^{-1}\bar{\chi}+a\lambda)\big{)}-\xi^{0}L_{a\bar{b}}\,,\\\
\delta\phi&=\dot{\xi}^{0}\,,\\\
\delta\varphi&=\dot{\Phi}\xi^{0}+\frac{1}{2\sqrt{2}}\big{(}-a^{-1}\partial^{2}(\lambda+\bar{\lambda})+a\tilde{\partial}^{2}(\chi-\bar{\chi})\big{)}\,,\\\
\delta\mathcal{A}_{a}&=\dot{\zeta}_{a}-L_{a}{}^{\bar{b}}\zeta_{\bar{b}}+\partial_{a}\bigg{(}\dot{\lambda}-H\bar{\lambda}-\frac{a^{-1}}{\sqrt{2}}(\tilde{\xi}_{0}+\xi^{0})\bigg{)}+\tilde{\partial}_{a}\bigg{(}\dot{\chi}-H\bar{\chi}+\frac{a}{\sqrt{2}}(\tilde{\xi}_{0}+\xi^{0})\bigg{)}\,,\\\
\delta\mathcal{A}_{\bar{a}}&=\dot{\zeta}_{\bar{a}}-L_{\bar{a}}{}^{b}\zeta_{b}+\partial_{\bar{a}}\bigg{(}\dot{\bar{\lambda}}-H\lambda+\frac{a^{-1}}{\sqrt{2}}(\tilde{\xi}_{0}-\xi^{0})\bigg{)}+\tilde{\partial}_{\bar{a}}\bigg{(}\dot{\bar{\chi}}-H\chi+\frac{a}{\sqrt{2}}(\tilde{\xi}_{0}-\xi^{0})\bigg{)}\,.\end{split}$
(4.21)

Comparing this with (4.8), (4.10) we can then read off the gauge
transformation of the SVT components of the fields:

$\begin{split}\delta\widehat{h}_{a\bar{b}}&=0\,,\qquad\delta E=-H\xi^{0}\,,\\\
\delta B_{a}&=\frac{a^{-1}}{\sqrt{2}}\zeta_{a}\,,\qquad\delta
B_{\bar{a}}=-\frac{a^{-1}}{\sqrt{2}}\zeta_{\bar{a}}\,,\qquad\delta\tilde{B}_{a}=\frac{a}{\sqrt{2}}\zeta_{a}\,,\qquad\delta\tilde{B}_{\bar{a}}=\frac{a}{\sqrt{2}}\zeta_{\bar{a}}\,,\\\
\delta
C&=-\frac{a^{-1}}{\sqrt{2}}(\lambda+\bar{\lambda})\,,\qquad\qquad\delta\tilde{C}=\frac{a}{\sqrt{2}}(\bar{\chi}-\chi)\,,\quad\\\
\delta
D&=\frac{1}{\sqrt{2}}(-a^{-1}\bar{\chi}-a\lambda)\,,\qquad\delta\tilde{D}=\frac{1}{\sqrt{2}}(a\bar{\lambda}-a^{-1}\chi)\,,\\\
\delta
A_{a}&=\dot{\zeta}_{a}-L_{a}{}^{\bar{b}}\zeta_{\bar{b}}\,,\qquad\qquad\delta
A_{\bar{a}}=\dot{\zeta}_{\bar{a}}-L_{\bar{a}}{}^{b}\zeta_{b}\,,\\\ \delta
A&=\dot{\lambda}-H\bar{\lambda}-\frac{a^{-1}}{\sqrt{2}}(\tilde{\xi}_{0}+\xi^{0})\,,\qquad\delta\tilde{A}=\dot{\chi}-H\bar{\chi}+\frac{a}{\sqrt{2}}(\tilde{\xi}_{0}+\xi^{0})\,,\\\
\delta\bar{A}&=\dot{\bar{\lambda}}-H\lambda+\frac{a^{-1}}{\sqrt{2}}(\tilde{\xi}_{0}-\xi^{0})\,,\qquad\delta\bar{\tilde{A}}=\dot{\bar{\chi}}-H\chi+\frac{a}{\sqrt{2}}(\tilde{\xi}_{0}-\xi^{0})\,.\end{split}$
(4.22)

Note that the tensor mode $\widehat{h}_{a\bar{b}}$ is gauge invariant.

###  4.2 Gauge fixing

In order to verify that the number of gauge independent fields is as expected
(i.e. equal to the number of off-shell degrees of freedom minus the number of
gauge redundancies) we can impose simple gauge fixing conditions, as we do in
the following.

From the second line in (4.22) we see that $\zeta_{a}$ and $\zeta_{\bar{a}}$
can be used to gauge fix two vectors to zero, e.g.,

$\tilde{B}_{a}=\tilde{B}_{\bar{a}}=0\;.$ (4.23)

This fixes the gauge invariance under $\zeta_{a}$, $\zeta_{\bar{a}}$
completely and does not require any compensating gauge transformations. Next,
we observe with the last two lines in (4.22) that $\xi^{0}$ and
$\tilde{\xi}_{0}$ can be used to gauge away two scalars, say:

$\tilde{A}=\bar{\tilde{A}}=0\;.$ (4.24)

This again fixes the gauge invariance under $\xi^{0}$, $\tilde{\xi}_{0}$
completely, but now we require compensating gauge transformations that
determine these parameters in terms of the remaining scalar gauge parameters:

$\frac{a}{\sqrt{2}}(\tilde{\xi}_{0}+\xi^{0})=-\dot{\chi}+H\bar{\chi}\;,\qquad\frac{a}{\sqrt{2}}(\tilde{\xi}_{0}-\xi^{0})=-\dot{\bar{\chi}}+H\chi\;.$
(4.25)

We are now left with four scalar gauge parameters ($\lambda$, $\chi$,
$\bar{\lambda}$, $\bar{\chi}$), so naively we would expect that we can gauge
away four more scalar field modes, but it turns out that, due to the gauge for
gauge symmetry, only three more scalar modes can be gauged away. Let us pick
the gauge that, say,

$\tilde{C}=D=\tilde{D}=0\;.$ (4.26)

The first condition fixes, say, the parameter $\bar{\chi}=\chi$. The second
gauge condition can then be achieved by means of $\lambda$, which in turn
fixes the compensating gauge transformation to be
$\lambda=-a^{-2}\bar{\chi}=-a^{-2}\chi$. Finally, the third gauge condition
can be achieved by means of $\bar{\lambda}$, which in turn fixes the
compensating gauge transformation to be $\bar{\lambda}=a^{-2}\chi$. In total
we have reduced the gauge redundancy to one, with independent parameter $\chi$
and compensating transformations in terms of $\chi$:

$\bar{\chi}=\chi\;,\qquad\lambda=-a^{-2}\chi\;,\qquad\bar{\lambda}=a^{-2}\chi\;.$
(4.27)

However, from the third line of (4.22) we then infer that $C$ is gauge
invariant and thus cannot be set to zero. Similarly, using (4.27) in (4.25)
yields

$\tilde{\xi}_{0}=-\sqrt{2}a^{-1}(\dot{\chi}-H{\chi})\;,\qquad\xi^{0}=0\;,$
(4.28)

and with this it follows that the other remaining scalar modes ($E$, $A$ and
$\bar{A}$) are all gauge invariant and can thus not be gauged away. Thus, the
apparent remaining parameter $\chi$ in fact does not act at all on the
remaining fields. This is just a consequence of there being a scalar gauge-
for-gauge symmetry. Thus, we have fixed the gauge redundancy completely.

Summarizing, the gauge independent fields can be chosen to be, for instance:

  * •

tensor modes: $\widehat{h}_{a\bar{b}}$ $[d^{2}-4d+3]$ ,

  * •

vector modes: $A_{a}$ , $A_{\bar{a}}$ , $B_{a}$ , $B_{\bar{a}}$ $[4(d-2)]$ ,

  * •

scalar modes: $E$ , $C$ , $A$ , $\bar{A}$ , $\phi$, $\varphi$ $[6]$ ,

where we also included the two scalar modes $\phi$, $\varphi$ that were
present from the beginning. We displayed in parenthesis the number of degrees
of freedom, which adds to $d^{2}+1$. This is equal to the total number of off-
shell degrees of freedom of the fields (4.7), given by $d^{2}+2d+2$, minus the
number of gauge redundancies for parameters $\xi_{a}$, $\xi_{\bar{a}}$,
$\tilde{\xi}_{0}$, $\xi^{0}$, which is given by $2d+1$ (taking into account
the one gauge-for-gauge redundancy).

###  4.3 Gauge invariant variables

We now aim to rewrite the field variables and the theory in terms of gauge
invariant combinations of the SVT components. This is essentially equivalent
to fixing a gauge in the sense that the resulting gauge invariant variables
are subject to constraints that are formally identical to gauge fixing
conditions. A gauge invariant formulation may always be reconstructed from a
gauge fixed one. We have already noted that $\widehat{h}_{a\bar{b}}$ gauge
invariant. In addition, one can build the following gauge invariant variables:

$\begin{split}\widehat{\phi}&=\phi+\frac{1}{2\sqrt{2}}\nabla_{t}\big{(}a(A+\bar{A}+\sqrt{2}a\dot{C})-a^{-1}(\tilde{A}-\bar{\tilde{A}}+\sqrt{2}a^{-1}\dot{\tilde{C}})\big{)}\,,\\\
\widehat{A}_{a}&=A_{a}-\frac{1}{\sqrt{2}}\nabla_{t}(aB_{a}+a^{-1}\tilde{B}_{a})-\frac{1}{\sqrt{2}}L_{a}{}^{\bar{b}}(aB_{\bar{b}}-a^{-1}\tilde{B}_{\bar{b}})\,,\\\
\widehat{A}_{\bar{a}}&=A_{\bar{a}}+\frac{1}{\sqrt{2}}\nabla_{t}(aB_{\bar{a}}-a^{-1}\tilde{B}_{\bar{a}})+\frac{1}{\sqrt{2}}L_{\bar{a}}{}^{b}(aB_{b}+a^{-1}\tilde{B}_{b})\,,\\\
\widehat{A}&=a(A+\bar{A}+\sqrt{2}a\dot{C})+a^{-1}(\tilde{A}-\bar{\tilde{A}}+\sqrt{2}a^{-1}\dot{\tilde{C}})\,,\\\
\widehat{\tilde{A}}&=a(A-\bar{A}-\sqrt{2}a\dot{C}-2\sqrt{2}aHC)+a^{-1}(\bar{\tilde{A}}+\tilde{A}+\sqrt{2}a^{-1}\dot{\tilde{C}}-2\sqrt{2}a^{-1}H\tilde{C})+2\sqrt{2}\dot{D}\,,\\\
\widehat{\bar{A}}&=a(\bar{A}-A-\sqrt{2}a\dot{C}-2\sqrt{2}aHC)-a^{-1}(\bar{\tilde{A}}+\tilde{A}-\sqrt{2}a^{-1}\dot{\tilde{C}}+2\sqrt{2}a^{-1}H\tilde{C})-2\sqrt{2}\dot{\tilde{D}}\,,\\\
\widehat{C}&=D-\tilde{D}-a^{2}C+a^{-2}\tilde{C}\,,\\\
\widehat{B}_{a}&=\frac{1}{2}(aB_{a}-a^{-1}\tilde{B}_{a})\,,\\\
\widehat{B}_{\bar{a}}&=\frac{1}{2}(aB_{\bar{a}}+a^{-1}\tilde{B}_{\bar{a}})\,,\\\
\widehat{\varphi}&=\varphi+\frac{1}{2\sqrt{2}}\dot{\Phi}\big{(}a(A+\bar{A}+\sqrt{2}a\dot{C})-a^{-1}(\tilde{A}-\bar{\tilde{A}}+\sqrt{2}a^{-1}\dot{\tilde{C}})\big{)}+\frac{1}{2}(-\partial^{2}C+\tilde{\partial}^{2}\tilde{C})\,,\\\
\widehat{E}&=E-\frac{1}{2\sqrt{2}}H\big{(}a(A+\bar{A}+\sqrt{2}a\dot{C})-a^{-1}(\tilde{A}-\bar{\tilde{A}}+\sqrt{2}a^{-1}\dot{\tilde{C}})\big{)}\,,\end{split}$
(4.29)

as follows with simple computations using (4.22). These gauge invariant
variables are of course not unique, as any linear combination of gauge
invariant fields is also gauge invariant. Moreover, there is one relation
among them: the scalars $\widehat{\tilde{A}}$ and $\widehat{\bar{A}}$ satisfy

$\widehat{\tilde{A}}+\widehat{\bar{A}}=2\sqrt{2}\dot{\widehat{C}}\,.$ (4.30)

We next aim to rewrite the quadratic double field theory action directly in
terms of gauge invariant variables that are linear combinations of (4.29).
Given that the latter take a rather complicated form, a priori this seems to
be a difficult task that, however, is simplified by the following trick: We
use that the original fields can be expressed in terms of specific
combinations of the gauge invariant fields plus terms that take the form of an
infinitesimal gauge transformation (2.43). Specifically, we claim that

$\begin{split}h_{a\bar{b}}\ &=\
\bar{h}_{a\bar{b}}+D_{a}F_{\bar{b}}-D_{\bar{b}}F_{a}-F^{0}L_{a\bar{b}}\,,\\\
\mathcal{A}_{a}\ &=\
\bar{\mathcal{A}}_{a}+\dot{F}_{a}-L_{a}{}^{\bar{b}}F_{\bar{b}}+D_{a}\big{(}\tilde{F}_{0}+F^{0}\big{)}\\\
\mathcal{A}_{\bar{a}}\ &=\
\bar{\mathcal{A}}_{\bar{a}}+\dot{F}_{\bar{a}}-L_{\bar{a}}{}^{b}F_{b}+D_{\bar{a}}(\tilde{F}_{0}-F^{0})\,,\\\
\varphi\ &=\
\bar{\varphi}+\dot{\Phi}F^{0}-\tfrac{1}{2}D_{a}F^{a}-\tfrac{1}{2}D_{\bar{a}}F^{\bar{a}}\,,\\\
\phi\ &=\ \bar{\phi}-\dot{F}^{0}\,,\end{split}$ (4.31)

where the gauge invariant fields, denoted by a bar, are given by

$\displaystyle\bar{h}_{a\bar{b}}$
$\displaystyle=\widehat{h}_{a\bar{b}}+(a^{-1}\partial_{a}+a\tilde{\partial}_{a})\widehat{B}_{\bar{b}}-(a^{-1}\partial_{\bar{b}}-a\tilde{\partial}_{\bar{b}})\widehat{B}_{a}+\widehat{E}g_{a\bar{b}}$
$\displaystyle\quad+\frac{1}{2}(a^{-2}\partial_{a}\partial_{\bar{b}}+2\partial_{a}\tilde{\partial}_{\bar{b}}-2\tilde{\partial}_{a}\partial_{\bar{b}}-a^{2}\tilde{\partial}_{a}\tilde{\partial}_{\bar{b}})\widehat{C}\,,$
(4.32) $\displaystyle\bar{\mathcal{A}}_{a}$
$\displaystyle=\widehat{\mathcal{A}}_{a}+\frac{a^{-1}}{4}\partial_{a}\big{(}\widehat{A}-\widehat{\bar{A}}+2\sqrt{2}H\widehat{C}\big{)}+\frac{a}{4}\tilde{\partial}_{a}\big{(}\widehat{A}+\widehat{\tilde{A}}+2\sqrt{2}H\widehat{C}\big{)}\,,$
(4.33) $\displaystyle\bar{\mathcal{A}}_{\bar{a}}$
$\displaystyle=\widehat{\mathcal{A}}_{\bar{a}}+\frac{a^{-1}}{4}\partial_{\bar{a}}\big{(}\widehat{A}-\widehat{\tilde{A}}+2\sqrt{2}H\widehat{C}\big{)}-\frac{a}{4}\tilde{\partial}_{\bar{a}}\big{(}\widehat{A}+\widehat{\bar{A}}+2\sqrt{2}H\widehat{C}\big{)}\,,$
(4.34) $\displaystyle\bar{\varphi}$
$\displaystyle=\widehat{\varphi}+\frac{1}{4}\Delta\widehat{C}\,,$ (4.35)
$\displaystyle\bar{\phi}$ $\displaystyle=\widehat{\phi}\,,$ (4.36)

while the ‘effective gauge parameters’, denoted by $F$, are given by

$\begin{split}F^{0}&=-\frac{1}{2\sqrt{2}}\big{(}a(A+\bar{A}+\sqrt{2}a\dot{C})-a^{-1}(\tilde{A}-\bar{\tilde{A}}+\sqrt{2}a^{-1}\dot{\tilde{C}})\big{)}\,,\\\
F_{0}&=\frac{1}{2\sqrt{2}}\big{(}a^{-1}(\tilde{A}+\bar{\tilde{A}})-a(A-\bar{A})\big{)}\,,\\\
F_{a}&=\frac{1}{\sqrt{2}}(aB_{a}+a^{-1}\tilde{B}_{a})+\frac{\sqrt{2}a^{-1}}{4}\partial_{a}(-3a^{2}C-2\tilde{D}+a^{-2}\tilde{C})+\frac{\sqrt{2}a}{4}\tilde{\partial}_{a}(-3a^{-2}\tilde{C}-2D+a^{2}C)\,,\\\
F_{\bar{a}}&=-\frac{1}{\sqrt{2}}(aB_{\bar{a}}-a^{-1}\tilde{B}_{\bar{a}})+\frac{\sqrt{2}a^{-1}}{4}\partial_{\bar{a}}(-3a^{2}C+2D+a^{-2}\tilde{C})+\frac{\sqrt{2}a}{4}\tilde{\partial}_{\bar{a}}(3a^{-2}\tilde{C}-2\tilde{D}-a^{2}C)\,.\end{split}$
(4.37)

These quantities transform under gauge transformations as

$\begin{split}\delta F^{0}&=\xi^{0}\;,\\\ \delta
F_{0}&=\tilde{\xi}_{0}-\dot{\eta}\;,\\\ \delta
F_{a}&=\zeta_{a}+\partial_{a}\big{(}\lambda-\tfrac{1}{\sqrt{2}}a^{-1}\eta\big{)}+\tilde{\partial}_{a}\big{(}\chi+\tfrac{1}{\sqrt{2}}a\eta\big{)}=\xi_{a}-\tfrac{1}{\sqrt{2}}a^{-1}\partial_{a}\eta+\tfrac{1}{\sqrt{2}}a\tilde{\partial}_{a}\eta\;,\\\
\delta
F_{\bar{a}}&=\zeta_{\bar{a}}+\partial_{\bar{a}}\big{(}\bar{\lambda}+\tfrac{1}{\sqrt{2}}a^{-1}\eta\big{)}+\tilde{\partial}_{\bar{a}}\big{(}\bar{\chi}+\tfrac{1}{\sqrt{2}}a\eta\big{)}=\xi_{\bar{a}}+\frac{1}{\sqrt{2}}a^{-1}\partial_{\bar{a}}\eta+\frac{1}{\sqrt{2}}a\tilde{\partial}_{\bar{a}}\eta\;,\\\
\end{split}$ (4.38)

where the gauge-for-gauge parameter $\eta$ is given by

$\eta=\frac{\sqrt{2}}{4}\left(a\lambda-a\bar{\lambda}-a^{-1}\chi-a^{-1}\bar{\chi}\right)\;.$
(4.39)

Using the expressions (4.37) and those for the gauge invariant variables
(4.29) one may verify that (4.31) are just identities. These identities are
very useful, however, since they decompose the fields into their gauge
invariant parts plus terms of the ‘pure gauge form’ (2.43).

We can now replace each appearance of a field in the quadratic action by its
right-hand side in (4.31). Gauge invariance then implies that the pure gauge
terms drop out, so that in effect we may simply replace each field by its
gauge invariant version, i.e., we just put a bar on each field. Afterwards we
use the expressions in terms of the SVT components, which in turn allows us to
express each divergence, trace, etc., of a tensor or vector in terms of
appropriate scalar modes, thereby achieving complete decoupling.

In order to perform this computation it is convenient to first display the
quadratic Lagrangian (2.31) in terms of the derivatives $\partial$ and
$\tilde{\partial}$:

$\begin{split}\mathcal{L}=&\,ne^{-2\Phi}\bigg{\\{}-4\dot{\varphi}^{2}+8\dot{\Phi}\phi\dot{\varphi}+H(2{\phi}+4{\varphi})g_{\bar{b}}{}^{a}\dot{h}_{a}{}^{\bar{b}}+\dot{h}_{a\bar{b}}\dot{h}^{a\bar{b}}+2H^{2}h^{a\bar{b}}h_{a\bar{b}}-2H^{2}g_{a}{}^{\bar{c}}g_{\bar{d}}{}^{b}h^{a\bar{d}}h_{b\bar{c}}\\\
&+\frac{1}{\sqrt{2}}(a\tilde{\partial}^{\bar{a}}\mathcal{A}_{\bar{a}}+a^{-1}\partial^{\bar{a}}\mathcal{A}_{\bar{a}}-a\tilde{\partial}^{a}\mathcal{A}_{a}+a^{-1}\partial^{a}\mathcal{A}_{a})(-4\dot{\varphi}+4\dot{\Phi}\phi-2Hg_{\bar{b}}{}^{a}h_{a}{}^{\bar{b}})\\\
&+\frac{1}{\sqrt{2}}H(4\varphi+2\phi)(-a\tilde{\partial}^{\bar{a}}\mathcal{A}_{\bar{a}}+a^{-1}\partial^{\bar{a}}\mathcal{A}_{\bar{a}}+a\tilde{\partial}^{a}\mathcal{A}_{a}+a^{-1}\partial^{a}\mathcal{A}_{a})\\\
&-\frac{1}{2}\mathcal{A}_{\bar{a}}\Delta\mathcal{A}^{\bar{a}}-\frac{1}{2}\mathcal{A}^{a}\Delta\mathcal{A}_{a}-\frac{1}{2}(a\tilde{\partial}^{a}\mathcal{A}_{a}-a^{-1}\partial^{a}\mathcal{A}_{a})^{2}-\frac{1}{2}(a\tilde{\partial}^{\bar{a}}\mathcal{A}_{\bar{a}}+a^{-1}\partial^{\bar{a}}\mathcal{A}_{\bar{a}})^{2}\\\
&+\sqrt{2}\mathcal{A}^{\bar{b}}(a\tilde{\partial}^{a}-a^{-1}\partial^{a})\dot{h}_{a\bar{b}}-\sqrt{2}\mathcal{A}^{a}(a\tilde{\partial}^{\bar{b}}+a^{-1}\partial^{\bar{b}})\dot{h}_{a\bar{b}}-\sqrt{2}Hg_{b}{}^{\bar{d}}\mathcal{A}^{b}(a\tilde{\partial}^{a}h_{a\bar{d}}-a^{-1}\partial^{a}h_{a\bar{d}})\\\
&+\sqrt{2}Hg_{\bar{b}}{}^{c}\mathcal{A}^{\bar{b}}(a\tilde{\partial}^{\bar{a}}h_{c\bar{a}}+a^{-1}\partial^{\bar{a}}h_{c\bar{a}})-\sqrt{2}H\mathcal{A}^{\bar{b}}(a\tilde{\partial}^{a}h_{a\bar{b}}+a^{-1}\partial^{a}h_{a\bar{b}})\\\
&+\sqrt{2}H\mathcal{A}^{a}(a\tilde{\partial}^{\bar{b}}h_{a\bar{b}}-a^{-1}\partial^{\bar{b}}h_{a\bar{b}})\\\
&+4\phi\Delta\varphi-4\varphi\Delta\varphi+(4\varphi-2\phi)(a^{-2}\partial^{a}\partial^{\bar{b}}-\tilde{\partial}^{a}\partial^{\bar{b}}+\partial^{a}\tilde{\partial}^{\bar{b}}-a^{2}\tilde{\partial}^{a}\tilde{\partial}^{\bar{b}})h_{a\bar{b}}+h^{a\bar{b}}\Delta
h_{a\bar{b}}\\\
&+(a^{-1}\partial^{a}-a\tilde{\partial}^{a})h_{a}{}^{\bar{b}}(a^{-1}\partial^{c}-a\tilde{\partial}^{c})h_{c\bar{b}}+(a^{-1}\partial^{\bar{b}}+a\tilde{\partial}^{\bar{b}})h_{a\bar{b}}(a^{-1}\partial_{\bar{c}}+a\tilde{\partial}_{\bar{c}})h^{a\bar{c}}\bigg{\\}}\;.\end{split}$
(4.40)

Following the above procedure one obtains the decoupled quadratic action in
terms of gauge invariant variables:

$\begin{split}S=\int&dt\int d^{2d}{\bf
X}\,n\,e^{-2\Phi}\big{(}\mathcal{L}_{T}+\mathcal{L}_{V}+\mathcal{L}_{S}\big{)}\,,\end{split}$
(4.41)

where $\mathcal{L}_{T}$, $\mathcal{L}_{V}$ and $\mathcal{L}_{S}$ denote the
Lagrangians for the tensor, vector and scalar modes, respectively. As
emphasized before, the action should completely decouple among these modes, as
indeed it does. Specifically, the action for the tensor modes is given by

$\mathcal{L}_{T}\ =\
\dot{\widehat{h}}_{a\bar{b}}\,\dot{\widehat{h}}^{a\bar{b}}+2H^{2}\,\widehat{h}_{a\bar{b}}\,\widehat{h}^{a\bar{b}}-2H^{2}g_{a}{}^{\bar{c}}g_{\bar{d}}{}^{b}\,\widehat{h}^{a\bar{d}}\,\widehat{h}_{b\bar{c}}+\widehat{h}_{a\bar{b}}\Delta\widehat{h}^{a\bar{b}}\,,$
(4.42)

while the action for the vector modes reads

$\begin{split}\mathcal{L}_{V}\,=\,&-\dot{\widehat{B}}_{\bar{a}}\Delta\dot{\widehat{B}}^{\bar{a}}-\dot{\widehat{B}}^{a}\Delta\dot{\widehat{B}}_{a}-H^{2}\widehat{B}_{a}\Delta\widehat{B}^{a}-H^{2}\widehat{B}_{\bar{a}}\Delta\widehat{B}^{\bar{a}}\\\
&+4H^{2}g_{a}{}^{\bar{b}}\widehat{B}^{a}\Diamond\widehat{B}_{\bar{b}}-\widehat{B}_{a}(\Delta^{2}-\Diamond^{2})\widehat{B}^{a}-\widehat{B}_{\bar{a}}(\Delta^{2}-\Diamond^{2})\widehat{B}^{\bar{a}}\\\
&-\frac{1}{2}\widehat{A}_{\bar{a}}\Delta\widehat{A}^{\bar{a}}-\frac{1}{2}\widehat{A}_{a}\Delta\widehat{A}^{a}-\sqrt{2}\widehat{A}^{a}\Diamond(\dot{\widehat{B}}_{a}+Hg_{a}{}^{\bar{b}}\widehat{B}_{\bar{b}})+\sqrt{2}\widehat{A}^{\bar{a}}\Diamond(\dot{\widehat{B}}_{\bar{a}}+Hg_{\bar{a}}{}^{b}\widehat{B}_{b})\,.\end{split}$
(4.43)

Note that while this action carries two more vector modes than in standard
gravity, the $A$ modes are actually auxiliary as they may be eliminated by
their own equations of motions. Indeed, varying with respect to
$\widehat{A}_{a}$ and $\widehat{A}_{\bar{a}}$ and assuming $\Delta$ is
invertible, we obtain

$\begin{split}\widehat{A}_{a}&=-\sqrt{2}\Delta^{-1}\Diamond(\dot{\widehat{B}}_{a}+Hg_{a}{}^{\bar{b}}\widehat{B}_{\bar{b}})\,,\\\
\widehat{A}_{\bar{a}}&=\sqrt{2}\Delta^{-1}\Diamond(\dot{\widehat{B}}_{\bar{a}}+Hg_{\bar{a}}{}^{b}\widehat{B}_{b})\,,\end{split}$
(4.44)

which may be reinserted into the action to obtain:

$\begin{split}{\cal
L}_{V}(B)=&-\dot{\widehat{B}}_{\bar{a}}\Delta\dot{\widehat{B}}^{\bar{a}}-\dot{\widehat{B}}^{a}\Delta\dot{\widehat{B}}_{a}-H^{2}\widehat{B}_{a}\Delta\widehat{B}^{a}-H^{2}\widehat{B}_{\bar{a}}\Delta\widehat{B}^{\bar{a}}\\\
&+4H^{2}g_{\bar{b}}{}^{a}\widehat{B}^{\bar{b}}\Diamond\widehat{B}_{a}-\widehat{B}_{\bar{a}}(\Delta^{2}-\Diamond^{2})\widehat{B}^{\bar{a}}-\widehat{B}_{a}(\Delta^{2}-\Diamond^{2})\widehat{B}^{a}\\\
&+(\dot{\widehat{B}}_{\bar{a}}+Hg_{\bar{a}}{}^{b}\widehat{B}_{b})\Delta^{-1}\Diamond^{2}(\dot{\widehat{B}}^{\bar{a}}+Hg^{\bar{a}c}\widehat{B}_{c})+(\dot{\widehat{B}}_{a}+Hg_{a}{}^{\bar{b}}\widehat{B}_{\bar{b}})\Delta^{-1}\Diamond^{2}(\dot{\widehat{B}}^{a}+Hg^{a\bar{c}}\widehat{B}_{\bar{c}})\,.\end{split}$
(4.45)

The action for the scalar modes is

$\begin{split}\mathcal{L}_{S}=&-4\dot{\widehat{\varphi}}^{2}+8\dot{\Phi}\phi\dot{\widehat{\varphi}}+d\dot{\widehat{E}}^{2}+2dH(\widehat{\phi}+2\widehat{\varphi})\dot{\widehat{E}}+\frac{1}{4}\dot{\widehat{C}}(\Delta^{2}-\Diamond^{2})\dot{\widehat{C}}+\frac{1}{2}H^{2}\widehat{C}(\Delta^{2}-\Diamond^{2})\widehat{C}\\\
&+\frac{\sqrt{2}}{2}\widehat{A}\Diamond(2\dot{\widehat{\varphi}}-2\dot{\Phi}\widehat{\phi}+dH\widehat{E})+\frac{\sqrt{2}}{2}\widehat{A}\Delta\big{(}H(\widehat{\phi}+2\widehat{\varphi})+\dot{\widehat{E}}\big{)}\\\
&-\frac{\sqrt{2}}{4}H\widehat{A}(\Delta^{2}-\Diamond^{2})\widehat{C}+\frac{1}{16}\widehat{A}(\Delta^{2}-\Diamond^{2})\widehat{A}-\frac{\sqrt{2}}{8}\widehat{\bar{A}}(\Delta^{2}-\Diamond^{2})\dot{\widehat{C}}+\frac{1}{16}\widehat{\bar{A}}(\Delta^{2}-\Diamond^{2})\widehat{\bar{A}}\\\
&+4\widehat{\phi}\Delta\widehat{\varphi}-4\widehat{\varphi}\Delta\widehat{\varphi}+(d-2)\widehat{E}\Delta\widehat{E}+2(\widehat{\phi}-2\widehat{\varphi})\Diamond\widehat{E}-\frac{1}{2}(\widehat{\phi}-2\widehat{\varphi})(\Delta^{2}-\Diamond^{2})\widehat{C}\,.\end{split}$
(4.46)

Here, $\widehat{\bar{A}}$, and $\widehat{A}$ are auxiliary fields. Varying the
action with respect to these fields yields

$(\Delta^{2}-\Diamond^{2})(-\sqrt{2}\dot{\widehat{C}}+\widehat{\bar{A}})=0\,,$
(4.47)
$\frac{1}{8}(\Delta^{2}-\Diamond^{2})(\widehat{A}-2\sqrt{2}H\widehat{C})+\frac{\sqrt{2}}{2}\Diamond(2\dot{\widehat{\varphi}}-2\dot{\Phi}\widehat{\phi}+dH\widehat{E})+\frac{\sqrt{2}}{2}\Delta\big{(}H(\widehat{\phi}+2\widehat{\varphi})+\dot{\widehat{E}}\big{)}=0\,.$
(4.48)

Since $\Delta^{2}-\Diamond^{2}=4\partial^{2}\tilde{\partial}^{2}$ and
$\partial^{2}$ and $\tilde{\partial}^{2}$ are invertible, we can solve for
$\widehat{\bar{A}}$ and $\widehat{A}$,

$\displaystyle\widehat{\bar{A}}$ $\displaystyle=\sqrt{2}\dot{\widehat{C}}\,,$
(4.49) $\displaystyle\widehat{A}$
$\displaystyle=2\sqrt{2}H\widehat{C}-\sqrt{2}\partial^{-2}\tilde{\partial}^{-2}\big{(}\Diamond(2\dot{\widehat{\varphi}}-2\dot{\Phi}\widehat{\phi}+dH\widehat{E})+\Delta\big{(}H(\widehat{\phi}+2\widehat{\varphi})+\dot{\widehat{E}}\big{)}\,,$
(4.50)

and re-insert these into the action, which yields

$\begin{split}\mathcal{L}_{S}=&\
(d-1)\dot{\widehat{E}}^{2}+2(d-1)H(\widehat{\phi}+2\widehat{\varphi})\dot{\widehat{E}}+\frac{1}{8}\dot{\widehat{C}}(\Delta^{2}-\Diamond^{2})\dot{\widehat{C}}\\\
&+2\widehat{C}\Diamond(2\dot{\widehat{\varphi}}-2\dot{\Phi}\widehat{\phi}+dH\widehat{E})+2\widehat{C}\Delta\big{(}H(\widehat{\phi}+2\widehat{\varphi})+\dot{\widehat{E}}\big{)}\\\
&+(d-1)H^{2}\widehat{\phi}^{2}-4H^{2}\widehat{\phi}\widehat{\varphi}-4H^{2}\widehat{\varphi}^{2}+d^{2}H^{2}\widehat{E}^{2}+4dH\dot{\widehat{\varphi}}\widehat{E}-4dH\dot{\Phi}\widehat{E}\widehat{\phi}\\\
&+4\widehat{\phi}\Delta\widehat{\varphi}-4\widehat{\varphi}\Delta\widehat{\varphi}+(d-2)\widehat{E}\Delta\widehat{E}+2(\widehat{\phi}-2\widehat{\varphi})\Diamond\widehat{E}-\frac{1}{2}(\widehat{\phi}-2\widehat{\varphi})(\Delta^{2}-\Diamond^{2})\widehat{C}\\\
&-\frac{1}{2}(2\dot{\widehat{\varphi}}-2\dot{\Phi}\widehat{\phi}+dH\widehat{E})\partial^{-2}\tilde{\partial}^{-2}(a^{4}\tilde{\partial}^{4}+a^{-4}\partial^{4})(2\dot{\widehat{\varphi}}-2\dot{\Phi}\widehat{\phi}+dH\widehat{E})\\\
&-\frac{1}{2}\big{(}H(\widehat{\phi}+2\widehat{\varphi})+\dot{\widehat{E}}\big{)}\partial^{-2}\tilde{\partial}^{-2}(a^{4}\tilde{\partial}^{4}+a^{-4}\partial^{4})\big{(}H(\widehat{\phi}+2\widehat{\varphi})+\dot{\widehat{E}}\big{)}\\\
&-(2\dot{\widehat{\varphi}}-2\dot{\Phi}\widehat{\phi}+dH\widehat{E})\partial^{-2}\tilde{\partial}^{-2}(a^{4}\tilde{\partial}^{4}-a^{-4}\partial^{4})\big{(}H(\widehat{\phi}+2\widehat{\varphi})+\dot{\widehat{E}}\big{)}\,.\end{split}$
(4.51)

We close this section by pointing out that the definition of gauge invariant
variables can be extended to cubic order, using the results of [32]. This
approach uses the insight that the passing over to gauge invariant variables
has a natural interpretation in the framework of $L_{\infty}$-algebras as
homotopy transfer, which in turn yields an algorithmic procedure to extend the
gauge invariant functions to higher order. As before, the cubic action in
terms of these variables takes the same form as the gauge fixed action, but
with the fields being the fully gauge invariant ones. We will leave the
details to future work.

##  5 Summary and Outlook

In this paper we have analyzed general aspects of a genuine double field
theory on time-dependent (cosmological) backgrounds with zero spatial
curvature, with a particular emphasis on the presence of winding modes when
the spatial topology is a torus. Indeed, to quadratic and cubic order the
theory is fully consistent in a weakly constrained sense, meaning that the
fields genuinely depend on doubled coordinates, hence encoding both momentum
and winding modes. We presented a self-contained discussion of the resolution
of the consistency issues related to the weak constraint, including the role
of cocycle factors. Finally, we analyzed, for FRW backgrounds with a single
scale factor $a(t)$, the independent gauge invariant modes under a scalar-
vector-tensor decomposition. This analysis is significantly more involved
compared to standard cosmology, due to the doubled derivatives, and also shows
some striking new features. Most importantly, the nature of the gauge
invariant modes changes relative to standard cosmology in that there are more
vector and scalar modes, while the number of tensor modes is reduced (leaving,
of course, the total number of off-shell degrees of freedom unchanged).

In this paper we have just taken the first steps in developing a full-fledged
cosmological perturbation theory for genuine double field theory. It remains
to extend this framework in a way that allows for (semi-)realistic
cosmological scenarios, so that a confrontation with observation may
eventually become feasible. Concretely, the following research problems
present themselves:

  * •

In this paper we have not included matter other than the scalar dilaton and
Kalb-Ramond B-field that are automatically encoded in double field theory. The
latter fields are insufficient in order to provide semi-realistic cosmologies,
so it is important to add further matter, perhaps in the form of a generic
duality invariant energy-momentum tensor.

  * •

In string theory there is an infinite number of higher-derivative
$\alpha^{\prime}$ corrections, which plausibly are relevant in the very early
universe but which have not been included here. While the complete
$\alpha^{\prime}$ corrections remain of course unknown, remarkably, it is
possible to give a complete classification of all corrections compatible with
$O(d,d)$ duality invariance for purely time-dependent cosmological backgrounds
[33]. In order to set up the corresponding $\alpha^{\prime}$-complete
cosmological perturbation theory one might hope that for the quadratic
approximation one can bypass the hard problem of defining the full double
field theory with all $\alpha^{\prime}$ corrections included. Assuming (as on
flat space [34]) that there is a field basis for which the quadratic terms are
two-derivative one may try to write directly a quadratic action that is gauge
invariant, but using now the known $\alpha^{\prime}$-completed background
Friedmann equations, which include only first-order time derivatives [33].

  * •

Recently, an intriguingly simple off-shell and gauge invariant double copy
relation has been uncovered between Yang-Mills theory and double field theory
on flat space [35], at least to cubic order in fields. The natural question
arises whether there are similar double copy relations for double field theory
on general backgrounds or at least on the purely time-dependent backgrounds
explored here.

  * •

Arguably the most important open research project is the computation of
cosmological correlation functions, as these may eventually relate to the CMB
and hence allow one to discriminate between different cosmological scenarios.
In particular, since the independent gauge invariant modes have a different
character in a genuine double field theory, meaning the degrees of freedom are
distributed differently among scalar, vector and tensor modes, it seems quite
promising that we will eventually be able to distinguish it observationally
from general relativity coupled to ordinary matter. Since cosmological
perturbation theory in general, and the computation of cosmological
correlation functions in particular, is infamously opaque [15, 23], to this
end it would be important to develop more systematic techniques. We believe
that a most promising framework is that of the homotopy algebra formulation of
gauge theories, see e.g. [36], because here there are natural ‘homological’
approaches to the definition of gauge invariant variables [32] and to the
computation of quantum mechanical expectation values [37].

### Acknowledgements

We are grateful to Heliudson Bernardo, Roberto Bonezzi, Robert Brandenberger,
Christoph Chiaffrino, Chris Hull, Matt Kleban, Jean-Luc Lehners and Barton
Zwiebach for useful discussions, correspondence and related collaborations.

This work is funded by the European Research Council (ERC) under the European
Union’s Horizon 2020 research and innovation programme (grant agreement No
771862) and by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation), “Rethinking Quantum Field Theory”, Projektnummer
417533893/GRK2575.

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