$\beta$ symmetry in type II Supergravities

Walter H. Baron1, Nahuel A. Yazbek2

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 Matemática Luis Santaló, (CONICET-UBA)

Ciudad Universitaria, Pabellón I, CABA, C1428ZAA, Argentina
<EMAIL_ADDRESS>

Abstract

> A non geometric sector of the duality group emerging in Kaluza-Klein
> reductions is realized as an effective symmetry in the low energy action of
> uncompactified type II theories. This is achieved by extending the so called
> $\beta$ symmetry of the universal NS-NS sector to the R-R sector of type
> IIA, IIB and massive type IIA.

##  1 Introduction

Symmetries and dualities in String theory play a fundamental role. At the
perturbative level, the very definition of the theory, including the field
content, the interactions and even the space-time dimension is determined by
symmetry arguments.

The same holds by transitivity in the effective actions, at low energy. The
symmetries inherited from string theory: diffeomorphisms, Lorentz, gauge and
supersymmetry impose strong constraints on the allowed interactions at the
supergravity level. However, these symmetries alone are not sufficient to
fully fix them. Fortunately, in addition to symmetries, dualities come into
play. Dualities establish connections between seemingly different quantum
field theories in backgrounds with opposite features ($e.g.$, large/small
compact spaces) or different regimes (for instance, at strong/weak couplings).

This article deals with T-duality, whose origin can be traced back to the
extended nature of the strings. At quantum level, it is related to a symmetry
in the string spectrum that involves exchanging Kaluza-Klein (KK) modes and
winding modes around closed directions in d-dimensional toroidal
compactifications. This symmetry is described by the action of a discrete
$O(d,d,{\mathbb{Z}})$ group, but it is enhanced into $O(d,d,{\mathbb{R}})$ at
low energy, when all massive fields are truncated.

This symmetry was explicitly realized at low energy, in the two-derivative
effective actions of string theory, by Meissner and Veneziano in [1] on
1-dimensional (cosmological) reductions. Subsequently, it was further extended
to arbitrary dimensions by Maharana and Schwarz in [2] by considering general
KK reductions. Interestingly, although $O(d,d)$ manifests itself in
supergravity only after toroidal compactifications, the fact that this
symmetry emerges after a proper reduction already constrains the allowed
couplings in the parent higher dimensional theory in an indirect way.

This simple observation has profound implications, specially after the work of
Sen [3], where closed string field theory arguments were used to promote a
tree level $O(d)\times O(d)$ symmetry111Actually, this $O(d)\times O(d)$ is
further completed into $O(d,d)$ when it is combined with $GL(d)$ and
$b$-shifts. emerging after KK to all orders in derivatives.

Besides these arguments, the explicit verification of this symmetry beyond
two-derivatives is not a straightforward task and it took some time until an
explicit verification was achieved [4]. The complexity arises from the fact
that while both T-duality and Lorentz are symmetries of the compactified
theory, they cannot be simultaneously made manifest. An explicit realization
of $O(d,d)$ requires Lorentz non covariant field redefinitions in the internal
space.

An alternative approach to address T-duality is given by Double Field Theory
(DFT), where the $O(d,d)$ is realized already before compactification at the
cost of introducing extra coordinates, dual to winding modes [5]. This is an
interesting approach because the constraints on the effective action arising
from T-duality are more easily implemented. Some interesting applications
concern the uplifting of non-geometric gaugings and the derivation of general
formulae for non linear uplift ansatze on gauged supergravities, leading to
consistent truncations in lower dimensions, both followings from the so called
generalized Scherk Schwarz compactifications [6].

Other relevant applications are the determination of
$\alpha^{\prime}$-corrections in the low energy limit, where an iterative
method was developed [7] leading to an infinite tower of couplings (all order
in derivatives) of the so called biparametric deformation of DFT [8]. Despite
some interesting consistency checks [9] showing agreement with the bosonic and
heterotic supergravities at sixth order in derivatives, it is known that extra
couplings, not captured by this method, appear at eighth order in derivative
and higher ($e.g.$ those corresponding to type II supergravities). Actually,
it is not even clear so far if such missing interactions admit an embedding in
a T-dual formulation, like DFT [10].

Recently, a different approach was introduced in [11], [12] where the non
geometric sector of the T-duality group is promoted as an effective symmetry
in the canonical formulation of supergravity, avoiding the standard procedure
of the KK reduction or the introduction of extra coordinates and so offering
an alternative method, free of the obstructions raised in [10] and provides a
new perspective on incorporating the non-geometric aspects of T-duality into
the framework of supergravity.

This program focuses on infinitesimal duality transformations,222We are
therefore not considering the so called Buscher rules. For instance, the
consequences of the discrete $O(1,1,{\mathbb{Z}})$ symmetry on string theory
due to circular compactifications is therefore not covered. See [13] for some
recent progress on $\alpha^{\prime}$-corrections, after exploiting this
symmetry. so it deals with the connected component of $O(d,d)$, containing the
identity element of the group. This is organized in two sectors dubbed
geometric and non geometric, respectively. The former is generated by $GL(d)$
and ${\mathbb{R}}^{d(d-1)/2}$ (referred to as b-shifts), which are already
symmetries of the higher dimensional theory, while the latter is a symmetry
only after compactification. Notwithstanding, it was shown in [11],[12] that
also the non-geometric sector, parameterized by a constant bivector
$\beta^{\mu\nu}$, can be considered as an effective symmetry of the universal
NS-NS Lagrangian prior to any compactification, with the following
transformation rules

$\displaystyle\delta_{\beta}e_{\mu}{}^{a}$ $\displaystyle=$ $\displaystyle-
e_{\mu}{}^{b}b_{bc}\beta^{ca}\;,$ (1.1)
$\displaystyle\delta_{\beta}b_{\mu\nu}$ $\displaystyle=$
$\displaystyle-\beta_{\mu\nu}-b_{\mu\rho}\beta^{\rho\sigma}b_{\sigma\nu}\;,$
(1.2) $\displaystyle\delta_{\beta}\phi$ $\displaystyle=$
$\displaystyle\frac{1}{2}\beta^{\mu\nu}b_{\mu\nu}\;,$ (1.3)

when these are supplemented with the constraint

$\displaystyle\beta^{\mu\nu}\partial_{\nu}=0.$ (1.4)

Interestingly, while diffeomorphisms, Lorentz and gauge symmetry constrain the
allowed interactions up to an arbitrary function of the dilaton and some
undetermined coefficients

$\displaystyle{\cal
L}_{f,c,d,e}=f(\phi)\left(R+c\Box\phi+d(\nabla\phi)^{2}+eH^{2}\right)\;,$
(1.5)

the requirement of beta symmetry fully fixes the dynamics at two derivative
order

$\displaystyle 0$ $\displaystyle=$ $\displaystyle\delta_{\beta}[\sqrt{-g}{\cal
L}_{f,c,d,e}]$ (1.6) $\displaystyle=$
$\displaystyle\sqrt{-g}f(\phi)\left[(c-4)\nabla^{a}\left(\beta^{bc}H_{abc}\right)-(12e+1)\nabla^{a}\beta^{bc}H_{abc}+2(c+d)\nabla^{d}\phi\beta^{bc}H_{abc}\right]$
(1.7) $\displaystyle+$
$\displaystyle\frac{1}{2}\sqrt{-g}\beta^{ab}b_{ab}\left(2f(\phi)+f^{\prime}(\phi)\right)\left[R+c\Box\phi+d(\nabla\phi)^{2}+eH^{2}\right]\;,$
(1.8)

leading to the NS-NS universal Lagrangian of string supergravity

$\displaystyle{\cal
L}_{NS}=e^{-2\phi}\left(R+4\Box\phi-4(\nabla\phi)^{2}-\frac{1}{12}H^{2}\right)\;.$
(1.9)

This Lagrangian admits a two-parameter family of deformations consistent with
string dualities at fourth derivative order. Specific choices in the parameter
space correspond to the bosonic and heterotic supergravities333The trivial
case is identified with the type II supergravities, whose first corrections
appear at eighth order in derivatives.. In [11] the
$\alpha^{\prime}$-corrections of the transformation rules (1.3) were obtained
for the whole family of deformations by requiring invariance of the
Lagrangian. An independent derivation (from DFT) was presented in [12].

The $O(d,d)$ sector described by $\beta$ was also discussed in a related
context in [14], within the so called $\beta$-supergravity scheme. There, the
manifest symmetries differ from the standard formulation of supergravity and
the $\beta$ symmetry becomes geometric, while the b-shifts are interpreted as
non-geometric.

Another interesting method that achieves $O(d)\times O(d)$ symmetry in
supergravity in 10 dimensions is discussed in reference [15]. In this
approach, duality transformations of the dimensionally reduced fields are used
to formally promote this symmetry to higher dimensions. This extension enables
the determination of interactions in the uncompactified theory by construction
of $O(d)\times O(d)$ invariant blocks.

In this article we will extend the $\beta$ symmetry to the R-R sector at two
derivative order. This extension will be carried out within the democratic
formulation of type II theories, which we will review in section 2, in order
to set the notation and to ensure a self-contained discussion. In section 3 we
analyze Kaluza Klein compactifications and exhibit how duality acts on $GL(d)$
multiplets. Moving on to section 4, we will utilize the previously derived
expressions to establish the non-geometric sector of the duality group as an
effective symmetry of the R-R sector in 10 dimensions. We will present a
detailed verification of the invariance of the Lagrangian ${\cal L}_{R}$ for
type IIB and IIA. In addition, we also display the extension to the massive
type IIA case. Compatibility with duality relations of the democratic
formulation, closure of the symmetry algebra and consistency with equations of
motion ($e.o.m.$) are also discussed. Finally, we present the conclusions in
section 5.

##  2 Democratic formulation of type II theories

The natural framework to study the $O(d,d)$ symmetry enhancement on KK
reductions of type II theories is the so called democratic formulation [16].
This approach is preferred because it naturally accommodates R-R potentials in
an $O(d,d)$ multiplet, allowing for a linear action of the duality group (see
$e.g.$ [17]). Additionally, a democratic like version of DFT was used to
incorporate R-R potentials within the Double Field Theory framework [18]-[20].
Therefore, it is reasonable to explore the extension of $\beta$ symmetry to
the R-R sector in such a scheme.

In the standard formulation of type II supergravities (see for instance [21]),
p-form potentials are present for $p$ ranging from 0 to 4, with even (odd) $p$
for type IIB (IIA). The Lagrangian includes kinetic terms $|F^{(p+1)}|^{2}$ as
well as Chern-Simons interactions. In the case of IIB, the curvature $F^{(5)}$
is constrained to be self-dual, a condition that must be imposed only after
obtaining the equations of motion. The type IIA case admits a further
deformation, known as massive type IIA, which includes a non-dynamical
curvature $F^{(0)}$. It is worth noting that in $D=10$ the theory actually
allows for the inclusion of higher p-forms: $F^{(6)}$ to $F^{(10)}$ and indeed
these fields are included in the theory, but as Hodge duals of the former
ones.

In the democratic formulation[16], in contrast, the Lagrangian contains
quadratic Kinetic terms but not Chern-Simons. The field content now includes
all allowed p-form in $D=10$,

$\displaystyle{\cal S}_{R}=\int d^{10}x\sqrt{-g}\;{\cal
L}_{R}=-\frac{1}{4}\int
d^{10}x\sqrt{-g}\;\sum_{n}|F^{(n)}|^{2}=\frac{1}{4}\int\;\sum_{n}F^{(n)}\wedge*F^{(n)}\;,$
(2.1)

where $n$ runs over $1,3,5,7,9$ for type IIB and $2,4,6,8$ for type IIA,
respectively. Here we used standard notation
$\displaystyle{F^{(n)}=\frac{1}{n!}F^{(n)}_{\mu_{1}...\mu_{n}}\,dx^{\mu_{1}}\wedge...\wedge
dx^{\mu_{n}}}$ and
$\displaystyle{|F^{(n)}|^{2}=\frac{1}{n!}F^{(n)}_{\mu_{1}...\mu_{n}}F^{(n)}{}^{\mu_{1}...\mu_{n}}}$.

The field strengths are defined in a very compact way by means of formal sums
of forms444Being $e^{-b}$ a formal sum of even forms, it then commutes with
respect to the wedge product: $e^{-b}\wedge G=G\wedge e^{-b}$ for any form G.
Notice also that $e^{b}\wedge e^{-b}=1$, so that (2.3) is easily inverted as
$\displaystyle dD=e^{b}\wedge F\;.$ (2.2)

$\displaystyle F=e^{-b}\wedge dD\;.$ (2.3)

As an example, the projection of (2.3) to degree 5 is
$F^{(5)}=dD^{(4)}-b\wedge dD^{(2)}+\frac{1}{2}b\wedge b\wedge dD^{(0)}$. This
simplification is achieved after a particular choice for the potentials
$D^{(n)}$ which are defined as a convenient combination of R-R potentials
$C^{(n)}$ and the Kalb Ramond 2-form

$\displaystyle D^{(0)}=C^{(0)}\;,\;\;\;\;D^{(2)}=C^{(2)}+b\wedge
C^{(0)}\;,\;\;\;\;D^{(4)}=C^{(4)}+\frac{1}{2}b\wedge
C^{(2)}+\frac{1}{2}b\wedge b\wedge C^{(0)}\;,$ (2.4) $\displaystyle
D^{(1)}=C^{(1)}\;,\;\;\;\;\;\;\;\;\;D^{(3)}=C^{(3)}+b\wedge
C^{(1)}\;,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$
(2.5)

while $D^{(p)}$ of higher orders are introduced as the electromagnetic duals
of the previous ones, leading to the following duality relations on the
corresponding curvatures

$\displaystyle*F^{(n)}=(-)^{\frac{n(n-1)}{2}}F^{(10-n)}\;,$ (2.6)

with

$\displaystyle*F^{(n)}=\frac{1}{(10-n)!\,n!}\epsilon_{\nu_{1}...\nu_{n}\mu_{1}...\mu_{10-n}}F^{(n)}{}^{\nu_{1}...\nu_{n}}dx^{\mu_{1}}\wedge\dots\wedge
dx^{\mu_{10-n}}\,,$ (2.7)

which implies (for 10D Minkowski signature) $**F^{(n)}=(-)^{n+1}F^{(n)}$. So
defined, the field strengths are subject to the following $BI$

$\displaystyle d\left(e^{b}\wedge F\right)=0\;.$ (2.8)

The compatibility of both formulations works at the level of $e.o.m.$ and
$BI$. It requires to vary the action by considering the full set of forms as
independent fields, which leads to

$\displaystyle\delta\left(\int d^{10}x\sqrt{-g}\;{\cal L}_{R}\right)$
$\displaystyle=$ $\displaystyle-\frac{1}{2}\int
d^{10}x\sqrt{-g}\;\sum_{p}\frac{1}{p!}\delta
D^{(p)}_{\mu_{1}\dots\mu_{p}}\left(*\left[d\left(e^{-b}\wedge*F\right)\right]\right)^{\mu_{1}\dots\mu_{p}}+...$
(2.9) $\displaystyle=$ $\displaystyle-\frac{1}{2}\int\;\sum_{p}(-)^{p}\delta
D^{(p)}\wedge d\left(e^{-b}\wedge*F\right)+...$ (2.10)

where ellipsis stand for total derivatives and terms proportional to $\delta
g_{\mu\nu}$ or $\delta b_{\mu\nu}$. Hence, vanishing of the $e.o.m.$ requires

$\displaystyle d(e^{-b}\wedge*F)=0\;,$ (2.11)

It is exactly at this point (after varying the action) when the degrees of
freedom ($d.o.f.$) are truncated by imposing (2.6).

The duality relation (2.6) switches $*F\to F$ in (2.11) up to a the sign
$(-)^{\frac{n(n-2)}{2}}$, which can be absorbed in the exponential of the Kalb
Ramond by flipping its sign. Then, $e.o.m\longleftrightarrow BI$ after (2.6)
which reduces by half the number of independent equations, matching precisely
the $e.o.m.$ and $BI$ of the standard formulation of type IIA

$\displaystyle d\left(*F^{(4)}\right)=-H\wedge
F^{(4)}\;,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;dF^{(4)}=-H\wedge
F^{(2)}\;,$ (2.12) $\displaystyle
d\left(*F^{(2)}\right)=-H\wedge*F^{(4)}\;,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;dF^{(2)}=0\;,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$
(2.13)

and type IIB, respectively

$\displaystyle d\left(*F^{(3)}\right)=H\wedge
F^{(5)}\;,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;dF^{(3)}=-H\wedge
F^{(1)}\;,$ (2.15) $\displaystyle
d\left(*F^{(1)}\right)=H\wedge*F^{(3)}\;,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;dF^{(1)}=0\;,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$
(2.16) $\displaystyle d\left(*F^{(5)}\right)=dF^{(5)}=-H\wedge
F^{(3)}\;.\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ (2.17)

##  3 Duality action in Kaluza Klein compactifications

In this section, we will discuss the emergence of the duality group in
toroidal compactifications of the bosonic sector of type II theories, as
outlined in [17]. We will proceed by performing a $GL(d)$ decomposition of the
duality multiplets to uncover how the geometric and non-geometric sectors
manifest in its components. This decomposition is crucial in order to
establish the role of $\beta$ as an effective symmetry in the 10-dimensional
framework, in next section.

###  3.1 $O(d,d)$ and the NS-NS sector

The T-duality group acts linearly on both NS-NS and R-R sectors after a KK
reduction. This is achieved after a particular organization of the $d.o.f.$
The effective action in a toroidal compactification is obtained by taking the
following decomposition of $GL(10)\to GL(n)\oplus GL(d)$, with $d=10-n$ in the
NS-NS sector

$\displaystyle\hat{g}_{\mu\nu}dx^{\mu}dx^{\nu}$ $\displaystyle=$
$\displaystyle
g_{mn}dx^{m}dx^{n}+G_{ij}\left(dy^{i}+A^{i}{}_{m}dx^{m}\right)\left(dy^{j}+A^{j}{}_{n}dx^{n}\right)$
(3.1) $\displaystyle\frac{1}{2}\hat{b}_{\mu\nu}dx^{\mu}\wedge dx^{\nu}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left(b_{mn}-B_{im}A^{i}{}_{n}\right)dx^{m}\wedge
dx^{n}+B_{im}dx^{m}\wedge\left(dy^{i}+A^{i}{}_{n}dx^{n}\right)$ (3.3)
$\displaystyle+\frac{1}{2}B_{ij}\left(dy^{i}+A^{i}_{m}dx^{m}\right)\wedge\left(dy^{j}+A^{j}_{n}dx^{n}\right)$
$\displaystyle\hat{\phi}$ $\displaystyle=$
$\displaystyle\phi-\frac{1}{4}\log(G)$ (3.4)

In this section (and only in section 3) we will use hatted notation to denote
10 dimensional tensors: $\hat{g},\hat{b},\hat{\phi}$. The field redefinitions
in (3.4) ensure nice properties under the local symmetries (diff + gauge) of
the effective theory in n-dimensions. On the other hand, global symmetry
($GL(d)$ and ${\mathbb{R}}^{d(d-1)/2}$) in the internal space is enhanced into
an $O(d,d)$ group, for toroidal compactifications, due to the appearance of
new symmetries induced by the isometries of the background. Making this
symmetry manifest requires to accommodate scalars and 1-forms in duality
multiplets $\left({\cal M}_{MN}\right)$ ${\cal A}_{Mn}$ transforming in the
(bi)fundamental representation

$\displaystyle{\cal M}=\left(\begin{matrix}G^{ij}&-G^{ik}B_{kj}\cr
B_{ik}G^{kj}&G_{ij}-B_{ik}G^{kl}B_{lj}\end{matrix}\right)\;,\;\;\;\;\;\;\;\;{\cal
A}=\left(\begin{matrix}-A^{i}{}_{m}\cr
B_{im}-B_{ik}A^{k}{}_{m}\end{matrix}\right)\;.$ (3.5)

The subgroup connected with the identity element, $SO^{+}(d,d)$ is generated
by exponentiation of the Lie algebra elements, in the fundamental
representation. Its generators can be organized by those leading to a $GL(d)$
subgroup, the b-shifts and the $\beta$-transformations respectively
$t_{\alpha}=\\{t^{i}{}_{j},t^{ij},t_{ij}\\}$, where the only non-vanishing
components of $(t_{\alpha})_{M}{}^{N}$ are

$\displaystyle\left(t^{i}{}_{j}\right)^{k}{}_{l}=-\delta^{i}_{l}\delta^{k}_{j}\;,\;\;\;\;\left(t^{i}{}_{j}\right)_{k}{}^{l}=\delta^{i}_{k}\delta_{j}^{l}\;;\;\;\;\;\left(t^{ij}\right)_{kl}=\delta^{ij}_{kl}\;;\;\;\;\;\left(t_{ij}\right)^{kl}=\delta_{ij}^{kl}\;.\;\;\;\;$
(3.6)

A general element of the $SO^{+}(d,d)$ group is constructed after proper
multiplication of $h_{a},h_{b}$ and $h_{\beta}$, where

$\displaystyle
h_{a}=e^{a_{i}{}^{j}t^{i}{}_{j}}=\left(\begin{matrix}e^{-a^{T}}&0\cr
0&e^{a}\end{matrix}\right)\;,\;\;\;\;\;\;\;h_{\mathfrak{b}}=e^{\mathfrak{b}_{ij}t^{ij}}=\left(\begin{matrix}1&0\cr\mathfrak{b}&1\end{matrix}\right)\;,\;\;\;\;\;\;\;h_{\beta}=e^{\beta^{ij}t_{ij}}=\left(\begin{matrix}1&\beta\cr
0&1\end{matrix}\right)\;,\;\;\;\;\;\;\;$ (3.7)

One readily confirms that the transformation ${\cal M}\to{\cal
M}^{\prime}=h{\cal M}h^{T}$ leads to

$\displaystyle\delta_{a}G_{ij}$ $\displaystyle=$ $\displaystyle
2a_{(i}{}^{k}G_{|k|j)}\;,\;\;\;\;\;\delta_{\mathfrak{b}}G_{ij}=0\;,\;\;\;\;\;\delta_{\beta}G_{ij}=-2B_{(i|k|}\beta^{k}{}_{j)}\;,\;$
(3.8) $\displaystyle\delta_{a}B_{ij}$ $\displaystyle=$ $\displaystyle
2a_{[i}{}^{k}B_{|k|j]}\;,\;\;\;\;\;\delta_{\mathfrak{b}}B_{ij}=\mathfrak{b}\;,\;\;\;\;\;\delta_{\beta}B_{ij}=-2\beta_{ij}-B_{ik}\beta^{kl}B_{lj}\;.\;\;\;\;\;$
(3.9)

Similarly, the transformation of ${\cal A}$ reads, in components

$\displaystyle\delta_{a}A^{i}{}_{m}$ $\displaystyle=$ $\displaystyle-
a_{j}{}^{i}A^{j}{}_{m}\;,\;\;\;\;\delta_{\mathfrak{b}}A^{i}{}_{m}=0\;,\;\;\;\;\delta_{\beta}A^{i}{}_{m}=-\beta^{ij}\left(B_{jm}-B_{ik}A^{k}{}_{m}\right)\;,\;\;\;\;$
(3.10) $\displaystyle\delta_{a}B_{i}{}_{m}$ $\displaystyle=$ $\displaystyle
a_{i}{}^{j}B_{j}{}_{m}\;,\;\;\;\;\delta_{\mathfrak{b}}B_{i}{}_{m}=0\;,\;\;\;\;\delta_{\beta}B_{i}{}_{m}=-\beta_{ik}A^{k}{}_{m}-B_{ik}\beta^{kj}B_{jm}\;,\;\;\;\;$
(3.11)

where indices in $\beta$ are lowered with the internal metric $G_{kl}$. It is
worth noticing that while the $GL(d)$ and b-shift of the compact space are
trivially promoted to the whole 10-dimensional space, the main reason for
$\beta$ to admit a promotion to higher dimensions upon (1.4), is because (3.9)
and (3.11) plugged with (3.4) implies the following embedding in 10d,

$\displaystyle\delta_{\beta}\hat{g}_{\mu\nu}=-\hat{b}_{\mu\rho}\hat{\beta}^{\rho\sigma}\hat{g}_{\sigma\nu}-\hat{b}_{\nu\rho}\hat{\beta}^{\rho\sigma}\hat{g}_{\sigma\mu}\;,\;\;\;\;\;\delta_{\beta}\hat{b}_{\mu\nu}=-\hat{g}_{\mu\rho}\hat{\beta}^{\rho\sigma}\hat{g}_{\sigma\nu}-\hat{b}_{\mu\rho}\hat{\beta}^{\rho\sigma}\hat{b}_{\sigma\nu}\;,\;\;\;\;\;$
(3.12)

where
$\hat{\beta}^{ij}=\beta^{ij},\hat{\beta}^{mi}=\hat{\beta}^{im}=\hat{\beta}^{mn}=0$
and this solution for $\hat{\beta}$ is precisely a consequence of the
condition (1.4).

###  3.2 $O(d,d)$ and the R-R sector

Let us move now to the R-R sector. The global symmetry in this sector admits a
linear realization because p-forms of the democratic formulation naturally fit
in a Majorana-Weyl representation of $SO(d,d)$ using fermionic operators
$\psi_{i}$.

Let us very briefly comment on this construction (for further details see
[17]). Let $\psi_{i}$, with $\psi^{i}{}^{\dagger}:=(\psi_{i})^{\dagger}$, be a
set of d dimensional fermionic operators satisfying an anti-commuting algebra

$\displaystyle\\{\psi_{i},\psi^{j}{}^{\dagger}\\}=\delta_{i}{}^{j}\;{\mathbb{I}}\;,\;\;\;\;\;\\{\psi_{i},\psi_{j}\\}=0\;,\;\;\;\;\;\\{\psi^{i}{}^{\dagger},\psi^{j}{}^{\dagger}\\}=0\;.$
(3.13)

An $O(d,d)$ Clifford algebra is built after identifying
$\Gamma_{i}=\sqrt{2}\psi_{i},\Gamma^{i}=\sqrt{2}\psi^{i}{}^{\dagger}$

$\displaystyle\\{\Gamma_{M},\Gamma_{N}\\}=2\eta_{MN}\;{\mathbb{I}}\;,\;\;\;\;\;\;\;\;\;\;\;\eta_{MN}=\left(\begin{matrix}0&\delta^{i}{}_{j}\cr\delta_{i}{}^{j}&0\end{matrix}\right)\;,$
(3.14)

where $\eta$ is the $O(d,d)$ invariant metric. Hence, there is a natural
isomorphism between the fermionic Fock space and p-forms of the internal space
(duality group does not act on the non-compact directions and so (10-d)
indices are implicit in $D$)555Here lower and upper index $(p)$ characterizes
the degree of the form in the compact and non-compact spaces, respectively.

$\displaystyle\ket{D}=\sum_{p}\frac{1}{p!}D_{(p)i_{1}\dots
i_{p}}\psi^{i_{1}}{}^{\dagger}\dots\psi^{i_{p}}{}^{\dagger}\ket{0}\;,$ (3.15)

$\ket{0}$ being the vacuum of the Fock space, satisfying

$\displaystyle\psi_{i}\ket{0}=0\;,\;\;\;\;\;\bra{0}\ket{0}=1\;.$ (3.16)

The sequence of field redefinitions in the R-R sector is as follows. First,
each p-form $f^{(p)}$ splits into a sum of q-forms $f^{(q)}_{(n)i_{1}\dots
i_{n}}$

$\displaystyle f^{(p)}=\sum_{q}\frac{1}{n!}f^{(q)}_{(n)i_{1}\dots
i_{n}}dy^{1}\wedge\dots\wedge dy^{n}\;,$ (3.17)

where $n=p-q$ and each $f^{(q)}_{(n)i_{1}\dots i_{n}}$ is a q-form in the non-
compact space

$\displaystyle f^{(q)}_{(n)i_{1}\dots i_{n}}=\frac{1}{q!}f^{(q)}_{m_{1}\dots
m_{q}i_{1}\dots i_{n}}dx^{m_{1}}\wedge\dots\wedge dx^{m_{q}}\;.$ (3.18)

Then, one introduces shifted curvatures $F^{\prime}$, formally defined as

$\displaystyle F^{\prime}=\left.F\vphantom{2^{\frac{1}{2}}}\right|_{dy^{i}\to
dy^{i}-A^{i}}\;.$ (3.19)

This redefinition simplifies the kinetic terms $\sum_{p}|F^{(p)}|^{2}$ after
the KK decomposition by absorbing the dependence on the metric 1-forms
$A^{i}=A^{i}{}_{m}dx^{m}$.

The action of the duality subgroup connected to the identity, the
$Spin^{+}(d,d)$ group,666$O(d,d)$ is the maximum subgroup of the U-duality
group that closes independently on the NS-NS and R-R sectors. Nevertheless,
the T-duality transformations around single circles (represented by matrices
with negative determinant) do not preserve chirality, so only the $SO(d,d)$
lift $Spin(d,d)$ is separately a symmetry of type IIA and type IIB, after
compactification. is generated with the spinorial representation of elements
in (3.7),

$\displaystyle
S_{a}=\frac{1}{\sqrt{det(A)}}e^{a_{i}{}^{j}\psi^{i}{}^{\dagger}\psi_{j}}\;,\;\;\;\;\;S_{\mathfrak{b}}=e^{\frac{1}{2}\mathfrak{b}_{ij}\psi^{i}{}^{\dagger}\psi^{j}{}^{\dagger}}\;,\;\;\;\;\;S_{\beta}=e^{-\frac{1}{2}\beta^{ij}\psi_{i}\psi_{j}}\;,\;\;\;\;\;$
(3.20)

where $A_{i}{}^{j}=(e^{a})_{i}{}^{j}$.

Therefore, after (3.15) we find the following transformation rules for the
internal p-forms $D_{(p)i_{1}\dots i_{p}}$

$\displaystyle\delta_{a}D_{(p)i_{1}\dots i_{p}}$ $\displaystyle=$
$\displaystyle pa_{[i_{1}}{}^{k}\;D_{(p)|k|i_{2}\dots
i_{p}]}-\frac{1}{2}a_{k}{}^{k}D_{(p)i_{1}\dots i_{p}}\;,$ (3.21)
$\displaystyle\delta_{\mathfrak{b}}D_{(p)i_{1}\dots i_{p}}$ $\displaystyle=$
$\displaystyle\left(\mathfrak{b}\wedge D_{(p-2)}\right)_{i_{1}\dots
i_{p}}=\frac{p!}{2!(p-2)!}\mathfrak{b}_{[i_{1}i_{2}}D_{(p-2)i_{3}\dots
i_{p}]}\;,$ (3.22) $\displaystyle\delta_{\beta}D_{(p)i_{1}\dots i_{p}}$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\left(i_{\beta}D_{(p+2)}\right)_{i_{1}\dots
i_{p}}=-\frac{1}{2}\beta^{kl}D_{(p+2)kli_{1}\dots i_{p}}\;,$ (3.23)

where in the last line we have introduced the interior product of $\beta$ with
(p+2)-forms.

##  4 $\beta$ symmetry of the RR sector in 10d

After analyzing how an infinitesimal duality transformation acts on the
internal p-forms after a Kaluza-Klein reduction, we proceed similarly to the
NS-NS sector. We promote these transformation rules as an effective symmetry
of the parent theory, subject to the constraint (1.4)

$\displaystyle\delta_{a}D^{(p)}_{\mu_{1}\dots\mu_{p}}$ $\displaystyle=$
$\displaystyle
pa_{[\mu_{1}}{}^{\nu}\;D^{(p)}_{|\nu|\mu_{2}\dots\mu_{p}]}\;-\frac{1}{2}a_{\mu}{}^{\mu}D^{(p)}_{\mu_{1}\dots\mu_{p}};,$
(4.1) $\displaystyle\delta_{\mathfrak{b}}D^{(p)}_{\mu_{1}\dots\mu_{p}}$
$\displaystyle=$ $\displaystyle\left(\mathfrak{b}\wedge
D^{(p-2)}\right)_{\mu_{1}\dots\mu_{p}}\;,$ (4.2)
$\displaystyle\delta_{\beta}D^{(p)}_{\mu_{1}\dots\mu_{p}}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\left(i_{\beta}D^{(p+2)}\right)_{\mu_{1}\dots\mu_{p}}\;.$
(4.3)

In this expression $D^{(p)}$ is assumed to be non vanishing only for
$p=0,...,8$. Global $GL(10)$ acts on RR p-forms with a non trivial density
contribution (last factor in (4.1)). While the linear piece (first term in the
$r.h.s$ of (4.1)) cancels the variation of the inverse metric in any coupling
with indices properly contracted, the density piece cancels the contribution
of the measure $\delta_{a}({\sqrt{-g}})$ as long as the couplings are
quadratic in $D^{(p)}$.777Interestingly, the field redefinition
$\tilde{D}^{(p)}=e^{\phi}D^{(p)}$ transforms linearly (without density factor)
under $GL(10)$ and the action written with these variables has the usual
(duality singlet) measure $\sqrt{-g}e^{-2\phi}$. This new scheme makes
explicit the dilaton dependence of the string loop expansion, at the cost of
introducing dilaton dependence on the gauge transformations and Bianchi
identities. b-shifts are also a manifest symmetry in 10 dimensions if the
action is written in terms of the curvatures (2.3). Indeed

$\displaystyle\delta_{\mathfrak{b}}F$ $\displaystyle=$
$\displaystyle\delta_{\mathfrak{b}}\left(e^{-b}\wedge
dD\right)=-\delta_{\mathfrak{b}}b\wedge e^{-b}\wedge dD+e^{-b}\wedge
d(\delta_{\mathfrak{b}}D)$ (4.4) $\displaystyle=$
$\displaystyle-\mathfrak{b}\wedge e^{-b}\wedge dD+e^{-b}\wedge
d(\mathfrak{b}\wedge D)=0.$ (4.5)

We focus now on the non geometric sector of $O(d,d)$ described by $\beta$. We
remark once again that while (4.1) and (4.2) are symmetries of the parent
theory, we expect (4.3) to be a symmetry only in an effective way, when it is
supplemented with the constraint (1.4).

An important observation is that the $\beta$-transformations of the NS-NS
sector in the KK reduction remain unchanged even when RR-potentials are
introduced. Based on this, we propose that equation (1.3) is not affected or
deformed. Consequently, equation (1.8) still holds, indicating that both
${\cal L}_{NS}$ and ${\cal L}_{R}$ should remain invariant independently.

In this section, we will provide a detailed verification that the $\beta$
transformations, as defined in equation (4.3), are indeed a symmetry of (2.1).
Based on the previous discussion, it is evident that any combination of
$\sqrt{-g}\;|F^{(p)}|^{2}$ is separately invariant under $GL(10)$ and b-shifts
in 10 dimensions. We will show that, in analogy with the NS-NS sector, $\beta$
transformations constrain the relative coefficients of the kinetic terms. To
accomplish this, we start by computing the induced transformation on the field
strengths $F^{(p)}$, due to (4.3),

$\displaystyle\delta_{\beta}F$ $\displaystyle=$
$\displaystyle-\delta_{\beta}b\wedge e^{-b}\wedge
dD-\frac{1}{2}e^{-b}\wedge[i_{\beta}(e^{b}\wedge F)]\;,$ (4.6)

where we have used

$\displaystyle\delta(dD)$ $\displaystyle=$
$\displaystyle\sum_{p}\delta\left(dD^{(p)}\right)=\sum_{p}\frac{1}{p!}\delta_{\beta}\left(\partial_{[\mu_{1}}D^{(p)}_{\mu_{2}\dots\mu_{p+1}]}\right)dx^{\mu_{1}}\wedge\dots\wedge
dx^{\mu_{p+1}}$ (4.7) $\displaystyle=$
$\displaystyle\sum_{p}-\frac{1}{2}\frac{1}{p!}\partial_{[\mu_{1}}\left(\beta^{\rho\sigma}D^{(p+2)}_{\rho\sigma\mu_{2}\dots\mu_{p+1}]}\right)dx^{\mu_{1}}\wedge\dots\wedge
dx^{\mu_{p+1}}$ (4.8) $\displaystyle=$
$\displaystyle\sum_{p}-\frac{1}{2}\frac{1}{p!}\beta^{\rho\sigma}\frac{(p+3)}{(p+1)}{}\partial_{[\rho}D^{(p+2)}_{\sigma\mu_{1}\dots\mu_{p+1}]}dx^{\mu_{1}}\wedge\dots\wedge
dx^{\mu_{p+1}}$ (4.9) $\displaystyle=$
$\displaystyle\sum_{p}-\frac{1}{2}i_{\beta}\left(dD^{(p+2)}\right)=-\frac{1}{2}i_{\beta}(dD)\;,$
(4.10)

where condition (1.4) was used in the third line. Hence, (4.6) follows from
(4.10), after inverting equation (2.3).

Expression (4.6) can be further manipulated by noting that

$i_{\beta}[A^{(p)}\wedge B^{(q)}]=i_{\beta}(A^{(p)})\wedge
B^{(q)}+A^{(p)}\wedge i_{\beta}(B^{(q)})+2[A\beta B]\;,$ (4.11)

where $A^{(p)}$ and $B^{(q)}$ are two arbitrary forms. Here we have defined

$\displaystyle[A\beta B]=\frac{1}{(p-1)!}\frac{1}{(q-1)!}A^{(p)}_{[i_{1}\dots
i_{p-1}|k}\beta^{kl}B^{(q)}_{l|i_{p}\dots
i_{p+q-2}]}dx^{i_{1}}\wedge\dots\wedge dx^{i_{p+q-2}}\;.$ (4.12)

Interestingly, we can use (4.11) inductively in order to obtain

$\displaystyle i_{\beta}(e^{b})$ $\displaystyle=$ $\displaystyle
i_{\beta}\left(\sum_{m\geq 0}\frac{1}{m!}b\wedge\dots\wedge b\right)$ (4.13)
$\displaystyle=$ $\displaystyle
e^{b}\wedge\left(\vphantom{\frac{1}{2}}i_{\beta}b+2[b\beta b]\right)\;.$
(4.14)

Similarly we can use (4.12) inductively to get

$\displaystyle[e^{b}\beta F]=e^{b}\wedge[b\beta F].$ (4.15)

So, the second term in (4.6) can therefore be expressed as

$\displaystyle-\frac{1}{2}e^{-b}\wedge[i_{\beta}(e^{b}\wedge F)]$
$\displaystyle=$
$\displaystyle-\frac{1}{2}e^{-b}\wedge\left(i_{\beta}(e^{b})\wedge
F+e^{b}\wedge i_{\beta}F+2[e^{b}\beta F]\right)$ (4.16) $\displaystyle=$
$\displaystyle-\frac{1}{2}\left(\vphantom{\frac{1}{2}}(i_{\beta}b)\,F+2[b\beta
b]\wedge F+i_{\beta}F+2[b\beta F]\right)\;,$ (4.17)

where we have used $e^{-b}\wedge e^{b}=1$. The first term in (4.6) can be
written instead as

$\displaystyle-\delta_{\beta}b\wedge e^{-b}\wedge dD=\left(\beta+[b\beta
b]\right)\wedge F\;.$ (4.18)

Here we have introduced

$\displaystyle\beta=\frac{1}{2}\beta_{\mu\nu}dx^{\mu}\wedge dx^{\nu}\;,$
(4.19)

with $\beta_{\mu\nu}=\beta^{\rho\sigma}g_{\mu\rho}g_{\nu\sigma}$. Plugging
(4.17) and (4.18) we finally obtain

$\displaystyle\delta F=-\frac{1}{2}(i_{\beta}b)\,F-[b\beta F]+\beta\wedge
F-\frac{1}{2}i_{\beta}F\;,$ (4.20)

or in components

$\displaystyle\delta F^{(p)}_{\mu_{1}\dots\mu_{p}}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\beta^{\rho\sigma}b_{\rho\sigma}F^{(p)}_{\mu_{1}\dots\mu_{p}}-p\;b_{[\mu_{1}|\rho|}\beta^{\rho\sigma}F^{(p)}_{|\sigma|\mu_{2}\dots\mu_{p}]}$
(4.21) $\displaystyle+$
$\displaystyle\frac{p(p-1)}{2}\beta{\vphantom{F^{(p)}_{\mu}}}_{[\mu_{1}\mu_{2}}F^{(p-2)}_{\mu_{3}\dots\mu_{p}]}-\frac{1}{2}\beta^{\rho\sigma}F^{(p+2)}_{\rho\sigma\mu_{1}\dots\mu_{p}}\;.$
(4.22)

In this expression, it is implicitly assumed that $F^{(p)}$ is non vanishing
only for $p=1,3,5,7,9$ in type IIB and $p=2,4,6,8$ for type IIA.

With these results we are ready to analyze the invariance of the Lagrangian
${\cal L}_{R}$.

###  4.1 $\beta$-invariance of the Lagrangian in type IIA and type IIB

The variation of the R-R sector of the Lagrangian due to a $\beta$
transformation is

$\displaystyle\delta_{\beta}\left[\sqrt{-g}{\cal L}_{R}\right]$
$\displaystyle=$
$\displaystyle\sqrt{-g}\left[\vphantom{{\frac{1}{2}}^{\frac{1}{2}}}\right.\frac{\delta_{\beta}\left(\sqrt{-g}\right)}{\sqrt{-g}}{\cal
L}_{R}+\sum_{p}\frac{p}{p!}\delta_{\beta}g^{\mu_{1}\nu_{1}}F^{(p)}_{\mu_{1}\mu_{2}\dots\mu_{p}}F^{(p)}_{\nu_{1}}{}^{\mu_{2}\dots\mu_{p}}$
(4.24)
$\displaystyle\;\;\;\;\;\;\;+\sum_{p}\frac{2}{p!}F^{(p)}{}^{\mu_{1}\dots\mu_{p}}\delta_{\beta}F^{(p)}_{\mu_{1}\dots\mu_{p}}\left.\vphantom{{\frac{1}{2}}^{\frac{1}{2}}}\right]\;,$

with the sum runing over $p=1,3,5,7,9$ ($p=2,4,6,8$) on type IIB (IIA).
Replacing the transformation of the volume measure and inverse metric in
(4.24):

$\displaystyle\delta\sqrt{-g}$ $\displaystyle=$
$\displaystyle\sqrt{-g}\;i_{\beta}b\;,$ (4.25) $\displaystyle\delta
g^{\mu\nu}$ $\displaystyle=$ $\displaystyle
2\beta^{\rho(\mu}b^{\nu)}{}_{\rho}\;,$ (4.26)

one readily verifies that first line in (4.24) exactly cancels terms in
$\delta F^{(p)}$ proportional to $F^{(p)}$ (first line in (4.22)), leaving
only mixed terms

$\displaystyle\delta_{\beta}\left[\sqrt{-g}{\cal L}_{R}\right]\;=\hskip
341.43306pt$ (4.27)
$\displaystyle\sqrt{-g}\sum_{p}\left(\frac{p(p-1)}{2\;p!}\beta{\vphantom{F^{(p)}_{\mu}}}_{\mu_{1}\mu_{2}}F^{(p-2)}_{\mu_{3}\dots\mu_{p}}F^{(p)}{}^{\mu_{1}\dots\mu_{p}}-\frac{1}{2\;p!}\beta^{\rho\sigma}F^{(p+2)}_{\rho\sigma\mu_{1}\dots\mu_{p}}F^{(p)}{}^{\mu_{1}\dots\mu_{p}}\right)=0\;,\;\;\;\;\;$
(4.28)

where we immediately note that terms cancel each other out in the sum over all
form orders. As we anticipated, it is now clear from this analysis that
invariance of ${\cal L}_{R}$ is broken if we modify the relative coefficients
of the kinetic terms.

###  4.2 $\beta$-invariance of the Lagrangian in massive type IIA

Massive type IIA is a deformation of type IIA supergravity containing extra
curvatures $F^{(0)}$ and $F^{(10)}$. The whole set of field strengths are
defined like in (2.3) by adding formally a 0-form to $dD$

$\displaystyle F=e^{b}\wedge(dD+M)\;.$ (4.29)

So, $F^{(0)}=M$ and the remaining forms contain an extra contribution
$e^{b}M$.

The Lagrangian of the RR sector is still (2.1) with the sum now running over
all allowed even degrees $n=0,2,\dots,10$. We remark here that while
$F^{(10)}$ can be interpreted as the curvature of a $D^{(9)}$ form potential,
the field $F^{(0)}$ does not have such an interpretation and is formally
introduced in (2.1) and finally interpreted as the hodge dual of the
former.888A formal (-1)-form potential for $F^{(0)}$ was introduced in [22].
Its mathematical interpretation was further clarified in [19], as a
contribution of the 1-form potential having a linear dependence on a
coordinate dual to a winding mode. Therefore, the new fields carry non
propagating degrees of freedom. Certainly, the $e.o.m.$ of $D^{(9)}$ is

$\displaystyle d(*F^{(10)})=0\;,$ (4.30)

but being $*F^{(10)}$ a 0-form, equation (4.30) implies $*F^{(10)}$
($=F^{(0)}$ after (2.6)) =constant. Despite this observation, it is worth
noting that massive type II supergravity does indeed arise in type IIA string.
The 9-form potential is not observed in the quantization of the type IIA
string precisely because it does not carry propagating $d.o.f.$, but it is
required by the very existence of D8-branes.

We now turn to explore whether $\beta$-symmetry can be extended in this case
as well. We immediately see that $\delta D^{(9)}$ is trivially invariant if
(4.3) holds, because it is defined through the contraction with an
hypothetical 11-dimensional form in 10d, but there is no interpretation at the
level of potential forms for $F^{(0)}$. For that reason we work directly at
the level of curvatures and extrapolate (4.22), instead of (4.3).

With this definition it readily follows that $\beta$ symmetry is consistent
with constant $F^{(0)}=M$, as (4.22) implies

$\displaystyle\delta_{\beta}F^{(0)}=-\frac{1}{2}\beta^{\rho\sigma}b_{\rho\sigma}F^{(0)}-\frac{1}{2}\beta^{\rho\sigma}F^{(2)}_{\rho\sigma}=-\frac{1}{2}\beta^{\rho\sigma}\partial_{\rho}D^{(1)}_{\sigma}=0\;.$
(4.31)

The M-dependent deformation on the curvatures $F^{(p)}$, for $p=2,4,6,8$
(4.29) and the introduction of curvatures $F^{(0)}$ and $F^{(10)}$ in (2.1) do
not spoil the proof of invariance of the action of the previous section, which
only depends on the formal expression (4.22).

We would like to emphasize here that massive type IIA can be considered as a
consistent truncation of type IIA with respect to infinitesimal $O(d,d)$
transformations. This is straightforwardly verified for $GL(d)$ and b-shift.
Regarding $\beta$ transformation we see that $M=0$ is also consistent due to
(4.31), while $F^{(10)}$ transforms non trivially

$\displaystyle\delta_{\beta}F^{(10)}_{\mu_{1}\dots\mu_{10}}=-\frac{1}{2}\beta^{\rho\sigma}b_{\rho\sigma}F^{(10)}_{\mu_{1}\dots\mu_{10}}-10b_{[\mu_{1}|\rho|}\beta^{\rho\sigma}F^{(10)}_{|\sigma|\mu_{2}\dots\mu_{10}]}+45\beta_{[\mu_{1}\mu_{2}}F^{(8)}_{\mu_{3}\dots\mu_{10}]}$
(4.32)

and so it could rise doubts about the consistency of the truncation as the
condition $F^{(10)}=0$, $\delta_{\beta}F^{(10)}=0$ requires vanishing of the
last factor in (4.32), or equivalently

$\displaystyle\beta^{[\mu_{1}\mu_{2}}F^{(8)}{}^{\mu_{3}\dots\mu_{10}]}=0\;,$
(4.33)

a condition which is not satisfied, in general. Actually, there is no conflict
here as $D^{(9)}=0$ does not imply $F^{(10)}=0$. We can still introduce a non
vanishing $F^{(10)}$ in the democratic formulation of (massless) type IIA
SUGRA through (2.3), which transforms non trivially. Of course, this curvature
is therefore interpreted as a particular combination of Kalb-Ramond and
curvatures $F^{(p)}$, for $p=2,4,6,8$ and so it carries no independent
$d.o.f.$ The condition $F^{(10)}=0$ is only a consequence of $F^{(0)}=M=0$,
after and only after the relation $F^{(10)}=*F^{(0)}$ is imposed. Indeed, when
duality relations are required to hold, (4.33) turns into

$\displaystyle
0=\beta^{[\mu_{1}\mu_{2}}\epsilon^{\mu_{3}\dots\mu_{10}]\rho\sigma}F^{(2)}_{\rho\sigma}=\beta^{[\mu_{1}\mu_{2}}\epsilon^{\mu_{3}\dots\mu_{10}]\rho\sigma}\left(\partial_{\rho}D^{(1)}_{\sigma}-Mb_{\rho\sigma}\right)\;\;.$
(4.34)

Last factor cancels for $M=0$, while the first one cancels due to (1.4),
because the index $\rho$ in the derivative necessarily matches one of the
indices in $\beta$.

###  4.3 Consistency with duality relations

In this section we will analyze compatibility of beta transformations and
duality relations. Concretely, we want to verify that both sides in (2.6)
transform consistently. The dual curvatures transform as

$\displaystyle\delta[*F^{(q)}]_{\mu_{1}\dots\mu_{10-q}}$ $\displaystyle=$
$\displaystyle\delta\left[\frac{1}{q!}\epsilon_{\nu_{1}\dots\nu_{q}\mu_{1}\dots\mu_{10-q}}F^{(q)}{}^{\nu_{1}\dots\nu_{q}}\right]$
(4.35) $\displaystyle=$
$\displaystyle\frac{1}{q!}\epsilon_{\nu_{1}\dots\nu_{q}\mu_{1}\dots\mu_{10-q}}\left[\vphantom{\frac{e^{2}}{2}}\right.\frac{\delta\sqrt{-g}}{\sqrt{-g}}g^{\nu_{1}\rho_{1}}\dots
g^{\nu_{q}\rho_{q}}F^{(q)}_{\rho_{1}\dots\rho_{q}}$ (4.37)
$\displaystyle+\;q\left(\delta g^{\nu_{1}\rho_{1}}\right)\dots
g^{\nu_{q}\rho_{q}}F^{(q)}_{\rho_{1}\dots\rho_{q}}+\;g^{\nu_{1}\rho_{1}}\dots
g^{\nu_{q}\rho_{q}}\delta
F^{(q)}_{\rho_{1}\dots\rho_{q}}\left.\vphantom{\frac{e^{2}}{2}}\right]\;.$

The contribution in the second line in (4.37) originates from the variation of
the Levi-Civita tensor and it leads to

$\displaystyle\delta[*F^{(q)}]_{\mu_{1}\dots\mu_{10-q}}\supset(i_{\beta}b)[*F^{(q)}]_{\mu_{1}\dots\mu_{10-q}}\;.$
(4.38)

First factor in the third line in (4.37) is instead

$\displaystyle\delta[*F^{(q)}]_{\mu_{1}\dots\mu_{10-q}}\supset\frac{q}{q!}\epsilon_{\nu_{1}\dots\nu_{q}\mu_{1}\dots\mu_{10-q}}\left(\beta^{\tau\nu_{1}}b_{\sigma\tau}+\beta^{\tau}{}_{\sigma}b^{\nu_{1}}{}_{\tau}\right)F^{(q)}{}^{\sigma\nu_{2}\dots\nu_{q}}\;,$
(4.39)

while last contribution in (4.37) is, after use of (4.22)

$\displaystyle\delta[*F^{(q)}]_{\mu_{1}\dots\mu_{10-q}}$
$\displaystyle\supset$
$\displaystyle-\frac{1}{2}(i_{\beta}b)[*F^{(q)}]_{\mu_{1}\dots\mu_{10-q}}+\frac{1}{2}[i_{\beta}(*F^{(q-2)})]_{\mu_{1}\dots\mu_{10-q}}$
(4.42)
$\displaystyle-\;\frac{q}{q!}\epsilon_{\nu_{1}\dots\nu_{q}\mu_{1}\dots\mu_{10-q}}\;\;\beta^{\tau}{}_{\sigma}b^{\nu_{1}}{}_{\tau}F^{(q)}{}^{\sigma\nu_{2}\dots\nu_{q}}$
$\displaystyle-\;\frac{1}{2}\frac{1}{q!}\epsilon^{\nu_{1}\dots\nu_{q}}{}_{\mu_{1}\dots\mu_{10-q}}\;\beta^{\rho\sigma}F^{(q+2)}_{\rho\sigma\nu_{1}\dots\nu_{p}}\;.$

Plugging (4.38)-(4.42) with (4.37) we get

$\displaystyle\delta[*F^{(q)}]_{\mu_{1}\dots\mu_{10-q}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\beta^{\rho\sigma}b_{\rho\sigma}[*F^{(q)}]_{\mu_{1}\dots\mu_{10-q}}+\frac{1}{2}\beta^{\rho\sigma}[*F^{(q-2)}]_{\rho\sigma\mu_{1}\dots\mu_{10-q}}$
(4.45)
$\displaystyle+\frac{q}{q!}\epsilon^{\nu_{1}\dots\nu_{q}}{}_{\mu_{1}\dots\mu_{10-q}}\beta^{\tau\nu_{1}}b^{\sigma}{}_{\tau}F^{(q)}{}_{\sigma\nu_{2}\dots\nu_{q}}$
$\displaystyle-\;\frac{1}{2}\frac{1}{q!}\epsilon^{\nu_{1}\dots\nu_{q}}{}_{\mu_{1}\dots\mu_{10-q}}\;\beta^{\rho\sigma}F^{(q+2)}_{\rho\sigma\nu_{1}\dots\nu_{q}}$

or equivalently

$\displaystyle\delta[*F^{(q)}]_{\mu_{1}\dots\mu_{10-q}}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\beta^{\rho\sigma}b_{\rho\sigma}[*F^{(q)}]_{\mu_{1}\dots\mu_{10-q}}+\frac{1}{2}\beta^{\rho\sigma}[*F^{(q-2)}]_{\rho\sigma\mu_{1}\dots\mu_{10-q}}$
(4.48)
$\displaystyle-(10-q)b_{[\mu_{1}|\tau|}\beta^{\tau\nu_{1}}[*F^{(q)}]_{|\nu_{1}|\mu_{2}\dots\mu_{10-q}]}$
$\displaystyle-\;\frac{(10-q)(9-q)}{2}\beta_{[\mu_{1}\mu_{2}}[*F^{(q+2)}]_{\mu_{3}\dots\mu_{10-q}]}\;,$

where we have used

$\displaystyle F^{(n)}_{\rho_{1}\dots\rho_{n}}$ $\displaystyle=$
$\displaystyle(-)^{n+1}[**F^{(n)}]_{\rho_{1}\dots\rho_{n}}$ (4.49)
$\displaystyle=$
$\displaystyle(-)^{n+1}\frac{1}{(10-n)!}\epsilon_{\sigma_{1}\dots\sigma_{10-n}\rho_{1}\dots\rho_{n}}[*F^{(n)}]^{\sigma_{1}\dots\sigma_{10-n}}$
(4.50)

and

$\displaystyle\epsilon^{\rho_{1}\dots\rho_{10-n}\mu_{1}\dots\mu_{n}}\epsilon_{\rho_{1}\dots\rho_{10-n}\nu_{1}\dots\nu_{n}}=-n!(10-n)!\delta^{[\mu_{1}}_{\nu_{1}}\dots\delta^{\mu_{n}]}_{\nu_{n}}\;,$
(4.51)

to write the right hand side in 4.48 entirely in terms of dual forms. It is
now straightforward to verify that $\beta$ symmetry of curvatures (4.22) and
dual curvatures (4.48) are consistent with the duality relation (2.6). Indeed,
using it on both sides of (4.48) and identifying $p=10-q$ one easily recovers
(4.22).

###  4.4 Equations of motion

We will now focus on the behavior of the $e.o.m.$ under the non geometric
sector of the duality group. The variation of the action is, after integrating
by parts

$\displaystyle\delta S=\int d^{10}x\left(-2\sqrt{-g}{\cal L}_{NS}\;\delta
d+\tilde{\cal E}^{ab}\;\delta e_{ab}+\tilde{\cal B}^{ab}\;\delta
b_{ab}+\sum_{p}\tilde{\cal E}^{(p)}{}^{a_{1}\dots a_{p}}\;\delta
D^{(p)}_{a_{1}\dots a_{p}}\right)\;.$ (4.52)

Here we have considered the generalized dilaton
$d=\phi-\frac{1}{2}\log\sqrt{-g}$ as independent $d.o.f.$999This is a
convenient choice when dealing with T-duality because the generalized dilaton
$d$ is a singlet. For instance, first term in (4.52) does not contribute to
$\delta_{\beta}S$. and we have introduced $\delta e_{ab}=e^{\mu}{}_{a}\delta
e_{\mu b}$, $\delta b_{ab}=e^{\mu}{}_{a}e^{\nu}{}_{b}\delta b_{\mu\nu}$,
$\delta D^{(p)}_{a_{1}\dots a_{p}}=e^{\mu_{1}}{}_{a_{1}}\dots
e^{\mu_{p}}{}_{a_{p}}\delta D^{(p)}_{\mu_{1}\dots\mu_{p}}$ and

$\displaystyle\tilde{\cal E}_{ab}$ $\displaystyle=$
$\displaystyle-2\sqrt{-g}e^{-2\phi}\left(\vphantom{\frac{1}{2}}R_{ab}+2\nabla_{a}\nabla_{b}\phi-\frac{1}{4}H_{acd}H_{b}{}^{cd}\right)$
$\displaystyle\;\;\;\;\;\;\;\;\;\;-\frac{1}{4}\sqrt{-g}\sum_{p}\left(g_{ab}|F^{(p)}|^{2}-\frac{2}{(p-1)!}F^{(p)}_{ac_{1}\dots
c_{p-1}}F^{(p)}_{b}{}^{c_{1}\dots c_{p-1}}\right)\,,$
$\displaystyle\tilde{\cal B}_{ab}$ $\displaystyle=$
$\displaystyle\sqrt{-g}e^{-2\phi}\left(\frac{1}{2}\nabla_{c}H_{ab}{}^{c}{}-\nabla_{c}\phi
H_{ab}{}^{c}\right)+\frac{1}{4}\sqrt{-g}\sum_{p}\frac{1}{(p-2)!}F^{(p)}_{abc_{1}\dots
c_{p-2}}F^{(p-2)}{}^{c_{1}\dots c_{p-2}}\,\;,$ $\displaystyle\tilde{\cal
E}^{(p)}_{a_{1}\dots a_{p}}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\frac{1}{p!}\sqrt{-g}\;\left(*d\left[e^{-b}\wedge*F\right]\right)_{a_{1}\dots
a_{p}}\;,$ (4.54)

$\nabla_{c}$ denotes the flattened (torsion free) covariant derivative.
Interestingly, the non geometric sector of $O(d,d)$ transforms the vielbein
equations of motion into the Kalb-Ramond equations of motion and vice versa

$\displaystyle\delta_{\beta}\tilde{\cal E}_{ab}=-4\beta_{c(a}\tilde{\cal
B}_{b)}{}^{c}\;,\;\;\;\;\;\;\delta_{\beta}\tilde{\cal
B}_{ab}=\beta_{c[a}\tilde{\cal E}_{b]}{}^{c}\;.$ (4.55)

It coincides exactly with the transformation rules found in [12],
corresponding to the case where all p-form potentials have been truncated. As
discussed in section 3.2 of loc.cit the transformations of the $e.o.m.$ play
an important role on the description of deformations of
$\beta$-transformations leaving invariant the action, but not the Lagrangian.
In addition, a covariant transformation rule at the level of $e.o.m.$
guarantees $\beta$ symmetry transforms solutions into new solutions and so can
be used as a solution generating technique in 10d.

Regarding the equations of motion of the p-forms, we start by noticing that

$\displaystyle\delta_{\beta}[d(e^{-b}\wedge*F)]$ $\displaystyle=$
$\displaystyle
d[\delta_{\beta}(e^{-b})\wedge*F+e^{-b}\wedge\delta_{\beta}(*F)]$ (4.56)
$\displaystyle=$ $\displaystyle
d\left[\vphantom{\frac{1}{2}}\right.(\beta+[b\beta b])\wedge e^{-b}\wedge*F$
(4.58)
$\displaystyle\;\;+\;e^{-b}\wedge\left(-\frac{1}{2}(i_{\beta}b)*F+\frac{1}{2}i_{\beta}(*F)-[b\beta*F]-\beta\wedge*F\right)\left.\vphantom{\frac{1}{2}}\right]$
$\displaystyle=$
$\displaystyle\frac{1}{2}d\left[(i_{\beta}(e^{-b})\wedge*F+e^{-b}\wedge
i_{\beta}(*F)-2e^{-b}\wedge[b\beta*F]\right]$ (4.59) $\displaystyle=$
$\displaystyle\frac{1}{2}d[i_{\beta}(e^{-b}\wedge*F)]\;.$ (4.60)

In the third equality we have used identity (4.14) with $b\to-b$, in the last
line we used instead (4.11). Hence, we obtain

$\displaystyle\delta_{\beta}[d(e^{-b}\wedge*F)]=\frac{1}{2}i_{\beta}[d(e^{-b}\wedge*F)]\;.$
(4.61)

This readily follows from (4.60) and the identity

$\displaystyle
d\left(i_{\beta}A^{(q)}\right)=i_{\beta}\left(dA^{(q)}\right)\;,$ (4.62)

which is valid for any $q$-form $A^{(q)}$ and is a consequence of (1.4) and
the constancy of $\beta^{\mu\nu}$.

With (4.61) at hand we can then proceed by following similar steps than those
of section 4.3 to obtain
$\delta_{\beta}\left[*\left(d\left[e^{-b}\wedge*F\right]\right)\right]$. We
find

$\displaystyle\delta\tilde{\cal E}^{(p)}_{a_{1}\dots a_{p}}=(\tilde{\cal
E}^{(p-2)}\wedge\beta)_{a_{1}\dots a_{p}}-[\beta b\tilde{\cal
E}^{(p)}]_{a_{1}\dots a_{p}}\;,$ (4.63)

a formal expression where it is implicitly assumed that $\tilde{\cal
E}^{(p-2)}$ vanishes for $p<2$. Here $[\beta b\tilde{\cal E}^{(p)}]$ is
defined in analogous fashion as (4.12),

$\displaystyle[\beta b\tilde{\cal
E}^{(p)}]=\frac{1}{(p-1)!}\beta_{a_{1}c}\;b^{cd}\;\tilde{\cal
E}^{(p)}_{da_{2}\dots a_{p}}e^{a_{1}}\wedge\dots\wedge
e^{a_{p}}\;,\;\;\;\;\;\;e^{a}=e_{\mu}{}^{a}dx^{\mu}\;.$ (4.64)

###  4.5 Closure

We close the analysis of consistency of the $\beta$-transformations as an
effective symmetry in 10 dimensions by verifying closure of the algebra. The
local transformations of the NS-NS sector combined with $\beta$ symmetry and
its associated bracket was studied in [11] and [12]. Here we extend it by
considering also the local transformations in the R-R sector

$\displaystyle\delta_{\xi}D^{(p)}_{\mu_{1}\dots\mu_{p}}$ $\displaystyle=$
$\displaystyle\xi^{\rho}\partial_{\rho}D^{(p)}_{\mu_{1}\dots\mu_{p}}+p\partial_{[\mu_{1}}\xi^{\rho}D^{(p)}_{|\rho|\mu_{2}\dots\mu_{p}]}$
(4.65) $\displaystyle\delta_{\Lambda}D^{(p)}_{\mu_{1}\dots\mu_{p}}$
$\displaystyle=$ $\displaystyle 0\;,\;$ (4.66)
$\displaystyle\delta_{\lambda}D^{(p)}_{\mu_{1}\dots\mu_{p}}$ $\displaystyle=$
$\displaystyle
p(p-1)\partial_{[\mu_{1}}\lambda_{\mu_{2}}D^{(p-2)}_{\mu_{3}\dots\mu_{p}]}$
(4.67) $\displaystyle\delta_{\chi}D^{(p)}_{\mu_{1}\dots\mu_{p}}$
$\displaystyle=$ $\displaystyle
p\partial_{[\mu_{1}}\chi^{(p-1)}_{\mu_{2}\dots\mu_{p}]}.$ (4.68)

where $\delta_{\xi}\;,\delta_{\Lambda},\;\delta_{\lambda},\;\delta_{\chi}$
stand for diffeomorphisms, Lorentz, gauge transformations of the Kalb-Ramond
and p-form potentials, respectively. Indeed, one finds that the condition
(1.4) leads to the following bracket

$\displaystyle[\delta_{1},\delta_{2}]=-\delta_{12}\;,$ (4.69)

with

$\displaystyle\Lambda_{12}{}_{ab}$ $\displaystyle=$ $\displaystyle
2\,\beta_{\underline{1}[a}{}^{c}\beta{\vphantom{\beta^{a}}}_{\underline{2}\,b]c}+2\,\Lambda_{\underline{1}[a}{}^{c}\Lambda{\vphantom{\Lambda^{a}}}_{\underline{2}\,b]c}+2\,\xi_{[\underline{1}}^{\mu}\partial_{\mu}\Lambda{\vphantom{\xi^{\mu}}}_{\underline{2}]ab}$
(4.70) $\displaystyle\beta_{12}$ $\displaystyle=$ $\displaystyle 0\;,$ (4.71)
$\displaystyle\xi_{12}^{\mu}$ $\displaystyle=$ $\displaystyle
2\xi^{\nu}_{[\underline{1}}\partial_{\nu}\xi_{\underline{2}]}^{\mu}+\beta_{[\underline{1}}^{\mu\nu}\lambda{\vphantom{\xi^{\mu}}}_{\underline{2}]\nu}\;,$
(4.72) $\displaystyle\lambda_{12\mu}$ $\displaystyle=$ $\displaystyle
4\,\xi_{[\underline{1}}^{\nu}\partial_{[\nu}\lambda_{\underline{2}]\mu]}$
(4.73) $\displaystyle\chi^{(p)}_{12}{}_{\mu_{1}\dots\mu_{p}}$ $\displaystyle=$
$\displaystyle-\beta^{\rho\sigma}_{[\underline{1}}\chi^{(p+2)}_{\underline{2}]\rho\sigma\mu_{1}\dots\mu_{p}}+2p(p-1)\lambda{\vphantom{\chi^{\mu}}}_{[\underline{1}}{}_{\mu_{1}}\partial{\vphantom{\chi^{\mu}}}_{\mu_{2}}\chi^{(p-2)}_{\underline{2}]\;\mu_{3}\dots\mu_{p}]}+2(p+1)\xi_{[\underline{1}}^{\rho}\partial{\vphantom{\chi^{\mu}}}_{[\rho}\chi^{(p)}_{\underline{2}]\;\mu_{1}\dots\mu_{p]}}\;.\;\;\;\;$
(4.74)

This completes the studies of consistency for the $\beta$-symmetry in the
bosonic sector of type II theories.

##  5 Conclusions

In a recent article [11], it was shown that $SO(d,d)$ symmetry of KK
reductions in the NS-NS universal sector of supergravity admits an uplift in
the standard 10d formulation through the inclusion of $\beta$-transformations
(1.3), provided that the condition (1.4) is satisfied. The main advantage of
this approach is that it enables the consideration of constraints on the
effective action due to T-duality within the standard scheme of supergravity,
without the need of performing compactifications, Lorentz non-covariant or
non-geometric field redefinitions or the introduction of additional
coordinates.

This work extends this symmetry to the RR sector of (massive) type IIA and
type IIB. The transformations of the RR-potentials and curvatures are
displayed in equations (4.3) and (4.22), respectively. We explicitly verified
the exact invariance of the Lagrangian (without neglecting any total
derivative terms) in sections 4.1 and 4.2. Additionally, we conducted various
consistency studies in sections 4.3, 4.4 and 4.5.

There are some interesting extensions of this work, both in the NS-NS and the
R-R sectors. Concerning the latter, it remains unclear whether there exists a
consistent definition of $\beta$-symmetry within the standard formulation of
type II theories, as there is no an off-shell mechanism to relate it to the
democratic formulation considered in this paper.

Regarding the NS-NS case, it is worth to investigate higher derivative
deformations. It was argued in [10] that there exists a universal sector of
bosonic, heterotic and type II effective actions at eight order in derivatives
(those proportional to $\zeta(3)$ ) that cannot be embedded in a Double Field
Theory formulation101010At least, by using the standard (fundamental)
representation of $O(d,d)$ and ignoring Chern-Simons like interactions.. The
approach presented here is free from these obstructions, thus it is worthwhile
to explore whether there is a $\beta$-deformation constraining these
couplings. If so, it could shed light on a potential DFT-like deformation,
circumventing the obstructions mentioned in the aforementioned work. First
corrections to ${\cal L}_{NS}$ of type II theories are precisely $\sim{\cal
O}(\alpha^{\prime}{}^{3})$, making it the simplest scenario to study this
problem.

Acknowledgements: We warmly thank C. Núñez and D. Marqués for discussions and
comments on the manuscript. Support by Consejo Nacional de Investigaciones
Científicas y Técnicas (CONICET), through the Grant PIP-11220200100981CO,
Universidad Nacional de La Plata (UNLP) and Universidad de Buenos Aires (UBA)
are also gratefully acknowledged.

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