boxsize=1.1em,centertableaux

# U-duality and $\alpha^{\prime}$ corrections in three dimensions

Camille Eloy  Theoretische Natuurkunde, Vrije Universiteit Brussel, and the
International Solvay Institutes, Pleinlaan 2, B-1050 Brussels, Belgium  Olaf
Hohm  Institute for Physics, Humboldt University Berlin, Zum Großen Windkanal
6, D-12489 Berlin, Germany  Henning Samtleben  ENSL, CNRS, Laboratoire de
physique, F-69342 Lyon, France  Institut Universitaire de France (IUF)

###### Abstract

We consider the target space theory of bosonic and heterotic string theory to
first order in $\alpha^{\prime}$ compactified to three dimensions, using a
formulation that is manifestly T-duality invariant under
$\mathrm{O}(d,d,\mathbb{R})$ with $d=23$ and $d=7$, respectively. While the
two-derivative supergravity exhibits a symmetry enhancement to the U-duality
group $\mathrm{O}(d+1,d+1)$, the continuous group is known to be broken to
$\mathrm{O}(d,d,\mathbb{R})$ by the first $\alpha^{\prime}$ correction. We
revisit this observation by computing the full effective actions in three
dimensions to first order in $\alpha^{\prime}$ by dualizing the vector gauge
fields. We give a formally $\mathrm{O}(d+1,d+1)$ invariant formulation by
invoking a vector compensator, and we observe a chiral pattern that allows one
to reconstruct the bosonic action from the heterotic action. Furthermore, we
obtain a particular massive deformation by integrating out the external $B$
field. This induces a novel Chern-Simons term based on composite connections
that, remarkably, is $\mathrm{O}(d+1,d+1)$ invariant to leading order in the
deformation parameter.

###### Contents

  1. 1 Introduction
  2. 2 Duality Enhancement in Three Dimensions
    1. 2.1 Scaling symmetries
    2. 2.2 $\mathrm{O}(d+1,d+1)$ enhancement
    3. 2.3 Field redefinitions
  3. 3 Four-derivative Action in Three Dimensions
    1. 3.1 Dualisation of the vector fields
    2. 3.2 $\mathrm{O}(d+1,d+1)$-covariant formulation
    3. 3.3 Comments on symmetry group
  4. 4 A massive deformation
    1. 4.1 Two-derivative action
    2. 4.2 Four-derivative action
  5. 5 Conclusions
  6. A $\mathrm{O}(d,d)$ decompositions of $\mathrm{O}(d+1,d+1)$
  7. B Four-derivative action with mass deformation
    1. B.1 Field redefinitions
    2. B.2 Einstein frame
    3. B.3 Integrations by part and field redefinitions
    4. B.4 Dualisation of the vector fields

## 1 Introduction

Our goal in this paper is to explore some features arising in the interplay of
higher-derivative $\alpha^{\prime}$ corrections of the effective actions of
string and M-theory and the duality properties that these theories are
expected to exhibit for certain backgrounds. Arguably the simplest duality
property of string theory is T-duality, which states that theories
compactified on toroidal backgrounds Td related by
$\mathrm{O}(d,d,\mathbb{Z})$ transformations are physically equivalent, even
though these backgrounds may be radically different as ordinary geometries.
This means that conventional Einstein-Hilbert gravity looks quite different on
these backgrounds, yet ‘stringy gravity’ supposedly cannot tell the
difference. Even larger duality groups arise for particular theories and
backgrounds. In this paper we consider compactifications to three spacetime
dimensions, which are particularly interesting since in the supergravity limit
the corresponding T-duality group $\mathrm{O}(7,7)$ is enhanced to larger
groups, and discrete subgroups of these so-called U-duality groups are
conjectured to be dualities of the full string/M-theory [1, 2]. Concretely,
$\mathrm{O}(7,7)$ is generally enhanced to $\mathrm{O}(8,8)$, which for type
II string theory or M-theory is then further enhanced to E8,8 (a non-compact
form of the largest finite-dimensional exceptional Lie group). The effect of
$\alpha^{\prime}$ corrections on the E8,8 enhancement is difficult to study,
since in type II string theory the corrections start at $\alpha^{\prime\,3}$
with eight derivatives, but luckily in three dimensions there is the smaller
U-duality group $\mathrm{O}(8,8)$ that can be discussed in bosonic and
heterotic string theory whose $\alpha^{\prime}$ corrections start with four
derivatives. (For heterotic string theory the U-duality group is really
$\mathrm{O}(8,24)$ but we truncate the vector fields, which reduces the group
to $\mathrm{O}(8,8)$.)

At the level of the low-energy effective target space actions the T-duality
property manifests itself in dimensional reduction (Kaluza-Klein
compactification on Td and subsequent truncation to massless modes), by
exhibiting a global symmetry under the continuous group
$\mathrm{O}(d,d,\mathbb{R})$.111The enhancement to the continuous group can be
understood as follows: the T-duality group $\mathrm{O}(d,d,\mathbb{Z})$ is
discrete for purely geometrical reasons, because the symmetry transformations
of fields need to be compatible with the periodicity conditions of the torus,
but in dimensional reduction all memory of the torus has disappeared,
explaining the enhancement to the continuous group. In contrast, the discrete
T-duality group $\mathrm{O}(d,d,\mathbb{Z})$ is not visible in the low-energy
effective actions compactified on tori without truncation. This requires a
genuine double field theory [3, 4]. The effective target space theories also
receive an infinite number of higher-derivative corrections governed by the
inverse string tension $\alpha^{\prime}$, and it is known that the
$\alpha^{\prime}$ corrections preserve the continuous
$\mathrm{O}(d,d,\mathbb{R})$ [5, 6] (see ref. [7] for a review). Generalizing
previous work on cosmological reductions to one dimension (cosmic time) [8, 9,
10], we have recently determined the $\mathrm{O}(d,d,\mathbb{R})$ invariant
effective actions to first order in $\alpha^{\prime}$ for general reductions
along $d$ dimensions [11, 12]. We apply this effective action to
compactifications to three dimensions, with the goal to explore the fate of
the duality enhancement to $\mathrm{O}(d+1,d+1)$. (Here $d=7$ for heterotic
string theory and $d=23$ for bosonic string theory, but for our discussion $d$
is really a free parameter, and so we sometimes only speak of
$\mathrm{O}(8,8)$ for the sake of vividness.)

In contrast to T-duality, which is a feature of classical string theory and
preserved by all $\alpha^{\prime}$ corrections, the U-dualities capture
features of the quantum theory. Therefore, one should perhaps not expect
supergravity to exhibit U-duality symmetries beyond zeroth order in
$\alpha^{\prime}$ without also including quantum corrections. A simple
argument based on a scaling symmetry of the two-derivative theory in fact
shows that the continuous U-duality group is not preserved to higher order in
$\alpha^{\prime}$ [13, 14]. Indeed, the Einstein-Hilbert term in string frame,

$I_{\rm EH}=\int\mathrm{d}^{D}x\,\sqrt{g}\,e^{-\phi}R\,,$ (1.1)

has a global $\mathbb{R}\simeq\mathrm{O}(1,1)$ scaling symmetry with constant
parameter $\lambda$, which acts as $g_{\mu\nu}\rightarrow
e^{\lambda}g_{\mu\nu}$ and $\phi\rightarrow\phi+\frac{D-2}{2}\,\lambda$. This
$\mathrm{O}(1,1)$ becomes part of the U-duality group, but a typical higher-
derivative coupling of the form

$\alpha^{\prime}\int\mathrm{d}^{D}x\,\sqrt{g}\,e^{-\phi}R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\,,$
(1.2)

is not invariant, but rather scales with $e^{-\lambda}$, hence breaking the
symmetry. Intriguingly, however, the complete order $\alpha^{\prime}$ action,
including all matter couplings, scales homogeneously. There is hence a formal
scaling invariance if one declares $\alpha^{\prime}$ to scale with
$e^{+\lambda}$. Below we will employ a similar scheme to establish a formal
$\mathrm{O}(8,8)$ invariance.

The above scaling argument avoids the need to actually compute the effective
action in, say, three dimensions, but by itself it is not sufficient to show
that the discrete subgroup is not realized in supergravity, because for the
discrete group the scaling symmetry in fact trivializes, as we will discuss.
We therefore revisit the problem and compute the complete order
$\alpha^{\prime}$ effective action in three dimensions, starting from the
results of ref. [11, 12] and using, perturbatively in $\alpha^{\prime}$, on-
shell transformations to dualize the vector gauge fields into scalar fields.
In the two-derivative theory, this transformation shows the enhancement from
$\mathrm{O}(7,7)$ to $\mathrm{O}(8,8)$ (or E8,8), because the new scalars
organize into a larger coset matrix ${\cal M}_{\cal M\cal N}$, with
$\mathrm{O}(8,8)$ indices ${\cal M},{\cal N}\in\llbracket 1,16\rrbracket$. As
expected, to first order in $\alpha^{\prime}$ one finds that the continuous
$\mathrm{O}(8,8)$ is no longer present, or at least no longer manifest. The
manifest symmetry is $\mathrm{O}(7,7)$ times $14$-dimensional translations
that act as shifts on the scalars originating from dualization.

While $\mathrm{O}(8,8)$ is not a manifest symmetry we provide a formulation
that, as alluded to above, exhibits a formal invariance under this group upon
introducing a non-dynamical compensator. Specifically, for this we can choose
a constant vector $\mbox{\sf u}^{\cal M}$ in the fundamental representation,
in terms of which the effective action takes an $\mathrm{O}(8,8)$ invariant
form. The true theory then arises for a fixed vector $\mbox{\sf u}^{\cal M}$
pointing in a particular direction. Even though it is in principle always
possible to restore a broken symmetry in a formal manner by introducing an
unphysical tensor compensator whose fictitious transformations absorb the
failure of the actual theory to be invariant, this approach may still be
technically useful. For instance, it allows us to systematically determine the
actual symmetry group as the invariance group of $\mbox{\sf u}^{\cal M}$. Here
we indeed find the expected ‘geometric’ subgroup, without any hidden
enhancements. (Strictly speaking, this conclusion applies only to the
continuous case, since in this formalism it is difficult to test for discrete
symmetries without automatically testing for continuous symmetries.)

More importantly, we find an intriguing ‘chiral pattern’, in which only one
chiral projection of $\mbox{\sf u}^{\cal M}$ is needed to write the action. To
explain this, recall that given the $\mathrm{O}(8,8)$ coset matrix ${\cal M}$
and invariant metric $\eta$ one can define the projection operators

$P_{\cal MN}=\frac{1}{2}\left(\eta_{\cal MN}-\mathcal{M}_{\cal
MN}\right)\;,\quad\quad\bar{P}_{\cal MN}=\frac{1}{2}\left(\eta_{\cal
MN}+\mathcal{M}_{\cal MN}\right),$ (1.3)

onto two subspaces of opposite ‘chirality’. The four-derivative action turns
out to be fully determined by an $\mathrm{O}(8,8)$ invariant function ${\cal
F}$ of a 2-tensor and a vector argument, respectively, as follows

$I_{1}=\frac{1}{4}\int\mathrm{d}^{3}x\,\sqrt{-g_{\rm E}}\,\Big{\\{}a\,{\cal
F}[{\cal M},P\mbox{\sf u}]+b\,\left({\cal F}[{\cal M},{P}\mbox{\sf
u}]\right)^{\star}\Big{\\}}\;.$ (1.4)

Here ‘⋆’ denotes a $\mathbb{Z}_{2}$ action, which is implemented on the coset
matrix as ${\cal M}\rightarrow{\cal Z}^{\rm t}{\cal M}{\cal Z}$, where ${\cal
Z}$ obeys ${\cal Z}^{2}={\bf 1}$ but is not an $\mathrm{O}(8,8)$ matrix. Under
this $\mathbb{Z}_{2}$ the projectors (1.3) are interchanged:

$P\rightarrow{\cal Z}^{\rm t}\bar{P}{\cal Z}\;,\qquad\bar{P}\rightarrow{\cal
Z}^{\rm t}{P}{\cal Z}\;.$ (1.5)

The parameters $a,b$ determine the theory: the heterotic action is obtained
for $(a,b)=(-\alpha^{\prime},0)$ and the bosonic action for
$(a,b)=(-\alpha^{\prime},-\alpha^{\prime})$. The $\mathbb{Z}_{2}$ action,
which exchanges $a$ and $b$, has a higher-dimensional analogue, sending the
$B$ field $B\rightarrow-B$, which is a symmetry of the bosonic action but not
of the heterotic action. Since it is the same function ${\cal F}$ that
determines the ‘$\mathbb{Z}_{2}$ dual’ terms in the action, it follows that
the bosonic action can be reconstructed from the heterotic action.

The above parameterization in terms of $(a,b)$, together with the
$\mathbb{Z}_{2}$ action, mimic the structure of double field theory at order
$\alpha^{\prime}$ [15, 16]. The crucial ‘experimental’ observation provided by
our computation is that the compensator $\mbox{\sf u}^{\cal M}$ in eq. (1.4)
appears only in a ‘chiral’ or projected form or, alternatively, that there is
the formal ‘gauge invariance’ under $\mbox{\sf u}\rightarrow\mbox{\sf
u}+\eta\bar{P}\Lambda$. (Note that this comes very close to an actual symmetry
enhancement, with an $\mathrm{O}(8)$ acting only on indices with a ‘barred’
projection, but this viewpoint is not quite consistent as the projectors are
field dependent and hence not compatible with global symmetries.) We do not
have an explanation for this chiral pattern, but it is natural to conjecture
that it holds to higher orders in $\alpha^{\prime}$, hence providing indirect
constraints on the allowed higher-derivative couplings.

As the second main result of our paper we consider a particular massive
deformation, as a window into more general gauged supergravities in presence
of $\alpha^{\prime}$ corrections. The latter would be important in order to
study, for instance, the fate of Kaluza-Klein truncations on spheres in
presence of higher derivatives. The massive deformation we consider is
obtained by integrating out the external $B$ field. In three dimensions its
field strength is on-shell a constant that is usually set to zero in
dimensional reduction. Keeping instead this constant $m$ while integrating out
the $B$ field leads to a massive deformation, which for the two-derivative
theory includes a potential term for the dilaton and a Chern-Simons term for
the Kaluza-Klein vectors $A_{\mu}{}^{M}$ [17]. Including then the order
$\alpha^{\prime}$ corrections one obtains additional couplings, which include
a Chern-Simons term based on composite connections, with the latter
originating from the scalar-dependent Green-Schwarz deformation uncovered in
ref. [11, 12] (and given a worldsheet interpretation in ref. [18]).
Specifically, introducing an $\mathrm{O}(8,8)$ frame field ${\cal V}$, the
compact part of its Maurer-Cartan form ${\cal V}^{-1}\mathrm{d}{\cal V}={\cal
P}+{\cal Q}$ defines composite $\mathrm{O}(8)\times\mathrm{O}(8)$ connections
${\cal Q}$. The topological or Chern-Simons terms of the massive deformation
at order $\alpha^{\prime}$ and order $m$ then read

$I_{(1){\rm top}}=(a+b)\,m\int{\rm tr}\left({\cal Q}\wedge\mathrm{d}{\cal
Q}+\frac{2}{3}{\cal Q}\wedge{\cal Q}\wedge{\cal
Q}\right)-\frac{a-b}{4}\,m\int{\rm
tr}\left(\omega\wedge\mathrm{d}\omega+\frac{2}{3}\omega\wedge\omega\wedge\omega\right)+{\cal
O}(m^{2})\,,$ (1.6)

where $\omega$ denotes the Levi-Civita spin connection (so that for $a-b\neq
0$ this action includes topologically massive gravity as a subsector [19]).
Unexpectedly, the Chern-Simons terms are hence $\mathrm{O}(8,8)$ invariant to
leading order in $m$, although the full theory is not. This is remarkable, for
there are now two parameters expected to break U-duality, $\alpha^{\prime}$
and $m$, yet for the leading Chern-Simons terms $\mathrm{O}(8,8)$ is restored.
Again, we do not know what the physical significance of this observation is,
and it remains to explore more general gaugings.

The remainder of this paper is organized as follows. In sec. 2 we give a short
review of the duality enhancement in three dimensions for the two-derivative
theory, with a particular focus on the scaling symmetries before and after
dimensional reduction, since these feature prominently in the subsequent
discussion of $\alpha^{\prime}$ corrections. In sec. 3 we compute the
effective action in three dimensions to first order in $\alpha^{\prime}$ by
perturbatively dualizing the vector gauge fields into scalars, and we exhibit
the chiral pattern explained above. We then turn in sec. 4 to a massive
deformation, that is obtained by integrating out the external $B$ field, with
a focus on the resulting Chern-Simons terms for composite connections that
exhibit an enhancement to the full U-duality group to first order in the mass
parameter. We conclude with a short outlook in sec. 5, while various
identities and intermediate results are collected in appendices.

## 2 Duality Enhancement in Three Dimensions

In this section we discuss some general aspects of the bosonic and heterotic
string effective actions dimensionally reduced to three spacetime dimensions.
We review the scaling symmetries in higher-dimensions (prior to any
dimensional reduction) and in three dimensions, as a preparation for the
discussion of duality enhancement from $\mathrm{O}(d,d)$ to
$\mathrm{O}(d+1,d+1)$ that is expected to be a feature of string theory in
three dimensions. We close the section with a discussion of field
redefinitions, which are needed once higher-derivative corrections are
included.

### 2.1 Scaling symmetries

The bosonic parts of the bosonic and heterotic string effective actions
coincide at the two-derivative order upon truncating the Yang-Mills gauge
fields of the heterotic theory. They describe the dynamics of a metric
$\hat{g}_{\hat{\mu}\hat{\nu}}$, a two-form $\hat{B}_{\hat{\mu}\hat{\nu}}$ and
a dilaton $\hat{\phi}$ in $D=26$ and $D=10$ dimensions, respectively. In the
string frame, the two-derivative action is given by

$I_{0}^{(D)}=\int\mathrm{d}^{D}X\,\sqrt{-\hat{g}}\,e^{-\hat{\phi}}\left(\hat{R}+\partial_{\hat{\mu}}\hat{\phi}\,\partial^{\hat{\mu}}\hat{\phi}-\frac{1}{12}\,\hat{H}_{\hat{\mu}\hat{\nu}\hat{\rho}}\hat{H}^{\hat{\mu}\hat{\nu}\hat{\rho}}\right),$
(2.1)

with the Ricci scalar $\hat{R}$ and the field-strength
$\hat{H}_{\hat{\mu}\hat{\nu}\hat{\rho}}=3\,\partial_{[\hat{\mu}}\hat{B}_{\hat{\nu}\hat{\rho}]}$.
It features several global scaling symmetries, with group
$\mathbb{R}^{+}\simeq\mathrm{O}(1,1)$ and constant parameter $\lambda$, that
we list in the following.

  * •

Constant dilaton shift:

$\left({\rm
Dilaton}^{D}\right):\qquad\hat{\phi}\rightarrow\hat{\phi}+\lambda,\qquad\hat{g}_{\hat{\mu}\hat{\nu}}\rightarrow
e^{2\lambda/(D-2)}\,\hat{g}_{\hat{\mu}\hat{\nu}},\qquad\hat{B}_{\hat{\mu}\hat{\nu}}\rightarrow
e^{2\lambda/(D-2)}\,\hat{B}_{\hat{\mu}\hat{\nu}}.$ (2.2)

  * •

On-shell “trombone” symmetry:

$\left({\rm
Trombone}^{D}\right):\qquad\hat{\phi}\rightarrow\hat{\phi},\qquad\hat{g}_{\hat{\mu}\hat{\nu}}\rightarrow
e^{2\lambda}\,\hat{g}_{\hat{\mu}\hat{\nu}},\qquad\hat{B}_{\hat{\mu}\hat{\nu}}\rightarrow
e^{2\lambda}\,\hat{B}_{\hat{\mu}\hat{\nu}},$ (2.3)

which leaves invariant the equations of motion but rescales uniformly the
action.

  * •

Scaling of the internal volume: as we are interested in dimensional reductions
down to three dimensions, we consider a splitting of the $D$-dimensional
coordinates $X^{\hat{\mu}}$ into $\\{x^{\mu},y^{m}\\}$, with $\mu\in\llbracket
1,3\rrbracket$ and $m\in\llbracket 1,D-3\rrbracket$ and decompose the fields
as in ref. [20]:

$\displaystyle\hat{g}_{\hat{\mu}\hat{\nu}}$
$\displaystyle=\begin{pmatrix}g_{\mu\nu}+A_{\mu}^{(1)\,p}G_{pq}\,A_{\nu}^{(1)\,q}&A_{\mu}^{(1)\,p}G_{pn}\\\\[2.15277pt]
G_{mp}A_{\nu}^{(1)\,p}&G_{mn}\end{pmatrix},$ (2.4a)
$\displaystyle\hat{B}_{\hat{\mu}\hat{\nu}}$
$\displaystyle=\begin{pmatrix}B_{\mu\nu}-A_{[\mu}^{(1)\,m}A_{\nu]\,m}^{(2)}+A_{\mu}^{(1)\,m}\,B_{mn}\,A_{\nu}^{(1)\,n}&A^{(2)}_{\mu\,n}-B_{np}\,A_{\mu}^{(1)\,p}\\\
-A^{(2)}_{\nu\,m}+B_{mp}\,A_{\nu}^{(1)\,p}&B_{mn}\end{pmatrix},$ (2.4b)
$\displaystyle e^{\hat{\phi}}$
$\displaystyle=\sqrt{\det\left(G_{mn}\right)}\,e^{\Phi}.$ (2.4c)

With this decomposition, and keeping the dependence on all coordinates, the
action (2.1) features the additional scaling symmetry

$\left({\rm Volume}^{D-3}\right):\qquad\begin{cases}y^{m}\rightarrow
e^{-\lambda}\,y^{m},\\\ \Phi\rightarrow\Phi+(3-D)\lambda,\\\
g_{\mu\nu}\rightarrow g_{\mu\nu},\\\ B_{\mu\nu}\rightarrow B_{\mu\nu},\\\
\end{cases}\qquad\begin{cases}G_{mn}\rightarrow e^{2\lambda}\,G_{mn},\\\
B_{mn}\rightarrow e^{2\lambda}\,B_{mn},\\\ A_{\mu}^{(1)\,m}\rightarrow
e^{-\lambda}\,A_{\mu}^{(1)\,m},\\\ A_{\mu\,m}^{(2)}\rightarrow
e^{\lambda}\,A_{\mu\,m}^{(2)},\\\ \end{cases}$ (2.5)

corresponding to the ${\rm GL}(1)$ subgroup of the ${\rm GL}(D-3)$ action on
the internal coordinates.

These three scaling symmetries are summarized, in the three-dimensional
variables of eq. (2.4), in tab. 1. These scaling symmetries are at the origin
of three-dimensional symmetries essential to the duality enhancement.

| $e^{\Phi}$ | $g_{{\rm E}\,\mu\nu}$ | $B_{\mu\nu}$ | $G_{mn}$ | $B_{mn}$ | $A_{\mu}^{(1)\,m}$ | $A_{\mu\,m}^{(2)}$
---|---|---|---|---|---|---|---
$\left({\rm Dilaton}^{D}\right)$ | $1/(D-2)$ | $0$ | $2/(D-2)$ | $2/(D-2)$ | $2/(D-2)$ | $0$ | $2/(D-2)$
$\left({\rm Trombone}^{D}\right)$ | $3-D$ | $2D-4$ | $2$ | $2$ | $2$ | $0$ | $2$
$\left({\rm Volume}^{D-3}\right)$ | $3-D$ | $2D-6$ | $0$ | $2$ | $2$ | $-1$ | $1$
Tab. 1: Scaling behaviour of the fields (2.4) under the transformations (2.2),
(2.3) and (2.5). We display the charges $q$ of each field $\varphi$,
representing the transformation $\varphi\rightarrow e^{q\lambda}\,\varphi$.
For convenience, we have indicated the shifted dilaton $\Phi$ and Einstein
frame metric $g_{{\rm E}\,\mu\nu}=e^{-2\Phi}\,g_{\mu\nu}$ rather than the
dilaton $\hat{\phi}$ and the string frame metric $g_{\mu\nu}$.

### Scaling symmetries in three dimensions

We now consider the three-dimensional theory which follows from toroidal
compactification of the action (2.1) on ${\rm T}^{D-3}$ and a subsequent
truncation to the zero-modes. Using the parametrization (2.4) this amounts to
neglecting the dependence on the internal coordinates $y^{m}$. We furthermore
move to the Einstein frame by rescaling the metric, $g_{\mu\nu}\rightarrow
g_{{\rm E}\,\mu\nu}=e^{-2\Phi}g_{\mu\nu}$, which yields the action [20]

$\displaystyle I_{0}=$ $\displaystyle\int\mathrm{d}^{3}x\,\sqrt{-g_{\rm
E}}\,\bigg{(}R_{\rm
E}-\partial_{\mu}\Phi\,\partial^{\mu}\Phi-\frac{e^{-4\Phi}}{12}\,H_{\mu\nu\rho}H^{\mu\nu\rho}+\frac{1}{4}\,\mathrm{Tr}\left(\partial_{\mu}G\partial^{\mu}G^{-1}\right)$
(2.6)
$\displaystyle+\frac{1}{4}\mathrm{Tr}\left(G^{-1}\partial_{\mu}BG^{-1}\partial^{\mu}B\right)-\frac{e^{-2\Phi}}{4}\,F_{\mu\nu}^{(1)\,m}G_{mn}F^{(1)\,\mu\nu\,n}-\frac{e^{-2\Phi}}{4}\,H_{\mu\nu
m}G^{mn}{H^{\mu\nu}}_{n}\bigg{)}\,,$

with

$\begin{split}H_{\mu\nu\rho}&=3\,\partial_{[\mu}B_{\nu\rho]}-(3/2)\left(A^{(1)\,m}_{[\mu}F_{\nu\rho]\,m}^{(2)}+F_{[\mu\nu}^{(1)\,m}A_{\rho]\,m}^{(2)}\right)\,,\\\
H_{\mu\nu m}&=F_{\mu\nu\,m}^{(2)}-B_{mn}F_{\mu\nu}^{(1)\,n}\,,\end{split}$
(2.7)

where
$F_{\mu\nu}^{(1)\,m}=\partial_{\mu}A_{\nu}^{(1)\,m}-\partial_{\nu}A_{\mu}^{(1)\,m}$
is the abelian field strength for $A_{\mu}^{(1)\,m}$, and similarly for
$A_{\mu\,m}^{(2)}$. This action is invariant under the following scaling
symmetries:

  * •

Constant dilaton shift:

$\left({\rm
Dilaton}^{3}\right):\qquad\begin{cases}\Phi\rightarrow\Phi+\lambda,\\\ g_{{\rm
E}\,\mu\nu}\rightarrow g_{{\rm E}\,\mu\nu},\\\ B_{\mu\nu}\rightarrow
e^{2\lambda}\,B_{\mu\nu},\\\ \end{cases}\quad\begin{cases}G_{mn}\rightarrow
G_{mn},\\\ B_{mn}\rightarrow B_{mn},\\\ A_{\mu}^{(1)\,m}\rightarrow
e^{\lambda}\,A_{\mu}^{(1)\,m},\\\ A_{\mu\,m}^{(2)}\rightarrow
e^{\lambda}\,A_{\mu\,m}^{(2)}.\\\ \end{cases}$ (2.8)

  * •

On-shell “trombone” symmetry:

$\left({\rm Trombone}^{3}\right):\qquad\begin{cases}\Phi\rightarrow\Phi,\\\
g_{{\rm E}\,\mu\nu}\rightarrow e^{2\lambda}\,g_{{\rm E}\,\mu\nu},\\\
B_{\mu\nu}\rightarrow e^{2\lambda}\,B_{\mu\nu},\\\
\end{cases}\quad\begin{cases}G_{mn}\rightarrow G_{mn},\\\ B_{mn}\rightarrow
B_{mn},\\\ A_{\mu}^{(1)\,m}\rightarrow e^{\lambda}\,A_{\mu}^{(1)\,m},\\\
A_{\mu\,m}^{(2)}\rightarrow e^{\lambda}\,A_{\mu\,m}^{(2)}.\\\ \end{cases}$
(2.9)

  * •

Internal rescaling:

$\left(\text{T-duality}\right):\qquad\begin{cases}\Phi\rightarrow\Phi,\\\
g_{{\rm E}\,\mu\nu}\rightarrow g_{{\rm E}\,\mu\nu},\\\ B_{\mu\nu}\rightarrow
B_{\mu\nu},\\\ \end{cases}\qquad\begin{cases}G_{mn}\rightarrow
e^{2\lambda}\,G_{mn},\\\ B_{mn}\rightarrow e^{2\lambda}\,B_{mn},\\\
A_{\mu}^{(1)\,m}\rightarrow e^{-\lambda}\,A_{\mu}^{(1)\,m},\\\
A_{\mu\,m}^{(2)}\rightarrow e^{\lambda}\,A_{\mu\,m}^{(2)},\\\ \end{cases}$
(2.10)

which corresponds to the $\mathrm{O}(1,1)$ subgroup of the T-duality group
$\mathrm{O}(D-3,D-3)$.

Tab. 2 summarizes these symmetries. They do not directly arise from the
reduction of the higher-dimensional scaling symmetries of sec. 2.1. Rather,
the scaling symmetries in three dimensions originate from the mixing of the
higher-dimensional ones:

$\displaystyle\left({\rm Dilaton}^{3}\right)$ $\displaystyle=\left({\rm
Dilaton}^{D}\right)+\frac{D-3}{D-2}\,\left({\rm
Trombone}^{D}\right)-\left({\rm Volume}^{D-3}\right),$ (2.11)
$\displaystyle\left({\rm Trombone}^{3}\right)$ $\displaystyle=\left({\rm
Trombone}^{D}\right)-\left({\rm Volume}^{D-3}\right),$
$\displaystyle\left(\text{T-duality}\right)$ $\displaystyle=(D-3)\,\left({\rm
Dilaton}^{D}\right)-\frac{D-3}{D-2}\,\left({\rm
Trombone}^{D}\right)+\left({\rm Volume}^{D-3}\right).$

| $e^{\Phi}$ | $g_{{\rm E}\,\mu\nu}$ | $B_{\mu\nu}$ | $G_{mn}$ | $B_{mn}$ | $A_{\mu}^{(1)\,m}$ | $A_{\mu\,m}^{(2)}$
---|---|---|---|---|---|---|---
$\left({\rm Dilaton}^{3}\right)$ | $1$ | $0$ | $2$ | $0$ | $0$ | $1$ | $1$
$\left({\rm Trombone}^{3}\right)$ | $0$ | $2$ | $2$ | $0$ | $0$ | $1$ | $1$
$\left(\text{T-duality}\right)$ | $0$ | $0$ | $0$ | $2$ | $2$ | $-1$ | $1$
Tab. 2: Scaling behaviour of the fields (2.4) under the transformations
(2.8)–(2.10). We display the charges $q$ of each field $\varphi$, representing
the transformation $\varphi\rightarrow e^{q\lambda}\varphi$.

### 2.2 $\mathrm{O}(d+1,d+1)$ enhancement

#### T duality and $\mathrm{O}(d,d)$

As already mentioned, the scaling symmetry (2.10) is part of the bigger
T-duality symmetry group $\mathrm{O}(D-3,D-3)=\mathrm{O}(d,d)$ (with $d=23$
and $d=7$ in the bosonic and heterotic cases, respectively). The invariance
under $\mathrm{O}(d,d)$ is best displayed upon packaging the $d^{2}$ scalar
fields $G_{mn}$ and $B_{mn}$ into the $\mathrm{O}(d,d)$ matrix

${\cal
H}_{MN}=\begin{pmatrix}G_{mn}-B_{mp}G^{pq}B_{qn}&B_{mp}G^{pn}\\\\[2.15277pt]
-G^{mp}B_{pn}&G^{mn}\end{pmatrix},$ (2.12)

parametrizing the coset space
$\mathrm{O}(d,d)/\left(\mathrm{O}(d)\times\mathrm{O}(d)\right)$, and
regrouping the vector fields $A^{(1)\,m}_{\mu}$ and $A^{(2)}_{\mu\,m}$ into a
single $\mathrm{O}(d,d)$ vector

${\cal A}_{\mu}{}^{M}=\begin{pmatrix}A_{\mu}^{(1)\,m}\\\
A_{\mu\,m}^{(2)}\end{pmatrix}.$ (2.13)

The action (2.6) then takes the form [20]

$\begin{split}I_{0}=\int\mathrm{d}^{3}x\,\sqrt{-g_{\rm E}}\Big{(}&R_{\rm
E}-\partial_{\mu}\Phi\,\partial^{\mu}\Phi+\frac{1}{8}\,\mathrm{Tr}\left(\partial_{\mu}\mathcal{H}\,\partial^{\mu}\mathcal{H}^{-1}\right)\\\
&-\frac{1}{12}e^{-4\Phi}\,H_{\mu\nu\rho}H^{\mu\nu\rho}-\frac{1}{4}\,e^{-2\Phi}\,\mathcal{F}_{\mu\nu}{}^{M}\mathcal{H}_{MN}\mathcal{F}^{\mu\nu\,N}\Big{)}\,,\end{split}$
(2.14)

with the field-strengths
$\mathcal{F}_{\mu\nu}{}^{M}=2\,\partial_{[\mu}\mathcal{A}_{\nu]}{}^{M}$ and
$H_{\mu\nu\rho}=3\,\partial_{[\mu}B_{\nu\rho]}-(3/2)\,\mathcal{A}_{[\mu}{}^{M}\mathcal{F}_{\nu\rho]\,M}$.
We choose the metric signature to be $(-1,1,1)$. The $\mathrm{O}(d,d)$ indices
are raised and lowered using the $\mathrm{O}(d,d)$-invariant metric

$\eta^{MN}=\begin{pmatrix}0&{\delta^{m}}_{n}\\\
{\delta_{m}}^{n}&0\end{pmatrix}.$ (2.15)

In particular, $\mathcal{H}^{MN}=\eta^{MP}\mathcal{H}_{PQ}\eta^{QN}$ is the
inverse of the generalised metric $\mathcal{H}_{MN}$.

The action (2.14) is invariant under the $\mathrm{O}(d,d)$ transformation

$g_{{\rm E}\,\mu\nu}\rightarrow g_{{\rm
E}\,\mu\nu},\quad\Phi\rightarrow\Phi,\quad\mathcal{H}_{MN}\rightarrow
L_{M}{}^{P}L_{N}{}^{Q}\mathcal{H}_{PQ},\quad\mathcal{A}_{\mu}{}^{M}\rightarrow
L^{M}{}_{N}\mathcal{A}_{\mu}{}^{N},\quad B_{\mu\nu}\rightarrow B_{\mu\nu},$
(2.16)

with $L_{M}{}^{N}\in\mathrm{O}(d,d)$, i.e.
$L_{M}{}^{P}L_{N}{}^{Q}\eta_{PQ}=\eta_{MN}$. In three dimensions the three-
form field-strength $H_{\mu\nu\rho}$ is on-shell determined by a constant,
which we set to zero for now. In sec. 4 we will explore the massive
deformations arising for a non-vanishing three-form.

#### From $\mathrm{O}(d,d)$ to $\mathrm{O}(d+1,d+1)$

The action (2.14) hides a symmetry enhancement from $\mathrm{O}(d,d)$ to
$\mathrm{O}(d+1,d+1)$, thanks to the duality between vector and scalar fields
in three dimensions [1]. Contrary to T-duality, this enhanced symmetry does
not leave the dilaton invariant: it combines in particular the
$\mathrm{O}(1,1)$ scaling symmetry (2.8) to the $\mathrm{O}(d,d)$ symmetry
(2.16) as $\mathrm{O}(d,d)\times\mathrm{O}(1,1)\subset\mathrm{O}(d+1,d+1)$.
The enhancement is made manifest by dualising the 2-form field-strengths
$\mathcal{F}_{\mu\nu}{}^{M}$ into gradients $\partial_{\mu}\xi_{M}$ of scalar
fields through the introduction of a Lagrange multiplier term in the
action:222Here and in the following $\varepsilon_{\mu\nu\rho}$ denotes the
Levi-Civita symbol, and $\epsilon_{\mu\nu\rho}$ is the associated tensor.

$\widetilde{I}_{0}=I_{0}+\int\mathrm{d}^{3}x\,\frac{1}{2}\,\varepsilon^{\mu\nu\rho}\mathcal{F}_{\mu\nu}{}^{M}\partial_{\rho}\xi_{M}\,.$
(2.17)

The equations of motion of $\xi_{M}$ give the Bianchi identity for
$\mathcal{F}_{\mu\nu}{}^{M}$. There is therefore no need for the vector fields
${\cal A}_{\mu}{}^{M}$, and we can consider $\mathcal{F}_{\mu\nu}{}^{M}$ as
independent fields. Their equations of motion, given by

$\mathcal{F}_{\mu\nu}{}^{M}=e^{2\Phi}\,\epsilon_{\mu\nu\rho}\,\partial^{\rho}\xi_{N}\mathcal{H}^{NM},$
(2.18)

are algebraic: we can eliminate the 2-forms $\mathcal{F}_{\mu\nu}{}^{M}$ from
the action in favour of the scalars $\xi_{M}$. The action then reeds

$\widetilde{I}_{0}=\int\mathrm{d}^{3}x\,\sqrt{-g_{\rm E}}\left(R_{\rm
E}-\partial_{\mu}\Phi\,\partial^{\mu}\Phi+\frac{1}{8}\,\mathrm{Tr}\left(\partial_{\mu}\mathcal{H}\,\partial^{\mu}\mathcal{H}^{-1}\right)-\frac{1}{2}\,e^{2\Phi}\,\partial_{\mu}\xi_{M}\mathcal{H}^{MN}\partial^{\mu}\xi_{N}\right).$
(2.19)

The $\mathrm{O}(1,1)$ and $\mathrm{O}(d,d)$ transformations (2.8) and (2.16)
of $\mathcal{A}_{\mu}{}^{M}$ imply, by use of eq. (2.18), the following
transformations of $\xi_{M}$:

$\mathrm{O}(1,1):\ \xi_{M}\rightarrow
e^{-\lambda}\,\xi_{M},\qquad\mathrm{O}(d,d):\ \xi_{M}\rightarrow
L_{M}{}^{N}\xi_{N}.$ (2.20)

The action $\widetilde{I}_{0}$ depends on the metric and on
$1+d^{2}+2d=(d+1)^{2}$ scalar fields. The scalar fields can be organized into
the $\mathrm{O}(d+1,d+1)$ matrix

$\mathcal{M}_{\cal
MN}=\begin{pmatrix}\mathcal{H}_{MN}+e^{2\Phi}\xi_{M}\xi_{N}&e^{2\Phi}\xi_{M}&-\mathcal{H}_{MP}\xi^{P}-\dfrac{1}{2}e^{2\Phi}\xi_{M}\,\xi_{P}\xi^{P}\\\
e^{2\Phi}\xi_{N}&e^{2\Phi}&-\dfrac{1}{2}e^{2\Phi}\xi_{P}\xi^{P}\\\
-\mathcal{H}_{NP}\xi^{P}-\dfrac{1}{2}e^{2\Phi}\xi_{N}\,\xi_{P}\xi^{P}&-\dfrac{1}{2}e^{2\Phi}\xi_{P}\xi^{P}&e^{-2\Phi}+\xi_{P}\mathcal{H}^{PQ}\xi_{Q}+\dfrac{1}{4}e^{2\Phi}\left(\xi_{P}\xi^{P}\right)^{2}\end{pmatrix}\,.$
(2.21)

Here the $\mathrm{O}(d+1,d+1)$ indices are split as $\mathcal{M}=\\{M,+,-\\}$
with respect to $\mathrm{O}(d,d)\times\mathrm{O}(1,1)$, while the
$\mathrm{O}(d+1,d+1)$-invariant metric takes the form

$\eta_{\cal MN}=\begin{pmatrix}\eta_{MN}&0&0\\\ 0&0&1\\\ 0&1&0\end{pmatrix}.$
(2.22)

Then, the action (2.19) becomes

$\widetilde{I}_{0}=\int\mathrm{d}^{3}x\,\sqrt{-g_{\rm E}}\left(R_{\rm
E}+\frac{1}{8}\,\mathrm{Tr}\left(\partial_{\mu}\mathcal{M}\partial^{\mu}\mathcal{M}^{-1}\right)\right).$
(2.23)

It is manifestly invariant under the $\mathrm{O}(d+1,d+1)$ transformation

$g_{{\rm E}\,\mu\nu}\rightarrow g_{{\rm E}\,\mu\nu},\quad\mathcal{M}_{\cal
MN}\rightarrow L_{\cal M}{}^{\cal P}L_{\cal N}{}^{\cal Q}\mathcal{M}_{\cal
PQ},$ (2.24)

with $L_{\cal M}{}^{\cal N}\in\mathrm{O}(d+1,d+1)$.

####  $\mathrm{O}(d+1,d+1)$ transformations

Let us have a closer look at the $\mathrm{O}(d+1,d+1)$ symmetry. The
generators $T^{\cal MN}$ of $\mathrm{O}(d+1,d+1)$ can be decomposed into
$\mathrm{O}(d,d)$ components as

$T^{\cal MN}=\left\\{T^{MN},T^{M+},T^{M-},T^{+-}\right\\}.$ (2.25)

The $T^{MN}$ generate the $\mathrm{O}(d,d)$ transformation

$g_{{\rm E}\,\mu\nu}\rightarrow g_{{\rm
E}\,\mu\nu},\quad\Phi\rightarrow\Phi,\quad\mathcal{H}_{MN}\rightarrow
L_{M}{}^{P}L_{N}{}^{Q}\mathcal{H}_{PQ},\quad\xi_{M}\rightarrow
L_{M}{}^{N}\xi_{N},$ (2.26)

while $T^{+-}$ generates the $\mathrm{O}(1,1)$ scaling symmetry

$g_{{\rm E}\,\mu\nu}\rightarrow g_{{\rm
E}\,\mu\nu},\quad\Phi\rightarrow\Phi+\lambda,\quad\mathcal{H}_{MN}\rightarrow\mathcal{H}_{MN},\quad\xi_{M}\rightarrow
e^{-\lambda}\,\xi_{M}.$ (2.27)

The charged generators $T^{M+}$ generate the constant shifts

$\xi_{M}\rightarrow\xi_{M}+c_{M}$ (2.28)

and the components $T^{M-}$ lead to complicated non-linear transformations.

#### Frame formalism

For later convenience, let us define the frame fields

${\cal V}_{\cal M}{}^{\cal A}=\begin{pmatrix}E_{M}{}^{A}&e^{\Phi}\xi_{M}&0\\\
0&e^{\Phi}&0\\\
-\xi^{P}E_{P}{}^{A}&-\dfrac{1}{2}e^{\Phi}\xi_{P}\xi^{P}&e^{-\Phi}\end{pmatrix},$
(2.29)

so that $\mathcal{M}_{\cal MN}={\cal V}_{\cal M}{}^{\cal A}\delta_{\cal
AB}{\cal V}_{\cal N}{}^{\cal B}$, with the $(2d+2)\times(2d+2)$ identity
matrix $\delta_{\cal AB}$. $E_{M}{}^{A}$ is the frame field associated to the
$\mathrm{O}(d,d)$ generalized metric, i.e.
$\mathcal{H}_{MN}=E_{M}{}^{A}\delta_{AB}E_{N}{}^{B}$, and we denote the
inverse by $E_{A}{}^{M}$. As for the ‘curved’ $\mathrm{O}(d+1,d+1)$ indices
$\mathcal{M}$, we split the flat indices as ${\cal
A}=\\{A,\hat{+},\hat{-}\\}$. These flat indices are raised and lowered using
the flat version of the invariant tensor (2.22):

$\eta_{\cal AB}={\cal V}_{\cal A}{}^{\cal M}\eta_{\cal MN}{\cal V}_{\cal
B}{}^{\cal N}=\begin{pmatrix}\eta_{AB}&0&0\\\ 0&0&1\\\ 0&1&0\end{pmatrix},$
(2.30)

with ${\cal V}_{\cal A}{}^{\cal M}$ the inverse frame field and
$\eta_{AB}=E_{A}{}^{M}\eta_{MN}E_{B}{}^{N}$. The frame fields can be used to
define the Maurer-Cartan form $\left({\cal V}^{-1}\partial_{\mu}{\cal
V}\right)_{\cal A}{}^{\cal B}={\cal P}_{\mu\,{\cal A}}{}^{B}+{\cal
Q}_{\mu\,{\cal A}}{}^{B}$, where

$\displaystyle\mathcal{P}_{\mu\,{\cal A}}{}^{\cal B}$
$\displaystyle=\begin{pmatrix}P_{\mu\,A}{}^{B}&\dfrac{1}{2}e^{\Phi}E_{A}{}^{M}\partial_{\mu}\xi_{M}&-\dfrac{1}{2}e^{\Phi}\partial_{\mu}\xi^{M}E_{M}{}^{C}\delta_{CA}\\\
\dfrac{1}{2}e^{\Phi}\partial_{\mu}\xi_{M}E_{C}{}^{M}\delta^{CB}&\partial_{\mu}\Phi&0\\\
-\dfrac{1}{2}e^{\Phi}\partial_{\mu}\xi^{M}E_{M}{}^{B}&0&-\partial_{\mu}\Phi\end{pmatrix},$
(2.31) $\displaystyle{\cal Q}_{\mu\,{\cal A}}{}^{\cal B}$
$\displaystyle=\begin{pmatrix}Q_{\mu\,A}{}^{B}&\dfrac{1}{2}e^{\Phi}E_{A}{}^{M}\partial_{\mu}\xi_{M}&\dfrac{1}{2}e^{\Phi}\partial_{\mu}\xi^{M}E_{M}{}^{C}\delta_{CA}\\\
-\dfrac{1}{2}e^{\Phi}\partial_{\mu}\xi_{M}E_{C}{}^{M}\delta^{CB}&0&0\\\
-\dfrac{1}{2}e^{\Phi}\partial_{\mu}\xi^{M}E_{M}{}^{B}&0&0\end{pmatrix},$
(2.32)

such that $\left(\mathcal{P}_{\mu}\delta\right)_{\cal AB}$ and $\left({\cal
Q}_{\mu}\delta\right)_{\cal AB}$ are symmetric and antisymmetric,
respectively. Sometimes it is also convenient to use the basis in which
$\eta_{\cal AB}$ is diagonal and has the form

$\eta_{\cal AB}=\begin{pmatrix}-\delta_{ab}&0\\\
0&\delta_{\bar{a}\bar{b}}\end{pmatrix},$ (2.33)

with the index split ${\cal A}\longrightarrow\\{a,\bar{a}\\},\
a,\bar{a}\in\llbracket 0,d\rrbracket$. In this basis, the Maurer-Cartan
components ${\cal P}_{\mu}{}^{\cal AB}$ can be expressed in terms of an
$\mathrm{O}(d+1)$ bivector ${\mathsf{P}}_{\mu}{}^{a\bar{a}}$:

${\cal P}_{\mu}{}^{\cal AB}=\begin{pmatrix}0&\mathsf{P}_{\mu}{}^{a\bar{b}}\\\
\mathsf{P}_{\mu}^{{\rm t}}{}^{\bar{a}b}&0\end{pmatrix}.$ (2.34)

### 2.3 Field redefinitions

The presence of higher-derivative corrections makes it possible to perform
field redefinitions that are perturbative in $\alpha^{\prime}$, and previous
works showed that they are necessary to exhibit duality symmetries [8]. We
denote by $\widetilde{I}_{1}$ the part of the action of order
$\alpha^{\prime}$ (after dualisation of the vector fields), so that
$\widetilde{I}=\widetilde{I}_{0}+\widetilde{I}_{1}+{\cal
O}\left(\alpha^{\prime 2}\right)$ is the total action. In the same manner as
in ref. [9, 10], we consider field redefinitions of the form

$\varphi\rightarrow\varphi+\alpha^{\prime}\,\delta\varphi,$ (2.35)

where $\varphi$ denotes a generic field. Under such redefinitions, the
variation of $\widetilde{I}_{0}$ is

$\delta\widetilde{I}_{0}=\alpha^{\prime}\,\int\mathrm{d}^{3}x\,\sqrt{-g_{\rm
E}}\bigg{[}\delta\Phi\,E_{\Phi}+\delta g^{\mu\nu}_{\rm
E}E_{g\,\mu\nu}+\mathrm{Tr}\left(\delta\mathcal{H}^{-1}\,E_{\cal
H}\right)+E_{\xi}{}^{M}\delta\xi_{M}\bigg{]},$ (2.36)

where333Note that, as $\mathcal{H}\eta\mathcal{H}=\eta$, $\mathcal{H}$ is a
constrained field and that the derivation of $E_{\mathcal{H}}$ by variation of
the action must be done carefully [10].

$\displaystyle E_{\Phi}=2\,\Box\Phi-e^{2\Phi}\,\partial_{\mu}\xi_{M}{\cal
H}^{MN}\partial^{\mu}\xi_{N},$ (2.37a) $\displaystyle E_{g\,\mu\nu}=R_{{\rm
E}\,\mu\nu}-\partial_{\mu}\Phi\partial_{\nu}\Phi+\frac{1}{8}\,\mathrm{Tr}\left(\partial_{\mu}\mathcal{H}\,\partial_{\nu}\mathcal{H}^{-1}\right)-\frac{1}{2}\,e^{2\Phi}\,\partial_{\mu}\xi_{M}{\cal
H}^{MN}\partial_{\nu}\xi_{N}$ $\displaystyle\qquad\quad-\frac{1}{2}g_{{\rm
E}\,\mu\nu}\left(R_{\rm
E}-\partial_{\rho}\Phi\,\partial^{\rho}\Phi+\frac{1}{8}\,\mathrm{Tr}\left(\partial_{\rho}\mathcal{H}\,\partial^{\rho}\mathcal{H}^{-1}\right)-\frac{1}{2}\,e^{2\Phi}\,\partial_{\rho}\xi_{M}{\cal
H}^{MN}\partial^{\rho}\xi_{N}\right),$ (2.37b) $\displaystyle E_{{\cal
H}\,MN}=-\frac{1}{4}\,\bigg{[}\Box\,\mathcal{H}_{MN}+\left(\mathcal{H}\partial_{\mu}\mathcal{H}^{-1}\,\partial^{\mu}\mathcal{H}\right)_{MN}+e^{2\Phi}\,\partial_{\mu}\xi_{M}\partial^{\mu}\xi_{N}$
$\displaystyle\qquad\qquad-e^{2\Phi}\,\mathcal{H}_{MP}\partial_{\mu}\xi^{P}\partial^{\mu}\xi^{Q}\mathcal{H}_{QN}\bigg{]},$
(2.37c) $\displaystyle E_{\xi}{}^{M}=e^{2\Phi}\,\left(\Box\xi_{N}{\cal
H}^{NM}+2\,\partial_{\mu}\Phi\partial^{\mu}\xi_{N}{\cal
H}^{NM}+\partial_{\mu}\xi_{N}\partial^{\mu}{\cal H}^{NM}\right)\,,$ (2.37d)

and $\Box=\nabla_{\mu}\nabla^{\mu}$. As we will not consider orders in
$\alpha^{\prime}$ higher than one, there is no need to compute how the
redefinition affects corrections $\widetilde{I}_{1}$ to $\widetilde{I}_{0}$ of
order ${\cal O}(\alpha^{\prime})$: this variation will generate ${\cal
O}(\alpha^{\prime 2})$ terms. The expressions of the shifts $\delta\varphi$
can then be chosen to cancel given terms in $\widetilde{I}_{1}$. In the
following, we will use these redefinitions to cancel terms that contain as
factors the leading two-derivative contributions from the field equations, as
was done in ref. [12]. These factors can be replaced as follows:

$\displaystyle\Box\Phi$ $\displaystyle\longrightarrow
Q_{\Phi}=\frac{1}{2}\,e^{2\Phi}\,\partial_{\mu}\xi_{M}{\cal
H}^{MN}\partial^{\mu}\xi_{N},$ (2.38) $\displaystyle R_{{\rm E}\,\mu\nu}$
$\displaystyle\longrightarrow
Q_{g\,\mu\nu}=\partial_{\mu}\Phi\partial_{\nu}\Phi-\frac{1}{8}\,\mathrm{Tr}\left(\partial_{\mu}\mathcal{H}\,\partial_{\nu}\mathcal{H}^{-1}\right)+\frac{1}{2}\,e^{2\Phi}\,\partial_{\mu}\xi_{M}{\cal
H}^{MN}\partial_{\nu}\xi_{N},$ $\displaystyle\Box\,\mathcal{H}_{MN}$
$\displaystyle\longrightarrow
Q_{\mathcal{H}\,MN}=-\left(\mathcal{H}\,\partial_{\mu}\mathcal{H}^{-1}\partial^{\mu}\mathcal{H}\right)_{MN}-e^{2\Phi}\,\partial_{\mu}\xi_{M}\partial^{\mu}\xi_{N}+e^{2\Phi}\,\mathcal{H}_{MP}\partial_{\mu}\xi^{P}\partial^{\mu}\xi^{Q}\mathcal{H}_{QN},$
$\displaystyle\Box\xi_{M}$ $\displaystyle\longrightarrow
Q_{\xi\,M}=-2\,\partial_{\mu}\Phi\partial^{\mu}\xi_{M}-\partial_{\mu}\xi_{N}\left(\partial^{\mu}{\cal
H}^{-1}{\cal H}\right)^{N}{}_{M}.$

These replacements and the associated field redefinitions are summed up in
tab. 3.

Term in the action | Field redefinitions | Replacement
---|---|---
$\alpha^{\prime}\,X\Box\Phi$ | $\delta\Phi=-\dfrac{1}{2}\,X$ | $\alpha^{\prime}\,XQ_{\Phi}$
$\alpha^{\prime}\,X^{\mu\nu}R_{{\rm E}\,\mu\nu}$ | $\delta g_{\rm E}^{\mu\nu}=-X^{(\mu\nu)}+\,g_{\rm E}^{\mu\nu}\,X_{\rho}{}^{\rho}$ | $\alpha^{\prime}\,X^{\mu\nu}Q_{g\,\mu\nu}$
$\alpha^{\prime}\,\mathrm{Tr}\left(X\Box\mathcal{H}\right)$ | $\delta\mathcal{H}^{MN}=4\,X^{MN}$ | $\alpha^{\prime}\,\mathrm{Tr}\left(XQ_{\mathcal{H}}\right)$
$\alpha^{\prime}\,X^{M}\Box\xi_{M}$ | $\delta\xi_{M}=-e^{-2\Phi}{\cal H}_{MN}X^{N}$ | $\alpha^{\prime}\,X^{M}Q_{\xi\,M}$
Tab. 3: Replacement rules for the terms carrying the leading two-derivative
contribution from the field equations descending from the two-derivative
action (2.19) and associated field redefinitions. The explicit replacement
rules are given in eq. (2.38).

## 3 Four-derivative Action in Three Dimensions

We now revisit the symmetry enhancement of sec. 2 in the presence of first
order $\alpha^{\prime}$ corrections. To this end, we start from the manifestly
$\mathrm{O}(d,d)$ invariant four-derivative action of ref. [12] and perform
the dualization of the vector fields as in eq. (2.18) above. We then express
the resulting action in terms of $\mathrm{O}(d+1,d+1)$ quantities upon
introduction of a non-dynamical compensator which reveals an interesting
structure of the action. Higher order corrections obstruct the
$\mathrm{O}(d+1,d+1)$ symmetry enhancement of the two-derivative action as is
most straightforwardly seen by tracking the fate of the scaling symmetries
discussed in sec. 2. Typical four-derivative corrections to the action (2.1)
are of the form [21]

$I_{1}\propto\alpha^{\prime}\,\int\mathrm{d}^{D}X\,\sqrt{-\hat{g}}\,e^{-\hat{\phi}}\left(\hat{R}_{\hat{\mu}\hat{\nu}\hat{\rho}\hat{\sigma}}\hat{R}^{\hat{\mu}\hat{\nu}\hat{\rho}\hat{\sigma}}-\frac{1}{2}\,\hat{R}_{\hat{\mu}\hat{\nu}\hat{\rho}\hat{\sigma}}\hat{H}^{\hat{\mu}\hat{\nu}\hat{\lambda}}\hat{H}^{\hat{\rho}\hat{\sigma}}{}_{\hat{\lambda}}+\ldots\right).$
(3.1)

Under the transformations (2.2), (2.3) and (2.5), the action $I_{1}$ transform
homogeneously with the charges444In eq. (3.2) and (3.3), we give the charges
of $I_{1}$ under the trombone symmetries using as convention that the charges
of $I_{0}^{(D)}$ and $I_{0}$ under these transformations are $0$, i.e. we drop
the global factor under which the lowest order equations of motion are
rescaled.

$\begin{array}[]{c|ccc}&\left({\rm Dilaton}^{D}\right)&\left({\rm
Trombone}^{D}\right)&\left({\rm Volume}^{D-3}\right)\\\ \hline\cr
I_{1}&-2/(D-2)&-2&0\end{array}.$ (3.2)

Both the dilaton shift and the trombone symmetries are broken by
$\alpha^{\prime}$ corrections. With the translation (2.11), the charges of
$I_{1}$ under the three-dimensional symmetries (2.8)–(2.10) are

$\begin{array}[]{c|ccc}&\left({\rm Dilaton}^{3}\right)&\left({\rm
Trombone}^{3}\right)&\left(\text{T-duality}\right)\\\ \hline\cr
I_{1}&-2&-2&0\end{array}.$ (3.3)

In particular, the T-duality scaling $\mathrm{O}(1,1)\subset\mathrm{O}(d,d)$
is preserved in presence of first order $\alpha^{\prime}$ corrections, in
agreement with the arguments of ref. [5]. The presence of $\mathrm{O}(d,d)$
was explicitly verified in ref. [11, 12] at first order in $\alpha^{\prime}$.
The symmetry under the dilaton shift (2.8) however is broken, and so is the
symmetry enhancement from $\mathrm{O}(d,d)$ to $\mathrm{O}(d+1,d+1)$. In the
following, we formulate the $\alpha^{\prime}$ corrections to the three-
dimensional action (2.23) in terms of the $\mathrm{O}(d+1,d+1)$ objects
defined in the previous section, by introduction of a non-dynamical
compensator.

### 3.1 Dualisation of the vector fields

We will treat the cases of the bosonic and heterotic string effective actions
at the same time, using the notations of ref. [16]. Our starting point is the
manifestly $\mathrm{O}(d,d)$ invariant action of ref. [12]

$\displaystyle I=$
$\displaystyle\int\mathrm{d}^{3}x\,\sqrt{-g}\,e^{-\Phi}\Bigg{[}R+\partial_{\mu}\Phi\,\partial^{\mu}\Phi+\frac{1}{8}\,\mathrm{Tr}\left(\partial_{\mu}\mathcal{H}\,\partial^{\mu}\mathcal{H}^{-1}\right)-\frac{1}{4}\,\mathcal{F}_{\mu\nu}{}^{M}\,\mathcal{H}_{MN}\,\mathcal{F}^{\mu\nu\,N}$
(3.4)
$\displaystyle-\frac{a+b}{8}\Big{(}R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}+\frac{1}{16}\,\mathrm{Tr}\left(\partial_{\mu}\mathcal{H}\partial_{\nu}\mathcal{H}^{-1}\partial^{\mu}\mathcal{H}\partial^{\nu}\mathcal{H}^{-1}\right)-\frac{1}{32}\,\mathrm{Tr}\left(\partial_{\mu}\mathcal{H}\partial_{\nu}\mathcal{H}^{-1}\right)\mathrm{Tr}\left(\partial^{\mu}\mathcal{H}\partial^{\nu}\mathcal{H}^{-1}\right)$
$\displaystyle\qquad\qquad+\frac{1}{8}\,{\mathcal{F}_{\mu\nu}}^{M}\mathcal{H}_{MN}\mathcal{F}_{\rho\sigma}^{N}\mathcal{F}^{\mu\rho\,P}\mathcal{H}_{PQ}\mathcal{F}^{\nu\sigma\,Q}-\frac{1}{2}\,{\mathcal{F}_{\mu\nu}}^{M}\mathcal{H}_{MN}\mathcal{F}^{\mu\rho\,N}\mathcal{F}^{\nu\sigma\,P}\mathcal{H}_{PQ}\mathcal{F}_{\rho\sigma}{}^{Q}$
$\displaystyle\qquad\qquad+\frac{1}{8}\,{\mathcal{F}_{\mu\nu}}^{M}{\mathcal{F}_{\rho\sigma}}_{M}\mathcal{F}^{\mu\rho\,N}{\mathcal{F}^{\nu\sigma}}_{N}-\frac{1}{2}\,R_{\mu\nu\rho\sigma}\mathcal{F}^{\mu\nu\,M}\mathcal{H}_{MN}\mathcal{F}^{\rho\sigma\,N}$
$\displaystyle\qquad\qquad-\frac{1}{2}\,{\mathcal{F}_{\mu\nu}}^{M}\left(\mathcal{H}\partial_{\rho}\mathcal{H}^{-1}\partial^{\nu}\mathcal{H}\right)_{MN}\mathcal{F}^{\mu\rho\,N}+\frac{1}{4}\,\mathcal{F}^{\mu\rho\,M}\mathcal{H}_{MN}\mathcal{F}^{\nu}{}_{\rho}{}^{N}\mathrm{Tr}\left(\partial_{\mu}\mathcal{H}\partial_{\nu}\mathcal{H}^{-1}\right)\Big{)}$
$\displaystyle+\frac{a-b}{4}\Big{(}-\frac{1}{16}\,\mathrm{Tr}\left(\partial_{\mu}\mathcal{H}\partial^{\mu}\mathcal{H}^{-1}\partial_{\nu}\mathcal{H}\partial^{\nu}\mathcal{H}^{-1}\mathcal{H}\eta\right)-\frac{1}{16}\,\mathcal{F}_{\mu\nu}{}^{M}\mathcal{F}_{\rho\sigma\,M}\,\mathcal{F}^{\mu\nu\,P}\mathcal{H}_{PQ}\mathcal{F}^{\rho\sigma\,Q}$
$\displaystyle\qquad\qquad+\frac{1}{4}\,R^{\mu\nu\rho\sigma}\mathcal{F}_{\mu\nu}{}^{M}\mathcal{F}_{\rho\sigma\,M}+\frac{1}{8}\,\mathcal{F}_{\mu\nu}{}^{M}\left(\partial_{\rho}\mathcal{H}\partial^{\rho}\mathcal{H}^{-1}\right)_{M}{}^{N}\mathcal{F}^{\mu\nu}{}_{N}$
$\displaystyle\qquad\qquad+\frac{1}{4}\,\mathcal{F}_{\mu\nu}{}^{M}\left(\partial^{\mu}\mathcal{H}\partial_{\rho}\mathcal{H}^{-1}\right)_{M}{}^{N}\mathcal{F}^{\nu\rho}{}_{N}\Big{)}+{\cal
O}\left(\alpha^{\prime 2}\right)\Bigg{]},$

having eliminated the 2-form $B_{\mu\nu}$ by virtue of its three-dimensional
field equations as in (2.19) above. The bosonic and heterotic actions
correspond to $(a,b)=(-\alpha^{\prime},-\alpha^{\prime})$ and
$(a,b)=(-\alpha^{\prime},0)$, respectively [16].

In order to express this action in terms of $\mathrm{O}(d+1,d+1)$ covariant
objects, we first switch to the Einstein frame

$g_{\mu\nu}\rightarrow g_{{\rm E}\,\mu\nu}=e^{-2\Phi}g_{\mu\nu}\;,$ (3.5)

and use the fact that in three dimensions the Riemann tensor can be expressed
as

$R_{\,\mu\nu\rho\sigma}=S_{\mu\rho}g_{\nu\sigma}+S_{\nu\sigma}g_{\mu\rho}-S_{\mu\sigma}g_{\nu\rho}-S_{\nu\rho}g_{\mu\sigma},$
(3.6)

in terms of the Schouten tensor
$S_{\mu\nu}=R_{\mu\nu}-\dfrac{1}{4}\,R\,g_{\mu\nu}$.

Next, we dualise the vectorial degrees of freedom to scalar ones. To this end,
we proceed as we did for the two-derivative action and introduce a Lagrange
multiplier term to the action:

$\widetilde{I}=I+\frac{1}{2}\,\int\mathrm{d}^{3}x\,\varepsilon^{\mu\nu\rho}\mathcal{F}_{\mu\nu}{}^{M}\partial_{\rho}\xi_{M}.$
(3.7)

The equations of motion of $\xi_{M}$ still gives the Bianchi identities for
$\mathcal{F}_{\mu\nu}{}^{M}$. But, considering $\mathcal{F}_{\mu\nu}{}^{M}$ as
an independent field, its equations of motion
$\delta\widetilde{I}/\delta\mathcal{F}=0$ now contain corrections of order
$\alpha^{\prime}$. These equations are algebraic in $\mathcal{F}$ and can be
solved perturbatively in $\alpha^{\prime}$. The solution takes the form

$\mathcal{F}_{\mu\nu}{}^{M}=\mathcal{F}^{(0)}_{\mu\nu}{}^{M}+\alpha^{\prime}\,\mathcal{F}^{(1)}_{\mu\nu}{}^{M},$
(3.8)

where $\mathcal{F}^{(0)}_{\mu\nu}{}^{M}$ is solution of
$\delta\widetilde{I}_{0}/\delta\mathcal{F}=0$, as given in eq. (2.18). The
exact expression of $\mathcal{F}^{(1)}_{\mu\nu}{}^{M}$ is not necessary for
our purpose, as we now show. Eq. (3.8) is algebraic and can be introduced in
the action (3.7), which, schematically, takes the following form:

$\widetilde{I}\left(\mathcal{F}^{(0)}+\alpha^{\prime}\,\mathcal{F}^{(1)}\right)=\widetilde{I}_{0}\left(\mathcal{F}^{(0)}\right)+I_{1}\left(\mathcal{F}^{(0)}\right)+\alpha^{\prime}\,\mathcal{F}^{(1)}\frac{\delta\widetilde{I}_{0}}{\delta\mathcal{F}}\left(\mathcal{F}^{(0)}\right)+{\cal
O}\left(\alpha^{\prime 2}\right),$ (3.9)

where $I_{1}$ is the four-derivative action. The dependence on
$\mathcal{F}^{(1)}$ is thus proportional to the equation of motion of
$\mathcal{F}$ at order $\alpha^{\prime 0}$, evaluated at its solution
$\mathcal{F}^{(0)}$. Then, at first order in $\alpha^{\prime}$, the
corrections to the duality relation (2.18) cancel out in the action, and the
lowest order relation can be used to dualise the vectors.

Applying this procedure, after some computation, the four-derivative part of
the action (3.4) turns into

$\displaystyle\widetilde{I}_{1}=$
$\displaystyle\int\mathrm{d}^{3}x\,\sqrt{-g_{\rm E}}\,e^{-2\Phi}\,\Bigg{[}$
(3.10)
$\displaystyle-\frac{a+b}{8}\Big{(}\frac{1}{16}\,\mathrm{Tr}\left(\partial_{\mu}\mathcal{H}\partial_{\nu}\mathcal{H}^{-1}\partial^{\mu}\mathcal{H}\partial^{\nu}\mathcal{H}^{-1}\right)-\frac{1}{32}\,\mathrm{Tr}\left(\partial_{\mu}\mathcal{H}\partial_{\nu}\mathcal{H}^{-1}\right)\mathrm{Tr}\left(\partial^{\mu}\mathcal{H}\partial^{\nu}\mathcal{H}^{-1}\right)$
$\displaystyle\qquad+4\,\partial_{\mu}\Phi\partial^{\mu}\Phi\,\partial_{\nu}\Phi\partial^{\nu}\Phi+\frac{1}{4}\,e^{4\Phi}\,\partial_{\mu}\xi^{M}\partial_{\nu}\xi_{M}\partial^{\mu}\xi^{N}\partial^{\nu}\xi_{N}$
$\displaystyle\qquad-\frac{1}{4}\,e^{4\Phi}\,\partial_{\mu}\xi_{M}\mathcal{H}^{MN}\partial_{\nu}\xi_{N}\partial^{\mu}\xi_{P}\mathcal{H}^{PQ}\partial^{\nu}\xi_{Q}-\frac{1}{2}\,e^{4\Phi}\,\partial_{\mu}\xi_{M}\mathcal{H}^{MN}\partial^{\mu}\xi_{N}\partial_{\nu}\xi_{P}\mathcal{H}^{PQ}\partial^{\nu}\xi_{Q}$
$\displaystyle\qquad+\frac{1}{2}\,e^{2\Phi}\,\partial_{\mu}\xi_{M}\left(\mathcal{H}^{-1}\partial_{\nu}\mathcal{H}\partial^{\nu}\mathcal{H}^{-1}\right)^{MN}\partial^{\mu}\xi_{N}-\frac{1}{2}\,e^{2\Phi}\,\partial_{\mu}\xi_{M}\left(\mathcal{H}^{-1}\partial^{\mu}\mathcal{H}\partial_{\nu}\mathcal{H}^{-1}\right)^{MN}\partial^{\nu}\xi_{N}$
$\displaystyle\qquad-\frac{1}{4}\,e^{2\Phi}\,\mathrm{Tr}\left(\partial_{\mu}\mathcal{H}\partial^{\mu}\mathcal{H}^{-1}\right)\,\partial_{\nu}\xi_{M}\mathcal{H}^{MN}\partial^{\nu}\xi_{N}+\frac{1}{4}\,e^{2\Phi}\,\mathrm{Tr}\left(\partial_{\mu}\mathcal{H}\partial_{\nu}\mathcal{H}^{-1}\right)\,\partial^{\mu}\xi_{M}\mathcal{H}^{MN}\partial^{\nu}\xi_{N}$
$\displaystyle\qquad-2\,e^{2\Phi}\,\partial_{\mu}\Phi\partial_{\nu}\Phi\,\partial^{\mu}\xi_{M}\mathcal{H}^{MN}\partial^{\nu}\xi_{N}+4\,R_{{\rm
E}\,\mu\nu}R_{\rm E}{}^{\mu\nu}-R_{\rm
E}^{2}+4\,\Box\Phi\Box\Phi+8\,\Box\Phi\,\partial_{\mu}\Phi\partial^{\mu}\Phi$
$\displaystyle\qquad+4\,\nabla_{\mu}\nabla_{\nu}\Phi\nabla^{\mu}\nabla^{\nu}\Phi-8\,\nabla_{\mu}\nabla_{\nu}\Phi\partial^{\mu}\Phi\,\partial^{\nu}\Phi-8\,R_{{\rm
E}\,\mu\nu}\nabla^{\mu}\nabla^{\nu}\Phi+8\,R_{{\rm
E}\,\mu\nu}\,\partial^{\mu}\Phi\partial^{\nu}\Phi$
$\displaystyle\qquad-4\,R_{\rm
E}\,\partial_{\mu}\Phi\partial^{\mu}\Phi-2\,e^{2\Phi}R_{\rm
E\,\mu\nu}\,\partial^{\mu}\xi_{M}\mathcal{H}^{MN}\partial^{\nu}\xi_{N}+e^{2\Phi}R_{\rm
E}\,\partial_{\mu}\xi_{M}\mathcal{H}^{MN}\partial^{\mu}\xi_{N}$
$\displaystyle\qquad-2\,e^{2\Phi}\Box\Phi\,\partial_{\mu}\xi_{M}\mathcal{H}^{MN}\partial^{\mu}\xi_{N}+2\,e^{2\Phi}\nabla_{\mu}\nabla_{\nu}\Phi\,\partial^{\mu}\xi_{M}\mathcal{H}^{MN}\partial^{\nu}\xi_{N}\Big{)}$
$\displaystyle+\frac{a-b}{4}\Big{(}-\frac{1}{16}\,\mathrm{Tr}\left(\partial_{\mu}\mathcal{H}\partial^{\mu}\mathcal{H}^{-1}\partial_{\nu}\mathcal{H}\partial^{\nu}\mathcal{H}^{-1}\mathcal{H}\eta\right)-\frac{1}{4}\,e^{4\Phi}\,\partial_{\mu}\xi_{M}\partial_{\nu}\xi^{M}\,\partial^{\mu}\xi_{P}\mathcal{H}^{PQ}\partial^{\nu}\xi_{Q}$
$\displaystyle\qquad-\frac{1}{2}\,e^{2\Phi}\,R_{\rm
E}\,\partial_{\mu}\xi_{M}\partial^{\mu}\xi^{M}+e^{2\Phi}\,R_{\rm
E\,\mu\nu}\,\partial^{\mu}\xi_{M}\partial^{\nu}\xi^{M}-e^{2\Phi}\,\nabla_{\mu}\nabla_{\nu}\Phi\,\partial^{\mu}\xi_{M}\partial^{\nu}\xi^{M}$
$\displaystyle\qquad+e^{2\Phi}\,\Box\Phi\,\partial_{\mu}\xi_{M}\partial^{\mu}\xi^{M}+e^{2\Phi}\,\partial_{\mu}\Phi\partial_{\nu}\Phi\,\partial^{\mu}\xi_{M}\partial^{\nu}\xi^{M}-\frac{1}{4}\,e^{2\Phi}\,\partial_{\mu}\xi_{M}\left(\partial_{\nu}\mathcal{H}^{-1}\partial^{\mu}\mathcal{H}\right)^{M}{}_{N}\partial^{\nu}\xi^{N}\Big{)}\Bigg{]}.$

We now convert all second order derivatives in eq. (3.10) into products of
first order derivatives, to allow comparison with the basis of
$\mathrm{O}(d+1,d+1)$-invariant four-derivative terms of ref. [12]. To do so,
we first use integrations by parts so that all the second order derivatives
appear in the leading two-derivative contribution of the equations of motion
(2.37d). Using furthermore the field redefinitions of sec. 2.3, we obtain the
action

$\displaystyle\widetilde{I}_{1}$
$\displaystyle=\int\mathrm{d}^{3}x\,\sqrt{-g_{\rm E}}\,e^{-2\Phi}\,\Bigg{[}$
$\displaystyle-\frac{a+b}{8}\Big{(}-\partial_{\mu}\Phi\partial^{\mu}\Phi\,\partial_{\nu}\Phi\partial^{\nu}\Phi+\frac{1}{16}\,\mathrm{Tr}\left(\partial_{\mu}\mathcal{H}\partial_{\nu}\mathcal{H}^{-1}\partial^{\mu}\mathcal{H}\partial^{\nu}\mathcal{H}^{-1}\right)$
$\displaystyle\qquad+\frac{1}{32}\,\mathrm{Tr}\left(\partial_{\mu}\mathcal{H}\partial_{\nu}\mathcal{H}^{-1}\right)\mathrm{Tr}\left(\partial^{\mu}\mathcal{H}\partial^{\nu}\mathcal{H}^{-1}\right)-\frac{1}{64}\,\mathrm{Tr}\left(\partial_{\mu}\mathcal{H}\partial^{\mu}\mathcal{H}^{-1}\right)\mathrm{Tr}\left(\partial_{\nu}\mathcal{H}\partial^{\nu}\mathcal{H}^{-1}\right)$
$\displaystyle\qquad+\frac{1}{4}\,e^{4\Phi}\,\partial_{\mu}\xi^{M}\partial_{\nu}\xi_{M}\partial^{\mu}\xi^{N}\partial^{\nu}\xi_{N}-\frac{1}{4}\,e^{4\Phi}\,\partial_{\mu}\xi_{M}\mathcal{H}^{MN}\partial_{\nu}\xi_{N}\partial^{\mu}\xi_{P}\mathcal{H}^{PQ}\partial^{\nu}\xi_{Q}$
$\displaystyle\qquad+\frac{1}{4}\,e^{4\Phi}\,\partial_{\mu}\xi_{M}\mathcal{H}^{MN}\partial^{\mu}\xi_{N}\partial_{\nu}\xi_{P}\mathcal{H}^{PQ}\partial^{\nu}\xi_{Q}+\frac{1}{2}\,e^{2\Phi}\,\partial_{\mu}\xi_{M}\left(\mathcal{H}^{-1}\partial_{\nu}\mathcal{H}\partial^{\nu}\mathcal{H}^{-1}\right)^{MN}\partial^{\mu}\xi_{N}$
$\displaystyle\qquad-\frac{1}{2}\,e^{2\Phi}\,\partial_{\mu}\xi_{M}\left(\mathcal{H}^{-1}\partial^{\mu}\mathcal{H}\partial_{\nu}\mathcal{H}^{-1}\right)^{MN}\partial^{\nu}\xi_{N}-2\,e^{2\Phi}\,\partial_{\mu}\Phi\partial_{\nu}\Phi\,\partial^{\mu}\xi_{M}\mathcal{H}^{MN}\partial^{\nu}\xi_{N}$
$\displaystyle\qquad+\frac{1}{2}\,\partial_{\mu}\Phi\partial_{\nu}\Phi\,\mathrm{Tr}\left(\partial^{\mu}\mathcal{H}\partial^{\nu}\mathcal{H}^{-1}\right)-\frac{1}{4}\,\partial_{\mu}\Phi\partial^{\mu}\Phi\,\mathrm{Tr}\left(\partial_{\nu}\mathcal{H}\partial^{\nu}\mathcal{H}^{-1}\right)+e^{2\Phi}\,\partial_{\mu}\Phi\,\partial_{\nu}\xi_{M}\partial^{\mu}\mathcal{H}^{MN}\partial^{\nu}\xi_{N}\Big{)}$
$\displaystyle+\frac{a-b}{4}\Big{(}-\frac{1}{16}\,\mathrm{Tr}\left(\partial_{\mu}\mathcal{H}\partial^{\mu}\mathcal{H}^{-1}\partial_{\nu}\mathcal{H}\partial^{\nu}\mathcal{H}^{-1}\mathcal{H}\eta\right)+\frac{1}{4}\,e^{4\Phi}\,\partial_{\mu}\xi_{M}\partial_{\nu}\xi^{M}\,\partial^{\mu}\xi_{P}\mathcal{H}^{PQ}\partial^{\nu}\xi_{Q}$
$\displaystyle\qquad-\frac{1}{4}\,e^{2\Phi}\,\partial_{\mu}\xi_{M}\left(\partial_{\nu}\mathcal{H}^{-1}\partial^{\mu}\mathcal{H}\right)^{M}{}_{N}\partial^{\nu}\xi^{N}-\frac{1}{2}\,e^{2\Phi}\,\partial_{\mu}\Phi\partial^{\mu}\Phi\,\partial_{\nu}\xi_{M}\partial^{\nu}\xi^{M}-\frac{1}{8}\,\mathrm{Tr}\left(\partial_{\mu}\mathcal{H}\partial_{\nu}\mathcal{H}^{-1}\right)\,\partial^{\mu}\xi_{M}\partial^{\nu}\xi^{M}$
$\displaystyle\qquad+\frac{1}{16}\,\mathrm{Tr}\left(\partial_{\mu}\mathcal{H}\partial^{\mu}\mathcal{H}^{-1}\right)\,\partial_{\nu}\xi_{M}\partial^{\nu}\xi^{M}+e^{2\Phi}\,\partial_{\mu}\Phi\,\partial^{\mu}\xi_{M}\left(\partial_{\nu}\mathcal{H}^{-1}\mathcal{H}\right)^{M}{}_{N}\partial^{\nu}\xi^{N}\Big{)}\Bigg{]}.$
(3.11)

Explicitly, we have used the following order $\alpha^{\prime}$ field
redefinitions:

$\displaystyle\delta g_{\rm E}^{\mu\nu}$
$\displaystyle=-\dfrac{a+b}{8\,\alpha^{\prime}}\Big{(}-4\,e^{-2\Phi}\,R_{\rm
E}^{\mu\nu}+2\,e^{-2\Phi}\,g_{\rm E}^{\mu\nu}\,R_{\rm
E}+8\,e^{-2\Phi}\,\partial^{\mu}\Phi\partial^{\nu}\Phi-2\,e^{-2\Phi}\,g_{\rm
E}^{\mu\nu}\,\partial_{\rho}\Phi\partial^{\rho}\Phi$ (3.12)
$\displaystyle\quad\qquad+\dfrac{1}{2}\,e^{-2\Phi}\,\mathrm{Tr}\left(\partial^{\mu}\mathcal{H}\partial^{\nu}\mathcal{H}^{-1}\right)-\dfrac{1}{4}\,e^{-2\Phi}\,g_{\rm
E}^{\mu\nu}\,\mathrm{Tr}\left(\partial_{\rho}\mathcal{H}\partial^{\rho}\mathcal{H}^{-1}\right)-3\,g_{\rm
E}^{\mu\nu}\,\partial_{\rho}\xi_{M}\mathcal{H}^{MN}\partial^{\rho}\xi_{N}\Big{)}$
$\displaystyle\quad-\dfrac{a-b}{4\,\alpha^{\prime}}\,\partial^{\mu}\xi_{M}\partial^{\nu}\xi^{M},$
$\displaystyle\delta\Phi$
$\displaystyle=-\dfrac{a+b}{8\,\alpha^{\prime}}\left(2\,e^{-2\Phi}\,R_{\rm
E}-4\,e^{-2\Phi}\,\Box\Phi-\dfrac{3}{2}\,\partial_{\mu}\xi_{M}\mathcal{H}^{MN}\partial^{\mu}\xi_{N}\right)-\dfrac{a-b}{4\,\alpha^{\prime}}\,\dfrac{1}{4}\,\partial_{\mu}\xi_{M}\partial^{\mu}\xi^{M},$
$\displaystyle\delta\xi_{M}$
$\displaystyle=-\dfrac{a+b}{4\,\alpha^{\prime}}\,e^{-2\Phi}\,\partial_{\mu}\Phi\partial^{\mu}\xi_{M}-\dfrac{a-b}{4\,\alpha^{\prime}}\,e^{-2\Phi}\,\mathcal{H}_{MN}\,\partial_{\mu}\Phi\partial^{\mu}\xi^{N},$

in the notations of eq. (2.35).

### 3.2 $\mathrm{O}(d+1,d+1)$-covariant formulation

We now aim to express the action (3.11) in terms of
$\mathrm{O}(d+1,d+1)$-covariant objects. We define the $\mathrm{O}(d+1,d+1)$
currents $\mathcal{J}_{\mu}=\partial_{\mu}\mathcal{M}\mathcal{M}^{-1}$ for the
matrix $\mathcal{M}$ from eq. (2.21). Explicitly, this current takes the form

$\mathcal{J}_{\mu\,{\cal M}}{}^{{\cal
N}}=\begin{pmatrix}\mathcal{J}_{\mu\,M}{}^{N}&\mathcal{J}_{\mu\,M}{}^{+}&-e^{2\Phi}\partial_{\mu}\xi_{P}\mathcal{H}^{P}{}_{M}\\\
e^{2\Phi}\partial_{\mu}\xi_{P}\mathcal{H}^{PN}&2\partial_{\mu}\Phi-e^{2\Phi}\xi_{P}\mathcal{H}^{PQ}\partial_{\mu}\xi_{Q}&0\\\
\mathcal{J}_{\mu\,-}{}^{N}&0&-2\partial_{\mu}\Phi+e^{2\Phi}\xi_{P}\mathcal{H}^{PQ}\partial_{\mu}\xi_{Q}\end{pmatrix}\;,$
(3.13)

with

$\displaystyle\mathcal{J}_{\mu\,M}{}^{N}$
$\displaystyle=\partial_{\mu}\mathcal{H}_{MP}\mathcal{H}^{PN}+e^{2\Phi}\left(\xi_{M}\partial_{\mu}\xi_{P}\mathcal{H}^{PN}-\mathcal{H}_{MP}\partial_{\mu}{\xi^{P}}\xi^{N}\right),$
(3.14) $\displaystyle\mathcal{J}_{\mu\,M}{}^{+}$
$\displaystyle=\partial_{\mu}\xi_{M}-\partial_{\mu}\mathcal{H}_{MP}\mathcal{H}^{PQ}\xi_{Q}+2\,\partial_{\mu}\Phi\xi_{M}-e^{2\Phi}\xi_{P}\mathcal{H}^{PQ}\partial_{\mu}\xi_{Q}\xi_{M}+\frac{1}{2}\,e^{2\Phi}\xi_{P}\xi^{P}\partial_{\mu}\xi_{Q}\mathcal{H}^{Q}{}_{M}\
,$ $\displaystyle\mathcal{J}_{\mu\,-}{}^{N}$
$\displaystyle=-\partial_{\mu}\xi^{N}-\xi^{P}\partial_{\mu}\mathcal{H}_{PQ}\mathcal{H}^{QN}-2\partial_{\mu}\Phi\xi^{N}+e^{2\Phi}\xi_{P}\mathcal{H}^{PQ}\partial_{\mu}\xi_{Q}\xi^{N}-\frac{1}{2}e^{2\Phi}\xi_{P}\xi^{P}\partial_{\mu}\xi_{Q}\mathcal{H}^{QN}\
.$

In terms of this object and using the $\mathrm{O}(d,d)$ decomposition of app.
A, the four-derivative action (3.11) can be cast into the rather compact form

$\displaystyle\widetilde{I}_{1}$
$\displaystyle=\int\mathrm{d}^{3}x\,\sqrt{-g_{\rm E}}\,e^{-2\Phi}\,\Bigg{\\{}$
(3.15) $\displaystyle+\frac{a}{4}\Bigg{[}-\frac{1}{32}\mathrm{Tr}\left({\cal
J}_{\mu}{\cal J}_{\nu}{\cal J}^{\mu}{\cal
J}^{\nu}\right)-\frac{1}{16}\,\mathrm{Tr}\left(\mathcal{J}_{\mu}\mathcal{J}^{\mu}\mathcal{J}_{\nu}\mathcal{J}^{\nu}\mathcal{M}\eta\right)$
$\displaystyle\quad\ \,\quad-\frac{1}{64}\,\mathrm{Tr}\left({\cal
J}_{\mu}{\cal J}_{\nu}\right)\mathrm{Tr}\left({\cal J}^{\mu}{\cal
J}^{\nu}\right)+\frac{1}{128}\,\mathrm{Tr}\left({\cal J}_{\mu}{\cal
J}^{\mu}\right)\mathrm{Tr}\left({\cal J}_{\nu}{\cal J}^{\nu}\right)$
$\displaystyle\quad\ \,\quad+e^{-2\Phi}\bigg{(}-\frac{1}{2}\,\left(\mbox{\sf
u}P\eta{\cal J}_{\mu}{\cal J}_{\nu}{\cal J}^{\mu}{\cal J}^{\nu}P\mbox{\sf
u}\right)+\frac{1}{2}\,\left(\mbox{\sf u}P\eta{\cal J}_{\mu}{\cal
J}^{\mu}{\cal J}_{\nu}{\cal J}^{\nu}P\mbox{\sf
u}\right)-\frac{1}{2}\,\left(\mbox{\sf u}P\eta{\cal J}_{\mu}{\cal
J}_{\nu}{\cal J}^{\nu}{\cal J}^{\mu}P\mbox{\sf u}\right)$
$\displaystyle\qquad\ \,\qquad\qquad-\frac{1}{4}\,\mathrm{Tr}\left({\cal
J}_{\mu}{\cal J}_{\nu}\right)\left(\mbox{\sf u}P\eta{\cal J}^{\mu}{\cal
J}^{\nu}P\mbox{\sf u}\right)+\frac{1}{8}\,\mathrm{Tr}\left({\cal J}_{\mu}{\cal
J}^{\mu}\right)\left(\mbox{\sf u}P\eta{\cal J}_{\nu}{\cal J}^{\nu}P\mbox{\sf
u}\right)\bigg{)}$ $\displaystyle\quad\
\,\quad+e^{-4\Phi}\bigg{(}-2\,\left(\mbox{\sf u}P\eta{\cal J}_{\mu}{\cal
J}_{\nu}P\mbox{\sf u}\right)\left(\mbox{\sf u}P\eta{\cal J}^{\mu}{\cal
J}^{\nu}P\mbox{\sf u}\right)+\left(\mbox{\sf u}P\eta{\cal J}_{\mu}{\cal
J}^{\mu}P\mbox{\sf u}\right)\left(\mbox{\sf u}P\eta{\cal J}_{\nu}{\cal
J}^{\nu}P\mbox{\sf u}\right)\bigg{)}\Bigg{]}$
$\displaystyle+\frac{b}{4}\Bigg{[}-\frac{1}{32}\mathrm{Tr}\left({\cal
J}_{\mu}{\cal J}_{\nu}{\cal J}^{\mu}{\cal
J}^{\nu}\right)+\frac{1}{16}\,\mathrm{Tr}\left(\mathcal{J}_{\mu}\mathcal{J}^{\mu}\mathcal{J}_{\nu}\mathcal{J}^{\nu}\mathcal{M}\eta\right)$
$\displaystyle\quad\ \,\quad-\frac{1}{64}\,\mathrm{Tr}\left({\cal
J}_{\mu}{\cal J}_{\nu}\right)\mathrm{Tr}\left({\cal J}^{\mu}{\cal
J}^{\nu}\right)+\frac{1}{128}\,\mathrm{Tr}\left({\cal J}_{\mu}{\cal
J}^{\mu}\right)\mathrm{Tr}\left({\cal J}_{\nu}{\cal J}^{\nu}\right)$
$\displaystyle\quad\ \,\quad+e^{-2\Phi}\bigg{(}\frac{1}{2}\,\left(\mbox{\sf
u}\bar{P}\eta{\cal J}_{\mu}{\cal J}_{\nu}{\cal J}^{\mu}{\cal
J}^{\nu}\bar{P}\mbox{\sf u}\right)-\frac{1}{2}\,\left(\mbox{\sf
u}\bar{P}\eta{\cal J}_{\mu}{\cal J}^{\mu}{\cal J}_{\nu}{\cal
J}^{\nu}\bar{P}\mbox{\sf u}\right)+\frac{1}{2}\,\left(\mbox{\sf
u}\bar{P}\eta{\cal J}_{\mu}{\cal J}_{\nu}{\cal J}^{\nu}{\cal
J}^{\mu}\bar{P}\mbox{\sf u}\right)$ $\displaystyle\qquad\
\,\qquad\qquad+\frac{1}{4}\,\mathrm{Tr}\left({\cal J}_{\mu}{\cal
J}_{\nu}\right)\left(\mbox{\sf u}\bar{P}\eta{\cal J}^{\mu}{\cal
J}^{\nu}\bar{P}\mbox{\sf u}\right)-\frac{1}{8}\,\mathrm{Tr}\left({\cal
J}_{\mu}{\cal J}^{\mu}\right)\left(\mbox{\sf u}\bar{P}\eta{\cal J}_{\nu}{\cal
J}^{\nu}\bar{P}\mbox{\sf u}\right)\bigg{)}$ $\displaystyle\quad\
\,\quad+e^{-4\Phi}\bigg{(}-2\,\left(\mbox{\sf u}\bar{P}\eta{\cal J}_{\mu}{\cal
J}_{\nu}\bar{P}\mbox{\sf u}\right)\left(\mbox{\sf u}\bar{P}\eta{\cal
J}^{\mu}{\cal J}^{\nu}\bar{P}\mbox{\sf u}\right)+\left(\mbox{\sf
u}\bar{P}\eta{\cal J}_{\mu}{\cal J}^{\mu}\bar{P}\mbox{\sf
u}\right)\left(\mbox{\sf u}\bar{P}\eta{\cal J}_{\nu}{\cal
J}^{\nu}\bar{P}\mbox{\sf u}\right)\bigg{)}\Bigg{]}\Bigg{\\}}.$

Here, we have defined the projectors

$P_{\cal MN}=\frac{1}{2}\,\left(\eta_{\cal MN}-\mathcal{M}_{\cal
MN}\right)\quad{\rm and}\quad\bar{P}_{\cal MN}=\frac{1}{2}\,\left(\eta_{\cal
MN}+\mathcal{M}_{\cal MN}\right),$ (3.16)

and the $\mathrm{O}(d+1,d+1)$ compensator vector

$\mbox{\sf u}^{\cal M}=\\{0,1,0\\}\;,$ (3.17)

which parametrizes the breaking of the symmetry group $\mathrm{O}(d+1,d+1)$ at
the four-derivative order. The action (3.15) enjoys a formal
$\mathrm{O}(d+1,d+1)$ invariance, which is broken by the explicit choice
(3.17) for the vector $\mbox{\sf u}^{\cal M}$. This is manifest in the above
form for all terms except the explicit dilaton prefactor, but it even holds
for these factor thanks to the relations (3.22) below.

It is interesting to note that the above action (3.15) has the chiral
structure

$I_{1}=\frac{1}{4}\int\mathrm{d}^{3}x\,\sqrt{-g_{\rm E}}\,\Big{\\{}a\,{\cal
F}[{\cal M},P\mbox{\sf u}]+b\,\left({\cal F}[{\cal M},{P}\mbox{\sf
u}]\right)^{*}\Big{\\}}$ (3.18)

with a fixed function ${\cal F}$ that depends only on ${\cal M}$ and a
projection of u. The ‘∗’ in the second term indicates the $\mathbb{Z}_{2}$
action, under which ${\cal M}$ transforms as

${\cal M}\rightarrow{\cal Z}^{\rm t}{\cal M}{\cal Z},\qquad{\cal
Z}=\begin{pmatrix}Z&&0\\\ 0&&\sigma_{3}\end{pmatrix},\qquad
Z=\begin{pmatrix}{\bf 1}&&0\\\ 0&&-{\bf 1}\end{pmatrix},$ (3.19)

where

${\cal Z}^{\rm t}\eta{\cal Z}=-\eta\;.$ (3.20)

Consequently, the projectors transform as

$P\rightarrow{\cal Z}^{\rm t}\bar{P}{\cal Z}\;,\qquad\bar{P}\rightarrow{\cal
Z}^{\rm t}{P}{\cal Z}\;,$ (3.21)

while furthermore

$e^{2\Phi}=\mbox{\sf u}^{M}{\cal M}_{MN}\,\mbox{\sf u}^{N}=-2\mbox{\sf
u}^{M}{P}_{MN}\,\mbox{\sf u}^{N}=2\mbox{\sf u}^{M}\bar{P}_{MN}\,\mbox{\sf
u}^{N}\;.$ (3.22)

In particular, the last equation shows that the dilaton $\Phi$ is invariant
under the $\mathbb{Z}_{2}$ action. The chiral form (3.18) of the action
implies that the four-derivative action of the heterotic string ($b=0$)
encodes the function ${\cal F}$, and thereby the entire action. In particular,
the result for the bosonic string ($a=b$) follows or can be reconstructed from
the heterotic result. Moreover, in the heterotic case we note the formal
“gauge invariance”

$\mbox{\sf u}\rightarrow\mbox{\sf u}+\eta\bar{P}\Lambda,$ (3.23)

under which the action (3.15) (for $b=0$) is invariant.

We may also express the result in terms of the $\mathrm{O}(d+1,d+1)$ coset
currents (2.31). In the basis (2.34), the four-derivative action (3.15) takes
the form

$\displaystyle\widetilde{I}_{1}$
$\displaystyle=\frac{\alpha^{\prime}}{4}\int\mathrm{d}^{3}x\,\sqrt{-g}\,e^{-2\Phi}\,\Bigg{\\{}$
(3.24) $\displaystyle+\frac{a}{4}\bigg{[}\mathrm{Tr}\left({\sf P}_{\mu}{\sf
P}^{{\rm t}\,\mu}{\sf P}_{\nu}{\sf P}^{{\rm
t}\,\nu}\right)-\mathrm{Tr}\left({\sf P}_{\mu}^{\rm t}{\sf P}^{\mu}{\sf
P}_{\nu}^{\rm t}{\sf P}^{\nu}\right)-\mathrm{Tr}\left({\sf P}_{\mu}{\sf
P}^{\rm t}_{\nu}{\sf P}^{\mu}{\sf P}^{{\rm
t}\,\nu}\right)-\mathrm{Tr}\left({\sf P}_{\mu}{\sf P}^{\rm
t}_{\nu}\right)\mathrm{Tr}\left({\sf P}^{\mu}{\sf P}^{{\rm t}\,\nu}\right)$
$\displaystyle\quad\ \,\quad+\frac{1}{2}\,\mathrm{Tr}\left({\sf P}_{\mu}{\sf
P}^{{\rm t}\,\mu}\right)\mathrm{Tr}\left({\sf P}_{\nu}{\sf P}^{{\rm
t}\,\nu}\right)+4\,\left({\sf P}_{\mu}{\sf P}^{\rm t}_{\nu}{\sf P}^{\mu}{\sf
P}^{{\rm t}\,\nu}\right)^{00}-4\,\left({\sf P}_{\mu}{\sf P}^{{\rm t}\,\mu}{\sf
P}_{\nu}{\sf P}^{{\rm t}\,\nu}\right)^{00}+4\,\left({\sf P}_{\mu}{\sf P}^{\rm
t}_{\nu}{\sf P}^{\nu}{\sf P}^{{\rm t}\,\mu}\right)^{00}$ $\displaystyle\quad\
\,\quad+4\,\mathrm{Tr}\left({\sf P}_{\mu}{\sf P}^{\rm
t}_{\nu}\right)\left({\sf P}^{\mu}{\sf P}^{{\rm
t}\,\nu}\right)^{00}-2\,\mathrm{Tr}\left({\sf P}_{\mu}{\sf P}^{{\rm
t}\,\mu}\right)\left({\sf P}_{\nu}{\sf P}^{{\rm
t}\,\nu}\right)^{00}-8\,\left({\sf P}_{\mu}{\sf P}^{\rm
t}_{\nu}\right)^{00}\left({\sf P}^{\mu}{\sf P}^{{\rm t}\,\nu}\right)^{00}$
$\displaystyle\quad\ \,\quad+4\,\left({\sf P}_{\mu}{\sf P}^{{\rm
t}\,\mu}\right)^{00}\left({\sf P}_{\nu}{\sf P}^{{\rm
t}\,\nu}\right)^{00}\bigg{]}$
$\displaystyle+\frac{b}{4}\,\bigg{[}-\mathrm{Tr}\left({\sf P}_{\mu}{\sf
P}^{{\rm t}\,\mu}{\sf P}_{\nu}{\sf P}^{{\rm
t}\,\nu}\right)+\mathrm{Tr}\left({\sf P}_{\mu}^{\rm t}{\sf P}^{\mu}{\sf
P}_{\nu}^{\rm t}{\sf P}^{\nu}\right)-\mathrm{Tr}\left({\sf P}_{\mu}{\sf
P}^{\rm t}_{\nu}{\sf P}^{\mu}{\sf P}^{{\rm
t}\,\nu}\right)-\mathrm{Tr}\left({\sf P}_{\mu}{\sf P}^{\rm
t}_{\nu}\right)\mathrm{Tr}\left({\sf P}^{\mu}{\sf P}^{{\rm t}\,\nu}\right)$
$\displaystyle\quad\ \,\quad+\frac{1}{2}\,\mathrm{Tr}\left({\sf P}_{\mu}{\sf
P}^{{\rm t}\,\mu}\right)\mathrm{Tr}\left({\sf P}_{\nu}{\sf P}^{{\rm
t}\,\nu}\right)+4\,\left({\sf P}^{\rm t}_{\mu}{\sf P}_{\nu}{\sf P}^{{\rm
t}\,\mu}{\sf P}^{\nu}\right)^{\bar{0}\bar{0}}-4\,\left({\sf P}^{\rm
t}_{\mu}{\sf P}^{\mu}{\sf P}^{\rm t}_{\nu}{\sf
P}^{\nu}\right)^{\bar{0}\bar{0}}+4\,\left({\sf P}^{\rm t}_{\mu}{\sf
P}_{\nu}{\sf P}^{{\rm t}\,\nu}{\sf P}^{\mu}\right)^{\bar{0}\bar{0}}$
$\displaystyle\quad\ \,\quad+4\,\mathrm{Tr}\left({\sf P}_{\mu}{\sf P}^{\rm
t}_{\nu}\right)\left({\sf P}^{{\rm t}\,\mu}{\sf
P}^{\nu}\right)^{\bar{0}\bar{0}}-2\,\mathrm{Tr}\left({\sf P}_{\mu}{\sf
P}^{{\rm t}\,\mu}\right)\left({\sf P}^{{\rm t}}_{\nu}{\sf
P}^{\nu}\right)^{\bar{0}\bar{0}}-8\,\left({\sf P}^{\rm t}_{\mu}{\sf
P}_{\nu}\right)^{\bar{0}\bar{0}}\left({\sf P}^{{\rm t}\,\mu}{\sf
P}^{\nu}\right)^{\bar{0}\bar{0}}$ $\displaystyle\quad\ \,\quad+4\,\left({\sf
P}^{{\rm t}}_{\mu}{\sf P}^{\mu}\right)^{\bar{0}\bar{0}}\left({\sf P}^{\rm
t}_{\nu}{\sf P}^{\nu}\right)^{\bar{0}\bar{0}}\bigg{]}\Bigg{\\}}.$

Here, the ‘0’ and ‘${}^{\bar{0}}$’ components are defined by contracting out
$\mathrm{O}(d+1)\times\mathrm{O}(d+1)$ vectors as

${\sf P}_{\mu}^{0\bar{a}}=e^{-\Phi}\,\widetilde{\mbox{\sf u}}_{a}{\sf
P}_{\mu}^{a\bar{a}}\;,\qquad{\sf P}_{\mu}^{a\bar{0}}={\sf
P}_{\mu}^{a\bar{a}}\,\widetilde{\mbox{\sf u}}_{\bar{a}}\,e^{-\Phi}\;,$ (3.25)

with $\widetilde{\mbox{\sf u}}_{a}=\mbox{\sf u}^{M}{\cal V}_{Ma}$,
$\widetilde{\mbox{\sf u}}_{\bar{a}}=\mbox{\sf u}^{M}{\cal V}_{M\bar{a}}$.
Spelling out eq. (3.25) brings the action into manifestly $H$ gauge invariant
form. In these notations, ${\sf P}$ and ${\sf P}^{\rm t}$ are interchanged
under the $\mathbb{Z}_{2}$ action.

More compactly, and using the notations of refs. [22, 23], the result (3.24)
can be rewritten in the form

$\displaystyle\widetilde{I}_{1}=\frac{\alpha^{\prime}}{4}\int\mathrm{d}^{3}x\,\sqrt{-g}\,e^{-2\Phi}$
$\displaystyle\Bigg{[}\frac{a}{4}\,F_{abcd}\left(({\sf P}_{\mu}{\sf P}^{{\rm
t}\,\mu})^{ab}({\sf P}_{\nu}{\sf P}^{{\rm t}\,\nu})^{cd}-2\,({\sf P}_{\mu}{\sf
P}^{\rm t}_{\nu})^{ab}({\sf P}^{\mu}{\sf P}^{{\rm t}\,\nu})^{cd}\right)$
(3.26)
$\displaystyle+\frac{b}{4}{\bar{F}}_{\bar{a}\bar{b}\bar{c}\bar{d}}\left(({\sf
P}^{\rm t}_{\mu}{\sf P}^{\mu})^{\bar{a}\bar{b}}({\sf P}^{\rm t}_{\nu}{\sf
P}^{\nu})^{\bar{c}\bar{d}}-2\,({\sf P}^{\rm t}_{\mu}{\sf
P}_{\nu})^{\bar{a}\bar{b}}({\sf P}^{{\rm t}\,\mu}{{}\sf
P}^{\nu})^{\bar{c}\bar{d}}\right)\Bigg{]},$

with

$F_{abcd}=\frac{3}{2}\,\delta_{(ab}\delta_{cd)}-6\,\delta_{0(a}\delta_{bc}\delta_{d)0}+4\,\delta_{0a}\delta_{0b}\delta_{0c}\delta_{0d},$
(3.27)

and ${\bar{F}}_{\bar{a}\bar{b}\bar{c}\bar{d}}$ defined in the same way,
exchanging unbared indices for bared ones. In the case of the heterotic
supergravity, $(a,b)=(-\alpha^{\prime},0)$, eq. (3.26) consistently reproduces
the weak coupling limit of the U-duality invariant modular integrals
conjectured to describe the exact four-derivative couplings, c.f. ref. [22,
23, 24]. (See in particular eqs. (4.16) and (4.34) of ref. [23].555We thank
Guillaume Bossard for helpful explanations on this relation.)

### 3.3 Comments on symmetry group

We close this section with some general remarks on the symmetry group after
inclusion of the first $\alpha^{\prime}$ correction, confirming our above
conclusion that the continuous $\mathrm{O}(8,8)$ duality group is broken to
its ‘geometric subgroup’ and making some general remarks about the issue of
discrete symmetries.

We begin by determining the manifest symmetry group. Since the above
formulation is manifestly $\mathrm{O}(8,8)$ invariant if we declare the
compensator $\mbox{\sf u}^{\cal M}$ to transform as a vector it follows that
the actual, or at least manifest, symmetry group is given by the invariance
group of $\mbox{\sf u}^{\cal M}=(0,1,0)$. Recalling the index split ${\cal
M}=(M,+,-)$, an $\mathrm{O}(8,8)$ matrix reads

$L_{{\cal M}}{}^{\cal N}=\begin{pmatrix}L_{M}{}^{N}&L_{M}{}^{+}&L_{M}{}^{-}\\\
L_{+}{}^{N}&L_{+}{}^{+}&L_{+}{}^{-}\\\
L_{-}{}^{N}&L_{-}{}^{+}&L_{-}{}^{-}\end{pmatrix}\;\in\;\mathrm{O}(8,8)\;,$
(3.28)

and is subject to

$L_{{\cal M}}{}^{\cal K}\,\eta_{{\cal K}{\cal L}}\,L_{{\cal N}}{}^{\cal
L}=\eta_{{\cal M}{\cal N}}\;.$ (3.29)

The condition that $\mbox{\sf u}^{\cal M}$ is invariant, i.e. that $\mbox{\sf
u}^{\prime M}=0$, $\mbox{\sf u}_{+}^{\prime}=0$ and $\mbox{\sf
u}_{-}^{\prime}=1$, is then quickly seen to imply

$L_{M}{}^{-}=0\;,\qquad L_{+}{}^{-}=0\;,\qquad L_{-}{}^{-}=1\;.$ (3.30)

Using this in eq. (3.29) yields furthermore

$\begin{split}L_{M}{}^{N}\in\mathrm{O}(7,7)\;,\quad
L_{+}{}^{M}&=0\;,\quad(L^{-1})_{N}{}^{M}L_{-}{}^{N}=-\eta^{MN}L_{N}{}^{+}\;,\\\
L_{+}{}^{+}&=1\;,\qquad
L_{-}{}^{+}=-\frac{1}{2}L_{-}{}^{M}\eta_{MN}L_{-}{}^{N}\;.\end{split}$ (3.31)

We infer that a general $\mathrm{O}(8,8)$ matrix leaving u invariant is
parametrized in terms of a general $\mathrm{O}(7,7)$ matrix $L_{M}{}^{N}$ and
a vector $c_{M}$:

$L_{\cal M}{}^{\cal N}=\begin{pmatrix}L_{M}{}^{N}&L_{M}{}^{K}c_{K}&0\\\
0&1&0\\\ -c^{N}&-\frac{1}{2}\,c_{K}c^{K}&1\end{pmatrix}\,.$ (3.32)

One may verify that the $c_{M}$, for $L_{M}{}^{N}=\delta_{M}{}^{N}$, act as

$\xi_{M}^{\prime}=\xi_{M}+c_{M}\;,$ (3.33)

and thus precisely parameterize the constant shifts of the scalars dual to
vectors. As expected, we recovered precisely the ‘geometric subgroup’ of
$\mathrm{O}(8,8)$ consisting of the semi-direct product of $\mathrm{O}(7,7)$
and $14$-dimensional translations.

We thus conclude that among the continuous transformations there is no
enhancement to the full duality group, but this still leaves open the fate of
the discrete group $\mathrm{O}(8,8,\mathbb{Z})$, which is conjectured to be
present in the full string/M-theory. Ultimately we see no evidence of this
true U-duality in the supergravity limit considered here once
$\alpha^{\prime}$ corrections are switched on, but we will briefly discuss
some general features.

We first observe that the scaling symmetry denoted (Dilaton3) above, which
acts as $\Phi\rightarrow\Phi+\lambda$, $\lambda\in\mathbb{R}$, on the dilaton
and is hence manifestly broken due to the explicit $e^{-2\Phi}$ prefactors in
the action at order $\alpha^{\prime}$, trivializes for the discrete subgroup.
In fact, these transformations are embedded into $\mathrm{O}(8,8)$ as

$\lambda\;\mapsto\;L_{(\lambda)}=\begin{pmatrix}{\bf 1}_{7}&0&0\\\
0&e^{\lambda}&0\\\ 0&0&e^{-\lambda}\end{pmatrix}\ \in\
\mathrm{O}(8,8;\mathbb{R})\;,$ (3.34)

but the requirement that this transformation actually belongs to
$\mathrm{O}(8,8,\mathbb{Z})$, i.e. that all its matrix entries are integers
then implies that $e^{\lambda}=e^{-\lambda}=1$ or $\lambda=0$, reducing
(Dilaton3) to the trivial group. Thus, by itself the absence of the continuous
scaling symmetry at order $\alpha^{\prime}$ is not in conflict with an
$\mathrm{O}(8,8,\mathbb{Z})$ symmetry.

Is then the $\mathrm{O}(8,8,\mathbb{Z})$ perhaps realized in supergravity
after all? While it is very difficult to exclude the possibility that certain
discrete symmetries are realized in a hidden fashion, we will argue now that
there is no evidence for any enhancement to the discrete U-duality group in
supergravity at order $\alpha^{\prime}$. To this end one may use, as discussed
by Sen in ref. [1], that $\mathrm{O}(8,8,\mathbb{Z})$ can be generated by
$\mathrm{O}(7,7,\mathbb{Z})$ and the S-duality group
$\mathrm{SL}(2,\mathbb{Z})$. As the invariance under
$\mathrm{O}(7,7,\mathbb{Z})$ is manifest, it is then sufficient to study the
invariance under $\mathrm{SL}(2,\mathbb{Z})$, which consists of matrices

$\begin{pmatrix}c&d\\\ e&f\end{pmatrix},\quad c,d,e,f\in\mathbb{Z}\,,\;\;{\rm
where}\quad cf-de=1\;.$ (3.35)

These are embedded into $\mathrm{O}(8,8,\mathbb{Z})$ as follows:

$L_{\cal M}{}^{\cal N}=\begin{pmatrix}c&0&0&0&d&0\\\
0&\mathds{1}_{d-1}&0&0&0&0\\\ 0&0&f&0&0&-\,e\\\ 0&0&0&\mathds{1}_{d-1}&0&0\\\
e&0&0&0&f&0\\\ 0&0&-d&0&0&c\end{pmatrix}.$ (3.36)

Under such transformations, a generic term $\left(\mbox{\sf u}X\mbox{\sf
u}\right)$ of the action (3.15) transforms as

$\left(\mbox{\sf u}X\mbox{\sf u}\right)=X_{++}\longrightarrow
f^{2}\,X_{++}+2\,ef\,X_{(+1)}+e^{2}\,X_{11}.$ (3.37)

Thus, each term is independently invariant only if $c=f=1$ and $e=0$, leaving
only integer shifts (2.28). Therefore, non-trivial $\mathrm{SL}(2,\mathbb{Z})$
transformations can only leave the action invariant if the different anomalous
terms cancel each other. Since the traces in eq. (3.15) are invariant, and the
transformation rule (3.37) cannot generate trace factors, transformations of
terms featuring traces can only cancel among themselves. Considering the pure
trace terms in the first and sixth lines of eq. (3.15), one infers that due to
the global dilaton prefactor, these terms are only invariant under
transformations that leave the dilaton invariant, and so with eq. (2.21),

$e^{2\Phi}=\left(\mbox{\sf u}\mathcal{M}\mbox{\sf
u}\right)\longrightarrow\left(f+e\,\xi_{1}\right)^{2}e^{2\Phi}+e^{2}\,\mathcal{H}_{11}\,,$
(3.38)

implying as before $f=1$ and $e=0$, and hence $c=1$ by the
$\mathrm{SL}(2,\mathbb{Z})$ constraint (3.35). Even discarding dilaton
prefactors we have found no evidence of any hidden discrete symmetries,
supporting the above conclusion that only the continuous geometric subgroup of
$\mathrm{O}(8,8)$ is a genuine symmetry.

## 4 A massive deformation

So far the three form field-strength $H_{\mu\nu\rho}$ has been set to zero. We
now integrate out the $B$ field explicitly and explore the deformations
induced by a non-vanishing three-form flux. At the two-derivative level, this
results in a topological mass for the vectors and a potential for the dilaton
[17]. We review this massive deformation and show how it fits in the more
general framework of gauged supergravity. We then extend the analysis to the
four-derivative corrections. In particular, we show that the resulting massive
deformation induces Chern-Simons terms for composite connections featuring an
enhancement to the full $\mathrm{O}(d+1,d+1)$ to first order in the mass
parameter.

### 4.1 Two-derivative action

We first consider the two-derivative action (2.14), that we rewrite in the
string frame as

$I_{0}=\int\mathrm{d}^{3}x\,\sqrt{-g}\,e^{-\Phi}\left({\cal
L}^{(0)}\left(g_{\mu\nu},\Phi,\mathcal{H},\mathcal{A}_{\mu}{}^{M}\right)-\frac{1}{12}\,H_{\mu\nu\rho}H^{\mu\nu\rho}\right),$
(4.1)

with
$H_{\mu\nu\rho}=3\,\partial_{[\mu}B_{\nu\rho]}+\,\Omega^{\rm(MS)}_{\mu\nu\rho}$
and
$\Omega^{\rm(MS)}_{\mu\nu\rho}=-(3/2)\,\mathcal{A}_{[\mu}{}^{M}\mathcal{F}_{\nu\rho]\,M}$
the abelian Chern-Simons deformation of the three-form field-strength [20]. In
three dimensions, we can rewrite the field-strength as666The global factor is
chosen so that $h=\epsilon^{\mu\nu\rho}H_{\mu\nu\rho}$, with $g_{\mu\nu}$ of
signature $(-1,1,1)$.

$H_{\mu\nu\rho}=-\frac{1}{6}\,h\,\epsilon_{\mu\nu\rho},$ (4.2)

and $I_{0}$ becomes

$I_{0}=\int\mathrm{d}^{3}x\,\sqrt{-g}\,e^{-\Phi}\left({\cal
L}^{(0)}\left(g_{\mu\nu},\Phi,\mathcal{H},\mathcal{A}_{\mu}{}^{M}\right)+\frac{1}{72}\,h^{2}\right).$
(4.3)

In the following, we do not specify explicitly the fields on which ${\cal
L}^{(0)}$ depends. To dualise the degrees of freedom related to $B_{\mu\nu}$,
we introduce an auxiliary field $f$ and consider the action

$I^{\prime}_{0}=\int\mathrm{d}^{3}x\,\sqrt{-g}\,e^{-\Phi}\left({\cal
L}^{(0)}-\frac{1}{2}\,f^{2}-\frac{1}{6}\,fh\right).$ (4.4)

Taking its variation with respect to $f$, we get the algebraic equation of
motion

$f=-\frac{1}{6}\,h,$ (4.5)

which can be used in eq. (4.4) to get back eq. (4.3), hence the equivalence
between $I_{0}$ and $I^{\prime}_{0}$. We now consider $I^{\prime}_{0}$, which
gives the following equation of motion after varying with respect to
$B_{\mu\nu}$:

$\nabla_{\mu}\left(e^{-\Phi}f\right)=0\quad\implies\quad f=m\,e^{\Phi},\
m\in\mathbb{R}.$ (4.6)

Unlike eq. (4.5), this equation is not algebraic and, in general, could not
simply be inserted back into the action. One needs to work at the level of the
equations of motion. It can however be checked that, in the particular case we
are considering here, it is consistent to use eq. (4.6) directly in the action
(4.4) to get

$I^{\prime\prime}_{0}=\int\mathrm{d}^{3}x\left[\sqrt{-g}\left(e^{-\Phi}\,{\cal
L}^{(0)}-\frac{1}{2}\,m^{2}\,e^{\Phi}\right)-\frac{1}{6}\,m\,\varepsilon^{\mu\nu\rho}\,\Omega^{\rm(MS)}_{\mu\nu\rho}\right],$
(4.7)

where we ignored a total derivative. $I^{\prime\prime}_{0}$ is equivalent to
$I_{0}$, with the degrees of freedom of the two-form $B_{\mu\nu}$ dualised
into $m$. For $m=0$, we recover the action we started with in sec. 2.2.

Let us move to the Einstein frame to discuss further the properties of the
action:

$\displaystyle I^{\prime\prime}_{0}=\int\mathrm{d}^{3}x\bigg{[}\sqrt{-g_{\rm
E}}\,\bigg{(}$ $\displaystyle R_{\rm
E}-\partial_{\mu}\Phi\,\partial^{\mu}\Phi+\frac{1}{8}\,\mathrm{Tr}\left(\partial_{\mu}{\cal
H}^{-1}\,\partial^{\mu}{\cal
H}\right)-\frac{1}{4}\,e^{-2\Phi}\,\mathcal{F}_{\mu\nu}{}^{M}\mathcal{H}_{MN}\mathcal{F}^{\mu\nu\,N}$
(4.8)
$\displaystyle-\frac{1}{2}\,m^{2}\,e^{4\Phi}\bigg{)}+\frac{1}{4}\,m\,\varepsilon^{\mu\nu\rho}\,\mathcal{A}_{\mu}{}^{M}\mathcal{F}_{\nu\rho\,M}\bigg{]}.$

For $m\neq 0$, the $\mathrm{O}(d+1,d+1)$ symmetry of the action (2.23) is
broken down to $\mathrm{O}(d,d)$ and constant shifts in $\xi_{M}$ (the scaling
symmetry (2.8) is broken). The term quadratic in $m$ acts as a potential, and
the vectors $\mathcal{A}_{\mu}{}^{M}$ acquire a topological mass proportional
to $m$, leading to a topologically massive Yang-Mills theory [25, 19] for
${\rm U}(1)^{d}$, with equations of motion

$\nabla_{\mu}\Big{(}e^{-2\Phi}\,\mathcal{F}^{\mu\nu\,N}\mathcal{H}_{NM}-m\,\epsilon^{\mu\nu\rho}\,\mathcal{A}_{\rho\,M}\Big{)}=0.$
(4.9)

The Chern-Simons coupling prevents us from dualizing the vectors as done in
sec. 2.2. We can however rewrite the Yang-Mills gauging of eq. (4.8) as a pure
Chern-Simons type gauging with gauge group ${\rm U}(1)^{d}\ltimes T_{d}$ using
the on-shell equivalence of ref. [26], where $T_{d}$ is a $d$-dimensional
translation group. Consider the action

$\displaystyle\widetilde{I}^{\prime\prime}_{0}=\int\mathrm{d}^{3}x\bigg{[}$
$\displaystyle\sqrt{-g_{\rm E}}\,\Big{(}R_{\rm
E}+\frac{1}{8}\,\mathrm{Tr}\left(\partial_{\mu}\mathcal{M}^{-1}\,\partial^{\mu}\mathcal{M}\right)-\frac{1}{2}\,m^{2}\,e^{2\Phi}\,\mathcal{A}_{\mu}{}^{M}\mathcal{H}_{MN}\mathcal{A}^{\mu\,N}$
(4.10)
$\displaystyle-m\,e^{2\Phi}\,\mathcal{A}_{\mu}{}^{M}\mathcal{H}_{MN}\partial^{\mu}\xi^{N}-\frac{1}{2}\,m^{2}\,e^{4\Phi}\Big{)}-\frac{1}{4}\,m\,\varepsilon^{\mu\nu\rho}\,\mathcal{A}_{\mu}{}^{M}\mathcal{F}_{\nu\rho\,M}\bigg{]},$

The induced equations of motion for the vectors are

$\mathcal{F}_{\mu\nu}{}^{M}=e^{2\Phi}\,\epsilon_{\mu\nu\rho}\left(\partial^{\rho}\xi^{N}+m\,\mathcal{A}^{\rho\,N}\right)\mathcal{H}_{N}{}^{M},$
(4.11)

which imply eq. (4.9). It can be checked similarly that all the equations of
motion of $\widetilde{I}^{\prime\prime}_{0}$ are identical on-shell to those
of $I^{\prime\prime}_{0}$, upon systematically eliminating
$\partial_{\mu}\xi_{M}$ using eq. (4.11). Thus, $I^{\prime\prime}_{0}$ and
$\widetilde{I}^{\prime\prime}_{0}$ are equivalent.

The action (4.10) can be nicely rewritten using the embedding tensor formalism
[27, 28] of three-dimensional half-maximal gauged supergravity [29, 30] (see
also ref. [31] for a review and the notations used here). The bosonic part of
the action describes the dynamics of the metric $g_{\rm E\,\mu\nu}$, scalars
$\mathcal{M}_{\cal MN}\in\mathrm{O}(d+1,d+1)$ and vectors
$A_{\mu}{}^{\cal[MN]}$ via the action

$\int\mathrm{d}^{3}x\left[\sqrt{-g_{\rm E}}\,\Big{(}R_{\rm
E}+\frac{1}{8}\,\mathrm{Tr}\left(D_{\mu}\mathcal{M}^{-1}\,D^{\mu}\mathcal{M}\right)-V\Big{)}+{\cal
L}_{\rm CS}\right].$ (4.12)

The covariant derivative on $\mathcal{M}_{\cal MN}$ is

$D_{\mu}\mathcal{M}_{\cal MN}=\partial_{\mu}\mathcal{M}_{\cal
MN}+4\,A_{\mu}{}^{\cal PQ}\Theta_{\cal PQ|(M}{}^{\cal K}\mathcal{M}_{\cal
N)K},$ (4.13)

with the gauging given by the embedding tensor

$\Theta_{\cal MN|PQ}=\frac{1}{2}\left(\eta_{\cal M[P}\,\theta_{\cal
Q]N}-\eta_{\cal N[P}\,\theta_{\cal Q]M}\right),$ (4.14)

where $\theta_{\cal MN}$ is symmetric. In full generality, the embedding
tensor contains more representations that we will not need here. The vectors
are described by a Chern-Simons term of the form

${\cal L}_{\rm CS}=-\varepsilon^{\,\mu\nu\rho}\,\Theta_{\cal
MN|PQ}\,A_{\mu}{}^{\cal MN}\left(\partial_{\nu}\,A_{\rho}{}^{\cal
PQ}+\frac{1}{3}\,\Theta_{\cal RS|UV}\,f^{\cal PQ,RS}{}_{\cal
XY}\,A_{\nu}{}^{\cal UV}A_{\rho}{}^{\cal XY}\right),$ (4.15)

with $f^{\cal MN,PQ}{}_{\cal KL}=4\,\delta_{[\cal K}{}^{[\cal M}\eta^{\cal
N][P}\delta_{\cal L]}{}^{\cal Q]}$ the structure constants of
$\mathfrak{so}(d+1,d+1)$. Finally, the potential is given by

$V=\frac{1}{8}\,\theta_{\cal MN}\theta_{\cal PQ}\left(2\,\mathcal{M}^{\cal
MP}\mathcal{M}^{\cal NQ}-2\,\eta^{\cal MP}\eta^{\cal NQ}-\mathcal{M}^{\cal
MN}\mathcal{M}^{\cal PQ}\right).$ (4.16)

The action (4.10) then results from the restriction to an embedding tensor
with only non-vanishing component $\theta_{--}$. More precisely, a formally
$\mathrm{O}(d+1,d+1)$-covariant form of $I^{\prime\prime}_{0}$ is given by

$\displaystyle\widetilde{I}^{\prime\prime}_{0}=\int\mathrm{d}^{3}x\bigg{[}$
$\displaystyle-\varepsilon^{\,\mu\nu\rho}\,\theta_{\cal MN}\,A_{\mu\,{\cal
P}}{}^{\cal N}\partial_{\nu}\,A_{\rho}{}^{\cal PM}+\sqrt{-g_{\rm
E}}\,\bigg{(}R_{\rm
E}+\frac{1}{8}\,\mathrm{Tr}\left(D_{\mu}\mathcal{M}^{-1}\,D^{\mu}\mathcal{M}\right)$
(4.17) $\displaystyle-\frac{1}{8}\,\theta_{\cal MN}\theta_{\cal
PQ}\left(2\,\mathcal{M}^{\cal MP}\mathcal{M}^{\cal NQ}-2\,\eta^{\cal
MP}\eta^{\cal NQ}-\mathcal{M}^{\cal MN}\mathcal{M}^{\cal
PQ}\right)\bigg{)}\bigg{]},$

with the specific parametrization

$A_{\mu}{}^{M-}=\frac{1}{2}\,\mathcal{A}_{\mu}{}^{M},\quad\theta_{--}=2\,m.$
(4.18)

The covariant derivative is then given by
$D_{\mu}\xi_{M}=\partial_{\mu}\xi_{M}+m\,\mathcal{A}_{\mu\,M}$,
$D_{\mu}\Phi=\partial_{\mu}\Phi$ and
$D_{\mu}\mathcal{H}_{MN}=\partial_{\mu}\mathcal{H}_{MN}$. The symmetry
breaking induced by $m\neq 0$ is now translated into the choice of embedding
tensor with $\theta_{--}$ as the only non-vanishing component, that breaks
$\mathrm{O}(d+1,d+1)$ to $\mathrm{O}(d,d)$ and shifts in $\xi_{M}$.

### 4.2 Four-derivative action

The 2-form degrees of freedom can be integrated out in the four-derivative
action using the same procedure as the one used in sec. 4.1. With eq. (4.2),
the action (7.16) of ref. [12] is given by

$\widetilde{I}=I+\int\mathrm{d}^{3}x\,\sqrt{-g}\,e^{-\Phi}\left[\frac{1}{72}\,h^{2}+\alpha^{\prime}\left(-\frac{1}{6}\,\mathfrak{a}\,h+\frac{1}{36}\,\mathfrak{b}\,h^{2}+\frac{1}{6^{4}}\,\mathfrak{c}\,h^{4}\right)\right],$
(4.19)

where $I$ is the action (3.4) and777Remember that $a$ and $b$ are of order
$\alpha^{\prime}$.

$\displaystyle\mathfrak{a}$
$\displaystyle=\epsilon^{\mu\nu\rho}\bigg{[}-\frac{a+b}{8\,\alpha^{\prime}}\left(\frac{2}{3}\,\Omega_{\mu\nu\rho}^{\rm(GS)}-\frac{1}{2}\,\mathcal{F}_{\mu\sigma}{}^{M}\left(\mathcal{H}\partial_{\nu}\mathcal{H}^{-1}\right)_{M}{}^{N}\mathcal{F}_{\rho}{}^{\sigma}{}_{N}\right)$
(4.20)
$\displaystyle\qquad\qquad+\frac{a-b}{4\,\alpha^{\prime}}\left(\Omega_{\mu\nu\rho}^{(\omega)}+\frac{1}{4}\,\mathcal{F}_{\mu\sigma}{}^{M}\partial^{\sigma}\mathcal{H}_{MN}\mathcal{F}_{\nu\rho}{}^{N}\right)\bigg{]},$
$\displaystyle\mathfrak{b}$
$\displaystyle=-\frac{a+b}{8\,\alpha^{\prime}}\left(R-\frac{1}{4}\,\mathrm{Tr}\left(\partial_{\mu}\mathcal{H}\partial^{\mu}\mathcal{H}^{-1}\right)+\frac{3}{4}\,\mathcal{F}_{\mu\nu}{}^{M}\mathcal{H}_{MN}\mathcal{F}^{\mu\nu\,N}\right)+\frac{a-b}{4\,\alpha^{\prime}}\,\frac{1}{8}\,\mathcal{F}_{\mu\nu}{}^{M}\mathcal{F}^{\mu\nu}{}_{M},$
$\displaystyle\mathfrak{c}$
$\displaystyle=\frac{a+b}{8\,\alpha^{\prime}}\,\frac{5}{4}.$

$\Omega^{(\omega)}_{\mu\nu\rho}$ is the gravitational Chern-Simons form of
heterotic supergravity, and $\Omega^{\rm(GS)}_{\mu\nu\rho}$ is the three-form
needed in the Green-Schwarz type mechanism of ref. [11], which
satisfies888Normalised as in eq. (14) of ref. [11], i.e.
$\widetilde{H}_{\mu\nu\rho}=H_{\mu\nu\rho}+\frac{a+b}{2}\,\Omega^{\rm(GS)}_{\mu\nu\rho}$.

$4\,\partial_{[\mu}\Omega^{\rm(GS)}_{\nu\rho\sigma]}=\frac{3}{8}\,\mathrm{Tr}\left(\partial_{[\mu}\mathcal{H}\partial_{\nu}\mathcal{H}^{-1}\partial_{\rho}\mathcal{H}\partial_{\sigma]}\mathcal{H}^{-1}\mathcal{H}\eta\right).$
(4.21)

As previously, we can equivalently write $\widetilde{I}$ as

$\widetilde{I}^{\prime}=I+\int\mathrm{d}^{3}x\,\sqrt{-g}\,e^{-\Phi}\left[-\frac{1}{2}\,f^{2}-\frac{1}{6}\,fh+\alpha^{\prime}\left(\mathfrak{a}\,f+\mathfrak{b}\,f^{2}+\mathfrak{c}\,f^{4}\right)\right],$
(4.22)

with an auxiliary field $f$. The equations of motion for $f$ are

$f=-\frac{1}{6}\,h+\alpha^{\prime}\left(\mathfrak{a}+2\,\mathfrak{b}\,f+4\,\mathfrak{c}\,f^{3}\right),$
(4.23)

and can be solved perturbatively in $\alpha^{\prime}$:

$f=f^{(0)}+\alpha^{\prime}\,f^{(1)},\ {\rm with}\ \begin{cases}\displaystyle
f^{(0)}=-\frac{1}{6}\,h,\\\ \displaystyle
f^{(1)}=\mathfrak{a}-\frac{1}{3}\,\mathfrak{b}\,h-\frac{1}{54}\,\mathfrak{c}\,h^{3}.\end{cases}$
(4.24)

As in the two-derivative case, eq. (4.22) with eq. (4.24) gives
$\widetilde{I}$. The equations of motion for $B_{\mu\nu}$ are unchanged and
given by eq. (4.6) and, again, it is consistent to use them directly in the
action, leading to

$\widetilde{I}^{\prime\prime}=I+\int\mathrm{d}^{3}x\,\sqrt{-g}\left[-\frac{1}{6}\,m\,\epsilon^{\mu\nu\rho}\,\Omega^{\rm(MS)}_{\mu\nu\rho}-\frac{1}{2}\,m^{2}\,e^{\Phi}+\alpha^{\prime}\left(\mathfrak{a}\,m\\!+\\!\mathfrak{b}\,m^{2}\,e^{\Phi}+\mathfrak{c}\,m^{4}\,e^{3\Phi}\right)\right].$
(4.25)

As a by product, observe that we can safely consider the case $m=0$ and
recover the actions considered in sec. 2 and 3.

Writing this action in terms of $\mathrm{O}(d+1,d+1)$ fields and an embedding
tensor breaking the symmetry to $\mathrm{O}(d,d)\times\mathrm{O}(1,1)$, as we
did for the two-derivative action in sec. 4.1, requires to reproduce the
analysis of sec. 3 with modified rules for the field redefinitions (given the
modified two-derivative equations of motion) and with the additional terms in
eq. (4.25), as detailed in app. B. All computations done, we get

$\displaystyle\widetilde{I}^{\prime\prime}_{1}$
$\displaystyle=\widehat{I}_{1}+\int\mathrm{d}^{3}x\,\Bigg{\\{}\sqrt{-g_{\rm
E}}\,\frac{a+b}{8}\,\bigg{(}4\,m^{2}\,e^{2\Phi}\,\partial_{\mu}\Phi\partial^{\mu}\Phi+m^{4}\,e^{6\Phi}\bigg{)}$
(4.26)
$\displaystyle+m\,\varepsilon^{\mu\nu\rho}\bigg{[}-\frac{a+b}{8}\bigg{(}\frac{2}{3}\,\Omega_{\mu\nu\rho}^{(\rm
GS)}-\frac{1}{2}\,e^{2\Phi}\,D_{\mu}\xi_{M}\left(\mathcal{H}^{-1}\partial_{\nu}\mathcal{H}\right)^{M}{}_{N}D_{\rho}\xi^{N}\bigg{)}+\frac{a-b}{4}\,\Omega_{\mu\nu\rho}^{(\omega_{\rm
E})}\bigg{]}\Bigg{\\}},$

where $\widehat{I}_{1}$ is given by the action (3.15) with all currents
covariantized:
$\widehat{\mathcal{J}}_{\mu}=D_{\mu}\mathcal{M}\mathcal{M}^{-1}$. Observe that
the Green-Schwarz type mechanism of ref. [11] generates, once restricted to
three dimensions, a Chern-Simons term based on composite gauge fields. The
properties of this term are best displayed using the frame formalism of sec.
2.2. In ref. [11], the Green-Schwarz three-from
$\Omega_{\mu\nu\rho}^{\rm(GS)}$ has been written as a Chern-Simons form for
$\mathrm{O}(d)\times O(d)$ composite gauge fields:

$\Omega_{\mu\nu\rho}^{\rm(GS)}=\frac{3}{2}\,{\rm
CS}_{\mu\nu\rho}\left(Q\right)=\frac{3}{2}\,\mathrm{Tr}\left(Q_{[\mu}\partial_{\nu}Q_{\rho]}\delta\eta+\frac{2}{3}\,Q_{[\mu}Q_{\nu}Q_{\rho]}\delta\eta\right),$
(4.27)

with $Q_{\mu\,A}{}^{B}$ defined in eq. (2.32) and $\delta_{AB}$ the identity
matrix.999The $\delta\eta$ in eq. (4.27) is needed to reproduce the right
relative sign in eq. (34) of ref. [11]. Generalizing this definition to the
covariantization $\widehat{\cal Q}_{\mu\,{\cal A}}{}^{\cal B}$ of the
$\mathrm{O}(d+1)\times\mathrm{O}(d+1)$ connection ${\cal Q}_{\mu\,{\cal
A}}{}^{\cal B}$ of eq. (2.32) (i.e. defining $\widehat{\cal Q}_{\mu\,{\cal
A}}{}^{\cal B}$ from the covariantized Maurer-Cartan form ${\cal
V}^{-1}D_{\mu}{\cal V}$), we get

${\rm CS}_{\mu\nu\rho}\left(\widehat{\cal
Q}\right)=\frac{2}{3}\,\Omega_{\mu\nu\rho}^{(\rm
GS)}-\frac{1}{2}\,e^{2\Phi}\,D_{[\mu}\xi_{M}\left(\mathcal{H}^{-1}\partial_{\nu}\mathcal{H}\right)^{M}{}_{N}D_{\rho]}\xi^{N}-\frac{m}{2}\,e^{2\Phi}\,D_{[\mu}\xi_{M}\mathcal{F}_{\nu\rho]}{}^{M},$
(4.28)

of exterior derivative101010Here, we used that
$\left[D_{\mu},D_{\nu}\right]\mathcal{M}_{\cal MN}=4\,F_{\mu\nu}{}^{\cal
PQ}\Theta_{\cal PQ|(M}{}^{\cal K}\mathcal{M}_{\cal N)K}$ and
$F_{\mu\nu}{}^{\cal MN}=2\,\partial_{[\mu}A_{\nu]}{}^{\cal
MN}+4\,A_{[\mu}{}^{\cal PQ}\Theta_{\cal PQ|K}{}^{\cal[M}A_{\nu]}{}^{\cal
N]K}$.

$\displaystyle\partial_{[\mu}{\rm CS}_{\nu\rho\sigma]}\left(\widehat{\cal
Q}\right)$
$\displaystyle=\frac{1}{16}\,\mathrm{Tr}\left(D_{[\mu}\mathcal{M}D_{\nu}\mathcal{M}^{-1}D_{\rho}\mathcal{M}D_{\sigma]}\mathcal{M}^{-1}\mathcal{M}\eta\right)$
(4.29) $\displaystyle\quad-2\,\Theta_{\cal MN|PQ}\,F_{[\mu\nu}{}^{\cal
MN}\left(D_{\rho}\mathcal{M}^{-1}D_{\sigma]}\mathcal{M}\mathcal{M}^{-1}\right)^{\cal
PQ}$ $\displaystyle\quad+16\,\Theta_{\cal MN|PQ}\Theta_{\cal KL|R}{}^{\cal
P}\,F_{[\mu\nu}{}^{\cal MN}F_{\rho\sigma]}{}^{\cal KL}\mathcal{M}^{\cal RQ}.$

Putting this term in the action and eliminating the field-strength using the
dualisation relation (4.11) gives

$\displaystyle\int\mathrm{d}^{3}x\,m\,\varepsilon^{\mu\nu\rho}\,{\rm
CS}_{\mu\nu\rho}\left(\widehat{\cal
Q}\right)=\int\mathrm{d}^{3}x\,m\,\bigg{[}$
$\displaystyle\varepsilon^{\mu\nu\rho}\,\left(\frac{2}{3}\,\Omega_{\mu\nu\rho}^{(\rm
GS)}-\frac{1}{2}\,e^{2\Phi}\,D_{\mu}\xi_{M}\left(\mathcal{H}^{-1}\partial_{\nu}\mathcal{H}\right)^{M}{}_{N}D_{\rho}\xi^{N}\right)$
(4.30) $\displaystyle+\sqrt{-g_{\rm
E}}\,m^{2}\,e^{4\Phi}\,D_{\mu}\xi_{M}\mathcal{H}^{MN}D^{\mu}\xi_{N}\bigg{]}.$

Thus, we can write the action (4.26) in terms of the three-form (4.28):

$\displaystyle\widetilde{I}^{\prime\prime}_{1}=\widehat{I}_{1}+$
$\displaystyle\int\mathrm{d}^{3}x\,\Bigg{[}\sqrt{-g_{\rm
E}}\,\frac{a+b}{8}\,\bigg{(}m^{2}\Big{(}4\,e^{2\Phi}\,\partial_{\mu}\Phi\partial^{\mu}\Phi+e^{4\Phi}\,D_{\mu}\xi_{M}\mathcal{H}^{MN}D^{\mu}\xi_{N}\Big{)}+m^{4}\,e^{6\Phi}\bigg{)}$
(4.31) $\displaystyle+m\,\varepsilon^{\mu\nu\rho}\bigg{(}-\frac{a+b}{8}\,{\rm
CS}_{\mu\nu\rho}\left(\widehat{\cal
Q}\right)+\frac{a-b}{4}\,\Omega_{\mu\nu\rho}^{(\omega_{\rm
E})}\bigg{)}\Bigg{]}.$

We can furthermore express the first line in terms of the
$\mathrm{O}(d+1,d+1)$ currents, as done in sec. 3.2, and the embedding tensor
(4.18):

$\displaystyle\widetilde{I}^{\prime\prime}_{1}=\widehat{I}_{1}+$
$\displaystyle\int\mathrm{d}^{3}x\,m\,\Bigg{[}\sqrt{-g_{\rm
E}}\,\frac{a+b}{8}\,\left(\frac{1}{2}\,\theta_{\cal
MN}\Big{(}\widehat{\mathcal{J}}_{\mu}\widehat{\mathcal{J}}^{\mu}\mathcal{M}\Big{)}^{\cal
MN}+\frac{1}{8}\left(\theta_{\cal MN}\mathcal{M}^{\cal MN}\right)^{3}\right)$
(4.32) $\displaystyle+\varepsilon^{\mu\nu\rho}\bigg{(}-\frac{a+b}{8}\,{\rm
CS}_{\mu\nu\rho}\left(\widehat{\cal
Q}\right)+\frac{a-b}{4}\,\Omega_{\mu\nu\rho}^{(\omega_{\rm
E})}\bigg{)}\Bigg{]}.$

Thus, the massive deformation following from integrating out the $B$ field
induces a gauging of the action (3.15) and additional couplings. Remarkably,
these couplings break the $\mathrm{O}(d+1,d+1)$ symmetry only due to the
gauging (4.18) of the shift symmetry; no vector compensator is needed, in
contrast to the term $\widehat{I}_{1}$. The first line in eq. (4.32) features
a deformation of the two-derivative action, and a potential. The second line
of eq. (4.32) is given by Chern-Simons terms based on composite and spin
connections. Most interestingly, to leading order in $m$, these Chern-Simons
terms are given by

$\displaystyle\int\mathrm{d}^{3}x\,m\,\varepsilon^{\mu\nu\rho}\bigg{(}-\frac{a+b}{8}$
$\displaystyle{\rm CS}_{\mu\nu\rho}\left(\widehat{\cal
Q}\right)+\frac{a-b}{4}\,\Omega_{\mu\nu\rho}^{(\omega_{\rm E})}\bigg{)}$
(4.33)
$\displaystyle=\int\mathrm{d}^{3}x\,m\,\varepsilon^{\mu\nu\rho}\bigg{(}-\frac{a+b}{8}\,{\rm
CS}_{\mu\nu\rho}\left({\cal
Q}\right)+\frac{a-b}{4}\,\Omega_{\mu\nu\rho}^{(\omega_{\rm E})}\bigg{)}+{\cal
O}\left(m^{2}\right),$

which is invariant under $\mathrm{O}(d+1,d+1)$. Although the full theory
exhibits a breaking of the $\mathrm{O}(d+1,d+1)$ U-duality both by the higher-
derivative parameter $\alpha^{\prime}$ and the mass deformation $m$, for the
leading Chern-Simons terms the full $\mathrm{O}(d+1,d+1)$ is restored. The
physical meaning of this observation has to be investigated, as well as its
extension to more general gaugings.

## 5 Conclusions

In this paper we have computed the effective action of bosonic and heterotic
string theory in three dimensions to first order in $\alpha^{\prime}$ starting
from the known four-derivative result in manifestly
$\mathrm{O}(d,d)$-invariant form and perturbatively dualizing the vector gauge
fields into scalars. We have cast the result into a formally
$\mathrm{O}(d+1,d+1)$-invariant form upon introduction of a non-dynamical
compensator $\mbox{\sf u}^{\cal M}$. The resulting action reveals the
intriguing chiral pattern (3.18), showing that in particular the action of the
bosonic string can be reconstructed from the heterotic action. One may expect
that an extension of this structure to higher orders in $\alpha^{\prime}$ will
constrain the potential higher order corrections in a similar way.

Another interesting application of the formally
$\mathrm{O}(d+1,d+1)$-invariant formulation will be the study of
$\mathrm{O}(4,4)$ triality rotations on the resulting action. This would for
instance allow one to relate the inequivalent higher order corrections
obtained from $T^{3}$ compactification of chiral ${\cal N}=(2,0)$ and non-
chiral ${\cal N}=(1,1)$ supergravity in six dimensions, respectively. With the
latter corresponding to the standard heterotic corrections (see [32] for a
recent discussion) this may be turned into a prediction of the
$\alpha^{\prime}$ corrections of the chiral theory in six dimensions. As this
theory arises from ten-dimensional type IIB supergravity compactified on the
complex surface K3, this computation would give profitable insights into the
higher-derivative corrections in ten dimensions.

The other main result of this paper is the massive deformation identified in
sec. 4 upon integrating out the $B$ field in three dimensions, keeping a
constant three-form flux. As exhibited in eq. (4.32) above, this gives rise to
a deformation of the target space metric and the scalar potential, as well as
to new Chern-Simons terms for composite connections which remarkably exhibit
an enhancement to the full $\mathrm{O}(d+1,d+1)$ to first order in the mass
parameter. Within the general framework of gauged supergravities, such massive
deformations take the form of a particular and somewhat degenerate example of
a general gauging. In this respect, our results may be viewed as a glimpse
into the structure of the $\alpha^{\prime}$ corrections of more general gauged
supergravities in three dimensions. Of particular interest for holographic
applications would be the study of these deformations around the
AdS${}_{3}\times S^{3}$ background. We hope to come back to these issues in
the future.

## Acknowledgements

We thank Guillaume Bossard and Massimo Porrati for comments and discussions.
CE is supported by the FWO-Vlaanderen through the project G006119N and by the
Vrije Universiteit Brussel through the Strategic Research Program “High-Energy
Physics”. OH is supported by the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation program (grant agreement
No 771862).

##  Appendix A $\mathrm{O}(d,d)$ decompositions of $\mathrm{O}(d+1,d+1)$

We list in the following the $\mathrm{O}(d,d)$ decomposition of the
$\mathrm{O}(d+1,d+1)$ terms used in sec. 3.2 and of the basis of
$\mathrm{O}(d+1,d+1)$-invariant terms carrying four derivatives [11].

$\mathrm{Tr}\left(\mathcal{J}_{\mu}\mathcal{J}_{\nu}\right)=-\mathrm{Tr}\left(\partial_{\mu}{\cal
H}\partial_{\nu}{\cal
H}^{-1}\right)+4\,e^{2\Phi}\,\partial_{\mu}\xi_{M}\mathcal{H}^{MN}\partial_{\nu}\xi_{N}+8\,\partial_{\mu}\Phi\partial_{\nu}\Phi.$
(A.1)
$\left(\mathcal{J}_{\mu}\mathcal{J}_{\nu}\mathcal{M}\right)_{++}=e^{4\Phi}\,\partial_{\mu}\xi_{M}\mathcal{H}^{MN}\partial_{\nu}\xi_{N}+4\,e^{2\Phi}\,\partial_{\mu}\Phi\partial_{\nu}\Phi.$
(A.2)

$\left(\mathcal{J}_{\mu}\mathcal{J}_{\nu}\eta\right)_{++}=-e^{4\Phi}\,\partial_{\mu}\xi_{M}\partial_{\nu}\xi^{M}.$
(A.3)

$\displaystyle\mathrm{Tr}\left(\mathcal{J}_{\mu}\mathcal{J}_{\nu}\mathcal{J}^{\mu}\mathcal{J}^{\nu}\right)=\mathrm{Tr}\left(\partial_{\mu}{\cal
H}\partial_{\nu}{\cal H}^{-1}\partial^{\mu}{\cal H}\partial^{\nu}{\cal
H}^{-1}\right)-8\,e^{2\Phi}\,\partial_{\mu}\xi_{M}\left(\mathcal{H}^{-1}\partial_{\nu}\mathcal{H}\partial^{\mu}\mathcal{H}^{-1}\right)^{MN}\partial^{\nu}\xi_{N}$
(A.4)
$\displaystyle\qquad\quad-16\,e^{2\Phi}\,\partial_{\mu}\Phi\partial_{\nu}\xi_{M}\partial^{\mu}\mathcal{H}^{MN}\partial^{\nu}\xi_{N}+32\,\partial_{\mu}\Phi\partial^{\mu}\Phi\partial_{\nu}\Phi\partial^{\mu}\Phi+4\,e^{4\Phi}\,\partial_{\mu}\xi_{M}\partial_{\nu}\xi^{M}\partial^{\mu}\xi_{N}\partial^{\nu}\xi^{N}$
$\displaystyle\qquad\quad+4\,e^{4\Phi}\,\partial_{\mu}\xi_{M}\mathcal{H}^{MN}\partial_{\nu}\xi_{N}\partial^{\mu}\xi_{P}\mathcal{H}^{PQ}\partial^{\nu}\xi^{Q}+32\,e^{2\Phi}\,\partial_{\mu}\Phi\partial_{\nu}\Phi\partial^{\mu}\xi_{M}\mathcal{H}^{MN}\partial^{\nu}\xi_{N}.$
$\displaystyle\mathrm{Tr}\left(\mathcal{J}_{\mu}\mathcal{J}^{\mu}\mathcal{J}_{\nu}\mathcal{J}^{\nu}\right)=\mathrm{Tr}\left(\partial_{\mu}{\cal
H}\partial^{\mu}{\cal H}^{-1}\partial_{\nu}{\cal H}\partial^{\nu}{\cal
H}^{-1}\right)-4\,e^{2\Phi}\,\partial_{\mu}\xi_{M}\left(\mathcal{H}^{-1}\partial_{\nu}\mathcal{H}\partial^{\nu}\mathcal{H}^{-1}\right)^{MN}\partial^{\mu}\xi_{N}$
(A.5)
$\displaystyle\qquad\quad-4\,e^{2\Phi}\,\partial_{\mu}\xi_{M}\left(\mathcal{H}^{-1}\partial^{\mu}\mathcal{H}\partial_{\nu}\mathcal{H}^{-1}\right)^{MN}\partial^{\nu}\xi_{N}-16\,e^{2\Phi}\,\partial_{\mu}\Phi\partial^{\mu}\xi_{M}\partial_{\nu}\mathcal{H}^{MN}\partial^{\nu}\xi_{N}$
$\displaystyle\qquad\quad+32\,\partial_{\mu}\Phi\partial^{\mu}\Phi\partial_{\nu}\Phi\partial^{\mu}\Phi+2\,e^{4\Phi}\,\partial_{\mu}\xi_{M}\partial^{\mu}\xi^{M}\partial_{\nu}\xi_{N}\partial^{\nu}\xi^{N}+2\,e^{4\Phi}\,\partial_{\mu}\xi_{M}\partial_{\nu}\xi^{M}\partial^{\mu}\xi_{N}\partial^{\nu}\xi^{N}$
$\displaystyle\qquad\quad+2\,e^{4\Phi}\,\partial_{\mu}\xi_{M}\mathcal{H}^{MN}\partial^{\mu}\xi_{N}\partial_{\nu}\xi_{P}\mathcal{H}^{PQ}\partial^{\nu}\xi^{Q}+2\,e^{4\Phi}\,\partial_{\mu}\xi_{M}\mathcal{H}^{MN}\partial_{\nu}\xi_{N}\partial^{\mu}\xi_{P}\mathcal{H}^{PQ}\partial^{\nu}\xi^{Q}$
$\displaystyle\qquad\quad+16\,e^{2\Phi}\,\partial_{\mu}\Phi\partial^{\mu}\Phi\partial_{\nu}\xi_{M}\mathcal{H}^{MN}\partial^{\nu}\xi_{N}+16\,e^{2\Phi}\,\partial_{\mu}\Phi\partial_{\nu}\Phi\partial^{\mu}\xi_{M}\mathcal{H}^{MN}\partial^{\nu}\xi_{N}.$
$\displaystyle\mathrm{Tr}\left(\mathcal{J}_{\mu}\mathcal{J}^{\mu}\mathcal{J}_{\nu}\mathcal{J}^{\nu}{\cal
M}\eta\right)=\mathrm{Tr}\left(\partial_{\mu}{\cal H}\partial^{\mu}{\cal
H}^{-1}\partial_{\nu}{\cal H}\partial^{\nu}{\cal H}^{-1}{\cal
H}\eta\right)-4\,e^{2\Phi}\,\partial_{\mu}\xi_{M}\left(\partial_{\nu}\mathcal{H}^{-1}\partial^{\nu}\mathcal{H}\right)^{M}{}_{N}\partial^{\mu}\xi^{N}$
(A.6)
$\displaystyle\qquad\quad+4\,e^{2\Phi}\,\partial_{\mu}\xi_{M}\left(\partial^{\mu}\mathcal{H}^{-1}\partial_{\nu}\mathcal{H}\right)^{M}{}_{N}\partial^{\nu}\xi^{N}-16\,e^{2\Phi}\,\partial_{\mu}\Phi\partial^{\mu}\xi_{M}\left(\mathcal{H}^{-1}\partial_{\nu}\mathcal{H}\right)^{M}{}_{N}\partial^{\nu}\xi^{N}$
$\displaystyle\qquad\quad+4\,e^{4\Phi}\,\partial_{\mu}\xi_{M}\mathcal{H}^{MN}\partial_{\nu}\xi_{N}\partial^{\mu}\xi_{P}\partial^{\nu}\xi^{P}-4\,e^{4\Phi}\,\partial_{\mu}\xi_{M}\mathcal{H}^{MN}\partial^{\mu}\xi_{N}\partial_{\nu}\xi_{P}\partial^{\nu}\xi^{P}$
$\displaystyle\qquad\quad-16\,e^{2\Phi}\,\partial_{\mu}\Phi\partial^{\mu}\Phi\partial_{\nu}\xi_{M}\partial^{\nu}\xi^{M}+16\,e^{2\Phi}\,\partial_{\mu}\Phi\partial_{\nu}\Phi\partial^{\mu}\xi_{M}\partial^{\nu}\xi^{M}.$

$\displaystyle\left(\mathcal{J}_{\mu}\mathcal{J}_{\nu}\mathcal{J}^{\mu}\mathcal{J}^{\nu}\mathcal{M}\right)_{++}$
$\displaystyle=-\,e^{4\Phi}\,\partial_{\mu}\xi_{M}\left(\mathcal{H}^{-1}\partial_{\nu}\mathcal{H}\partial^{\mu}\mathcal{H}^{-1}\right)^{MN}\partial^{\nu}\xi_{N}-4\,e^{4\Phi}\,\partial_{\mu}\Phi\,\partial_{\nu}\xi_{M}\partial^{\mu}\mathcal{H}^{MN}\partial_{\nu}\xi_{N}$
(A.7)
$\displaystyle\quad+16\,e^{2\Phi}\,\partial_{\mu}\Phi\partial^{\mu}\Phi\partial_{\nu}\Phi\partial^{\nu}\Phi+e^{6\,\Phi}\,\partial_{\mu}\xi_{M}\mathcal{H}^{MN}\partial_{\nu}\xi_{N}\,\partial^{\mu}\xi_{P}\mathcal{H}^{PQ}\partial^{\nu}\xi_{Q}$
$\displaystyle\quad+e^{6\Phi}\,\partial_{\mu}\xi_{M}\partial_{\nu}\xi^{M}\,\partial^{\mu}\xi_{N}\partial^{\nu}\xi^{N}+12\,e^{4\Phi}\,\partial_{\mu}\Phi\partial_{\nu}\Phi\partial^{\mu}\xi_{M}\mathcal{H}^{MN}\partial^{\nu}\xi_{N}.$

$\displaystyle\left(\mathcal{J}_{\mu}\mathcal{J}_{\nu}\mathcal{J}^{\mu}\mathcal{J}^{\nu}\eta\right)_{++}$
$\displaystyle=e^{4\Phi}\,\partial_{\mu}\xi_{M}\left(\partial_{\nu}\mathcal{H}^{-1}\partial^{\mu}\mathcal{H}\right)^{M}{}_{N}\partial^{\nu}\xi^{N}-2\,e^{6\Phi}\,\partial_{\mu}\xi_{M}\partial_{\nu}\xi^{M}\,\partial^{\mu}\xi_{P}\mathcal{H}^{PQ}\partial^{\nu}\xi_{Q}$
(A.8)
$\displaystyle\quad-4\,e^{4\Phi}\,\partial_{\mu}\Phi\partial_{\nu}\Phi\partial^{\mu}\xi_{M}\partial^{\nu}\xi^{M}.$

$\displaystyle\left(\mathcal{J}_{\mu}\mathcal{J}^{\mu}\mathcal{J}_{\nu}\mathcal{J}^{\nu}\mathcal{M}\right)_{++}$
$\displaystyle=-\,e^{4\Phi}\,\partial_{\mu}\xi_{M}\left(\mathcal{H}^{-1}\partial^{\mu}\mathcal{H}\partial^{\nu}\mathcal{H}^{-1}\right)^{MN}\partial_{\nu}\xi_{N}-4\,e^{4\Phi}\,\partial_{\mu}\Phi\,\partial^{\mu}\xi_{M}\partial_{\nu}\mathcal{H}^{MN}\partial^{\nu}\xi_{N}$
(A.9)
$\displaystyle\quad+16\,e^{2\Phi}\,\partial_{\mu}\Phi\partial^{\mu}\Phi\partial_{\nu}\Phi\partial^{\nu}\Phi+e^{6\Phi}\,\partial_{\mu}\xi_{M}\mathcal{H}^{MN}\partial^{\mu}\xi_{N}\,\partial_{\nu}\xi_{P}\mathcal{H}^{PQ}\partial^{\nu}\xi_{Q}$
$\displaystyle\quad+e^{6\Phi}\,\partial_{\mu}\xi_{M}\partial^{\mu}\xi^{M}\,\partial_{\nu}\xi_{N}\partial^{\nu}\xi^{N}+4\,e^{4\Phi}\,\partial_{\mu}\Phi\partial_{\nu}\Phi\,\partial^{\mu}\xi_{M}\mathcal{H}^{MN}\partial^{\nu}\xi_{N}$
$\displaystyle\quad+8\,e^{4\Phi}\,\partial_{\mu}\Phi\partial^{\mu}\Phi\,\partial_{\nu}\xi_{M}\mathcal{H}^{MN}\partial^{\nu}\xi_{N}.$

$\displaystyle\left(\mathcal{J}_{\mu}\mathcal{J}^{\mu}\mathcal{J}_{\nu}\mathcal{J}^{\nu}\eta\right)_{++}$
$\displaystyle=e^{4\Phi}\,\partial_{\mu}\xi_{M}\left(\partial^{\mu}\mathcal{H}^{-1}\partial_{\nu}\mathcal{H}\right)^{M}{}_{N}\partial^{\nu}\xi^{N}+4\,e^{4\Phi}\,\partial_{\mu}\Phi\,\partial^{\mu}\xi_{M}\left(\partial_{\nu}\mathcal{H}^{-1}\mathcal{H}\right)^{M}{}_{N}\partial^{\nu}\xi^{N}$
(A.10)
$\displaystyle\quad-2\,e^{6\Phi}\,\partial_{\mu}\xi_{M}\partial^{\mu}\xi^{M}\,\partial_{\nu}\xi_{P}\mathcal{H}^{PQ}\partial^{\nu}\xi_{Q}+4\,e^{4\Phi}\,\partial_{\mu}\Phi\partial_{\nu}\Phi\partial^{\mu}\xi_{M}\partial^{\nu}\xi^{M}$
$\displaystyle\quad-8\,e^{4\Phi}\,\partial_{\mu}\Phi\partial^{\mu}\Phi\partial_{\nu}\xi_{M}\partial^{\nu}\xi^{M}.$

$\displaystyle\left(\mathcal{J}_{\mu}\mathcal{J}_{\nu}\mathcal{J}^{\nu}\mathcal{J}^{\mu}\mathcal{M}\right)_{++}$
$\displaystyle=-\,e^{4\Phi}\,\partial_{\mu}\xi_{M}\left(\mathcal{H}^{-1}\partial_{\nu}\mathcal{H}\partial^{\nu}\mathcal{H}^{-1}\right)^{MN}\partial^{\mu}\xi_{N}-4\,e^{4\Phi}\,\partial_{\mu}\Phi\,\partial^{\mu}\xi_{M}\partial_{\nu}\mathcal{H}^{MN}\partial^{\nu}\xi_{N}$
(A.11)
$\displaystyle\quad+16\,e^{2\Phi}\,\partial_{\mu}\Phi\partial^{\mu}\Phi\partial_{\nu}\Phi\partial^{\nu}\Phi+e^{6\Phi}\,\partial_{\mu}\xi_{M}\mathcal{H}^{MN}\partial_{\nu}\xi_{N}\,\partial^{\mu}\xi_{P}\mathcal{H}^{PQ}\partial^{\nu}\xi_{Q}$
$\displaystyle\quad+e^{6\Phi}\,\partial_{\mu}\xi_{M}\partial_{\nu}\xi^{M}\,\partial^{\mu}\xi_{N}\partial^{\nu}\xi^{N}+8\,e^{4\Phi}\,\partial_{\mu}\Phi\partial_{\nu}\Phi\,\partial^{\mu}\xi_{M}\mathcal{H}^{MN}\partial^{\nu}\xi_{N}$
$\displaystyle\quad+4\,e^{4\Phi}\,\partial_{\mu}\Phi\partial^{\mu}\Phi\,\partial_{\nu}\xi_{M}\mathcal{H}^{MN}\partial^{\nu}\xi_{N}.$

$\displaystyle\left(\mathcal{J}_{\mu}\mathcal{J}_{\nu}\mathcal{J}^{\nu}\mathcal{J}^{\mu}\eta\right)_{++}$
$\displaystyle=e^{4\Phi}\,\partial_{\mu}\xi_{M}\left(\partial_{\nu}\mathcal{H}^{-1}\partial^{\nu}\mathcal{H}\right)^{M}{}_{N}\partial^{\mu}\xi^{N}-4\,e^{4\Phi}\,\partial_{\mu}\Phi\,\partial^{\mu}\xi_{M}\left(\partial_{\nu}\mathcal{H}^{-1}\mathcal{H}\right)^{M}{}_{N}\partial^{\nu}\xi^{N}$
(A.12)
$\displaystyle\quad-2\,e^{6\Phi}\,\partial_{\mu}\xi_{M}\partial_{\nu}\xi^{M}\,\partial^{\mu}\xi_{P}\mathcal{H}^{PQ}\partial^{\nu}\xi_{Q}-8\,e^{4\Phi}\,\partial_{\mu}\Phi\partial_{\nu}\Phi\partial^{\mu}\xi_{M}\partial^{\nu}\xi^{M}$
$\displaystyle\quad+4\,e^{4\Phi}\,\partial_{\mu}\Phi\partial^{\mu}\Phi\partial_{\nu}\xi_{M}\partial^{\nu}\xi^{M}.$

##  Appendix B Four-derivative action with mass deformation

We detail here the computation of the action (4.26). We first give the rules
for field redefinitions as modified by the mass deformation of the two-
derivative action. We then move to Einstein frame, use field redefinitions to
convert all second order derivatives into product of first order derivatives
and finally dualise the vector fields.

### B.1 Field redefinitions

The two-derivative action in Einstein frame with $B$ field integrated out is
given in eq. (4.8). Its equations of motion are

$\delta
I^{\prime\prime}_{0}=\alpha^{\prime}\,\int\mathrm{d}^{3}x\,\sqrt{-g_{\rm
E}}\bigg{[}\delta\Phi\,E_{\Phi}+\delta g^{\mu\nu}_{\rm
E}E_{g\,\mu\nu}+\mathrm{Tr}\left(\delta\mathcal{H}^{-1}\,E_{\cal
H}\right)+\delta\mathcal{A}_{\mu}{}^{M}E_{\mathcal{A}}{}^{\mu}{}_{M}\bigg{]},$
(B.1)

where

$\displaystyle
E_{\Phi}=2\,\Box\Phi+\frac{1}{2}e^{-2\Phi}\,\mathcal{F}_{\mu\nu}{}^{M}{\cal
H}_{MN}\mathcal{F}^{\mu\nu\,N}-2\,m^{2}\,e^{4\Phi},$ (B.2a) $\displaystyle
E_{g\,\mu\nu}=R_{{\rm
E}\,\mu\nu}-\partial_{\mu}\Phi\partial_{\nu}\Phi+\frac{1}{8}\,\mathrm{Tr}\left(\partial_{\mu}\mathcal{H}\,\partial_{\nu}\mathcal{H}^{-1}\right)-\frac{1}{2}\,e^{-2\Phi}\,\mathcal{F}_{\mu\rho}{}^{M}{\cal
H}_{MN}\mathcal{F}_{\nu}{}^{\rho\,N}$
$\displaystyle\qquad\quad-\frac{1}{2}g_{{\rm E}\,\mu\nu}\left(R_{\rm
E}-\partial_{\rho}\Phi\,\partial^{\rho}\Phi+\frac{1}{8}\,\mathrm{Tr}\left(\partial_{\rho}\mathcal{H}\,\partial^{\rho}\mathcal{H}^{-1}\right)-\frac{1}{4}\,e^{-2\Phi}\,\mathcal{F}_{\rho\sigma}{}^{M}{\cal
H}_{MN}\mathcal{F}^{\rho\sigma\,N}-\frac{1}{2}\,m^{2}\,e^{4\Phi}\right),$
(B.2b) $\displaystyle E_{{\cal
H}\,MN}=-\frac{1}{4}\,\bigg{[}\Box\,\mathcal{H}_{MN}+\left(\mathcal{H}\partial_{\mu}\mathcal{H}^{-1}\,\partial^{\mu}\mathcal{H}\right)_{MN}+\frac{1}{2}\,e^{-2\Phi}\,\mathcal{F}_{\mu\nu\,M}\mathcal{F}^{\mu\nu}{}_{N}-\frac{1}{2}\,e^{-2\Phi}\,\mathcal{F}_{\mu\nu}{}^{P}\mathcal{H}_{PM}\mathcal{F}^{\mu\nu\,Q}\mathcal{H}_{QN}\bigg{]},$
(B.2c) $\displaystyle
E_{\mathcal{A}}{}^{\mu}{}_{M}=e^{-2\Phi}\,\nabla_{\nu}\mathcal{F}^{\nu\mu\,N}\mathcal{H}_{NM}-2\,e^{-2\Phi}\,\nabla_{\nu}\Phi\,\mathcal{F}^{\nu\mu\,N}\mathcal{H}_{NM}+e^{-2\Phi}\,\mathcal{F}^{\nu\mu\,N}\nabla_{\nu}\mathcal{H}_{NM}+\frac{m}{2}\,\epsilon^{\mu\nu\rho}\mathcal{F}_{\nu\rho\,M}.$
(B.2d)

The resulting field redefinitions are as follows:

$\displaystyle\Box\Phi$ $\displaystyle\longrightarrow
Q_{\Phi}=-\frac{1}{4}e^{-2\Phi}\,\mathcal{F}_{\mu\nu}{}^{M}{\cal
H}_{MN}\mathcal{F}^{\mu\nu\,N}+m^{2}\,e^{4\Phi},$ (B.3) $\displaystyle R_{{\rm
E}\,\mu\nu}$ $\displaystyle\longrightarrow
Q_{g\,\mu\nu}=\partial_{\mu}\Phi\partial_{\nu}\Phi-\frac{1}{8}\,\mathrm{Tr}\left(\partial_{\mu}\mathcal{H}\,\partial_{\nu}\mathcal{H}^{-1}\right)+\frac{1}{2}\,e^{-2\Phi}\,\mathcal{F}_{\mu\rho}{}^{M}{\cal
H}_{MN}\mathcal{F}_{\nu}{}^{\rho\,N}$
$\displaystyle\qquad\quad\qquad-\frac{1}{4}\,g_{{\rm
E}\,\mu\nu}\,e^{-2\Phi}\,\mathcal{F}_{\rho\sigma}{}^{M}{\cal
H}_{MN}\mathcal{F}^{\rho\sigma\,N}+\frac{1}{2}\,g_{{\rm
E}\,\mu\nu}\,m^{2}\,e^{4\Phi},$ $\displaystyle\Box\,\mathcal{H}_{MN}$
$\displaystyle\longrightarrow
Q_{\mathcal{H}\,MN}=-\left(\mathcal{H}\,\partial_{\mu}\mathcal{H}^{-1}\partial^{\mu}\mathcal{H}\right)_{MN}-\frac{1}{2}\,e^{-2\Phi}\,\mathcal{F}_{\mu\nu\,M}\mathcal{F}^{\mu\nu}{}_{N}$
$\displaystyle\qquad\qquad\qquad+\frac{1}{2}\,e^{-2\Phi}\,\mathcal{F}_{\mu\nu}{}^{P}\mathcal{H}_{PM}\mathcal{F}^{\mu\nu\,Q}\mathcal{H}_{QN},$
$\displaystyle\nabla_{\nu}\mathcal{F}^{\nu\mu\,M}$
$\displaystyle\longrightarrow Q_{\mathcal{A}}{}^{\mu
M}=2\,\partial_{\nu}\Phi\,\mathcal{F}^{\nu\mu\,M}-\mathcal{F}^{\nu\mu\,N}\left(\partial_{\nu}{\cal
H}{\cal
H}^{-1}\right)_{N}{}^{M}-\frac{m}{2}\,e^{2\Phi}\,\epsilon^{\mu\nu\rho}\,\mathcal{F}_{\nu\rho\,N}\mathcal{H}^{NM}.$

### B.2 Einstein frame

We write the action (4.25) in Einstein frame ($g_{\mu\nu}\rightarrow g_{{\rm
E}\,\mu\nu}=e^{-2\Phi}g_{\mu\nu}$):

$\displaystyle I^{\prime\prime}_{1}=$
$\displaystyle\int\mathrm{d}^{3}x\,\sqrt{-g_{\rm E}}\Bigg{[}$ (B.4)
$\displaystyle-\frac{a+b}{8}\,e^{-2\Phi}\,\bigg{(}4\,R_{{\rm E}\mu\nu}R_{\rm
E}{}^{\mu\nu}-R_{\rm E}^{2}-8\,R_{{\rm
E}\,\mu\nu}\nabla^{\mu}\nabla^{\nu}\Phi+8\,R_{{\rm
E}\,\mu\nu}\,\partial^{\mu}\Phi\partial^{\nu}\Phi-4\,R_{\rm
E}\,\partial_{\mu}\Phi\partial^{\mu}\Phi$
$\displaystyle\qquad+4\,\nabla_{\mu}\nabla_{\nu}\Phi\nabla^{\mu}\nabla^{\nu}\Phi+4\,\Box\Phi\Box\Phi-8\,\nabla_{\mu}\nabla_{\nu}\Phi\partial^{\mu}\Phi\partial^{\nu}\Phi+8\,\Box\Phi\partial_{\mu}\Phi\partial^{\mu}\Phi+4\,\partial_{\mu}\Phi\partial^{\mu}\Phi\partial_{\nu}\Phi\partial^{\nu}\Phi$
$\displaystyle\qquad+\frac{1}{16}\,\mathrm{Tr}\left(\partial_{\mu}\mathcal{H}\partial_{\nu}\mathcal{H}^{-1}\partial^{\mu}\mathcal{H}\partial^{\nu}\mathcal{H}^{-1}\right)-\frac{1}{32}\,\mathrm{Tr}\left(\partial_{\mu}\mathcal{H}\partial_{\nu}\mathcal{H}^{-1}\right)\mathrm{Tr}\left(\partial^{\mu}\mathcal{H}\partial^{\nu}\mathcal{H}^{-1}\right)$
$\displaystyle\qquad+\frac{1}{8}\,e^{-4\Phi}\,{\mathcal{F}_{\mu\nu}}^{M}{\mathcal{F}_{\rho\sigma}}_{M}\mathcal{F}^{\mu\rho\,N}{\mathcal{F}^{\nu\sigma}}_{N}+\frac{1}{8}\,e^{-4\,\Phi}\,{\mathcal{F}_{\mu\nu}}^{M}\mathcal{H}_{MN}\mathcal{F}_{\rho\sigma}{}^{N}\mathcal{F}^{\mu\rho\,P}\mathcal{H}_{PQ}\mathcal{F}^{\nu\sigma\,Q}$
$\displaystyle\qquad-\frac{1}{2}\,e^{-4\Phi}\,{\mathcal{F}_{\mu\nu}}^{M}\mathcal{H}_{MN}\mathcal{F}^{\mu\rho\,N}\mathcal{F}^{\nu\sigma\,P}\mathcal{H}_{PQ}\mathcal{F}_{\rho\sigma}{}^{Q}+\frac{1}{2}\,e^{-2\Phi}\left(R_{\rm
E}+2\,\partial_{\mu}\Phi\partial^{\mu}\Phi\right)\mathcal{F}_{\rho\sigma}{}^{M}\mathcal{H}_{MN}\mathcal{F}^{\rho\sigma\,N}$
$\displaystyle\qquad-2\,e^{-2\Phi}\left(R_{{\rm
E}\,\mu\nu}-\nabla_{\mu}\nabla_{\nu}\Phi+\partial_{\mu}\Phi\partial_{\nu}\Phi\right)\mathcal{F}^{\mu\rho\,M}\mathcal{H}_{MN}\mathcal{F}^{\nu}{}_{\rho}{}^{N}$
$\displaystyle\qquad-\frac{1}{2}\,e^{-2\Phi}\,{\mathcal{F}_{\mu\nu}}^{M}\left(\mathcal{H}\partial_{\rho}\mathcal{H}^{-1}\partial^{\nu}\mathcal{H}\right)_{MN}\mathcal{F}^{\mu\rho\,N}+\frac{1}{4}\,e^{-2\Phi}\,\mathcal{F}^{\mu\rho\,M}\mathcal{H}_{MN}\mathcal{F}^{\nu}{}_{\rho}{}^{N}\,\mathrm{Tr}\left(\partial_{\mu}\mathcal{H}\partial_{\nu}\mathcal{H}^{-1}\right)$
$\displaystyle\qquad+m\,e^{2\Phi}\,\epsilon^{\mu\nu\rho}\left(\frac{2}{3}\,\Omega_{\mu\nu\rho}^{\rm(GS)}-\frac{1}{2}\,e^{-2\Phi}\,\mathcal{F}_{\mu\sigma}{}^{M}\left(\mathcal{H}\partial_{\nu}\mathcal{H}^{-1}\right)_{M}{}^{N}\mathcal{F}_{\rho}{}^{\sigma}{}_{N}\right)-\frac{5}{4}\,m^{4}\,e^{8\Phi}$
$\displaystyle\qquad+m^{2}\,e^{4\Phi}\left(R_{\rm
E}-4\,\Box\Phi-2\,\partial_{\mu}\Phi\partial^{\mu}\Phi-\frac{1}{4}\,\mathrm{Tr}\left(\partial_{\mu}\mathcal{H}\partial^{\mu}\mathcal{H}^{-1}\right)+\frac{3}{4}\,e^{-2\Phi}\,\mathcal{F}_{\mu\nu}{}^{M}\mathcal{H}_{MN}\mathcal{F}^{\mu\nu\,N}\right)\bigg{)}$
$\displaystyle+\frac{a-b}{4}\,e^{-2\Phi}\bigg{(}-\frac{1}{16}\,\mathrm{Tr}\left(\partial_{\mu}\mathcal{H}\partial^{\mu}\mathcal{H}^{-1}\partial_{\nu}\mathcal{H}\partial^{\nu}\mathcal{H}^{-1}\mathcal{H}\eta\right)-\frac{1}{16}\,e^{-4\Phi}\,\mathcal{F}_{\mu\nu}{}^{M}\mathcal{F}_{\rho\sigma\,M}\,\mathcal{F}^{\mu\nu\,P}\mathcal{H}_{PQ}\mathcal{F}^{\rho\sigma\,Q}$
$\displaystyle\qquad-\frac{1}{4}\,e^{-2\Phi}\,R_{\rm
E}\,\mathcal{F}_{\mu\nu}{}^{M}\mathcal{F}^{\mu\nu}{}_{M}+e^{-2\Phi}\,R_{\rm
E\,\mu\nu}\,\mathcal{F}^{\mu\rho\,M}\mathcal{F}^{\nu}{}_{\rho\,M}-e^{-2\Phi}\,\nabla_{\mu}\nabla_{\nu}\Phi\,\mathcal{F}^{\mu\rho\,M}\mathcal{F}^{\nu}{}_{\rho\,M}$
$\displaystyle\qquad-\frac{1}{2}\,e^{-2\Phi}\,\partial_{\rho}\Phi\partial^{\rho}\Phi\,\mathcal{F}_{\mu\nu}{}^{M}\mathcal{F}^{\mu\nu}{}_{M}+e^{-2\Phi}\,\partial_{\mu}\Phi\partial_{\nu}\Phi\,\mathcal{F}^{\mu\rho\,M}\mathcal{F}^{\nu}{}_{\rho\,M}$
$\displaystyle\qquad+\frac{1}{8}\,e^{-2\Phi}\,\mathcal{F}_{\mu\nu}{}^{M}\left(\partial_{\rho}\mathcal{H}\partial^{\rho}\mathcal{H}^{-1}\right)_{M}{}^{N}\mathcal{F}^{\mu\nu}{}_{N}+\frac{1}{4}\,e^{-2\Phi}\,\mathcal{F}_{\mu\nu}{}^{M}\left(\partial^{\mu}\mathcal{H}\partial_{\rho}\mathcal{H}^{-1}\right)_{M}{}^{N}\mathcal{F}^{\nu\rho}{}_{N}$
$\displaystyle\qquad+m\,e^{2\Phi}\,\epsilon^{\mu\nu\rho}\left(\Omega_{\mu\nu\rho}^{(\omega_{\rm
E})}+\frac{1}{4}\,e^{-2\Phi}\,\mathcal{F}_{\mu\sigma}{}^{M}\partial^{\sigma}\mathcal{H}_{MN}\mathcal{F}_{\nu\rho}{}^{N}\right)+\frac{m^{2}}{8}\,e^{2\Phi}\,\mathcal{F}_{\mu\nu}{}^{M}\mathcal{F}^{\mu\nu}{}_{M}\bigg{)}\Bigg{]}.$

### B.3 Integrations by part and field redefinitions

We now convert all second order derivatives in eq. (B.4) into products of
first order derivatives: we first use integrations by part so that all the
second order derivatives appear in the leading two-derivative contribution of
the equations of motion (B.2d), and obtain111111Note that here, contrary to
what we did in sec. 3, we use partial integrations and field redefinitions
before dualising the vector fields.

$\displaystyle I^{\prime\prime}_{1}=$
$\displaystyle\int\mathrm{d}^{3}x\,\sqrt{-g_{\rm E}}\Bigg{[}$ (B.5)
$\displaystyle-\frac{a+b}{8}\,e^{-2\Phi}\,\bigg{(}4\,\partial_{\mu}\Phi\partial^{\mu}\Phi\partial_{\nu}\Phi\partial^{\nu}\Phi+\frac{1}{16}\,\mathrm{Tr}\left(\partial_{\mu}\mathcal{H}\partial_{\nu}\mathcal{H}^{-1}\partial^{\mu}\mathcal{H}\partial^{\nu}\mathcal{H}^{-1}\right)$
$\displaystyle\qquad-\frac{1}{32}\,\mathrm{Tr}\left(\partial_{\mu}\mathcal{H}\partial_{\nu}\mathcal{H}^{-1}\right)\mathrm{Tr}\left(\partial^{\mu}\mathcal{H}\partial^{\nu}\mathcal{H}^{-1}\right)+\frac{1}{8}\,e^{-4\,\Phi}\,{\mathcal{F}_{\mu\nu}}^{M}\mathcal{H}_{MN}\mathcal{F}_{\rho\sigma}{}^{N}\mathcal{F}^{\mu\rho\,P}\mathcal{H}_{PQ}\mathcal{F}^{\nu\sigma\,Q}$
$\displaystyle\qquad+\frac{1}{8}\,e^{-4\Phi}\,{\mathcal{F}_{\mu\nu}}^{M}{\mathcal{F}_{\rho\sigma}}_{M}\mathcal{F}^{\mu\rho\,N}{\mathcal{F}^{\nu\sigma}}_{N}-\frac{1}{2}\,e^{-4\Phi}\,{\mathcal{F}_{\mu\nu}}^{M}\mathcal{H}_{MN}\mathcal{F}^{\mu\rho\,N}\mathcal{F}^{\nu\sigma\,P}\mathcal{H}_{PQ}\mathcal{F}_{\rho\sigma}{}^{Q}$
$\displaystyle\qquad-e^{-2\Phi}\,\partial_{\mu}\Phi\partial^{\mu}\Phi\,\mathcal{F}_{\rho\sigma}{}^{M}\mathcal{H}_{MN}\mathcal{F}^{\rho\sigma\,N}+6\,e^{-2\Phi}\,\partial_{\mu}\Phi\partial_{\nu}\Phi\,\mathcal{F}^{\mu\rho\,M}\mathcal{H}_{MN}\mathcal{F}^{\nu}{}_{\rho}{}^{N}$
$\displaystyle\qquad-\frac{1}{2}\,e^{-2\Phi}\,{\mathcal{F}_{\mu\nu}}^{M}\left(\mathcal{H}\partial_{\rho}\mathcal{H}^{-1}\partial^{\nu}\mathcal{H}\right)_{MN}\mathcal{F}^{\mu\rho\,N}+\frac{1}{4}\,e^{-2\Phi}\,\mathcal{F}^{\mu\rho\,M}\mathcal{H}_{MN}\mathcal{F}^{\nu}{}_{\rho}{}^{N}\,\mathrm{Tr}\left(\partial_{\mu}\mathcal{H}\partial_{\nu}\mathcal{H}^{-1}\right)$
$\displaystyle\qquad-2\,e^{-2\Phi}\,\partial_{\nu}\Phi\,\mathcal{F}^{\mu\rho\,M}\partial_{\mu}\mathcal{H}_{MN}\mathcal{F}^{\nu}{}_{\rho}{}^{N}+\frac{1}{2}\,e^{-2\Phi}\,\partial_{\rho}\Phi\,\mathcal{F}_{\mu\nu}{}^{M}\partial^{\rho}\mathcal{H}_{MN}\mathcal{F}^{\mu\nu\,N}$
$\displaystyle\qquad+m\,e^{2\Phi}\,\epsilon^{\mu\nu\rho}\left(\frac{2}{3}\,\Omega_{\mu\nu\rho}^{\rm(GS)}-\frac{1}{2}\,e^{-2\Phi}\,\mathcal{F}_{\mu\sigma}{}^{M}\left(\mathcal{H}\partial_{\nu}\mathcal{H}^{-1}\right)_{M}{}^{N}\mathcal{F}_{\rho}{}^{\sigma}{}_{N}\right)-\frac{5}{4}\,m^{4}\,e^{8\Phi}$
$\displaystyle\qquad+m^{2}\,e^{4\Phi}\left(-2\,\partial_{\mu}\Phi\partial^{\mu}\Phi-\frac{1}{4}\,\mathrm{Tr}\left(\partial_{\mu}\mathcal{H}\partial^{\mu}\mathcal{H}^{-1}\right)+\frac{3}{4}\,e^{-2\Phi}\,\mathcal{F}_{\mu\nu}{}^{M}\mathcal{H}_{MN}\mathcal{F}^{\mu\nu\,N}\right)$
$\displaystyle\qquad+4\,R_{{\rm E}\mu\nu}R_{\rm E}{}^{\mu\nu}-R_{\rm
E}^{2}-12\,R_{{\rm E}\,\mu\nu}\,\partial^{\mu}\Phi\partial^{\nu}\Phi+4\,R_{\rm
E}\,\partial_{\mu}\Phi\partial^{\mu}\Phi-4\,R_{\rm
E}\,\Box\Phi+8\,\Box\Phi\Box\Phi$
$\displaystyle\qquad+\frac{1}{2}\,e^{-2\Phi}\,R_{\rm
E}\,\mathcal{F}_{\mu\nu}{}^{M}\mathcal{H}_{MN}\mathcal{F}^{\mu\nu\,N}-2\,e^{-2\Phi}\,R_{{\rm
E}\,\mu\nu}\,\mathcal{F}^{\mu\rho\,M}\mathcal{H}_{MN}\mathcal{F}^{\nu}{}_{\rho}{}^{N}+m^{2}\,e^{4\Phi}\left(R_{\rm
E}-4\,\Box\Phi\right)$
$\displaystyle\qquad-2\,e^{-2\Phi}\,\partial_{\nu}\Phi\,\nabla_{\mu}\mathcal{F}^{\mu\rho\,M}\mathcal{H}_{MN}\mathcal{F}^{\nu}{}_{\rho}{}^{N}+\frac{1}{2}\,e^{-2\Phi}\,\Box\Phi\,\mathcal{F}_{\mu\nu}{}^{M}\mathcal{H}_{MN}\mathcal{F}^{\mu\nu\,N}\bigg{)}$
$\displaystyle+\frac{a-b}{4}\,e^{-2\Phi}\bigg{(}-\frac{1}{16}\,\mathrm{Tr}\left(\partial_{\mu}\mathcal{H}\partial^{\mu}\mathcal{H}^{-1}\partial_{\nu}\mathcal{H}\partial^{\nu}\mathcal{H}^{-1}\mathcal{H}\eta\right)-\frac{1}{16}\,e^{-4\Phi}\,\mathcal{F}_{\mu\nu}{}^{M}\mathcal{F}_{\rho\sigma\,M}\,\mathcal{F}^{\mu\nu\,P}\mathcal{H}_{PQ}\mathcal{F}^{\rho\sigma\,Q}$
$\displaystyle\qquad-3\,e^{-2\Phi}\,\partial_{\mu}\Phi\partial_{\nu}\Phi\,\mathcal{F}^{\mu\rho\,M}\mathcal{F}^{\nu}{}_{\rho\,M}+\frac{1}{2}\,e^{-2\Phi}\,\partial_{\rho}\Phi\partial^{\rho}\Phi\,\mathcal{F}_{\mu\nu}{}^{M}\mathcal{F}^{\mu\nu}{}_{M}$
$\displaystyle\qquad+\frac{1}{8}\,e^{-2\Phi}\,\mathcal{F}_{\mu\nu}{}^{M}\left(\partial_{\rho}\mathcal{H}\partial^{\rho}\mathcal{H}^{-1}\right)_{M}{}^{N}\mathcal{F}^{\mu\nu}{}_{N}+\frac{1}{4}\,e^{-2\Phi}\,\mathcal{F}_{\mu\nu}{}^{M}\left(\partial^{\mu}\mathcal{H}\partial_{\rho}\mathcal{H}^{-1}\right)_{M}{}^{N}\mathcal{F}^{\nu\rho}{}_{N}$
$\displaystyle\qquad+m\,e^{2\Phi}\,\epsilon^{\mu\nu\rho}\,\left(\Omega_{\mu\nu\rho}^{(\omega_{\rm
E})}+\frac{1}{4}\,e^{-2\Phi}\,\mathcal{F}_{\mu\sigma}{}^{M}\partial^{\sigma}\mathcal{H}_{MN}\mathcal{F}_{\nu\rho}{}^{N}\right)+\frac{m^{2}}{8}\,e^{2\Phi}\,\mathcal{F}_{\mu\nu}{}^{M}\mathcal{F}^{\mu\nu}{}_{M}$
$\displaystyle\qquad-\frac{1}{4}\,e^{-2\Phi}\,R_{\rm
E}\,\mathcal{F}_{\mu\nu}{}^{M}\mathcal{F}^{\mu\nu}{}_{M}+e^{-2\Phi}\,R_{\rm
E\,\mu\nu}\,\mathcal{F}^{\mu\rho\,M}\mathcal{F}^{\nu}{}_{\rho\,M}-\frac{1}{4}\,e^{-2\Phi}\,\Box\Phi\,\mathcal{F}_{\mu\nu}{}^{M}\mathcal{F}^{\mu\nu}{}_{M}$
$\displaystyle\qquad+e^{-2\Phi}\,\nabla_{\mu}\mathcal{F}^{\mu\rho}\,\partial_{\nu}\Phi\,\mathcal{F}^{\nu}{}_{\rho\,M}\bigg{)}\Bigg{]}.$

Using the field redefinitions (B.3), we get

$\displaystyle I^{\prime\prime}_{1}=$
$\displaystyle\int\mathrm{d}^{3}x\,\sqrt{-g_{\rm E}}\Bigg{[}$ (B.6)
$\displaystyle-\frac{a+b}{8}\,e^{-2\Phi}\,\bigg{(}\frac{1}{16}\,\mathrm{Tr}\left(\partial_{\mu}\mathcal{H}\partial_{\nu}\mathcal{H}^{-1}\partial^{\mu}\mathcal{H}\partial^{\nu}\mathcal{H}^{-1}\right)+\frac{1}{32}\,\mathrm{Tr}\left(\partial_{\mu}\mathcal{H}\partial_{\nu}\mathcal{H}^{-1}\right)\mathrm{Tr}\left(\partial^{\mu}\mathcal{H}\partial^{\nu}\mathcal{H}^{-1}\right)$
$\displaystyle\qquad-\frac{1}{64}\,\mathrm{Tr}\left(\partial_{\mu}\mathcal{H}\partial^{\mu}\mathcal{H}^{-1}\right)\mathrm{Tr}\left(\partial_{\nu}\mathcal{H}\partial^{\nu}\mathcal{H}^{-1}\right)+\frac{1}{8}\,e^{-4\,\Phi}\,{\mathcal{F}_{\mu\nu}}^{M}\mathcal{H}_{MN}\mathcal{F}_{\rho\sigma}{}^{N}\mathcal{F}^{\mu\rho\,P}\mathcal{H}_{PQ}\mathcal{F}^{\nu\sigma\,Q}$
$\displaystyle\qquad+\frac{1}{8}\,e^{-4\Phi}\,{\mathcal{F}_{\mu\nu}}^{M}{\mathcal{F}_{\rho\sigma}}_{M}\mathcal{F}^{\mu\rho\,N}{\mathcal{F}^{\nu\sigma}}_{N}-\frac{1}{2}\,e^{-4\Phi}\,{\mathcal{F}_{\mu\nu}}^{M}\mathcal{H}_{MN}\mathcal{F}^{\mu\rho\,N}\mathcal{F}^{\nu\sigma\,P}\mathcal{H}_{PQ}\mathcal{F}_{\rho\sigma}{}^{Q}$
$\displaystyle\qquad+\frac{3}{16}\,e^{-4\Phi}\,\mathcal{F}_{\mu\nu}{}^{M}\mathcal{H}_{MN}\mathcal{F}^{\mu\nu\,N}\mathcal{F}_{\rho\sigma}{}^{P}\mathcal{H}_{PQ}\mathcal{F}^{\rho\sigma\,Q}-\partial_{\mu}\Phi\partial^{\mu}\Phi\partial_{\nu}\Phi\partial^{\nu}\Phi$
$\displaystyle\qquad+\frac{1}{2}\,\partial_{\mu}\Phi\partial_{\nu}\Phi\,\mathrm{Tr}\left(\partial^{\mu}\mathcal{H}\partial^{\nu}\mathcal{H}^{-1}\right)-\frac{1}{4}\,\partial_{\mu}\Phi\partial^{\mu}\Phi\,\mathrm{Tr}\left(\partial_{\nu}\mathcal{H}\partial^{\nu}\mathcal{H}^{-1}\right)$
$\displaystyle\qquad+e^{-2\Phi}\,\partial_{\mu}\Phi\partial^{\mu}\Phi\,\mathcal{F}_{\rho\sigma}{}^{M}\mathcal{H}_{MN}\mathcal{F}^{\rho\sigma\,N}-2\,e^{-2\Phi}\,\partial_{\mu}\Phi\partial_{\nu}\Phi\,\mathcal{F}^{\mu\rho\,M}\mathcal{H}_{MN}\mathcal{F}^{\nu}{}_{\rho}{}^{N}$
$\displaystyle\qquad-\frac{1}{2}\,e^{-2\Phi}\,{\mathcal{F}_{\mu\nu}}^{M}\left(\mathcal{H}\partial_{\rho}\mathcal{H}^{-1}\partial^{\nu}\mathcal{H}\right)_{MN}\mathcal{F}^{\mu\rho\,N}+\frac{1}{2}\,e^{-2\Phi}\,\partial_{\rho}\Phi\,\mathcal{F}_{\mu\nu}{}^{M}\partial^{\rho}\mathcal{H}_{MN}\mathcal{F}^{\mu\nu\,N}$
$\displaystyle\qquad+m\,e^{2\Phi}\,\epsilon^{\mu\nu\rho}\left(\frac{2}{3}\,\Omega_{\mu\nu\rho}^{\rm(GS)}-\frac{1}{2}\,e^{-2\Phi}\,\mathcal{F}_{\mu\sigma}{}^{M}\left(\mathcal{H}\partial_{\nu}\mathcal{H}^{-1}\right)_{M}{}^{N}\mathcal{F}_{\rho}{}^{\sigma}{}_{N}+e^{-2\Phi}\,\partial_{\sigma}\Phi\,\mathcal{F}_{\mu\nu}{}^{M}\mathcal{F}^{\sigma}{}_{\rho\,M}\right)$
$\displaystyle\qquad-4\,m^{2}\,e^{4\Phi}\,\partial_{\mu}\Phi\partial^{\mu}\Phi-m^{4}\,e^{8\Phi}\bigg{)}$
$\displaystyle+\frac{a-b}{4}\,e^{-2\Phi}\bigg{(}-\frac{1}{16}\,\mathrm{Tr}\left(\partial_{\mu}\mathcal{H}\partial^{\mu}\mathcal{H}^{-1}\partial_{\nu}\mathcal{H}\partial^{\nu}\mathcal{H}^{-1}\mathcal{H}\eta\right)-\frac{1}{16}\,e^{-4\Phi}\,\mathcal{F}_{\mu\nu}{}^{M}\mathcal{F}_{\rho\sigma\,M}\,\mathcal{F}^{\mu\nu\,P}\mathcal{H}_{PQ}\mathcal{F}^{\rho\sigma\,Q}$
$\displaystyle\qquad-\frac{1}{8}\,e^{-4\Phi}\,\mathcal{F}_{\mu\nu}{}^{M}\mathcal{F}^{\mu\nu}{}_{M}\,\mathcal{F}_{\rho\sigma}{}^{P}\mathcal{H}_{PQ}\mathcal{F}^{\rho\sigma\,Q}+\frac{1}{2}\,e^{-4\Phi}\,\mathcal{F}_{\mu\rho}{}^{M}\mathcal{F}_{\nu}{}^{\rho}{}_{M}\,\mathcal{F}^{\mu\sigma\,P}\mathcal{H}_{PQ}\mathcal{F}^{\nu}{}_{\sigma}{}^{Q}$
$\displaystyle\qquad+\frac{1}{4}\,e^{-2\Phi}\,\partial_{\rho}\Phi\partial^{\rho}\Phi\,\mathcal{F}_{\mu\nu}{}^{M}\mathcal{F}^{\mu\nu}{}_{M}+\frac{1}{32}\,e^{-2\Phi}\,\mathrm{Tr}\left(\partial_{\rho}\mathcal{H}\partial^{\rho}\mathcal{H}^{-1}\right)\,\mathcal{F}_{\mu\nu}{}^{M}\mathcal{F}^{\mu\nu}{}_{M}$
$\displaystyle\qquad-\frac{1}{8}\,e^{-2\Phi}\,\mathrm{Tr}\left(\partial_{\mu}\mathcal{H}\partial_{\nu}\mathcal{H}^{-1}\right)\,\mathcal{F}^{\mu\rho\,M}\mathcal{F}^{\nu}{}_{\rho\,M}+\frac{1}{8}\,e^{-2\Phi}\,\mathcal{F}_{\mu\nu}{}^{M}\left(\partial_{\rho}\mathcal{H}\partial^{\rho}\mathcal{H}^{-1}\right)_{M}{}^{N}\mathcal{F}^{\mu\nu}{}_{N}$
$\displaystyle\qquad+\frac{1}{4}\,e^{-2\Phi}\,\mathcal{F}_{\mu\nu}{}^{M}\left(\partial^{\mu}\mathcal{H}\partial_{\rho}\mathcal{H}^{-1}\right)_{M}{}^{N}\mathcal{F}^{\nu\rho}{}_{N}-e^{-2\Phi}\,\partial_{\mu}\Phi\,\mathcal{F}_{\nu\rho}{}^{M}\left(\partial^{\nu}\mathcal{H}\mathcal{H}^{-1}\right)_{M}{}^{N}\mathcal{F}^{\mu\rho}{}_{N}$
$\displaystyle\qquad+m\,e^{2\Phi}\,\epsilon^{\mu\nu\rho}\,\left(\Omega_{\mu\nu\rho}^{(\omega_{\rm
E})}+\frac{1}{4}\,e^{-2\Phi}\,\mathcal{F}_{\mu\sigma}{}^{M}\left(\partial^{\sigma}\mathcal{H}_{MN}+2\,\partial^{\sigma}\Phi\,\mathcal{H}_{MN}\right)\mathcal{F}_{\nu\rho}{}^{N}\right)\bigg{)}\Bigg{]}.$

### B.4 Dualisation of the vector fields

We know dualise the vector fields into scalars by using the two-derivative
dualisation eq. (4.11) in the form

$\mathcal{F}_{\mu\nu}{}^{M}=e^{2\Phi}\epsilon_{\mu\nu\rho}D^{\rho}\xi_{N}\mathcal{H}^{NM},$
(B.7)

with $D_{\mu}\xi_{M}=\partial_{\mu}\xi_{M}+m\,\mathcal{A}_{\mu\,M}$. We get

$\displaystyle\widetilde{I}^{\prime\prime}_{1}$
$\displaystyle=\int\mathrm{d}^{3}x\,\sqrt{-g_{\rm E}}\,e^{-2\Phi}\,\Bigg{[}$
$\displaystyle-\frac{a+b}{8}\bigg{(}-\partial_{\mu}\Phi\partial^{\mu}\Phi\,\partial_{\nu}\Phi\partial^{\nu}\Phi+\frac{1}{16}\,\mathrm{Tr}\left(\partial_{\mu}\mathcal{H}\partial_{\nu}\mathcal{H}^{-1}\partial^{\mu}\mathcal{H}\partial^{\nu}\mathcal{H}^{-1}\right)$
$\displaystyle\qquad+\frac{1}{32}\,\mathrm{Tr}\left(\partial_{\mu}\mathcal{H}\partial_{\nu}\mathcal{H}^{-1}\right)\mathrm{Tr}\left(\partial^{\mu}\mathcal{H}\partial^{\nu}\mathcal{H}^{-1}\right)-\frac{1}{64}\,\mathrm{Tr}\left(\partial_{\mu}\mathcal{H}\partial^{\mu}\mathcal{H}^{-1}\right)\mathrm{Tr}\left(\partial_{\nu}\mathcal{H}\partial^{\nu}\mathcal{H}^{-1}\right)$