Exploring the $\beta$ symmetry of supergravity

Walter H. Baron1, Diego Marqués2,3 and Carmen A. Núñez2

1 Instituto de Física de La Plata, (CONICET-UNLP)

Departamento de Matemática, Universidad Nacional de La Plata.
C. 115 s/n, B1900, La Plata, Argentina
<EMAIL_ADDRESS>
2 Instituto de Astronomía y Física del Espacio, (CONICET-UBA)

Ciudad Universitaria, Pabellón IAFE, CABA, C1428ZAA, Argentina
<EMAIL_ADDRESS><EMAIL_ADDRESS>
3 Departamento de Física, Universidad de Buenos Aires.
Ciudad Universitaria, Pabellón 1, CABA, C1428ZAA, Argentina

Abstract

> Kaluza-Klein reductions of low energy string effective actions possess a
> continuous $O(d,d)$ symmetry. The non-geometric elements of this group,
> parameterized by a bi-vector $\beta$, are not inherited from the symmetries
> of the higher-dimensional theory, but constitute instead a symmetry
> enhancement produced by the isometries of the background. The realization of
> this enhancement in the parent theory was recently defined as $\beta$
> symmetry, a powerful tool that allows to avoid the field reparameterizations
> of the Kaluza-Klein procedure. In this paper we further explore this
> symmetry and its impact on the first order $\alpha^{\prime}$-corrections. We
> derive the $\beta$ transformation rules from the frame formulation of Double
> Field Theory (DFT), and connect them to the corresponding rules in the
> Metsaev-Tseytlin and Bergshoeff-de Roo supergravity schemes. While $\beta$
> transformations that descend from DFT leave the Lagrangian invariant and
> define a symmetry algebra that closes off-shell, we find a four-parameter
> deformation that preserves the action and leads to on-shell closure. It
> follows from our results that $\beta$ symmetry is a necessary condition for
> the uplift of string $\alpha^{\prime}$-expansions to DFT.

###### Contents

  1. 1 Introduction
  2. 2 $\beta$ transformations from Double Field Theory
  3. 3 Scheme independent statements
    1. 3.1 $\beta$ symmetry at leading order
    2. 3.2 A four-parameter redundancy of first order transformations
    3. 3.3 Invariance of the action implies closure of the symmetry algebra
    4. 3.4 Effect of field and parameter redefinitions on the symmetry algebra
  4. 4 Bergshoeff-de Roo scheme
  5. 5 Metsaev-Tseytlin scheme
    1. 5.1 Invariance of the Metsaev-Tseytlin Lagrangian
      1. 5.1.1 Invariance of the even parity sector
      2. 5.1.2 Invariance of the odd parity sector
  6. 6 Conclusions
  7. A Notation and definitions
  8. B Bracket in MT scheme after reparameterization
  9. C Leading order variation of $L^{(1)}$ in MT scheme
    1. C.1 Variation of $L^{(1)}_{+}$
    2. C.2 Variation of $L^{(1)}_{-}$

##  1 Introduction

The low energy limits of $D$-dimensional string theories compactified on
$T^{d}$ are invariant with respect to the global $O(d,d;R)$ group [1]. There
are two distinct types of elements in this group. On the one hand, the
geometric subgroup $GL(d)\times R^{\frac{d(d-1)}{2}}$ descends from the
diffeomorphisms and two-form shifts of the higher dimensional theory. On the
other hand, the complement of this subgroup is not associated to a symmetry of
the higher dimensional theory, and so it constitutes a symmetry enhancement
produced by the toroidal truncation.

The compactification from the higher to the lower dimensional action is
achieved through Kaluza-Klein (KK) reduction. When only the massless modes are
kept, KK reductions involve two steps. First, the isometries of the background
are imposed on the fields, which simply amounts to removing their dependence
on the toroidal coordinates. The second step is a rearrangement of degrees of
freedom, namely field redefinitions. The fields of the parent action, which
constitute the multiplets of the higher dimensional diffeomorphisms, are
decomposed into multiplets of the lower dimensional diffeomorphisms and gauge
symmetries. While it is in the first step that the symmetry is enhanced to
$O(d,d;R)$, the second step is intended to make this symmetry (and others)
manifest.

Recently, it was shown in [2] that the non-geometric elements of $O(d,d;R)$,
which are parameterized by a rigid bi-vector $\beta$, act covariantly on the
multiplets of the higher dimensional diffeomorphisms. In other words, the
$\beta$ transformations preserve the structure of the higher-dimensional
fields, which permits to avoid the second step of the KK procedure. This in
turn allows to act with $\beta$ transformations on the higher dimensional
action, and to assess/verify its invariance under the assumption of
isometries. Even if one is actually simply dealing with the $O(d,d;R)$
invariance of the lower dimensional theory, effectively this looks like a
symmetry of the higher-dimensional action, and so we dubbed it $\beta$
symmetry of supergravity. For all practical purposes, this can be conceived as
a symmetry of the higher dimensional theory, whose effect in lower dimensions
serves to enhance $GL(d)\times R^{\frac{d(d-1)}{2}}$ to the full $O(d,d;R)$
group. Interestingly, since this symmetry mixes the NSNS fields, it fixes
their relative couplings and might then become a powerful tool to compute
string theory interactions.

The $\beta$-invariance of the string effective actions to first order in
$\alpha^{\prime}$ was proved in [2]. We presented the $\beta$ transformation
rules, their first order corrections, the closure of the symmetry algebra and
the invariance of the Lagrangian, in the so called generalized Bergshoeff-de
Roo scheme [3]. In this paper, we further explore several features of the
$\beta$ symmetry of string supergravities.

In Section 2 we derive $\beta$ symmetry from Double Field Theory (DFT) [4]. We
use the background independent frame-like formulation of DFT [5, 6],
restricted by the so-called strong constraint. The formalism is manifestly
invariant under the global $O(D,D;R)$ duality group, which contains the
$\beta$ elements already embedded into $GL(D)$, ready to act on the
$D$-dimensional fields that parameterize the duality multiplets. Of course,
they are precisely the elements that determine the $\beta$ symmetry of
supergravity. Here we derive the $\beta$ transformation rules to first order
in $\alpha^{\prime}$ in the DFT scheme of supergravity, following the standard
procedure of solving the strong constraint by annihilating the dual
derivatives, and gauge-fixing the double-Lorentz symmetry. In other words,
since it is implied by DFT, $\beta$ symmetry is a necessary condition for the
uplift of supergravity to DFT, at least to this formulation. As such, it can
serve as a tool to understand potential obstructions to manifest duality
covariant uplifts of supergravity interactions.

Before moving from the DFT scheme of supergravity to other more conventional
schemes, we discuss various scheme-independent results in Section 3. $\beta$
transformations to lowest order are unique, and preserve the Lagrangian
exactly. The first order corrections derived from DFT also leave the
Lagrangian invariant. Interestingly, there is a four-parameter deformation of
the first order transformations that leave the action invariant (i.e. the
variation of the Lagrangian leads to a total derivative). While the symmetries
that leave the Lagrangian invariant close off-shell, we show that the
transformations that preserve the action but not the Lagrangian also close,
but on-shell. These, and other scheme independent ambiguities in the structure
of the first order corrections to $\beta$ symmetry are discussed in detail in
this section.

While Section 4 is devoted to recover all the results of [2], by performing
the field redefinitions that connect the DFT scheme with the generalized
Bergshoeff-de Roo scheme, in Section 5 we show how $\beta$ symmetry works in
the Metsaev-Tseytlin scheme. In all sections the results are presented for a
bi-parametric theory, such that the bosonic and heterotic strings are obtained
for specific choices of the parameters. In particular, we show that $\beta$
symmetry in the Metsaev-Tseytlin scheme of the bosonic string admits a purely
metric formulation, as expected. Instead, the $\beta$-invariance of the Chern-
Simons form in the heterotic string is necessarily frame-like. We discuss both
cases separately, providing the transformation rules that leave the Lagrangian
invariant, as well as the deformations that preserve the action, and the
brackets of the symmetry algebra.

We end with some conclusions in Section 6. Notation, definitions and side
computations are contained in three Appendices.

##  2 $\beta$ transformations from Double Field Theory

In this section we derive the $\beta$ transformations of supergravity to first
order in $\alpha^{\prime}$ from their action in the strong constrained and
background independent frame formulation of Double Field Theory (DFT) [5, 6].
The procedure requires solving the strong constraint in the supergravity
section, and applying a gauge fixing of the double Lorentz symmetry, after
which it systematically provides the $\beta$ transformation rules of the
supergravity fields in the DFT scheme. Other supergravity schemes can then be
reached from this one through field redefinitions.

The field content of DFT involves a pair of generalized frames
$E_{M}{}^{\underline{a}}$ and $E_{M}{}^{\overline{a}}$ and a generalized
dilaton $d$. The symmetries that we will deal with in this paper are $O(D,D)$,
with invariant metric $\eta_{MN}$, and two independent Lorentz symmetries with
invariant metrics $\eta_{\underline{ab}}$ and $\eta_{\overline{ab}}$, each one
acting on its respective frame. All indices are raised and lowered with their
corresponding $\eta$ invariant metric. The generalized frames are constrained
to satisfy the identities

$E_{M\underline{a}}E^{M}{}_{\underline{b}}=\eta_{\underline{ab}}\ ,\ \ \
E_{M\underline{a}}E^{M}{}_{\overline{b}}=0\ ,\ \ \
E_{M\overline{a}}E^{M}{}_{\overline{b}}=\eta_{\overline{ab}}\,.$ (2.1)

The fields are formally defined on a doubled space and so are acted on by
derivatives $\partial_{M}$, which are strong constrained, meaning that the
$O(D,D)$ invariant contraction of derivatives vanishes.

Infinitesimally, $O(D,D)$ acts through a constant antisymmetric matrix
$h_{MN}$, and each Lorentz group through independent local antisymmetric
parameters $\Lambda_{\underline{ab}}$ and $\Lambda_{\overline{ab}}$. While the
generalized dilaton is invariant, the two generalized frames transform as

$\displaystyle\delta E_{M}{}^{\underline{a}}\ $ $\displaystyle=\
h_{M}{}^{N}E_{N}{}^{\underline{a}}+E_{M}{}^{\underline{b}}\Lambda_{\underline{b}}{}^{\underline{a}}+E_{M}{}^{\overline{b}}\Delta_{\overline{b}}{}^{\underline{a}}\,,$
(2.2a) $\displaystyle\delta E_{M}{}^{\overline{a}}\ $ $\displaystyle=\
h_{M}{}^{N}E_{N}{}^{\overline{a}}+E_{M}{}^{\overline{b}}\Lambda_{\overline{b}}{}^{\overline{a}}-E_{M}{}^{\underline{b}}\Delta^{\overline{a}}{}_{\underline{b}}\
.$ (2.2b)

The last terms in these expressions account for the generalized Green-Schwarz
transformation [3, 10, 7, 8, 9], consisting on all-order higher-derivative
deformations of the double Lorentz transformations. In particular, the first
order deformation is given by

$\Delta^{(1)}_{\overline{a}\underline{b}}=\frac{a}{2}F_{\overline{a}\underline{cd}}D_{\underline{b}}\Lambda^{\underline{cd}}+\frac{b}{2}D_{\overline{a}}\Lambda^{\overline{cd}}F_{\underline{b}\overline{cd}}\,,$
(2.3)

where we have defined

$D_{\overline{a}}=E^{M}{}_{\overline{a}}\partial_{M}\ ,\ \ \
D_{\underline{a}}=E^{M}{}_{\underline{a}}\partial_{M}\ ,$ (2.4)

and

$\displaystyle F_{\overline{a}\underline{bc}}$
$\displaystyle=D_{\overline{a}}E_{M}{}_{[\underline{b}}E^{M}{}_{\underline{c}]}+2D_{[\underline{b}}E^{M}{}_{\underline{c}]}E_{M\overline{a}}\,,$
(2.5a) $\displaystyle\ F_{\underline{a}\overline{bc}}$
$\displaystyle=D_{\underline{a}}E_{M}{}_{[\overline{b}}E^{M}{}_{\overline{c}]}+2D_{[\overline{b}}E^{M}{}_{\overline{c}]}E_{M\underline{a}}\,.$
(2.5b)

The parameters $a$ and $b$ interpolate between generalized Green-Schwarz
transformations with respect to the two different Lorentz factors. They were
originally found in [11], and give rise to a two-parameter family of theories
which contains the bosonic string $(a,b)=(-\alpha^{\prime},-\alpha^{\prime})$,
the heterotic string $(a,b)=-(\alpha^{\prime},0)$ and more general
deformations like those in [12]. Interestingly, to first order in
$\alpha^{\prime}$ DFT is defined through a Lagrangian that is invariant under
all these symmetries [3], and transforms as a scalar with respect to
generalized diffeomorphisms. This property will be inherited by the
supergravity Lagrangian that descends from it, and also by all others
connected by field redefinitions.

Making contact with supergravity degrees of freedom requires performing a
$GL(D)$ decomposition of $O(D,D)$, with respect to which the indices split as
$M=({}^{\mu},\,{}_{\mu})$. Selecting the supergravity section as a solution of
the strong constraint, simply amounts to setting
$\partial_{M}=(0,\,\partial_{\mu})$. The decomposition of the invariant
metrics is given by

$\eta_{MN}=\left(\begin{matrix}0&\delta^{\mu}_{\nu}\\\
\delta^{\nu}_{\mu}&0\end{matrix}\right)\ ,\ \ \
\eta_{\underline{ab}}=-g_{\underline{ab}}\ ,\ \ \
\eta_{\overline{ab}}=g_{\overline{ab}}\,,$ (2.6)

where $g_{\underline{ab}}$ and $g_{\overline{ab}}$ are Minkowski metrics. The
most general $GL(D)$ covariant parameterization of the generalized fields is

${E}_{M}{}^{\underline{a}}=\frac{1}{\sqrt{2}}\left(\begin{matrix}\bar{e}^{\mu}{}_{\underline{c}}g^{\underline{ca}}\\\
(\bar{b}_{\mu\nu}-\bar{g}_{\mu\nu})\bar{e}^{\nu}{}_{\underline{c}}g^{\underline{ca}}\end{matrix}\right)\
,\ \ \
{E}_{M}{}^{\overline{a}}=\frac{1}{\sqrt{2}}\left(\begin{matrix}\bar{e}^{\mu}{}_{\overline{c}}g^{\overline{ca}}\\\
(\bar{g}_{\mu\nu}+\bar{b}_{\mu\nu})\bar{e}^{\nu}{}_{\overline{c}}g^{\overline{ca}}\end{matrix}\right)\
,\ \ \ e^{-2d}=\sqrt{-\bar{g}}e^{-2\bar{\phi}}\,,$

where we have introduced a two-form $\bar{b}_{\mu\nu}$, a dilaton $\bar{\phi}$
and a pair of vielbeins $\bar{e}_{\mu}{}^{\underline{a}}$ and
$\bar{e}_{\mu}{}^{\overline{a}}$ that generate the same metric
$\bar{g}_{\mu\nu}$

$\bar{g}_{\mu\nu}=\bar{e}_{\mu}{}^{\underline{a}}g_{\underline{ab}}\bar{e}_{\nu}{}^{\underline{b}}=\bar{e}_{\mu}{}^{\overline{a}}g_{\overline{ab}}\bar{e}_{\nu}{}^{\overline{b}}\,,$
(2.7)

which raises and lowers the $GL(D)$ indices. Finally, the element in the
algebra of $O(D,D)$ splits into constant components

$\displaystyle
h_{M}{}^{N}=\left(\begin{matrix}a^{\mu}{}_{\nu}&\beta^{\mu\nu}\\\
B_{\mu\nu}&-a^{\nu}{}_{\mu}\end{matrix}\right)\,,$ (2.8)

where $a^{\mu}{}_{\nu}$ generates rigid $GL(D)$ rotations, $B_{\mu\nu}$
produces rigid shifts of the two-form and $\beta^{\mu\nu}$ is a bi-vector that
mixes the gravitational and two-form sector. In this paper we will consider
$a^{\mu}{}_{\nu}=B_{\mu\nu}=0$ because their action in supergravity is
trivial, and focus only on $\beta$ transformations. Notice that in DFT the
derivatives belong to the fundamental representation of $O(D,D)$, and so
transform linearly under $O(D,D)$. In particular, the $\beta$ elements break
the choice of section

$\delta_{\beta}\partial_{M}\dots=h_{M}{}^{N}\partial_{N}\dots=(\beta^{\mu\nu}\partial_{\nu},\,0)\dots\,,$
(2.9)

which must be restored by imposing the constraint

$\beta^{\mu\nu}\partial_{\nu}\dots=0\,.$ (2.10)

To establish a connection with supergravity, we must break the double Lorentz
symmetry to a single Lorentz transformation and gauge fix the two vielbeins to
a single one

$\bar{e}_{\mu}{}^{a}=\bar{e}_{\mu}{}^{\underline{a}}\delta^{a}_{\underline{a}}=\bar{e}_{\mu}{}^{\overline{a}}\delta^{a}_{\overline{a}}\,.$
(2.11)

To this end we have introduced Kronecker deltas to force the two Lorentz
groups to carry the same set of indices $a,b,c,\dots$, which will be the
standard Lorentz indices in supergravity. The gauge fixing applied to the flat
derivatives yields

$D_{\underline{a}}=-\frac{1}{\sqrt{2}}\delta^{a}_{\underline{a}}\,D_{a}\ ,\ \
\ \ D_{\overline{a}}=\frac{1}{\sqrt{2}}\delta^{a}_{\overline{a}}\,D_{a}\ ,\ \
\ \ D_{a}=\bar{e}^{\mu}{}_{a}\partial_{\mu}\ ,$ (2.12)

and to the generalized fluxes (see Appendix A for definitions.)

$\displaystyle F_{\overline{a}\underline{bc}}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\delta^{a}_{\overline{a}}\delta^{b}_{\underline{b}}\delta^{c}_{\underline{c}}\left(w_{abc}-\frac{1}{2}H_{abc}\right)\,,$
(2.13) $\displaystyle F_{\underline{a}\overline{bc}}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\delta^{a}_{\underline{a}}\delta^{b}_{\overline{b}}\delta^{c}_{\overline{c}}\left(w_{abc}+\frac{1}{2}H_{abc}\right)\,.$
(2.14)

Because we have gauge fixed the double Lorentz symmetry to its diagonal
subgroup, the two Lorentz parameters are no longer independent. We must then
explore how they are related, and what is the most convenient way to express
them in terms of the Lorentz parameter in supergravity. To achive this, we
write the two Lorentz invariant metrics in terms of a single one

$g_{ab}=\delta_{a}^{\underline{a}}\delta_{b}^{\underline{b}}g_{\underline{ab}}=\delta_{a}^{\overline{a}}\delta_{b}^{\overline{b}}g_{\overline{ab}}\,,$
(2.15)

and also express all the Lorentz parameters (including the generalized Green-
Schwarz deformation $\Delta$) in terms of the same set of indices

$\Lambda_{\overline{ab}}=\delta^{a}_{\overline{a}}\delta^{b}_{\overline{b}}\overline{\Lambda}_{ab}\
,\ \ \ \
\Lambda_{\underline{ab}}=\delta^{a}_{\underline{a}}\delta^{b}_{\underline{b}}\underline{\Lambda}_{ab}\
,\ \ \ \
\Delta^{(1)}_{\overline{a}\underline{b}}=\delta^{a}_{\overline{a}}\delta^{b}_{\underline{b}}\Delta^{(1)}_{ab}\
.$ (2.16)

In this way, we end up with two different transformations for the gauge fixed
vielbein, each one coming from the transformation of each component of the
generalized frame

$\displaystyle\delta E^{\mu\underline{a}}\ \ \ \ \ \to\ \ \ \ \
\delta\bar{e}^{\mu}{}_{a}\ $ $\displaystyle=\ \bar{e}^{\mu
b}\left(\beta_{b}{}^{c}\bar{b}_{ca}-\beta_{ba}-\underline{\Lambda}_{ba}-\Delta^{(1)}_{ba}\right)\
,$ (2.17a) $\displaystyle\delta E^{\mu\overline{a}}\ \ \ \ \ \to\ \ \ \ \
\delta\bar{e}^{\mu}{}_{a}\ $ $\displaystyle=\ \bar{e}^{\mu
b}\left(\beta_{b}{}^{c}\bar{b}_{ca}+\beta_{ba}+\overline{\Lambda}_{ba}-\Delta^{(1)}_{ab}\right)\
,$ (2.17b)

where

$\Delta^{(1)}_{ab}=-\frac{a}{4}D_{b}\underline{\Lambda}^{{cd}}\left(w_{acd}-\frac{1}{2}H_{acd}\right)+\frac{b}{4}D_{a}\overline{\Lambda}^{{cd}}\left(w_{bcd}+\frac{1}{2}H_{bcd}\right)\,.$
(2.18)

These transformations must obviously coincide, which imposes the required
relation between the double Lorentz parameters

$\underline{\Lambda}_{ab}=-\overline{\Lambda}_{ab}-2\Delta^{(1)}_{[ab]}-2\beta_{ab}\
.$ (2.19)

Up to redefinitions of the standard Lorentz parameter of supergravity
$\Lambda_{ab}$, this equation is parametrically solved as

$\underline{\Lambda}_{ab}=-\Lambda_{ab}-\beta_{ab}-\Delta^{(1)}_{[ab]}\ ,\ \ \
\overline{\Lambda}_{ab}=\Lambda_{ab}-\beta_{ab}-\Delta^{(1)}_{[ab]}\,.$ (2.20)

Combining these expressions with (2.18) and keeping only terms up to first
order in $\alpha^{\prime}$, we find that Lorentz and $\beta$ transformations
of the gauge fixed fields in the DFT scheme of supergravity read

$\displaystyle\delta\bar{e}_{\mu a}\ $ $\displaystyle=\
-\bar{e}_{\mu}{}^{b}\left(\beta_{a}{}^{c}\bar{b}_{cb}+\Lambda_{ab}-\Delta^{(1)}_{(ab)}\right)\,,$
(2.21a) $\displaystyle\delta\bar{\phi}\ $ $\displaystyle=\
\frac{1}{2}\left(\beta^{\mu\nu}\bar{b}_{\mu\nu}+\Delta^{(1)}_{a}{}^{a}\right)\,,$
(2.21b) $\displaystyle\delta\bar{b}_{\mu\nu}\ $ $\displaystyle=\
-\beta_{\mu\nu}-\bar{b}_{\mu\rho}\beta^{\rho\sigma}\bar{b}_{\sigma\nu}-2\bar{e}_{\mu}{}^{a}\bar{e}_{\nu}{}^{b}\Delta^{(1)}_{[ab]}\,,$
(2.21c)

where

$\displaystyle\Delta^{(1)}_{ab}$ $\displaystyle\equiv$
$\displaystyle\frac{a+b}{4}\left(D_{(a}\Lambda^{{cd}}w_{b)cd}-\frac{1}{2}D_{(a}\beta^{{cd}}H_{b)cd}\right)+\frac{b-a}{4}\left(\frac{1}{2}D_{(a}\Lambda^{{cd}}H_{b)cd}-D_{(a}\beta^{{cd}}w_{b)cd}\right)\,$
(2.22) $\displaystyle+$
$\displaystyle\frac{a+b}{4}\left(\frac{1}{2}D_{[a}\Lambda^{{cd}}H_{b]cd}-D_{[a}\beta^{{cd}}w_{b]cd}\right)+\frac{b-a}{4}\left(D_{[a}\Lambda^{{cd}}w_{b]cd}+\frac{1}{2}D_{[a}\beta^{{cd}}H_{b]cd}\right)\,.\
\ \ \ \ \ \ $

In summary, we have derived the first order deformations of Lorentz and
$\beta$ transformations in the DFT scheme of supergravity, namely the scheme
in which the fields are given by a gauge fixing of the duality covariant
components of the generalized fields in DFT.

It is instructive to cast the result in the form given in [2], which allows to
find purely flattened expressions for the transformation rules. Defining111It
is important to note that, contrary to the conventions in the rest of the
paper, here $\delta b_{ab}$ is the flat version of $\delta b_{\mu\nu}$ and not
the variation of $b_{ab}$.

$\delta\bar{e}_{ab}=\bar{e}^{\mu}{}_{a}\,\delta\bar{e}_{\mu b}\ ,\ \ \ \
\delta\bar{\phi}=\frac{1}{2}\delta\bar{e}_{a}{}^{a}\ ,\ \ \ \
\delta\bar{b}_{ab}=\bar{e}^{\mu}{}_{a}\bar{e}^{\nu}{}_{b}\,\delta\bar{b}_{\mu\nu}\
,$ (2.23)

the leading order transformation rules are

$\displaystyle\delta^{(0)}\bar{e}_{ab}$
$\displaystyle=-\bar{b}_{a}{}^{c}\beta_{cb}+\Lambda_{ab}\,,$ (2.24a)
$\displaystyle\delta^{(0)}\bar{b}_{ab}$
$\displaystyle=-\beta_{ab}-\bar{b}_{ac}\beta^{cd}\bar{b}_{db}\,,$ (2.24b)

and the first order $\alpha^{\prime}$-corrections are

$\displaystyle\delta^{(1)}_{\Lambda}\bar{e}_{ab}$
$\displaystyle=\frac{a+b}{4}D_{(a}\Lambda^{cd}w_{b)cd}+\frac{b-a}{8}D_{(a}\Lambda^{cd}H_{b)cd}\,,$
(2.25a) $\displaystyle\delta^{(1)}_{\Lambda}\bar{b}_{ab}$
$\displaystyle=-\frac{a+b}{4}D_{[a}\Lambda^{cd}H_{b]cd}-\frac{b-a}{2}D_{[a}\Lambda^{cd}w_{b]cd}\,,$
(2.25b)

and

$\displaystyle\delta^{(1)}_{\beta}\bar{e}_{ab}$
$\displaystyle=-\frac{a+b}{8}D_{(a}\beta^{cd}H_{b)cd}-\frac{b-a}{4}D_{(a}\beta^{cd}w_{b)cd}\,,$
(2.26a) $\displaystyle\delta^{(1)}_{\beta}\bar{b}_{ab}$
$\displaystyle=\frac{a+b}{8}D_{[a}\beta^{cd}w_{b]cd}-\frac{b-a}{4}D_{[a}\beta^{cd}H_{b]cd}\,.$
(2.26b)

These transformations can be cast in other schemes, performing field
redefinitions $\\{\bar{e},\,\bar{\phi},\,\bar{b}\\}\to\\{e,\,\phi,\,b\\}$ of
the form

$\displaystyle\bar{e}_{\mu}{}^{a}\ $ $\displaystyle=\
e_{\mu}{}^{b}(\delta_{b}^{a}+F^{a}{}_{b})\ ,\ \ \ e_{\mu}{}^{a}\ =\
\bar{e}_{\mu}{}^{b}(\delta_{b}^{a}-F^{a}{}_{b})\,,$ (2.27a)
$\displaystyle\bar{\phi}\ $ $\displaystyle=\ \phi+\frac{1}{2}F_{a}{}^{a}\ ,\ \
\ \ \ \ \ \ \ \bar{b}_{\mu\nu}\ =\ b_{\mu\nu}-G_{\mu\nu}\,,$ (2.27b)

where both $F_{ab}$ and $G_{\mu\nu}$ start at first order in
$\alpha^{\prime}$. Therefore, for the purpose of obtaining the first order
deformations of the transformation rules, these functions can be expressed in
terms of either the new (unbarred) or the previous (barred) set of fields in
the DFT scheme. Since we are only interested in connecting with schemes that
preserve the generalized measure
$\sqrt{-\bar{g}}e^{-2\bar{\phi}}=\sqrt{-g}e^{-2\phi}$, the redefinition of the
dilaton is determined by that of the vielbein.

The functions $F_{ab}$ and $G_{\mu\nu}$ define the scheme, and the Lorentz and
$\beta$ transformations in different schemes are given, up to redefinitions of
the parameters, by

$\displaystyle\delta e_{ab}\ $ $\displaystyle=\
-(b_{a}{}^{c}-G_{a}{}^{c})\beta_{cb}+\Lambda_{ab}+\Delta^{(1)}_{(ab)}-\Delta
F_{ba}\,,$ (2.28a) $\displaystyle\delta b_{ab}\ $ $\displaystyle=\
-\beta_{ab}-b_{ac}\beta^{cd}b_{db}-2\Delta^{(1)}_{[ab]}-4\beta_{[a}{}^{c}F_{(b]c)}+2\beta^{cd}b_{c[a}G_{b]d}+e^{\mu}{}_{a}e^{\nu}{}_{b}\delta
G_{\mu\nu}\,,\ \ \ \ \ \ \ \ \ $ (2.28b) $\displaystyle\delta\phi\ $
$\displaystyle=\ \frac{1}{2}\delta e_{a}{}^{a}\,,$ (2.28c)

where all the fields belong to the new (unbarred) scheme, and

$\Delta F_{ab}=\delta F_{ab}-\Lambda^{c}{}_{a}F_{cb}-\Lambda^{c}{}_{b}F_{ac}\
,$ (2.29)

denotes the non-covariant transformation of the field redefinition.

In forthcoming sections we will specify the functions $F_{ab}$ and
$G_{\mu\nu}$ that connect the DFT scheme with those of Bergshoeff-de Roo (BdR)
and Metsaev-Tseytlin (MT).

##  3 Scheme independent statements

Although the first order corrections to the $\beta$ transformations depend on
the scheme, there are important issues that are independent of field
redefinitions. In this section we discuss some of these scheme independent
questions: $\beta$ symmetry at leading order, redundancies in the $\beta$
transformation rules at ${\cal O}(\alpha^{\prime})$, the relation between
closure of the symmetry algebra and the invariance of the action/Lagrangian,
and the effect of field and parameter redefinitions in the closure.

###  3.1 $\beta$ symmetry at leading order

The two-derivative DFT Lagrangian descends to the universal two derivative NS-
NS Lagrangian

$L^{(0)}=R-4(\nabla\phi)^{2}+4\Box\phi-\frac{1}{12}H^{2}\ ,$ (3.1)

which must then be invariant under the transformations (2.24),

$\delta_{\beta}e_{ab}=-b_{ac}\beta^{c}{}_{b}\ ,\ \ \
\delta_{\beta}b_{ab}=-\beta_{ab}-b_{ac}\beta^{cd}b_{db}\ ,\ \ \
\delta_{\beta}\phi=\frac{1}{2}\delta e_{a}{}^{a}\ ,$ (3.2)

whenever the constraint (2.10) holds, namely

$\displaystyle\beta^{\mu\nu}\partial_{\nu}\cdots=0\,.$ (3.3)

Using that $\beta^{\mu\nu}$ is constant and antisymmetric, the following
useful identities are obtained

$\displaystyle\nabla_{a}\beta^{bc}=2w_{da}{}^{[b}\beta^{c]d}\,,\qquad\beta^{ab}w_{abc}=0\,,\qquad\omega_{[ab]}{}^{c}\beta_{1}^{ad}\beta_{2}^{be}=0\,,$
(3.4)

where the last one is particularly useful for the study of the symmetry
algebra. It is instructive to compute the $\beta$ transformations of many of
the tensors and connections that appear in supergravity

$\displaystyle\delta_{\beta}\left(\sqrt{-g}e^{-2\phi}\right)\ $
$\displaystyle=\ 0\,,$ (3.5a) $\displaystyle\delta_{\beta}w_{cab}\ $
$\displaystyle=\
\beta_{[a}{}^{d}H_{b]cd}-\frac{1}{2}\beta_{c}{}^{d}H_{abd}\,,$ (3.5b)
$\displaystyle\delta_{\beta}H_{abc}\ $ $\displaystyle=\
-3\nabla_{[a}\beta_{bc]}\,,$ (3.5c) $\displaystyle\delta_{\beta}(D_{a}\phi)\ $
$\displaystyle=\ \frac{1}{2}\beta^{cd}H_{acd}\,,$ (3.5d)
$\displaystyle\delta_{\beta}(\nabla_{a}\nabla_{b}\phi)\ $ $\displaystyle=\
\frac{1}{2}\nabla_{(a}\left(\beta^{cd}H_{b)cd}\right)-\beta^{c}{}_{(a}H_{b)cd}\nabla^{d}\phi\,,$
(3.5e) $\displaystyle\delta_{\beta}R_{ab}{}^{cd}\ $ $\displaystyle=\
-\nabla_{[a}\left(\beta_{b]}{}^{e}H_{e}{}^{cd}\right)-\nabla^{[c}\left(\beta^{d]e}H_{eab}\right)\,,$
$\displaystyle\ \ \ \
+\left(\nabla_{[a}\beta^{[ce}-\nabla^{[c}\beta_{[a}{}^{e}-\nabla^{e}\beta_{[a}{}^{[c}\right)H^{d]}{}_{b]e}\,,$
(3.5f) $\displaystyle\delta_{\beta}R_{ab}\ $ $\displaystyle=\
-\nabla^{c}\left(\beta^{d}{}_{(a}H_{b)cd}\right)-\nabla_{(a}\left(\beta^{cd}H_{b)cd}\right)-\frac{1}{2}\nabla_{(a}\beta^{cd}H_{b)cd}\,,$
(3.5g) $\displaystyle\delta_{\beta}R\ $ $\displaystyle=\
-2\nabla^{a}\left(\beta^{bc}H_{abc}\right)-\frac{1}{2}\nabla^{a}\beta^{bc}H_{abc}\;.$
(3.5h)

These identities can now be used to fix the couplings of a generic combination
of gauge invariant terms, by requiring $\beta$ symmetry. Indeed,

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\delta_{\beta}\left[\sqrt{-g}f(\phi)\left(R+c\,\Box\phi+d\left(\nabla\phi\right)^{2}+e\,H^{2}\right)\right]$
(3.8) $\displaystyle=$
$\displaystyle\sqrt{-g}f(\phi)\left[(c-4)\nabla^{a}\left(\beta^{bc}H_{abc}\right)-(12e+1)\nabla^{a}\beta^{bc}H_{abc}+2(c+d)\nabla^{a}\phi\beta^{bc}H_{abc}\right]$
$\displaystyle+\frac{1}{2}\sqrt{-g}\beta^{ab}b_{ab}\left(\vphantom{2^{\frac{1}{2}}}2f(\phi)+f^{\prime}(\phi)\right)\;\left[R+c\,\Box\phi+d\left(\nabla\phi\right)^{2}+e\,H^{2}\right]\;,$

determines $c=-d=4$, $e=-\frac{1}{12}$ and $f(\phi)=e^{-2\phi}$, in perfect
agreement with supergravity (3.1). Finally, we notice that $\beta$ and Lorentz
transformation rules close off-shell to lowest order as follows

$\left[\delta_{1}\,,\,\delta_{2}\right]=-\delta_{12}\ ,\ \ \
\Lambda_{12ab}=2\,\beta_{1[a}{}^{c}\beta_{2\,b]c}+2\,\Lambda_{1[a}{}^{c}\Lambda_{2\,b]c}\
.$ (3.9)

###  3.2 A four-parameter redundancy of first order transformations

When the variation of the dilaton leaves the measure $\sqrt{-g}e^{-2\phi}$
invariant, i.e. $\delta\phi=\frac{1}{2}\delta e_{a}{}^{a}$, the variation of
$L^{(0)}$ can be decomposed as

$\delta{L}^{(0)}={\cal E}{}^{ab}\;\delta e_{ab}+{\cal B}^{ab}\;\delta
b_{ab}+{\cal D}_{a}\left[2\;{\cal D}_{b}\delta e^{ab}-\frac{1}{2}H^{abc}\delta
b_{bc}\right]\,.$ (3.10)

Here we defined the operator

${\cal
D}_{a}X={e^{2\phi}}\nabla_{a}\left(e^{-2\phi}X\right)=\nabla_{a}X-2\nabla_{a}\phi
X\ ,$ (3.11)

acting on a generic tensor $X$. Although it is not formally a derivative as it
does not obey the Leibnitz rule, it leads to a total derivative when inserted
into the action, and so can be discarded in such context. Additionally, we
introduced the tensors that define the equations of motion of the frame ${\cal
E}_{ab}$ and the two-form ${\cal B}_{ab}$

$\displaystyle{\cal E}_{ab}$ $\displaystyle=$
$\displaystyle-2\left(\vphantom{\frac{1}{2}}R_{ab}+2\nabla_{a}\nabla_{b}\phi-\frac{1}{4}H_{acd}H_{b}{}^{cd}\right)\,,$
(3.12) $\displaystyle{\cal B}_{ab}$ $\displaystyle=$
$\displaystyle\frac{1}{2}{\cal D}_{c}H_{ab}{}^{c}{}\ .$ (3.13)

An interesting observation is that, if the total derivative term in (3.10) is
ignored, namely if one focuses on invariance of the action instead of the
Lagrangian, $\delta L^{(0)}$ (3.10) vanishes up to total derivatives for
variations of the form

$\displaystyle\delta e_{ab}$ $\displaystyle=$ $\displaystyle
T^{1}_{[(ab)(cd)]}{\cal E}^{cd}+T^{2}_{(ab)[cd]}{\cal B}^{cd}\ ,$ (3.14)
$\displaystyle\delta b_{ab}$ $\displaystyle=$
$\displaystyle-T^{2}_{(cd)[ab]}{\cal E}^{cd}+T^{3}_{[[ab][cd]]}{\cal B}^{cd}\
.$ (3.15)

The $T$ tensors can depend on anything as long as their symmetries are
respected. In particular, if they are linear in the parameters of the
transformation, and contain no derivatives, they account for first order
deformations that leave the action invariant.

For $\beta$ transformations, there are only four possible deformations of this
type to first order in $\alpha^{\prime}$, namely

$\displaystyle T^{1}_{ab}{}^{cd}$
$\displaystyle=\kappa\beta_{(a}{}^{(c}g_{b)}{}^{d)}\,,$ (3.16a) $\displaystyle
T^{2}_{ab}{}^{cd}$ $\displaystyle=\gamma\beta_{(a}{}^{[c}g_{b)}{}^{d]}+\chi
g_{ab}\beta^{cd}\,,$ (3.16b) $\displaystyle T^{3}_{ab}{}^{cd}$
$\displaystyle=\rho\beta_{[a}{}^{[c}g_{b]}{}^{d]}\,,$ (3.16c)

which in turn define the following four-parameter ($\gamma,\chi,\kappa,\rho$)
family of first order transformations

$\displaystyle\delta_{\beta}^{(1)}e_{ab}$
$\displaystyle=\gamma\beta_{c(a}{\cal B}_{b)}{}^{c}+\chi g_{ab}\beta_{cd}{\cal
B}^{cd}+\kappa\beta_{(a}{}^{c}{\cal E}_{b)c}\,,$ (3.17a)
$\displaystyle\delta_{\beta}^{(1)}b_{ab}$
$\displaystyle=-\gamma\beta^{c}{}_{[a}{\cal E}_{b]c}-\chi\beta_{ab}{\cal
E}_{c}{}^{c}+\rho\beta_{c[a}{\cal B}^{c}{}_{b]}\,.$ (3.17b)

This result is scheme independent, as it simply amounts to a first order
invariance of the lowest order action. Then, in any scheme, adding these first
order deformations to the $\beta$ transformations, respects the invariance of
the action to first order, for whatever values of ($\gamma,\chi,\kappa,\rho$).
Let us emphasize that, when the transformations are deformed by these terms,
the invariance of the Lagrangian is only achieved up to total derivatives.

Besides these, there are still extra symmetries of the two-derivative action,
of the form

$\displaystyle\delta_{\beta}^{(1)}e_{ab}=$ $\displaystyle
f_{[ab]}(\beta,\psi_{i})+\nabla_{(a}h_{b)}(\beta,\psi_{i})\,,$ (3.18a)
$\displaystyle\delta_{\beta}^{(1)}b_{ab}=$
$\displaystyle\nabla_{[a}t_{b]}(\beta,\psi_{i})+H_{abc}h^{c}(\beta,\psi_{i})\,,$
(3.18b)

where $f,h$ and $t$ are arbitrary functions, that can be interpreted as due to
$\beta$ and $\psi_{i}$ field dependent reparameterizations of Lorentz,
diffeomorphisms and gauge parameters. It is worth noticing that, while $f$ and
$t$ describe an exact symmetry of the Lagrangian, $h$ is in general only a
symmetry of the action. Exceptions can occur, for instance when
$h_{a}\sim\beta_{ab}X^{b}$ for arbitrary functions $X^{b}$, because
diffeomorphisms with parameter $\xi^{\mu}=\beta^{\mu\nu}X_{\nu}$ leave the
Lagrangian exactly invariant. In fact in the conventional KK reduction, the
parameter $\xi^{\mu}=\beta^{\mu\nu}X_{\nu}$ has only internal components and
so represents a gauge transformation, which leaves the Lagrangian invariant.
We will go back to the role of reparameterizations in subsection 3.4.

In conclusion, any first order deformation of the $\beta$ transformation rules
that leave the four-derivative action invariant, is determined up to the four
parameters $\gamma,\kappa,\chi,\rho$ and three functions $f,h,t$.

###  3.3 Invariance of the action implies closure of the symmetry algebra

So far we have found the symmetries that leave the first order Lagrangian (in
the DFT scheme) invariant. Now we point out that invariance of the Lagrangian
with respect to a complete set of symmetries implies that the algebra of
transformations closes off-shell. Later, we will show that the deformations
discussed in the previous section, i.e. transformations that depend on the
equations of motion and leave the action (instead of the Lagrangian)
invariant, still respect closure, but only on-shell. Again, the contents of
this section are scheme independent. Similar results along these lines can be
found in [13].

The algebra of transformations that leave a Lagrangian invariant is expected
to close on general grounds. To see this, consider a generic Lagrangian
depending on fields $\psi_{i}$ and their derivatives up to an arbitrary order,
namely
$L(\psi_{i},\partial_{\mu}\psi_{i},\partial_{\mu}\partial_{\nu}\psi_{i}\,,...)$.
We assume that terms with derivatives of different orders are independent. If
$L$ is invariant under a complete set of local and global transformations,
infinitesimally parameterized by $\zeta_{A}$, then

$\displaystyle 0=\delta_{\zeta}L=\sum_{n=0}^{\infty}\frac{\partial
L}{\partial(\partial^{n}\psi_{i})}\partial^{n}\delta_{\zeta}\psi_{i}=\frac{\partial
L}{\partial\psi_{i}}\delta_{\zeta}\psi_{i}+\frac{\partial
L}{\partial(\partial_{\mu}\psi_{i})}\partial_{\mu}\delta_{\zeta}\psi_{i}+\frac{\partial
L}{\partial(\partial_{\mu}\partial_{\nu}\psi_{i})}\,\partial_{\mu}\partial_{\nu}\delta_{\zeta}\psi_{i}+...\,,\
\ \ \ $ (3.19)

where the dots stand for contributions depending on higher-order derivatives.
The above sum vanishes if and only if the variation is performed with respect
to a symmetry transformation.

Now we consider the commutator of two variations

$\displaystyle 0=[\delta_{\zeta_{1}},\delta_{\zeta_{2}}]L$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\frac{\partial
L}{\partial(\partial^{n}\psi_{i})}\partial^{n}\left(\vphantom{e^{\frac{1}{2}}}[\delta_{\zeta_{1}},\delta_{\zeta_{2}}]\psi_{i}\right)$
$\displaystyle+2\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}\frac{\partial^{2}L}{\partial(\partial^{n}\psi_{i})\partial(\partial^{m}\psi_{j})}\left(\partial^{n}\delta_{\zeta_{[1}}\psi_{i}\right)\left(\partial^{m}\delta_{\zeta_{2]}}\psi_{j}\right)\
.$

The last term here cancels due the symmetry under the exchange
$(n,i)\leftrightarrow(m,j)$ and the antisymmetry with respect to
$1\leftrightarrow 2$. Comparing the first term with (3.19), we conclude that
$[\delta_{\zeta_{1}},\delta_{\zeta_{2}}]L$ vanishes if and only if
$[\delta_{\zeta_{1}},\delta_{\zeta_{2}}]\psi_{i}$ amounts to a symmetry
transformation, namely

$[\delta_{\zeta_{1}},\delta_{\zeta_{2}}]\psi_{i}=-\delta_{\zeta_{12}}\psi_{i}\
,$ (3.21)

for some parameter (bracket) $\zeta_{12}$. Since we only assumed that the
Lagrangian is invariant, we conclude that invariance of the Lagrangian implies
that the symmetry algebra closes off-shell.

Now we deform the transformations that leave the Lagrangian invariant as
discussed in the previous subsection, i.e. we consider the deformations (3.17)
with parameters ($\gamma,\chi,\kappa,\rho$) that are first order in
$\alpha^{\prime}$. Then, the LHS of (3.21) becomes, to first order,

$\displaystyle[\delta_{\zeta_{1}}+\delta_{\zeta_{1}}^{(\gamma,\chi,\kappa,\rho)},\delta_{\zeta_{2}}+\delta_{\zeta_{2}}^{(\gamma,\chi,\kappa,\rho)}]\psi_{i}=[\delta_{\zeta_{1}},\delta_{\zeta_{2}}]\psi_{i}+2\delta^{(0)}_{\zeta_{[1}}\delta_{\zeta_{2]}}^{(\gamma,\chi,\kappa,\rho)}\psi_{i}+2\delta_{\zeta_{[1}}^{(\gamma,\chi,\kappa,\rho)}\delta^{(0)}_{\zeta_{2]}}\psi_{i}\,.$
(3.22)

Here, the first term in the RHS gives the bracket
$\delta_{\zeta_{12}}\psi_{i}$ discussed above. The last term vanishes on-shell
because the ($\gamma,\chi,\kappa,\rho$) variations depend on the lowest order
equations of motion. The second term is more subtle, because it involves
variations of the equations of motion. Under the standard symmetries they
transform covariantly, but we have to make sure that the same is true for the
lowest order $\beta$ transformations. Indeed, we find that $\beta$
transformations mix the equations of motion of the vielbein and two-form as
follows

$\delta_{\beta}^{(0)}{\cal E}_{ab}=-4\beta_{c(a}{\cal B}_{b)}{}^{c}\ ,\ \ \
\delta_{\beta}^{(0)}{\cal B}_{ab}=\beta_{c[a}{\cal E}_{b]}{}^{c}\,,$ (3.23)

which then implies that the second term in the RHS of (3.22) also vanishes on-
shell. In summary, the deformed transformations that leave the action
invariant, close on-shell

$\displaystyle[\delta_{\zeta_{1}}+\delta_{\zeta_{1}}^{(\gamma,\chi,\kappa,\rho)},\delta_{\zeta_{2}}+\delta_{\zeta_{2}}^{(\gamma,\chi,\kappa,\rho)}]\psi_{i}=-(\delta_{\zeta_{12}}+\delta_{\zeta_{12}}^{(\gamma,\chi,\kappa,\rho)})\psi_{i}+\dots\,,$
(3.24)

where the dots represent terms proportional to the equations of motion, and we
have the freedom to add the last term in (3.24) as it also vanishes on-shell.

###  3.4 Effect of field and parameter redefinitions on the symmetry algebra

In section 2 we examined the effect of field redefinitions on the
transformation rules, when moving from one scheme to another. Here we further
explore the role of field and parameter redefinitions on the bracket.

Even though field redefinitions preserve the closure of the symmetry algebra,
the precise form of the bracket could be altered. Indeed, consider fields
$\psi_{i}$ satisfying

$\displaystyle\left[\delta_{\zeta_{1}},\delta_{\zeta_{2}}\right]\psi_{i}=-\delta_{\zeta_{12}(\psi)}\psi_{i}\,,$
(3.25)

where we have made explicit that the bracket $\zeta_{12}(\psi)$ could
eventually be field dependent. Let us consider the field redefinition

$\displaystyle\psi_{i}\to\psi^{\prime}{}_{i}=\psi{}_{i}+\Delta\psi{}_{i}=\psi{}_{i}+\sum
a_{i}{}^{j_{1}j_{2}...}\partial^{n_{1}}\psi_{j_{1}}\partial^{n_{2}}\psi_{j_{2}}\dots$
(3.26)

Here $\partial^{n}$ denote derivatives of order $n=0,1,2,...$ and ellipsis
stand for extra field dependence. So, it parametrizes a generic field
redefinition with no restriction on the number of fields and derivatives
thereof. The bracket transforms the new field as

$\displaystyle\left[\delta_{\zeta_{1}},\delta_{\zeta_{2}}\right]\psi^{\prime}{}_{i}$
$\displaystyle=$
$\displaystyle\left[\delta_{\zeta_{1}},\delta_{\zeta_{2}}\right]\psi{}_{i}+\sum
a_{i}{}^{j_{1}j_{2}...}\partial^{n_{1}}\left(\left[\delta_{\zeta_{1}},\delta_{\zeta_{2}}\right]\psi_{j_{1}}\right)\partial^{n_{2}}\psi_{j_{2}}\dots$
(3.28) $\displaystyle+\ \sum
a_{i}{}^{j_{1}j_{2}...}\partial^{n_{1}}\psi_{j_{1}}\partial^{n_{2}}\left(\left[\delta_{\zeta_{1}},\delta_{\zeta_{2}}\right]\psi_{j_{2}}\right)\dots+\dots$
$\displaystyle=$ $\displaystyle-\delta_{\zeta_{12}(\psi)}\psi^{\prime}_{i}\;,$
(3.29)

where in the first equality we used a symmetry argument similar to the one
followed in (3.3), implying that terms where $\delta_{\zeta_{1}}$ and
$\delta_{\zeta_{2}}$ act on different fields cancel each other after
antisymmetrization of indices $(1\leftrightarrow 2)$.

Plugging the derivative expansion of the old bracket at ${\cal
O}(\alpha^{\prime})$

$\displaystyle\zeta_{12}(\psi)=\zeta^{(0)}_{12}(\psi)+\zeta^{(1)}_{12}(\psi)\;,$
(3.30)

into (3.26), we get the new bracket

$\displaystyle\left[\delta_{\zeta_{1}},\delta_{\zeta_{2}}\right]\psi^{\prime}{}_{i}=-\delta_{\zeta^{\prime}_{12}(\psi^{\prime})}\psi^{\prime}_{i}\;\;,\
\ \ \ \ \ \ \ \ \
\zeta^{\prime}_{12}(\psi^{\prime}):=\zeta_{12}(\psi^{\prime})-\frac{\partial\zeta^{(0)}_{12}}{\partial\psi_{i}}\Delta\psi_{i}\;,$
(3.31)

where we have considered that $\zeta^{(0)}_{12}$ eventually depends on the
fields $\psi_{i}$, but not on derivatives thereof. It is worth noticing here
that in general the bracket gets no corrections after a field redefinition,
because the leading order bracket $\zeta^{(0)}_{12}$ is in general field
independent. Although this is the case for diffeomorphisms, Lorentz and gauge
transformations, for $\beta$ transformations it is not (see Appendix B).

Redefinitions of the symmetry parameters are more subtle. Sometimes it is
convenient to redefine them so that the transformation rules look more
suitable. For instance, an antisymmetric piece in $\delta^{(1)}_{\beta}e_{ab}$
can be canceled with an appropriate choice of the function $f_{[ab]}$ in
(3.18). To see the impact on the bracket, take two equivalent transformations
differing in a reparameterization

$\displaystyle\tilde{\delta}_{\zeta}=\delta_{\zeta}+\delta_{\Delta\zeta}\,,$
(3.32)

where $\delta_{\zeta}$ is a symmetry transformation with a known bracket and
$\Delta\zeta=\ell(\zeta,\psi_{i})$ is a subleading shift depending on the old
parameters $\zeta$ as well as the fields $\psi_{i}$ of the invariant theory.
In our case $\zeta$ will represent the collective parameter
$\zeta_{n}=(\xi^{\mu},\lambda_{\nu},\Lambda^{a}{}_{b},\beta^{\mu\nu})$,
containing diffeomorphism, gauge, Lorentz and $\beta$ parameters. The new
bracket can be expressed in terms of the old one as

$\displaystyle\left[\tilde{\delta}_{\zeta_{1}},\tilde{\delta}_{\zeta_{2}}\right]\psi_{i}=\left[\delta_{\zeta_{1}},\delta_{\zeta_{2}}\right]\psi_{i}+2\delta_{\zeta_{[1}}\delta_{\Delta\zeta_{2]}}\psi_{i}+2\delta_{\Delta\zeta_{[1}}\delta_{\zeta_{2]}}\psi_{i}\,.$
(3.33)

The first term gives the old bracket, which is assumed to be known,

$\displaystyle\left[\delta_{\zeta_{1}},\delta_{\zeta_{2}}\right]\psi_{i}$
$\displaystyle=$
$\displaystyle-\delta_{\zeta_{12}}\psi_{i}=-\tilde{\delta}_{\zeta_{12}-\Delta\ell_{12}}\psi_{i}\,,$
(3.34)

where $\Delta\ell_{12}=\ell(\zeta_{12},\psi_{i})$ is written in terms of the
same set of functions $\ell$ introduced above, and

$\displaystyle\zeta_{12}=g(\zeta_{1},\zeta_{2},\psi_{i})\ ,$ (3.35)

defines the old bracket for certain known functions $g$.

The second and third factors in (3.33) can be written as

$\displaystyle
2\delta_{\zeta_{[1}}\delta_{\Delta\zeta_{2]}}\psi_{i}+2\delta_{\Delta\zeta_{[1}}\delta_{\zeta_{2]}}\psi_{i}$
$\displaystyle=$
$\displaystyle\left[\delta_{\zeta_{1}},\delta_{\Delta\zeta_{2}}\right]\psi_{i}-(1\leftrightarrow
2)\,.$ (3.36)

We can split
$\left[\delta_{\zeta_{1}},\delta_{\Delta\zeta_{2}}\right]\psi_{i}$ into a
variation with $\Delta\zeta$ fixed and $\delta_{\hat{\zeta}_{12}}\psi_{i}$
with $\hat{\zeta}_{12}=\delta_{\zeta_{1}}(\Delta\zeta_{2})$. The former can be
cast in terms of the old bracket as

$\displaystyle\left[\delta_{\zeta_{1}},\delta_{\Delta\zeta_{2}}\right]\psi_{i}\left.\vphantom{\frac{1}{2}}\right|_{\Delta\zeta}$
$\displaystyle=$ $\displaystyle-\delta_{\Delta g_{12}}\psi_{i}\,,$ (3.37)

where $\Delta g_{12}=g(\zeta_{1},\Delta\zeta_{2},\psi_{i})$, while
$\delta_{\hat{\zeta}_{12}}\psi_{i}$ is non trivial when $\Delta\zeta$ is field
dependent. Taking into account that $\delta_{\Delta
g_{12}}=\tilde{\delta}_{\Delta g_{12}}$ and
$\delta_{\hat{\zeta}_{12}}=\tilde{\delta}_{\hat{\zeta}_{12}}$ up to ${\cal
O}(\alpha^{\prime}{}^{2})$, the bracket after reparameterization turns out to
be

$\displaystyle\zeta^{(new)}_{12}=\zeta^{(old)}_{12}-\Delta\ell_{12}+2\Delta
g_{[12]}-2\hat{\zeta}_{[12]}.$ (3.38)

##  4 Bergshoeff-de Roo scheme

The gravitational sector in the DFT scheme is rather unconventional because
the metric is not Lorentz invariant. There is however a Lorentz non-covariant
redefinition of the vielbein that connects it to a scheme with a Lorentz
invariant metric. Actually, inserting in (2.27) the function

$\displaystyle
F_{ab}=\frac{a+b}{8}\left(w_{acd}w_{b}{}^{cd}+\frac{1}{4}H_{acd}H_{b}{}^{cd}\right)+\frac{b-a}{8}w_{(a}{}^{cd}H_{b)cd}\,,$
(4.1)

one obtaines the so-called generalized Bergshoeff-de Roo (BdR) scheme [3] with
this sole redefinition. Since $G_{\mu\nu}=0$, the two-form is not redefined
with respect to the DFT scheme, and so it maintains a non-trivial Lorentz
transformation for any value of $a$ and $b$ (2.25b). This might look
surprising, as for the bosonic string (which is obtained with $a=b$), the two-
form is normally Lorentz invariant. We will later clarify how this is resolved
by moving to the Metsaev-Tseytlin scheme.

The bi-parametric $(a\,,\,b)$ BdR action [3] generalizes that of [14] as
follows

$S_{\rm{BdR}}=\int
d^{D}x\sqrt{-g}e^{-2\phi}\left(L^{(0)}+aL^{(1)}_{a}+bL^{(1)}_{b}\right)\ ,$
(4.2)

where the lowest order Lagrangian $L^{(0)}$ is defined in (3.1), and the first
order one can be written in a flat index notation

$\displaystyle L^{(1)}_{a}$
$\displaystyle=\frac{1}{4}H^{abc}\Omega^{(-)}_{abc}-\frac{1}{8}R^{(-)}_{abcd}R^{(-)abcd}\
,$ (4.3a) $\displaystyle L^{(1)}_{b}$
$\displaystyle=-\frac{1}{4}H^{abc}\Omega^{(+)}_{abc}-\frac{1}{8}R^{(+)}_{abcd}R^{(+)abcd}\
.$ (4.3b)

Defining the torsionful connections
$w^{(\pm)}_{abc}=w_{abc}\pm\frac{1}{2}H_{abc}$, these expressions contain the
Chern-Simons three-forms $\Omega^{(\pm)}_{abc}$ and Riemann tensors
$R^{(\pm)}_{abcd}$,

$\displaystyle\Omega^{(\pm)}_{abc}$ $\displaystyle=$ $\displaystyle
w^{(\pm)}_{[\underline{a}d}{}^{e}D_{\underline{b}}w^{(\pm)}_{\underline{c}]e}{}^{d}+w^{(\pm)}_{[\underline{a}d}{}^{e}w^{(\pm)}_{fe}{}^{d}w_{\underline{bc}]}{}^{f}+\frac{2}{3}w^{(\pm)}_{[\underline{a}d}{}^{e}w^{(\pm)}_{\underline{b}e}{}^{f}w^{(\pm)}_{\underline{c}]f}{}^{d}\
,$ (4.4) $\displaystyle R^{(\pm)}_{abcd}$ $\displaystyle=$ $\displaystyle
2D_{[a}w^{(\pm)}_{b]cd}+2w_{[ab]}{}^{e}w^{(\pm)}_{ecd}+2w^{(\pm)}_{[\underline{a}c}{}^{e}w^{(\pm)}_{\underline{b}]ed}\
.$ (4.5)

Plugging (4.1) into (2.28) and using the identities (3.4-3.5), we find

$\displaystyle\delta^{(1)}e_{ab}$
$\displaystyle=\frac{a+b}{8}\beta_{(a}{}^{e}\left(w_{b)cd}H_{e}{}^{cd}+H_{b)cd}w_{e}{}^{cd}\right)+\frac{b-a}{4}\beta_{(a}{}^{e}\left(w_{b)cd}w_{e}{}^{cd}+\frac{1}{4}H_{b)cd}H_{e}{}^{cd}\right)\
,\ $ (4.6a) $\displaystyle\delta^{(1)}b_{ab}$
$\displaystyle=(a+b)\left[\beta^{ec}w_{e[a}{}^{d}w_{b]cd}-\beta^{ec}w_{[\underline{a}e}{}^{d}w_{\underline{b}]cd}-\frac{1}{2}\beta_{[a}{}^{c}w_{b]de}w_{c}{}^{de}-\frac{1}{8}\beta_{[a}{}^{c}H_{b]de}H_{c}{}^{de}\right]$
$\displaystyle\ \
+\,\frac{b-a}{2}\left[\beta^{ec}w_{e[a}{}^{d}H_{b]cd}-\beta^{ec}w_{[\underline{a}e}{}^{d}H_{\underline{b}]cd}-\frac{1}{2}\beta_{[a}{}^{c}w_{b]de}H_{c}{}^{de}-\frac{1}{2}\beta_{[a}{}^{c}H_{b]de}w_{c}{}^{de}\right]\
.$ (4.6b)

These expressions are in perfect agreement with our previous results in
equations (3.12) of [2], where they were obtained from the requirement of
invariance of the action. Actually, since $\delta^{(1)}L^{(0)}$ is of the form
(3.10), namely a sum of terms linear in the equations of motion plus total
derivatives, we took $\delta^{(0)}L^{(1)}$ precisely to that form, which
allowed us to read the first order transformations (4.6). We then verified
that the total derivatives in $\delta^{(1)}L^{(0)}$ are exactly canceled by
those in $\delta^{(0)}L^{(1)}$, implying
$\delta^{(0)}L^{(1)}+\delta^{(1)}L^{(0)}=0$ concluding that the
transformations (4.6) in fact leave the Lagrangian invariant.

We have seen in the previous section that a complete set of symmetries that
leave the Lagrangian invariant must necessarily close off-shell. In [2] we
confirmed this general statement, in particular for the transformations (3.2)
and (4.6), which close in combination with the local symmetries of the theory,
with the following $\alpha^{\prime}$-corrected brackets222 Here the bracket
for $\lambda_{12\mu}$ differs from the one in [2] by a trivial transformation
proportional to $\partial_{\mu}\left(\Lambda_{[1}^{ab}\beta_{2]ab}\right)$.

$\displaystyle\Lambda_{12}{}_{ab}$
$\displaystyle=2\,\beta_{1[a}{}^{c}\beta_{2\,b]c}+2\,\Lambda_{1[a}{}^{c}\Lambda_{2\,b]c}+2\,\xi_{[1}^{\mu}\partial_{\mu}\Lambda_{2]ab}$
(4.7a) $\displaystyle\ \ \
-4\,{F}_{c[a}\beta_{[1}{}_{b]}{}_{d}\beta_{2]}^{cd}-4\,{F}^{cd}{{\beta_{1}}}_{c[a}{{\beta_{2}}}_{b]d}-\left[\frac{a+b}{4}\,{H}_{ecd}+\frac{b-a}{2}\,{w}_{ecd}\right]\beta^{e}_{[1[a}\,{D}_{b]}\Lambda_{2]}^{cd}$
(4.7b) $\displaystyle\ \ \
-\left[(a+b)\,{w}^{ef}{}_{d}+\frac{b-a}{2}{H}^{ef}{}_{d}\right]\left({w}_{[a}{}^{cd}+{w}^{cd}{}_{[a}\right)\beta_{[1b]e}\beta_{2]cf}\
,$ (4.7c) $\displaystyle\lambda_{12\mu}$
$\displaystyle=4\,\xi_{[1}^{\nu}\partial_{[\nu}\lambda_{2]\mu]}-\,\frac{a+b}{4}\left(\partial_{\mu}\Lambda_{[1}^{ab}\;\beta_{2]}{}_{ab}-\Lambda_{[1}^{ab}\;\partial_{\mu}\beta_{2]}{}_{ab}\right)$
(4.7d) $\displaystyle\ \ \
-(b-a)\left(\Lambda_{[1}^{ab}\partial_{\mu}\Lambda_{2]}{}_{ab}+\beta_{[1}^{ab}\partial_{\mu}\beta_{2]}{}_{ab}\right)\
,$ (4.7e) $\displaystyle\xi_{12}^{\mu}$
$\displaystyle=2\xi^{\nu}_{[1}\partial_{\nu}\xi_{2]}^{\mu}+\beta_{[1}^{\mu\nu}\lambda_{2]\nu}\;,$
(4.7f)

where $\xi^{\mu}$ stands for the vector parameter of diffeomorphisms and
$\lambda_{\mu}$ for the one-form gauge parameter of $b_{\mu\nu}$. The presence
of diffeomorphisms in the closure of the algebra might appear in contradiction
with the results in Section 3.3, where we found that a complete set of
symmetries that leave a Lagrangian invariant must close among each other.
Indeed, the Lagrangian is not invariant but transforms as a scalar with
respect to the Lie derivative. However, this apparent contradiction is easily
resolved by noting that only the subset of trivial diffeomorphisms
$\xi^{\mu}=\beta^{\mu\nu}X_{\nu}$, i.e. those that leave the Lagrangian
invariant, are required to close the algebra.

We end this section by noting that the $\beta$ transformations (4.6) admit
first order deformations, as discussed in subsection 3.2. For any non
vanishing value of the set of parameters ($\gamma,\chi,\rho,\kappa$), the
Lagrangian would not to be invariant, but it would transform up to a total
derivative, which then implies that the action is invariant. Activating these
deformations would then require using the equations of motion to close the
symmetry algebra, as discussed in subsection 3.3.

##  5 Metsaev-Tseytlin scheme

The functions $F_{ab}$ and $G_{\mu\nu}$ that connect the DFT with the Metsaev-
Tseytlin scheme (MT) are

$\displaystyle F_{ab}\ $ $\displaystyle=\
\frac{a+b}{8}\left(w_{acd}w_{b}{}^{cd}+\frac{3}{4}H_{acd}H_{b}{}^{cd}\right)+\frac{b-a}{8}w_{(a}{}^{cd}H_{b)cd}\;,$
(5.1a) $\displaystyle G_{\mu\nu}\ $ $\displaystyle=-\ \frac{a+b}{4}\left({\cal
D}^{\rho}H_{\rho\mu\nu}+H_{[\mu}{}^{cd}w_{\nu]cd}\right)\,,$ (5.1b)

where ${\cal D}_{\mu}$ is the curved version of (3.11). These redefinitions
take the bi-parametric action in the DFT scheme to the following form [3]

$S_{\rm{MT}}=\int
d^{D}x\sqrt{-g}e^{-2\phi}\left(L^{(0)}+L^{(1)}_{-}+L^{(1)}_{+}+{\cal
D}_{\mu}V^{\mu}\right)\ ,$ (5.2)

where $L^{(0)}$ is the universal two-derivative Lagrangian (3.1), and the
first order corrections induced from DFT contain the standard MT first order
terms [15] (see Appendix A for definitions)

$\displaystyle L^{(1)}_{-}$ $\displaystyle=$
$\displaystyle\frac{a-b}{4}H^{\mu\nu\rho}\Omega_{\mu\nu\rho}\,,$ (5.3)
$\displaystyle L^{(1)}_{+}$ $\displaystyle=$
$\displaystyle-\frac{a+b}{8}\left[R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}-\frac{1}{2}H^{\mu\nu\rho}H_{\mu\sigma\lambda}R_{\nu\rho}{}^{\sigma\lambda}+\frac{1}{24}H^{4}-\frac{1}{8}H_{\mu\nu}^{2}H^{2\mu\nu}\right]\,,$
(5.4)

and includes in addition a total derivative ${\cal D}_{\mu}V^{\mu}$ with

$V^{\mu}=\frac{a+b}{8}H^{\mu\nu\rho}{\cal D}^{\sigma}H_{\sigma\nu\rho}\ .$
(5.5)

Of course, since this is what descends directly from DFT, we expect the
Lagrangian to be $\beta$ invariant only with the inclusion of this term.

The terms proportional to $a+b$, which are common to the bosonic and heterotic
strings, contain even powers of $H_{\mu\nu\rho}$ and are then invariant with
respect to an exchange of sign of the two-form, $b_{\mu\nu}\to-b_{\mu\nu}$. In
addition, they are purely metric, in the sense that the vielbein enters into
these terms only through the metric. Notice also that in this case, the two-
form is Lorentz invariant, as expected for the bosonic string in this scheme.

On the other hand, the term proportional to $a-b$ involves the Chern-Simons
form contained in the heterotic string, which is linear in $H_{\mu\nu\rho}$,
and hence is odd with respect to the exchange of sign of the two-form. This is
a frame-like contribution, as it cannot be written purely in terms of the
metric. Lorentz invariance of $L_{-}^{(1)}$ requires the non-standard Green-
Schwarz Lorentz transformation of the two-form [3]

$\displaystyle\delta_{\Lambda}b_{\mu\nu}=-\frac{a-b}{2}\partial_{[\mu}\Lambda_{c}{}^{d}w_{\nu]d}{}^{c}\,.$
(5.6)

$\beta$ transformations in the MT scheme follow from inserting (5.1) into
(2.28)

$\displaystyle\delta^{(1)}_{\beta}e_{ab}=\tau_{(ab)}+f_{[ab]}\,,\qquad\qquad\delta^{(1)}_{\beta}b_{ab}=\sigma_{ab}+\nabla_{[a}t_{b]}\,,\qquad\qquad\delta^{(1)}_{\beta}\phi=\frac{1}{2}\tau_{a}{}^{a}\;,$
(5.7)

with

$\displaystyle\tau_{ab}=$
$\displaystyle-\frac{a+b}{2}\left(\nabla{}_{d}\beta_{c(a}H_{b)}{}^{cd}+\frac{1}{2}\beta_{c(a}{\cal
D}{}_{d}{H_{b)}{}^{cd}}\right)-\frac{b-a}{4}\beta_{c(a}\left({w}_{b)de}{w}^{cde}+\frac{1}{4}H_{b)de}H{}^{cde}\right)\,,$
(5.8a) $\displaystyle f_{ab}=$ $\displaystyle\
-\frac{a+b}{8}\beta_{c[a}\left(\vphantom{\frac{1}{2}}{w}_{b]de}H^{cde}-H_{b]de}{w}^{cde}+2{\cal
D}^{d}{H_{b]d}{}^{c}}\right)\,,$ (5.8b) $\displaystyle\sigma_{ab}=$
$\displaystyle\
(a+b)\left[-\frac{1}{2}\beta{}^{cd}\left(R_{abcd}+\frac{1}{2}H_{ace}H_{bd}{}^{e}\right)+\beta_{c[a}\left(R_{b]}{}^{c}+2\nabla_{b]}\nabla^{c}\phi+\frac{1}{4}H_{b]de}H{}^{cde}\right)\right]$
$\displaystyle+\frac{b-a}{2}\left[\beta_{cd}H{}^{de}{}_{[a}\left({w}_{b]}{}^{c}{}_{e}-{w}{}^{c}{}_{b]e}\right)+\frac{1}{2}\beta_{c[a}\left({w}_{b]de}H{}^{cde}+H_{b]de}{w}{}^{cde}\right)\right]\;,$
(5.8c) $\displaystyle t_{a}=$ $\displaystyle\ (a+b)\;\beta{}^{bc}w_{abc}\;.$
(5.8d)

Written in this form, equations (5.7) highlight the observation made in
subsection 3.4, about redefinitions of the symmetry parameters. In this case,
we see that the $f_{[ab]}$ piece of the $\beta$ transformations can be
absorbed into redefinitions of the Lorentz parameter and $t_{a}$ into (the
flat version of) the one-form that parameterizes the gauge transformations of
the Kalb-Ramond field. We will assume this redefinition of the parameters from
now on. In particular, this means that we will have to take care of the
redefined brackets discussed in subsection 3.4 when verifying closure of the
symmetry algebra.

Since the terms in ${L}^{(1)}_{+}$ are purely metric, we expect the
corresponding $\beta$ transformations to be purely metric as well. Indeed,
performing the parameter redefinitions mentioned in the previous paragraph
(i.e. ignoring $f_{[ab]}$ and $t_{a}$), we obtain the following $\beta$
transformations for the metric and the two-form

$\displaystyle\delta^{(1)}_{\beta}g_{\mu\nu}\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!(a+b)\left(\nabla{}^{\sigma}\beta{}_{(\mu}{}^{\lambda}H{}_{\nu)\lambda\sigma}+\frac{1}{2}\beta{}_{(\mu}{}^{\lambda}{\cal
D}{}^{\sigma}{H{}_{\nu)\lambda\sigma}}\right)+\frac{a-b}{2}\beta{}^{\lambda}{}_{(\mu}\left({w}_{\nu)cd}{w}_{\lambda}{}^{cd}+\frac{1}{4}H{}_{\nu)\rho\sigma}H{}_{\lambda}{}{}^{\rho\sigma}\right)\,,$
$\displaystyle\delta^{(1)}_{\beta}b_{\mu\nu}\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\
(a+b)\left[-\frac{1}{2}\beta{}^{\rho\sigma}\left(R_{\mu\nu\rho\sigma}+\frac{1}{2}H_{\mu\rho\tau}H_{\nu\sigma}{}^{\tau}\right)+\beta_{\rho[\mu}\left(R_{\nu]}{}^{\rho}+2\nabla_{\nu]}\nabla^{\rho}\phi+\frac{1}{4}H_{\nu]\sigma\tau}H{}^{\rho\sigma\tau}\right)\right]$
(5.9)
$\displaystyle\\!\\!\\!\\!-\frac{a-b}{2}\left[\beta_{cd}H{}^{de}{}_{[\mu}\left({w}_{\nu]}{}^{c}{}_{e}-{w}{}^{c}{}_{\nu]e}\right)+\frac{1}{2}\beta_{c[\mu}\left({w}_{\nu]de}H{}^{cde}+H_{\nu]de}{w}{}^{cde}\right)\right]\
.$

As expected, the terms proportional to $a+b$ can be cast in purely metric
expressions, with spacetime greek indices. Instead, those proportional to
$a-b$, involve spin connections that necessarily require the vielbein, as they
cannot be expressed only in terms of the metric.

By construction, since we derived them from DFT, these $\beta$ transformations
leave the Lagrangian invariant. Hence, they close off-shell, as we proved in
subsection 3.3. On the other hand, as discussed in 3.2, if one is interested
in invariance of the action, instead of focusing on invariance of the
Lagrangian, then these transformations can be supplemented with the four-
parameter family of deformations (3.17), that leave the action invariant and
determine that the symmetry algebra closes on-shell.

After reparameterization, the brackets in this scheme can be computed from the
corresponding ones in the BdR scheme, following the discussion in subsection
3.4. Details of the computation can be found in Appendix B. The brackets
$\xi^{\mu}_{12}$ and $\beta^{\mu\nu}_{12}$ are not modified, while those for
gauge and Lorentz transformations turn into

$\displaystyle\lambda_{\mu}^{12}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!-4\,\xi_{{}_{[1}}{}^{\nu}\partial_{[\mu}\lambda_{{}_{2]}}{}_{\nu]}-(b-a)\left(\Lambda_{[1}^{ab}\partial_{\mu}\Lambda_{2]}{}_{ab}+\beta_{[1}^{ab}\partial_{\mu}\beta_{2]}{}_{ab}\right)-\,\frac{a+b}{4}\left(\partial_{\mu}\Lambda_{[1}^{ab}\;\beta_{2]ab}-\;\Lambda_{[1}^{ab}\;\partial_{\mu}\beta_{2]ab}\right)$
$\displaystyle\Lambda^{12}_{ab}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\Lambda^{(0)12}_{ab}-4\,{K}_{c[a}\beta^{[1}{}_{b]}{}_{d}\beta^{2]}{}^{cd}-4\,{K}^{cd}\beta^{[1}_{c[a}{{\beta^{2]}}}_{b]d}-4\Lambda^{[1}_{[a}{}^{c}f^{2]}_{b]c}-\frac{b-a}{2}\,{\omega}^{e}{}_{cd}{{\beta^{[1}}}_{e[a}\,{D}_{b]}{\Lambda^{2]cd}}$
(5.10)
$\displaystyle\\!\\!\\!-\left[2(a+b)\left({\omega}^{ef}{}_{d}\omega^{cd}{}_{[a}+\frac{1}{4}{H}^{ef}{}_{d}H{}^{cd}{}_{[a}\right)+\frac{b-a}{2}{H}^{ef}{}_{d}\left({\omega}_{[a}{}^{cd}+{\omega}^{cd}{}_{[a}\right)\right]\beta^{[1}_{b]e}\beta^{2]}_{cf}\;\;,$

where

$\displaystyle
K_{ab}=\frac{a+b}{4}\left(\vphantom{\frac{1}{2}}R_{ab}+2\nabla_{a}\nabla_{b}\phi+\frac{1}{4}H_{acd}H_{b}{}^{cd}\right)+\frac{b-a}{8}\omega_{(a}{}^{cd}H_{b)cd}\
.$ (5.11)

Despite having computed the brackets systematically from the one obtained in
[2], we have verified explicitly that they lead exactly to off-shell closure
of the symmetry algebra.

###  5.1 Invariance of the Metsaev-Tseytlin Lagrangian

Here we present an explicit derivation of the invariance of the bi-parametric
MT Lagrangian (5.2)-(5.5) with respect to the transformations (5.7)-(5.8d). We
will first focus on the terms proportional to $a+b$ (5.4), and then we deal
with those proportional to $a-b$ (5.3).

####  5.1.1 Invariance of the even parity sector

The deformations of the $\beta$ transformations that preserve the even sector
of the MT action were systematically derived from DFT, namely

$\displaystyle\delta^{(1)}_{\beta}e_{ab}=$
$\displaystyle-\frac{a+b}{2}\left(\nabla{}_{d}\beta_{c(a}H_{b)}{}^{cd}+\frac{1}{2}\beta_{c(a}{\cal
D}{}_{d}{H_{b)}{}^{cd}}\right)+\gamma\beta_{c(a}{\cal B}_{b)}{}^{c}+\chi
g_{ab}\beta_{cd}{\cal B}^{cd}$ (5.12a)
$\displaystyle\delta^{(1)}_{\beta}b_{ab}=$
$\displaystyle(a+b)\left[-\frac{1}{2}\beta{}^{cd}\left(R_{abcd}+\frac{1}{2}H_{ace}H_{bd}{}^{e}\right)+\beta_{c[a}\left(R_{b]}{}^{c}+2\nabla_{b]}\nabla^{c}\phi+\frac{1}{4}H_{b]de}H{}^{cde}\right)\right]$
$\displaystyle-\gamma\beta^{c}{}_{[a}{\cal E}_{b]c}-\chi\beta_{ab}{\cal
E}_{c}{}^{c}\ .$ (5.12b)

We consider here only the deformations proportional to the parameters $\gamma$
and $\chi$, as they preserve the expected parity of the transformations. Since
this sector is common to the bosonic and heterotic strings, the
transformations can be cast in a purely metric formulation. Indeed, as
expected, we find

$\displaystyle\delta^{(1)}_{\beta}g_{\mu\nu}=$
$\displaystyle-(a+b)\left(\nabla{}^{\sigma}\beta{}^{\lambda}{}_{(\mu}H{}_{\nu)\lambda\sigma}+\frac{1}{2}\beta^{\lambda}{}_{(\mu}{\cal
D}{}^{\sigma}{H{}_{\nu)\lambda\sigma}}\right)+2\gamma\beta_{\rho(\mu}{\cal
B}_{\nu)}{}^{\rho}+2\chi g_{\mu\nu}\beta_{\rho\sigma}{\cal B}^{\rho\sigma}\,,$
(5.13a) $\displaystyle\delta^{(1)}_{\beta}b_{\mu\nu}=$
$\displaystyle(a+b)\left[-\frac{1}{2}\beta{}^{\rho\sigma}\left(R_{\mu\nu\rho\sigma}+\frac{1}{2}H_{\mu\rho\tau}H_{\nu\sigma}{}^{\tau}\right)+\beta_{\rho[\mu}\left(R_{\nu]}{}^{\rho}+2\nabla_{\nu]}\nabla^{\rho}\phi+\frac{1}{4}H_{\nu]\sigma\tau}H{}^{\rho\sigma\tau}\right)\right]$
$\displaystyle-\gamma\beta^{\rho}{}_{[\mu}{\cal
E}_{\nu]\rho}-\chi\beta_{\mu\nu}{\cal E}_{\rho}{}^{\rho}\ .$ (5.13b)

By construction, for $\gamma=\chi=0$ these transformations must leave the even
parity Lagrangian invariant. Here we want to show this explicitly, and for
this we must verify that the following equation holds

$\delta_{\beta}^{(1)}L^{(0)}+\delta^{(0)}_{\beta}(L_{+}^{(1)}+{\cal
D}_{a}V^{a})=0\,.$ (5.14)

To this end, we first replace (3.10) in the first term and the fact that
$[\delta^{(0)}_{\beta}\,,\,{\cal D}_{a}]W^{a}=0$ for any $W^{a}$, to re-write
(5.14) as

$\delta^{(0)}_{\beta}L^{(1)}_{+}=\underbrace{-{\cal
E}{}^{ab}\;\delta^{(1)}_{\beta}e_{ab}-{\cal
B}^{ab}\;\delta^{(1)}_{\beta}b_{ab}\vphantom{\left[\frac{1}{2}\right]}}_{(A)}\underbrace{-{\cal
D}_{a}\left[2\;{\cal
D}_{b}\delta^{(1)}_{\beta}e^{ab}-\frac{1}{2}H^{abc}\delta^{(1)}_{\beta}b_{bc}+\delta^{(0)}_{\beta}V^{a}\right]}_{(B)}\
.$ (5.15)

Next, we explicitly compute the lowest order variation of $L^{(1)}_{+}$ (5.4)
in the LHS. For this it is useful to introduce the following identities

$\displaystyle\delta^{(0)}_{\beta}\left(R_{abcd}R^{abcd}\right)$
$\displaystyle=$
$\displaystyle\left[\vphantom{\frac{1}{2}}-4\nabla_{a}\left(\beta_{b}{}^{e}H_{ecd}\right)+\left(\nabla^{e}\beta_{ab}-2\nabla_{a}\beta_{b}{}^{e}\right)H_{ecd}\right]R^{abcd}\,,\
\ \ \ \ \ \ \ \ \ \ \ $ (5.16)
$\displaystyle\delta^{(0)}_{\beta}\left(-\frac{1}{2}H_{eab}H^{e}{}_{cd}R^{abcd}\right)$
$\displaystyle=$
$\displaystyle\;H_{e}{}^{ab}H^{ecd}\left[\nabla_{a}\left(\beta_{b}{}^{f}H_{fcd}\right)+\left(\nabla_{a}\beta_{c}{}^{f}-\frac{1}{2}\nabla^{f}\beta_{ac}\right)H_{bdf}\right]\,$
(5.17)
$\displaystyle+\left[\vphantom{\frac{1}{2}}\nabla^{e}\beta_{ab}+2\nabla_{a}\beta_{b}{}^{e}\right]H_{ecd}R^{abcd}\,,$
$\displaystyle\delta^{(0)}_{\beta}\left(-\frac{1}{8}H_{aef}H_{b}{}^{ef}H^{a}{}_{cd}H^{bcd}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left[\vphantom{\frac{1}{2}}\nabla_{a}\beta_{cd}+2\nabla_{c}\beta_{da}\right]H_{b}{}^{cd}H^{aef}H^{b}{}_{ef}\,,$
(5.18)
$\displaystyle\delta^{(0)}_{\beta}\left(\frac{1}{24}H_{abc}H^{ade}H^{b}{}_{ef}H^{cf}{}_{d}\right)$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\nabla_{a}\beta_{bc}H^{ade}H^{b}{}_{ef}H^{cf}{}_{d}\,,$
(5.19)

which readily lead to the following result

$\displaystyle\delta^{(0)}_{\beta}L_{+}^{(1)}$ $\displaystyle=$
$\displaystyle\frac{a+b}{2}R^{abcd}\left[\nabla_{a}\left(\beta_{b}{}^{e}H_{ecd}\right)-\frac{1}{2}\nabla_{e}\beta_{ab}H^{e}{}_{cd}\right]$
(5.20)
$\displaystyle-\frac{a+b}{8}\;H_{f}{}^{ab}H^{fcd}\left[\nabla_{a}\left(\beta_{b}{}^{e}H_{ecd}\right)+\frac{3}{2}\nabla_{[e}\beta_{ab]}H^{e}{}_{cd}\right]\,.$

Following the steps detailed in Appendix C, this can be rewritten as

$\delta^{(0)}_{\beta}L_{+}^{(1)}=(a+b)\left[(A^{\prime})+(B^{\prime})\right]\
,$ (5.21)

with

$\displaystyle(A^{\prime})$ $\displaystyle=-{\cal
E}_{ab}\nabla_{f}\beta^{ae}H_{e}{}^{bf}-{\cal
B}_{ab}\left(R^{abef}\beta_{ef}-\beta^{da}H^{2b}{}_{d}+\frac{1}{2}\beta_{ef}H^{bf}{}_{d}H^{aed}\right)-2\beta^{d}{}_{a}{\cal
B}^{ab}{\cal E}_{db}\ ,$ (5.22) $\displaystyle(B^{\prime})$
$\displaystyle=-{\cal
D}_{a}\left[\vphantom{\frac{1}{2}}\beta_{be}\left(H_{cd}{}^{e}R^{abcd}-\frac{1}{2}H_{cd}{}^{a}\left(R^{becd}-\frac{1}{2}H^{cef}H_{f}{}^{db}\right)+2\left(\vphantom{e^{\frac{1}{2}}}R^{b}{}_{c}+2\nabla^{b}\nabla_{c}\phi\right)H^{eca}\right)\right]\;,$

which is closer in structure to (5.15). It is now easy to recover the
conditions for invariance of the Lagrangian and the action by comparing (5.21)
with (5.15).

  * •

For the action to be invariant, it is enough to verify

$(A)=(a+b)(A^{\prime})\ .$ (5.23)

By definition, when inserting (5.12) into $(A)$, the terms depending on
$\gamma$ and $\chi$ simply drop out. The rest of the terms can be rapidly
shown to verify this identity. As a consequence, the action is invariant for
whatever choice of the parameters $\gamma$ and $\chi$.

  * •

For the Lagrangian to be invariant, on top of (5.23), also

$(B)=(a+b)(B^{\prime})\ ,$ (5.24)

must hold. This is the imposition that the total derivatives that emerge from
the variation of the lowest order Lagrangian are exactly canceled by those
arising in the variation of the first order Lagrangian. In this case, when
(5.12) are inserted in $(B)$, the identity (5.24) can be shown to hold for the
specific choice $\gamma=\chi=0$ (see Appendix C).

In summary, we have shown that the transformations (5.12) leave the even
parity sector of the MT action invariant for any choice of the parameters
$\gamma$ and $\chi$. Instead, the Lagrangian was shown to be invariant for the
specific choice $\gamma=\chi=0$, which is the case that descends directly from
DFT.

####  5.1.2 Invariance of the odd parity sector

We now repeat the analysis for the terms proportional to $a-b$, namely
$L^{(1)}_{-}$. We have seen that, up to parameter redefinitions, the first
order transformations that leave the odd parity piece of the action invariant,
and respect the parity of the fields accordingly, are

$\displaystyle\delta^{(1)}_{\beta}e_{ab}=$
$\displaystyle\frac{a-b}{4}\beta_{c(a}\left({w}^{2\ c}_{b)}+\frac{1}{4}H^{2\
c}_{b)}\right)+\kappa\beta_{(a}{}^{c}{\cal E}_{b)c}\,,$ (5.25a)
$\displaystyle\delta^{(1)}_{\beta}b_{ab}=$
$\displaystyle-\frac{a-b}{2}\left[\beta_{cd}H{}^{de}{}_{[a}\left({w}_{b]}{}^{c}{}_{e}-{w}{}^{c}{}_{b]e}\right)+\frac{1}{2}\beta_{c[a}\left(({wH})_{b]}{}^{c}+(Hw)_{b]}{}^{c}\right)\right]+\rho\beta_{c[a}{\cal
B}^{c}{}_{b]}\,.$ (5.25b)

To verify explicitly the invariance of the Lagrangian, we must check that the
following equation holds

$\delta_{\beta}^{(1)}L^{(0)}+\delta^{(0)}_{\beta}L_{-}^{(1)}=0\,.$ (5.26)

Replacing (3.10) in the first term leads to

$\delta^{(0)}_{\beta}L^{(1)}_{-}=\underbrace{-{\cal
E}{}^{ab}\;\delta^{(1)}_{\beta}e_{ab}-{\cal
B}^{ab}\;\delta^{(1)}_{\beta}b_{ab}\vphantom{\left(\frac{1}{2}\right)}}_{(C)}\underbrace{-{\cal
D}_{a}\left[2\;{\cal
D}_{b}\delta^{(1)}_{\beta}e^{ab}-\frac{1}{2}H^{abc}\delta^{(1)}_{\beta}b_{bc}\right]}_{(D)}\
.$ (5.27)

The explicit variation of $L^{(1)}_{-}$ in the LHS is shown in Appendix C to
take the form

$\delta_{\beta}^{(0)}L^{(1)}_{-}=(a-b)\left[(C^{\prime})+(D^{\prime})\right]\
,$ (5.28)

with

$\displaystyle(C^{\prime})$ $\displaystyle=$ $\displaystyle-\frac{1}{2}{\cal
E}{}^{ab}\beta_{ca}\left(w^{2}{}_{b}{}^{c}+\frac{1}{4}H^{2}{}_{b}{}^{c}\right)-\frac{1}{2}{\cal
B}{}^{ab}\left[\beta_{cd}H^{de}{}_{a}\left(w_{b}{}^{c}{}_{e}-w^{c}{}_{be}\right)+\frac{1}{2}\beta_{ca}(Hw+wH)_{b}{}^{c}\right]\,,$
$\displaystyle(D^{\prime})$ $\displaystyle=$ $\displaystyle\frac{1}{2}{\cal
D}_{b}\left[\vphantom{{\frac{1}{2}}^{\frac{1}{2}}}\right.H^{abc}\left(H_{aef}w_{[dc]}{}^{e}\beta^{fd}-\frac{1}{4}\beta_{cd}(Hw+wH)_{a}{}^{d}\right)+\beta_{ca}w^{ab}{}_{d}\left(\frac{1}{4}H^{2}{}^{cd}-w^{2}{}^{cd}\right)\left.\vphantom{{\frac{1}{2}}^{\frac{1}{2}}}\right]\,,$

which leaves us very close to our purpose. It is now easy to recover the
conditions for invariance of the Lagrangian and the action by comparing (5.28)
with (5.27).

  * •

For the action to be invariant, it is enough to verify

$(C)=(a-b)(C^{\prime})\ .$ (5.29)

By definition, when inserting (5.25) into $(C)$, the terms depending on
$\kappa$ and $\rho$ simply drop out. The rest of the terms can be rapidly
shown to verify this identity. As a consequence, the action is invariant for
whatever choice of the parameters $\kappa$ and $\rho$.

  * •

For the Lagrangian to be invariant, on top of (5.29), also

$(D)=(a-b)(D^{\prime})\ ,$ (5.30)

must hold. This is the imposition that the total derivatives that emerge from
the variation of the lowest order Lagrangian are exactly canceled by those
arising in the variation of the first order Lagrangian. In this case, when
(5.25) are inserted in $(D)$, the identity (5.30) can be shown to hold for the
specific choice $\kappa=\rho=0$ (see Appendix C).

In summary, we have shown that the transformations (5.25) leave the odd parity
sector of the MT action invariant for any choice of the parameters $\kappa$
and $\rho$, while the Lagrangian is left invariant for the specific choice
$\kappa=\rho=0$, which is the case that descends directly from DFT.

##  6 Conclusions

In this paper we derived the $\beta$ symmetry transformation rules of the
supergravity fields to first order in $\alpha^{\prime}$ from the frame-like
formulation of DFT. The fact that DFT implies $\beta$ symmetry, in turn means
that this symmetry is a necessary condition for DFT uplifts of supergravity
interactions. It is not sufficient because $\beta$ symmetry in supergravity
requires isometries, whereas in DFT it is realized through the introduction of
dual derivatives. The road from supergravity to DFT would require relaxing the
assumption of isometries, namely the condition
$\beta^{\mu\nu}\partial_{\nu}\dots=0$, and extending supergravity with
interactions containing dual derivatives, to compensate the failure. This
observation might become important when assessing the possibility of finding a
DFT formulation for the quartic Riemann terms in maximal supergravity.
Actually, at the moment, the uplift of the ${\cal O}(\alpha^{\prime 3})$
interactions of type II strings to the frame-like version of DFT, faces an
obstruction that was stated in [16]. It would then be interesting to
understand how $\beta$ symmetry works for these couplings.

Having derived the $\beta$ transformations systematically from DFT, we then
moved to more conventional supergravity schemes. We first showed that the
field redefinitions that connect DFT with the generalized Bergshoeff-de Roo
scheme of supergravity, allowed us to recover and confirm all the results in
[2]. Additionally, we performed further redefinitions to reach the Metsaev-
Tseytlin scheme. For completion, we showed through explicit computations in
this scheme that the MT Lagrangian is exactly $\beta$ invariant for the
bosonic and heterotic strings to order $\alpha^{\prime}$, and that the
transformations close off-shell, as expected.

Interestingly, we found a four-parameter deformation of the $\beta$
transformations that preserve the action, as they transform the Lagrangian
into a total derivative. These deformations depend on the equations of motion,
and then happen to close the symmetry algebra on-shell.

Acknowledgements: We warmly thank C. Hull and D. Waldram for hospitality and
discussions at Imperial College, where part of this project was developed. We
also thank O. Hohm for pointing out reference [13]. Support by Consejo
Nacional de Investigaciones Científicas y Técnicas (CONICET), Agencia Nacional
de Promoción Científica y Técnica (ANPCyT), Universidad de La Plata (UNLP) and
Universidad de Buenos Aires (UBA) is also gratefully acknowledged.

Note: Our results confirm those of our previous paper [2] and are in direct
contradiction with supposed obstructions pointed out in [17]-v5 and [18]-v2.
Contrary to what is stated in [18]-v2, we confirm here, through explicit
computations in subsection 5.1, that the ${\cal O}(\alpha^{\prime})$ bulk
Lagrangian of the heterotic string is exactly invariant under appropriate
deformations of the $\beta$ transformations. We also disagree with statements
in [17]-v5, where it is asserted that there is a unique $\beta$ transformation
at ${\cal O}(\alpha’)$ that leaves the bulk bosonic action invariant and that
it does not form a closed symmetry algebra. The transformation found in
equation (31) of [17]-v5, corresponds to the particular choice $a=b=-\alpha’$,
$\gamma=-\frac{\alpha’}{2}$ and $\chi=0$ in our equations (5.13a)-(5.13b).
Actually, $\beta$ transformations at ${\cal O}(\alpha’)$ turn out not to be
unique, but there is a two-parameter family of deformations in the bosonic
string that lead to on-shell closure of the symmetry algebra.

##  Appendix A Notation and definitions

We use $\mu,\nu,\rho,\dots$ and $a,b,c,\dots$ indices for space-time and
tangent space coordinates, respectively. The infinitesimal Lorentz
transformation of the vielbein is

$\delta_{\Lambda}e_{\mu}{}^{a}=e_{\mu}{}^{b}\Lambda_{b}{}^{a}\ \,.$ (A.1)

The spin connection

$w_{cab}=e^{\mu}{}_{c}\left(\partial_{\mu}e_{\nu
a}e^{\nu}{}_{b}-\Gamma^{\rho}_{\mu\nu}e_{\rho
a}e^{\nu}{}_{b}\right)\,,\qquad{\rm with}\ \ \ \ \
\Gamma^{\rho}_{\mu\nu}=\frac{1}{2}g^{\rho\sigma}\left(\partial_{\mu}g_{\sigma\nu}+\partial_{\nu}g_{\mu\sigma}-\partial_{\sigma}g_{\mu\nu}\right)\,,$
(A.2)

transforms as

$\delta_{\Lambda}w_{cab}=D_{c}\Lambda_{ab}+w_{dab}\Lambda^{d}{}_{c}+2w_{cd[b}\Lambda^{d}{}_{a]}\,,$
(A.3)

and hence, it turns flat derivatives $D_{a}$ into covariant flat derivatives
$\nabla_{a}$ as

$\nabla_{a}T_{b}{}^{c}=D_{a}T_{b}{}^{c}+w_{ab}{}^{d}T_{d}{}^{c}-w_{ad}{}^{c}T_{b}{}^{d}\
,\ \ \ D_{a}=e^{\mu}{}_{a}\partial_{\mu}\,.$ (A.4)

The Christoffel connection $\Gamma^{\rho}_{\mu\nu}$ turns spacetime partial
into covariant derivatives as

$\nabla_{\mu}T_{\rho}{}^{\sigma}=\partial_{\mu}T_{\rho}{}^{\sigma}-\Gamma_{\mu\rho}^{\lambda}T_{\lambda}{}^{\sigma}+\Gamma_{\mu\lambda}^{\sigma}T_{\rho}{}^{\lambda}\,.$
(A.5)

The Riemann tensor

$R^{\mu}{}_{\nu\rho\sigma}=\partial_{\rho}\Gamma^{\mu}_{\nu\sigma}-\partial_{\sigma}\Gamma^{\mu}_{\nu\rho}+\Gamma^{\mu}_{\rho\lambda}\Gamma^{\lambda}_{\nu\sigma}-\Gamma^{\mu}_{\sigma\lambda}\Gamma^{\lambda}_{\nu\rho}\,,$
(A.6)

with flat space indices is defined as

$R_{abcd}=2D_{[a}w_{b]cd}+2w_{[ab]}{}^{e}w_{ecd}+2w_{[\underline{a}c}{}^{e}w_{\underline{b}]ed}\,.$
(A.7)

While the symmetry $R_{abcd}=R_{[ab][cd]}$ is manifest, other symmetries of
the Riemann tensor are hidden and determine the Bianchi identities

$R_{abcd}=R_{cdab}\ ,\ \ \ \ R_{[abc]d}=0\\\ ,\ \ \ \ \nabla_{[a}R_{bc]de}=0.$
(A.8)

The Ricci tensor and scalar curvature are given by the traces

$R_{ab}=R^{c}{}_{acb}\ ,\ \ \ R=R_{a}{}^{a}\ .\ \ \ R_{[ab]}=0\ .$ (A.9)

In terms of the torsionful spin connection

$w^{(\pm)}_{abc}=w_{abc}\pm\frac{1}{2}H_{abc}\,,$ (A.10)

the Riemann tensor defined in (4.5) satisfies the Bianchi identities

$R^{(\pm)}_{abcd}\ =\ R^{(\mp)}_{cdab}\,,\qquad R^{(\pm)}_{[abc]d}\ =\
\pm\frac{1}{3}\nabla_{d}H_{abc}+\frac{1}{2}H_{[ab}{}^{e}H_{c]de}\,.$ (A.11)

The curvature of the 2-form

$H_{abc}=3\,e^{\mu}{}_{a}e^{\nu}{}_{b}e^{\rho}{}_{c}\,\partial_{[\mu}b_{\nu\rho]}\
,\ \ \ \ $ (A.12)

obeys the Bianchi identity

$\nabla_{[a}H_{bcd]}=0\ .$ (A.13)

We define contractions of powers of the 3-form and spin connection as follows:

$\displaystyle H^{2}$ $\displaystyle=$ $\displaystyle
H_{abc}H^{abc}\,,\qquad\qquad\ H^{2}_{ab}=H_{acd}H_{b}{}^{cd}\,,\qquad\ \ \
H^{4}=H^{abc}H_{ad}{}^{e}H_{be}{}^{f}H_{cf}{}^{d}\,,$ (A.14) $\displaystyle
w^{2}_{ab}$ $\displaystyle=$ $\displaystyle w_{acd}w_{b}{}^{cd}\,,\qquad\ \ \
(wH)_{ab}=w_{acd}H_{b}{}^{cd}\,,\qquad\ \ \ (Hw)_{ab}=H_{acd}w_{b}{}^{cd}\ .$
(A.15)

The Chern-Simons form

$\Omega_{\mu\nu\rho}=w_{[\mu
a}{}^{b}\partial_{\nu}w_{\rho]b}{}^{a}+\frac{1}{2}w_{\mu a}{}^{b}w_{\nu
b}{}^{c}w_{\rho c}{}^{a}\,,$ (A.16)

was defined in terms of the torsionful connection in flat index notation in
(4.4).

##  Appendix B Bracket in MT scheme after reparameterization

As discussed in Subsection 3.4, field redefinitions modify the brackets only
through field dependent contributions in the zero order bracket. Starting from
the symmetry algebra in the BdR scheme (4.7f) we observe that the only field
dependent parameter is in

$\displaystyle\Lambda^{(0)}_{12ab}\supset
2\,\beta_{1[a}{}^{c}\beta_{2\,b]c}=2\,e_{\mu a}\,e_{\nu
b}\,g_{\rho\sigma}\beta_{[1}^{\mu\rho}\;\beta_{2]}^{\sigma\nu}\;.$ (B.1)

Hence, equation (3.31) implies that the only modification in the bracket,
after the field redefinition that exchanges from the BdR to the MT scheme, is

$\displaystyle\Lambda^{\prime}_{12ab}$ $\displaystyle=$
$\displaystyle\Lambda^{(old)}_{12ab}-\frac{a+b}{4}\left(H^{2}_{c[a}\beta^{[1}_{b]d}\beta^{2]cd}+H^{2cd}\beta^{[1}_{c[a}\beta^{2]}_{b]d}\right)\;.$
(B.2)

Although it is not necessary a priori, in (5.7) we performed an extra
parameter redefinition, in order to put the $\beta$ transformations in a
minimal form, where

$\displaystyle\Delta\zeta=\ell(\zeta,\psi)\to\left(\Delta\xi^{\mu},\Delta\lambda_{a},\Delta\Lambda_{ab},\Delta\beta^{\mu\nu}\right)=\left(0,-t_{a},-f_{ab},0\right)\;,$
(B.3)

with $t_{a}$ and $f_{[ab]}$ defined in (5.8b) and (5.8d). $\Delta g_{12}$ is
instead

$\displaystyle(\Delta g_{12})^{\mu\nu}$ $\displaystyle=$ $\displaystyle 0\;,$
(B.4) $\displaystyle(\Delta g_{12})^{\mu}$ $\displaystyle=$
$\displaystyle-\frac{a+b}{2}\beta_{1}^{\mu\nu}t_{2\nu}=-\frac{1}{2}\beta_{1}^{\mu\nu}\beta_{2}^{ab}\omega_{\nu
ab}=0\;,$ (B.5) $\displaystyle(\Delta g_{12})_{\mu}$ $\displaystyle=$
$\displaystyle 2\xi^{\nu}_{1}\partial_{[\mu}t_{2\nu]}\;,$ (B.6)
$\displaystyle(\Delta g_{12})_{ab}$ $\displaystyle=$
$\displaystyle-2\Lambda^{1}_{[a}{}^{c}f_{2b]c}-\xi_{1}^{\mu}\partial_{\mu}f_{2ab}\;\
.$ (B.7)

Finally, we notice that
$\hat{\beta}^{\mu\nu}_{12}=0,\;\hat{\xi}^{\mu}_{12}=0$, while

$\displaystyle-2\hat{\Lambda}^{[12]}_{ab}$ $\displaystyle:=$
$\displaystyle\delta_{\xi_{1},\lambda_{1},\Lambda_{1},\beta_{1}}\left(f_{2ab}\right)-\delta_{\xi_{2},\lambda_{2},\Lambda_{2},\beta_{2}}\left(f_{1ab}\right)$
(B.8) $\displaystyle=$ $\displaystyle
2\;\xi^{[1\nu}\partial_{\nu}f_{2ab}-\frac{a+b}{4}\beta^{[2}_{c[a}D_{b]}\Lambda^{1]}_{de}H^{cde}$
(B.12)
$\displaystyle-(a+b)\;\beta_{[1}{}_{de}\beta_{2]c[a}\left(\vphantom{\frac{1}{2}}\right.\omega_{b]f}{}^{d}\omega^{ef}{}_{c}+\omega^{e}{}_{b]f}\omega_{c}{}^{fd}-\frac{1}{2}H_{b]f}{}^{e}H^{df}{}_{c}\left.\vphantom{\frac{1}{2}}\right)$
$\displaystyle+(a+b)\;\beta_{[1}{}_{c[a}\beta_{2]}{}^{cd}\left(\vphantom{\frac{1}{2}}\right.R_{b]d}+2\nabla_{b]}\nabla_{d}\phi-\frac{1}{8}H^{2}_{b]d}-\frac{1}{2}\omega^{2}_{b]d}\left.\vphantom{\frac{1}{2}}\right)$
$\displaystyle+(a+b)\;\beta_{[1}{}_{[a}{}^{c}\beta_{2]}{}_{b]}{}^{d}\left(\vphantom{\frac{1}{2}}\right.R_{cd}+2\nabla_{c}\nabla_{d}\phi-\frac{1}{8}H^{2}_{cd}-\frac{1}{2}\omega^{2}_{cd}\left.\vphantom{\frac{1}{2}}\right)\
,$

and

$\displaystyle-2\hat{\lambda}^{[12]}_{\nu}$ $\displaystyle=$
$\displaystyle\delta_{\xi_{1},\lambda_{1},\Lambda_{1},\beta_{1}}\left(t_{2\nu}\right)-\delta_{\xi_{2},\lambda_{2},\Lambda_{2},\beta_{2}}\left(t_{1\nu}\right)$
(B.13) $\displaystyle=$ $\displaystyle
4\xi_{[1}^{\nu}\partial_{[\nu}{\hat{t}}_{2]\mu]}+2(a+b)\;\left[\beta_{[2}^{bc}\partial_{\mu}\Lambda_{1]bc}+\partial_{\mu}\left(\xi_{[1}^{a}\beta_{2]}^{bc}\omega_{abc}\right)+H_{\mu
bc}\beta_{[1}^{bd}\beta_{2]}^{c}{}_{d}\right]\ .$ (B.14)

The first term in (B.14) cancels with $(\Delta g_{12})_{\mu}$ and the
remaining terms inside the bracket can be put in the form

$\displaystyle\left[\vphantom{\partial_{p}\beta^{mn}}\;\;\;\right]=\frac{1}{2}\left(\beta_{[2}^{bc}\partial_{\mu}\Lambda_{1]bc}-\partial_{\mu}\beta_{[2}^{bc}\;\Lambda_{1]bc}\right)+\partial_{\mu}\left(\frac{1}{2}\beta_{[2}^{bc}\Lambda_{1]bc}+\xi_{[1}^{a}\beta_{2]}^{bc}\omega_{abc}+b_{\nu\rho}\beta_{[1}^{\nu\sigma}\beta_{2]}^{\rho}{}_{\sigma}\right)\
,$ (B.15)

where the last factor is pure gauge and so can be ignored. Plugging these
expressions with (3.33) one readily finds (5.10).

##  Appendix C Leading order variation of $L^{(1)}$ in MT scheme

In this Appendix we present some details of the computations summarized in
section 5.1 . The route followed here is not unique, due to Bianchi identities
and other less familiar relations, whose origin can be traced back to the
constraints imposed after contractions with $\beta$. Following the structure
of subsection 5.1, we first deal with the even parity sector in C.1, and then
with the odd sector in C.2.

###  C.1 Variation of $L^{(1)}_{+}$

Here we present some details of the computations leading from equation (5.20),
namely

$\displaystyle\delta^{(0)}_{\beta}L_{+}^{(1)}$ $\displaystyle=$
$\displaystyle\frac{a+b}{2}R^{abcd}\left[\nabla_{a}\left(\beta_{b}{}^{e}H_{ecd}\right)-\frac{1}{2}\nabla_{e}\beta_{ab}H^{e}{}_{cd}\right]$
(C.1)
$\displaystyle-\frac{a+b}{8}\;H_{f}{}^{ab}H^{fcd}\left[\nabla_{a}\left(\beta_{b}{}^{e}H_{ecd}\right)+\frac{3}{2}\nabla_{[e}\beta_{ab]}H^{e}{}_{cd}\right]\,,$

to (5.21).

A chain of integration by parts in the first term of the r.h.s. of (C.1),
followed by the Bianchi identity $\nabla_{[a}R_{bc]d}=0$ and a further
integration by parts, leads to

$\displaystyle R^{abcd}\nabla_{a}\left(\beta_{b}{}^{e}H_{ecd}\right)$
$\displaystyle=$
$\displaystyle-2R_{ab}\;\nabla_{c}\beta^{ae}\;H_{e}{}^{bc}-2\beta^{e}{}_{a}R_{eb}{\cal
D}_{c}H^{abc}+2\nabla_{a}\phi R^{abcd}\beta_{b}{}^{e}H_{ecd}$ (C.3)
$\displaystyle+{\cal
D}_{a}\left(R^{abcd}\beta_{b}{}^{e}H_{ecd}+2R_{bc}\beta^{be}H_{e}{}^{ca}\right)\;,$

where, using $R_{abcd}\nabla^{d}\phi=2\nabla_{[a}\nabla_{b]}\nabla_{c}\phi$
and integrating by parts, the third term in the r.h.s. can be rewritten as

$\displaystyle 2\nabla_{a}\phi R^{abcd}\beta_{b}{}^{e}H_{ecd}$
$\displaystyle=$
$\displaystyle-4\nabla_{a}\nabla_{b}\phi\left(\nabla_{c}\beta^{ae}H_{e}{}^{bc}+{\cal
D}_{c}H^{ebc}\beta^{a}{}_{e}\right)+{\cal
D}_{c}\left(4\nabla_{a}\nabla_{b}\phi\beta^{ae}H_{e}{}^{bc}\right)\;.\ \ \ \ \
\ \ \ \ \ \ $ (C.4)

The second term in the r.h.s. of (C.1) can be expressed alternatively as

$\displaystyle R^{abcd}\nabla_{e}\beta_{ab}H^{e}{}_{cd}$ $\displaystyle=$
$\displaystyle-R^{abcd}\beta_{ab}{\cal D}_{e}H^{ecd}+{\cal
D}_{e}\left(R^{abcd}\beta_{ab}H^{e}{}_{cd}\right)\;,$ (C.5)

and the remaining terms as

$\begin{split}\frac{1}{4}H_{f}{}^{ab}H^{fcd}&\left[\nabla_{a}\left(\beta_{b}{}^{e}H_{ecd}\right)+\frac{3}{2}\nabla_{[e}\beta_{ab]}H^{e}{}_{cd}\right]=\\\
&=\frac{1}{2}H_{f}{}^{ab}H^{fcd}\left(\nabla_{a}\beta_{b}{}^{e}H_{ecd}+\frac{1}{2}\beta_{b}{}^{e}\nabla_{a}H_{ecd}+\frac{1}{4}\nabla_{e}\beta_{ab}H^{e}{}_{cd}\right)\;.\end{split}$
(C.6)

Finally, the last two terms in (C.6) can be recast in the form

$\displaystyle
H_{f}{}^{ab}H^{fcd}\left(\beta_{b}{}^{e}\nabla_{a}H_{ecd}+\frac{1}{2}\nabla_{e}\beta_{ab}H^{e}{}_{cd}\right)$
$\displaystyle=$
$\displaystyle-H^{abc}H_{bf}{}^{d}\left(2\beta^{ef}\nabla_{[e}H_{d]ca}-H_{aed}\nabla_{c}\beta^{ef}\right)$
$\displaystyle=$ $\displaystyle-{\cal
D}_{c}H^{abc}\beta^{ef}H_{bf}{}^{d}H_{aed}+{\cal
D}_{c}\left(\beta^{ef}H^{abc}H_{bf}{}^{d}H_{aed}\right)\,,$

where in the first equality we have used the relation

$\displaystyle\frac{1}{2}H_{f}{}^{ab}H^{fcd}\nabla_{e}\beta_{ab}H^{e}{}_{cd}=-H^{abc}H_{bf}{}^{d}\left(\beta^{ef}\nabla_{e}H_{dca}-H_{aed}\nabla_{c}\beta^{ef}\right)\;,$
(C.8)

while in the second line we have used the Bianchi identity
$\nabla_{[a}H_{bcd]}=0$ before integrating by parts. The relation (C.8) is not
a Bianchi identity, but a consequence of the $\beta$ constraints, and it
readily follows after explicit evaluation of the covariant derivatives.

Using (C.3)-(C.8), we obtain the expression displayed in (5.21).

Let us pursue the computation further, to see how (5.14) is satisfied by
(5.12) for $\gamma=\chi=0$. Combining the above results with (3.10) we obtain

$\displaystyle\delta_{\beta}^{(1)}{L}^{(0)}+\delta_{\beta}^{(0)}{L}^{(1)}_{+}={\cal
D}_{a}X^{a}\;,$ (C.9)

with

$\displaystyle X^{a}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!-\frac{a+b}{2}\left[\nabla_{b}\phi\left(3\nabla_{c}\beta^{b}{}_{d}H^{acd}-\nabla_{c}\beta^{a}{}_{d}H^{bcd}+\beta_{c}{}^{d}\nabla_{d}H^{abc}\right)-\beta_{be}H^{cd[e}R^{a]bcd}\right.$
$\displaystyle-$
$\displaystyle\\!\\!\\!\\!\left.2\beta_{bc}R^{b}{}_{d}H^{acd}+\frac{1}{2}\nabla_{d}\beta_{c}{}^{a}\nabla_{b}H^{bcd}-\nabla_{c}\beta_{d}{}^{b}\nabla_{b}H^{acd}+\frac{1}{2}\beta_{c}{}^{b}\nabla_{b}\nabla_{d}H^{acd}-\frac{1}{4}\beta_{bc}H^{abd}H^{cef}H_{def}\right]$

which can be cast in the form

$X^{a}=-\delta^{(0)}V^{a}+{\cal D}_{b}Z^{ab}\ ,$ (C.11)

where

$\displaystyle V^{a}=\frac{a+b}{8}H^{acd}{\cal
D}^{b}H_{bcd}\;,\;\;\;\;Z^{ab}=-\frac{a+b}{2}\beta_{cd}\nabla^{c}H^{dab}\;.$
(C.12)

To put (C.1) into the form (C.11) we used

$\displaystyle\delta^{(0)}_{\beta}\nabla^{c}H_{cab}=-4R^{c}{}_{[a}\beta_{b]c}+\left(\beta_{de}H_{abc}+\beta_{e[a}H_{b]cd}\right)H^{cde}\;,$
(C.13)

in addition to Riemann Bianchi identities. Noting that

$\displaystyle{\cal D}_{a}{\cal
D}_{b}Z^{ab}=\nabla_{a}\nabla_{b}Z^{ab}-2\left(\nabla_{a}\nabla_{b}\phi-2\nabla_{a}\phi\nabla_{b}\phi\right)Z^{ab}-2\left(\nabla_{a}\phi\nabla_{b}Z^{ab}+\nabla_{b}\phi\nabla_{a}Z^{ab}\right)\
,$ (C.14)

vanishes for an antisymmetric object $Z^{ab}$, we get

$\displaystyle{\cal D}_{a}X^{a}=-{\cal
D}_{a}\left(\delta^{(0)}_{\beta}V^{a}\right)=-\delta^{(0)}_{\beta}\left({\cal
D}_{a}V^{a}\right)\;.$ (C.15)

Finally, plugging this back into (C.9) we reproduce (5.14). This completes the
discussion on the invariance of ${L}^{(1)}_{+}$.

###  C.2 Variation of $L^{(1)}_{-}$

Now we focus on the variation of ${L}^{(1)}_{-}$. We start by analyzing the
terms that are independent of the 3-form, namely

$\displaystyle-\frac{1}{4}\;\Omega^{abc}\delta^{(0)}_{\beta}H_{abc}$
$\displaystyle=$
$\displaystyle-\frac{1}{8}\nabla_{a}\beta_{bc}w^{a}{}_{ef}\left(R^{bcef}+2\,w^{bf}{}_{d}w^{cd}{}_{e}\right)-\frac{1}{4}\nabla_{b}\beta_{ca}w^{a}{}_{ef}R^{bcef}\;,$
(C.16)

where we have used

$\displaystyle\Omega^{abc}=-\frac{1}{2}w^{[a}{}_{ef}R^{bc]ef}-\frac{1}{3}w^{ad}{}_{e}w^{be}{}_{f}w^{cf}{}_{d}\;.$
(C.17)

The first term in the r.h.s. can be rewritten as

$\displaystyle-\frac{1}{8}\nabla_{a}\beta_{bc}w^{a}{}_{ef}\left(R^{bcef}+2w^{bf}{}_{d}w^{cde}\right)$
$\displaystyle=$
$\displaystyle-\frac{1}{4}\beta^{bc}w^{a}{}_{ef}\left(\nabla_{b}R_{ca}{}^{ef}-2\nabla_{a}w_{b}{}^{fd}w_{cd}{}^{e}\right)$
(C.18) $\displaystyle=$
$\displaystyle\frac{1}{4}\beta_{c}{}^{a}w_{abd}w^{d}{}_{ef}\left(R^{bcef}+2w^{beg}w^{cf}{}_{g}\right)\;,$
(C.19)

where in the first equality we noted that

$\displaystyle
R^{bcef}+2w^{bf}{}_{d}w^{cde}=2D^{[b}w^{c]ef}+2w^{[bc]d}w_{d}{}^{ef}\ ,$
(C.20)

vanishes after contraction with $\beta_{bc}$, which allows to move the
covariant derivative to the factors inside the parenthesis. Upon implementing
Riemann Bianchi identities, the second equality in (C.19) follows after
applying the relations

$\displaystyle\beta^{bc}\nabla_{a}w_{bde}$ $\displaystyle=$
$\displaystyle\beta^{bc}\left(R_{abde}+w_{ba}{}^{f}w_{fde}\right)\;,\;\;\;\textbf{}$
(C.21) $\displaystyle\beta^{bc}\nabla_{b}R_{ca}{}^{ef}$ $\displaystyle=$
$\displaystyle-2\beta^{bc}w_{b}{}^{d[e}R_{cad}{}^{f]}\;.$ (C.22)

The second term in the r.h.s. of (C.16) can be cast as

$\displaystyle-\frac{1}{4}\nabla_{b}\beta_{ca}w^{a}{}_{ef}R^{bcef}=\frac{1}{4}\beta_{ca}\nabla_{b}w^{a}{}_{ef}R^{bcef}-\frac{1}{4}\nabla_{b}\left(\beta_{ca}w^{a}{}_{ef}\right)R^{bcef}\;.$
(C.23)

Using (C.21), the first term in the r.h.s. of this expression can be rewritten
as

$\displaystyle\frac{1}{4}\beta_{ca}\nabla_{b}w^{a}{}_{ef}R^{bcef}=\frac{1}{4}\beta_{c}{}^{a}w_{ab}{}^{d}w_{def}R^{bcef}\;,$
(C.24)

whereas the second one is

$\displaystyle-\frac{1}{4}\nabla_{b}\left(\beta_{ca}w^{a}{}_{ef}\right)R^{bcef}=-\frac{1}{4}{\cal
D}_{b}\left[\beta_{ca}\left(w^{a}{}_{ef}R^{bcef}-2w^{ab}{}_{d}(R^{cd}+2\nabla^{c}\nabla^{d}\phi)\right)\right]\;.$
(C.25)

Here we have followed similar steps to those performed in (C.3) and (C.4),
supplemented with the relation

$\displaystyle{\cal
D}_{b}\left(\beta_{ca}w^{ab}{}_{d}\right)\left(R^{cd}+2\nabla^{c}\nabla^{d}\phi\right)=\;\;\;\;\;\;\;\;\;\;\;\;$
(C.26)
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=\left[\beta_{c}{}^{a}\left(R^{a}{}_{d}+2\nabla^{a}\nabla_{d}\phi\right)+\beta_{ae}w^{ab}{}_{d}w^{e}{}_{bc}\right]\left(R^{cd}+2\nabla^{c}\nabla^{d}\phi\right)=0\;,\;\;\;$
(C.27)

which vanishes by symmetry arguments. Actually, using the relation

$\displaystyle{\cal
D}_{b}\left[2\beta_{ca}w^{ab}{}_{d}\nabla^{c}\nabla^{d}\phi\right]={\cal
D}_{b}\left[\nabla_{c}\left(\beta_{ae}w^{ac}{}_{d}w^{edb}\right)\right]\;,$
(C.28)

it can be shown that (C.25) vanishes.

Regarding terms quadratic in the 3-form, we get

$\displaystyle-\frac{1}{4}H_{abc}\delta^{(0)}\Omega^{abc}$ $\displaystyle=$
$\displaystyle\frac{1}{8}H^{abc}H^{e}{}_{af}\beta^{df}\;\left(R_{debc}-2w_{be}{}^{g}w_{cdg}\right)$
(C.29) $\displaystyle+$
$\displaystyle\frac{1}{8}H^{abc}H^{fde}\beta_{af}\;w_{be}{}^{g}w_{cdg}$ (C.30)
$\displaystyle-$
$\displaystyle\frac{1}{8}H^{abc}w_{a}{}^{de}\left[\nabla_{b}\left(\beta_{c}{}^{f}H_{fde}\right)+\nabla_{d}\left(\beta_{e}{}^{f}H_{fbc}\right)\right]$
(C.31) $\displaystyle+$
$\displaystyle\frac{1}{8}H^{abc}H_{ecf}w_{a}{}^{de}\left[\nabla_{b}\beta_{d}{}^{f}-\nabla_{d}\beta_{b}{}^{f}-\nabla^{f}\beta_{bd}\right]\;,$
(C.32)

where we applied

$\displaystyle\beta_{a}{}^{f}H^{abc}H_{f}{}^{de}R_{debc}=0\,.$ (C.33)

Using (C.21) and integrating by parts, the first line in (C.32) can be taken
to the form

$\displaystyle\frac{1}{8}H^{abc}H^{e}{}_{af}\beta^{df}\;\left(R_{debc}-2w_{be}{}^{f}w_{cdf}\right)={\cal
D}_{b}\left(-\frac{1}{8}H^{abc}H_{aef}\beta_{cd}w^{def}\right)$ (C.34)
$\displaystyle\;\;\;\;\;\;\;\;\;-\frac{1}{8}{\cal
D}_{b}H^{baf}\beta_{ca}H_{f}{}^{de}w_{cde}-\frac{1}{8}\nabla^{b}H^{adc}H_{fdc}\beta^{ef}w_{eba}+\frac{1}{2}H^{abc}H^{e}{}_{af}\beta^{df}w_{bd}{}^{g}w_{ceg}\;.\;\;\;\;\
\ \ \ \ \ \ \ \ $ (C.35)

The first term in the third line of (C.32) is

$\displaystyle-\frac{1}{8}H^{abc}w_{a}{}^{de}\;\nabla_{b}\left(\beta_{c}{}^{f}H_{fde}\right)$
$\displaystyle=$ $\displaystyle-\frac{1}{8}{\cal
D}_{f}H^{fab}\beta_{ca}w_{b}{}^{de}H^{c}{}_{de}+\frac{1}{8}\beta_{c}{}^{f}H^{abc}H_{fde}w_{a}{}^{dg}w_{b}{}^{e}{}_{g}$
(C.37) $\displaystyle{\cal
D}_{b}\left(-\frac{1}{8}H^{abc}w_{a}{}^{de}\beta_{c}{}^{f}H_{fde}\right)\;,$

and the second one is

$\displaystyle-\frac{1}{8}H^{abc}w_{a}{}^{de}\nabla_{d}\left(\beta_{e}{}^{f}H_{fbc}\right)=\frac{1}{8}H^{abc}w_{ea}{}^{d}\nabla_{d}\left(\beta^{ef}H_{fbc}\right)-\frac{1}{4}H^{abc}w_{[ea]}{}^{d}\nabla_{d}\left(\beta^{ef}H_{fbc}\right)\;.$
(C.38)

Performing similar steps to those followed in (C.3)-(C.4), the first factor in
(C.38) can be cast as

$\displaystyle\frac{1}{8}H^{abc}w_{ea}{}^{d}\nabla_{d}\left(\beta^{ef}H_{fbc}\right)$
$\displaystyle=$ $\displaystyle{\cal
D}_{d}\left(\frac{1}{8}\beta^{ef}w_{ea}{}^{d}H^{2}{}^{a}{}_{f}\right)+\frac{1}{8}\beta^{ac}H^{2}{}^{b}{}_{c}\left(R_{ab}+\nabla_{a}\nabla_{b}\phi\right)$
(C.40)
$\displaystyle-\frac{1}{8}\beta^{ef}H_{fbc}w_{ea}{}^{d}\nabla_{d}H^{abc}-\frac{1}{8}H^{2}{}^{a}{}_{f}\beta^{ef}w_{ed}{}^{g}w_{ga}{}^{d}\;.$

Operating with the covariant derivative, followed by Bianchi identities of the
3-form and integration by parts, the last component in (C.38) can be worked
out, leading to

$\displaystyle-\frac{1}{4}H^{abc}w_{[ea]}{}^{d}\nabla_{d}\left(\beta^{ef}H_{fbc}\right)$
$\displaystyle=$ $\displaystyle{\cal
D}_{b}\left(-\frac{1}{2}H^{abc}w_{[ae]}{}^{d}\beta^{ef}H_{cdf}\right)+\frac{1}{2}{\cal
D}_{f}H^{fab}H_{bcd}w_{[ae]}{}^{c}\beta^{de}$
$\displaystyle-\frac{1}{4}H^{abc}w_{[ea]d}\left(\beta^{g}{}_{f}w^{fde}H_{gbc}+2\beta^{ef}w_{fb}{}^{g}H^{d}{}_{gc}\right)$
$\displaystyle-H^{abc}H_{cdf}\left(w_{[ea]}{}^{d}w_{gb}{}^{[e}\beta^{f]g}+\frac{1}{4}\beta^{ef}(w_{ae}{}^{g}w_{bg}{}^{d}-w_{ea}{}^{g}w_{gb}{}^{d})\right).$

We notice that the last line in (C.32) is simply

$\displaystyle\frac{1}{8}H^{abc}H_{ecf}w_{a}{}^{de}\left[\nabla_{b}\beta_{d}{}^{f}-\nabla_{d}\beta_{b}{}^{f}-\nabla^{f}\beta_{bd}\right]$
$\displaystyle=$
$\displaystyle-\frac{1}{4}\beta^{f}{}_{g}H^{abc}H_{ecf}w_{a}{}^{de}w^{g}{}_{db}\;.$
(C.42)

Finally, plugging all these equations into (C.16) and (C.32), and after a
proper rescaling with $a-b$, we obtain (5.28). Note that we have included for
convenience a vanishing quartic factor

$\displaystyle-\frac{1}{32}\beta_{ca}H^{2}_{b}{}^{c}H^{2}{}^{ab}=0\,.$ (C.43)

We conclude this appendix by noticing that the total derivative from
$\delta^{(1)}_{\beta}L^{(0)}$ in (3.10) with the choice $\kappa=\rho=0$ in
(5.25) exactly cancels the total derivative in (5.28). This follows
straightforwardly after using the relations

$\displaystyle{\cal D}_{b}\left(\beta_{c}{}^{b}{\cal
D}_{a}H^{2}{}^{ac}\right)=0\;,\;\;\;\;\;\;{\cal
D}_{b}\left(\beta_{c}{}^{b}\nabla_{a}w^{2}{}^{ac}\right)=0\;,\;\;\ $ (C.44)

which are particular cases of the more general identity valid for any $W^{a}$

$\displaystyle{\cal D}_{b}\left(\beta_{c}{}^{b}W^{c}\right)=0\;.$ (C.45)

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