**institutetext: Charles University Prague, Faculty of Mathematics and Physics
Mathematical Institute, Prague 186 75, Czech Republic****institutetext: Jacobs
University Bremen, Department of Physics and Earth Sciences
Campus Ring 1, 28759 Bremen, Germany

# A gravitational action with stringy $Q$ and $R$ fluxes via deformed
differential graded Poisson algebras

Eugenia Boffo ** Peter Schupp<EMAIL_ADDRESS>p.schupp@jacobs-
university.de

###### Abstract

We study a deformation of a $2$-graded Poisson algebra where the functions of
the phase space variables are complemented by linear functions of parity odd
velocities. The deformation is carried by a $2$-form $B$-field and a bivector
$\Pi$, that we consider as gauge fields of the geometric and non-geometric
fluxes $H$, $f$, $Q$ and $R$ arising in the context of string theory
compactification. The technique used to deform the Poisson brackets is widely
known for the point particle interacting with a $U(1)$ gauge field
Tanimura_1992 , but not in the case of non-abelian or higher spin fields. The
construction is closely related to Generalized Geometry: With an element of
the algebra that squares to zero, the graded symplectic picture is equivalent
to an exact Courant algebroid over the generalized tangent bundle $E\cong
TM\oplus T^{*}M$, and to its higher gauge theory. A particular idempotent
graded canonical transformation is equivalent to the generalized metric.
Focusing on the generalized differential geometry side we construct an action
functional with the Ricci tensor of a connection on covectors, encoding the
dynamics of a gravitational theory for a contravariant metric tensor and $Q$
and $R$ fluxes. We also extract a connection on vector fields and determine a
non-symmetric metric gravity theory involving a metric and $H$-flux.

##  1 Introduction

An interacting theory for a particle or a field can be formulated starting
from the canonical Poisson algebra of functions for the free theory by
deforming this algebra along force-carrying fields. The technique essentially
consists in a local redefinition of phase space coordinates in the direction
of gauge and other fields. We will apply the method to gravitational
interactions and higher gauge fields. The symplectic $2$-form looks quite
different in the new chart. Coupling gauge fields to matter field(s) is
relatively easy and makes the theory covariant. Along these lines, Jackiw
Jackiw:1984rd  first noticed that in a quantum mechanical system unitary
representations of the $\mathfrak{u}(1)$ algebra do not associate when a
magnetic monopole is present. Another pioneering example where this technique
is applied is the $N=2$ superparticle with a background Dai:2008bh , where
however it was found that for spacetime symmetries the field strengths are
trivial and therefore do not propagate. In this article we show how this
problem can be overcome in the framework of differential graded Poisson
algebras. We will consider the degree-$2$ Poisson algebra of a point particle.
Our reference manifold is the (degree-)shifted cotangent space $T^{*}[2]T[1]M$
of the shifted tangent space to a manifold $M$. This is the minimal symplectic
realization of a given graded manifold $T[1]M$; for details on how to
construct the bundle and relations to other graded manifolds we refer to
1410.3346 . We couple the particle to $g-B$, where $g$ is a Riemannian metric
on the base manifold and $B$ a $2$-form, and to $\left(g+B\right)^{-1}\equiv
G^{-1}+\Pi$. The antisymmetric tensors $B$ and $\Pi$ can be regarded as the
gauge fields for some $3$-tensors (with covariant and contravariant indices)
arising in string theory compactification with fluxes. Behind the possibility
to pick up and study a closed non-degenerate $2$-form, path-connected to the
canonical symplectic form $\omega$, there is Moser lemma Moser1965 . More
precisely, the lemma states that two non-degenerate and closed $2$-forms on a
manifold, which can be smoothly deformed into each other, are linked by a
diffeomorphism $\varphi_{t}$, for each real $t\in[0,1]$. The forms are
cohomologous in $H_{\text{dR}}^{2}$ hence they can only differ by an exact
$2$-form $\mathrm{d}A_{t}$. The diffeomorphism is unique up to gauge
transformations of $A_{t}$. With slight adaption, as we shall see, Moser lemma
subsists also when the manifold is graded. $A_{t}$ comprises the gauge fields
(but also “higher” gauge fields in the geometric ladder), which are therefore
incorporated into the Poisson algebra.

Standard exact Courant algebroids are naturally associated with the graded
Poisson algebra on the graded symplectic manifold $T^{*}[2]T[1]M$, under the
condition that a nilpotent differential operator is also present. Algebroids
are the generalization of an algebra, extended over all the points of a
manifold; they intrinsically carry the notion of a fiber bundle
$E\xrightarrow{\pi}M$. The bracket for the Courant algebroid (CA) is generally
taken to be non-skewsymmetric, in order to fulfill Jacobi’s identity, and
there is also a fiberwise symmetric pairing that respects some compatibility
conditions with the bracket. See the article Gualtieri:2007ng  for a good
mathematical explanation of this concept. Furthermore, the bracket accounts
for the current algebra of closed bosonic strings Alekseev:2004np . When the
CA on $E$ is exact, it was shown Severa:2017oew  to be canonically isomorphic
to a CA on the bundle $TM\oplus T^{*}M$. If moreover the metric is the
standard pairing of vector fields with $1$-forms, $TM\oplus T^{*}M$ is seen as
a principal bundle with structure group $O(d,d)$, that takes into account the
stringy T-dualities, which constitute the Weyl group $O(d,d;\mathbb{Z})$. The
defining objects (alongside with their relations) of an exact CA on $TM\oplus
T^{*}M$ can be obtained from the derived bracket of the Poisson brackets, with
the canonically related degree-shifted Hamiltonian vector field Q. The vector
field can be used to define a cochain complex, because for the Hamiltonian
function $\Theta$, $\text{Q}=\\{\cdot,\Theta\\}$, the following master
equation holds:

$\\{\Theta,\Theta\\}=0.$

Introducing additional geometric data on the exact CA we manage to derive a
pair of low energy effective actions for string theory compactification,
involving T-dual stringy fluxes.

T-duality is a symmetry exhibited by the non-linear sigma-model of the
relativistic bosonic string: In the presence of isometry directions, the
sigma-model with background fields $(g,B)$ was shown to be equivalent to the
sigma-model based on another pair of backgrounds. This equivalence is dubbed
T-duality Buscher:1987sk . The famous Buscher rules regulate the
correspondence between the fields; in general, T-dual backgrounds do not share
the isometry symmetries of the original fields. Moreover, the T-duality
symmetry concerns just the effective field theory for the string (it holds at
$\left(\alpha^{\prime}\right)^{0}$ order). In type II string theories,
compactification with fluxes on a $\mathbb{T}_{3}$ complex torus was suggested
as an alternative to Calabi-Yau compactifications without fluxes, since the
additional structure carried by the fluxes can attempt to stabilize the
moduli, by making them massive through a potential that combines nicely the
superpotential and the Kähler potential. The fluxes were also believed to shed
light on the presence of de Sitter solutions, but research in this direction
has not been quite fortunate. In Shelton:2005cf  it was argued that the NS-NS
and R-R fields of IIA and those of IIB must be T-dual to each other, since the
two theories are already their respective T-dual when compactified without
fluxes. Successive T-duality transformations applied to the Neveu-Schwarz
$3$-form $H$ lead to the $f,Q,R$ fluxes according to the chain:

$H_{abc}\xrightarrow{\text{T}^{c}}f_{ab}{}^{c}\xrightarrow{\text{T}^{b}}Q_{a}{}^{bc}\xrightarrow{\text{T}^{a}}R^{abc},$

where for the last arrow one invokes Buscher rules despite the lack of an
isometry direction. This results in a rupture with standard geometry, since
now the local open sets of the base manifold cannot be patched together;
neither with general linear transformations as transition functions, nor with
a more general $O(d,d)$ transformation such as a T-duality (like when there is
a non-trivial $Q$-flux, which presents this type of monodromy). Despite this
unusual feature, the non-geometric flux $R$ is very interesting due to its
rôle as source for non-associativity of the phase space algebra of open
strings ending on D-branes. In more recent work, there are indications that
the geometry of Yang-Mills theory on a D3-brane in the background of a
$R$-flux can also be understood in a fully associative picture Hull:2019iuy .
However conjectures of non-associativity are still viable for closed strings.
For a study of non-geometric backgrounds from the string sigma-model
perspective a good reference is Halmagyi:2009te . Topological T-duality with
$H$ flux is explained in Bouwknegt:2003vb . Other types like non-abelian
T-duality and Poisson-Lie T-duality constitute an active research field. A
manifestly T-duality covariant formalism was developed in Hohm:2010pp  and led
to the formulation of the most general invariant action for the background
fields, dubbed Double Field Theory (DFT). The Supergravity action for the
gravity multiplet $(g,B,\phi)$ is included in the DFT action. From the latter
one can also extract an action functional with the $R$-flux as invariant
tensor and the $Q$-flux as connection coefficient Andriot:2012an . A further
equivalent action named $\beta$-supergravity appeared in Andriot:2013xca . It
was initially obtained from a field redefinition of the NS-NS action
functional limit of the DFT covariant action (with $H$-flux and dilaton) but
eventually it was expressed through the differential-geometric objects of a
Lie algebroid structure on $T^{*}M$ in Blumenhagen:2013aia , where $M$ is a
regular manifold. Besides Double Field Theory, the fluxes can also be analyzed
in the context of Generalized Geometry and algebroids. The bound between
T-dual non-geometric fluxes and the Courant bracket is discussed in
Grana:2008yw  which primarily stresses the importance of Generalized Geometry
for flux compactification. Reference Plauschinn:2018wbo  reviews this
particular aspect too and provides an exhaustive overview of non-geometric
fluxes. In the context of differential graded (dg) symplectic geometry, the
fluxes are usually understood as a twist of the differential, see for example
Heller:2016abk ; subsequently at the level of algebroids the bracket gets
twisted as well. The fluxes can also arise from an appropriate basis choice,
as done for example in Blumenhagen:2012nk . The net result is that the gauge
algebra comprises now the fluxes of the T-duality chain as structure
functions. The Jacobi identity leads to the Bianchi identity for the fluxes.
For a derivation of the string effective action with $H$-flux and dilaton in
the context of Generalized Geometry please refer to Jurco:2015bfs  and
Jurco:2017gii .

We are going to adopt the viewpoint of Blumenhagen:2012nk , but applied to a
deformed $2$-graded Poisson algebra. When switching to the Courant algebroid
picture, the Hamiltonian vector field will remain the untwisted differential
of the free theory. Although normally in dg-symplectic geometry the standard
prescription for the fluxes requires the differential to be twisted by them,
the local expression for $H$, $Q$ and $R$ will anyway pop up in our formulas,
thanks to other new geometrical data introduced at the level on the
generalized tangent bundle in section 4.

The article is organized as follows: we start in section 2 with an overview of
the features of the differential graded symplectic manifold
$(T^{*}[2]T[1]M,\omega,\text{Q})$. A general deformation is proposed and the
outcome is analyzed; the particular example of the deformation that involves
$g,B,G^{-1}$ and $\Pi$ is given at the end of this section. We then proceed in
section 3 by focusing on the structure of Courant algebroids on the
generalized tangent bundle $TM\oplus T^{*}M$, which is equivalent to the
graded Poisson structure previously considered. When the latter is deformed,
the former gets changed in the same way: We hence unveil the expressions for
the new objects in this section. Section 4 is the main body of the article:
There we make full use of the developments from the previous sections, in
particular the new deformed Dorfman bracket. Defining a generalized Lie
bracket alongside with a generalized torsion, naturally leads to an affine
connection with torsion. We compute the generalized Riemann tensor for vectors
and, separately, for $1$-forms. In this case, the Riemann tensor is a section
of $\bigotimes^{3}TM\otimes T^{*}M$, and expresses the curvature of the
Riemannian manifold $M$ with metric $G^{-1}\in\vee^{2}TM$. The connection on
forms, in light of the ansatz chosen for the deformation, depends on the local
expression of the non-geometric $R$ and $Q$, which are connection coefficients
and make up the torsion tensor. In the final part section 5 we construct a
curvature scalar that retains the antisymmetric part in a consistent way; the
resulting invariant action encodes a gravitational theory in the background of
$Q$ and $R$. We end in section 6 with comments, discussion and outlook.

##  2 Graded Poisson structure

Graded geometry and super geometry are popular in physics, primarily because
they are convenient for dealing with manifest supersymmetry. The significance
of the supersymmetry algebra was in particular noticed in Haag:1974qh  because
it gives a chance to circumvent the Coleman-Mandula theorem Coleman:1967ad , a
powerful no-go theorem that severely constrains a relativistic quantum field
theory to have only internal symmetries besides Poincaré symmetry. When a
nilpotent differential operator is also part of the geometrical data, there
are immediate applications in the context of BRST and BV-BFV quantization of
theories, whose action is degenerate because of the gauge redundancies, and in
the context of AKSZ sigma models. There has also been a recent proposal for a
formulation of T-duality in this framework, see Deser:2019pue . This section
aims at presenting the canonical Poisson structure of a specific graded
manifold, together with a naturally associated differential operator. We will
introduce the required underlying theory while trying to keep it accessible to
physicists with expertise in theoretical gravitational physics. By the end of
this section, we will propose a very general deformed graded Poisson algebra,
that can ultimately accommodate the geometrical data needed for a
gravitational action with stringy background fields.

###  2.1 Canonical ($T^{*}[2]T[1]M,\omega,\text{Q}$)

Let us consider the graded $4d$-dimensional manifold given by the shifted
cotangent of the shifted tangent space to a $d$-dimensional manifold $M$,
$T^{*}[2]T[1]M\equiv\mathcal{M}$. The integer numbers in the square brackets
refer to the degree of the coordinates of the corresponding space. According
to Batchelor’s theorem, the structure sheaf111Denoted as
$\mathcal{O}_{\mathcal{M}}$, it can be thought as the algebra of functions on
the graded manifold. is isomorphic to the sheaf of sections of the exterior
algebra of $E$, which is a vector bundle over $M$. The graded manifold
$\mathcal{M}$ admits a natural symplectic structure and thus a Darboux chart.
The coordinates of the Darboux chart in a small neighborhood of a point $q\in
M$, corresponding to a chart $\\{x^{i}\\}$ on $M$, are degree $1$ $(\chi_{i})$
in the fiber of $T[1]M$, degree $1$ $(\theta^{i})$ for the fiber of
$T^{*}[1]M$ and degree $2$ $(p_{i})$ in $T^{*}[2]M$. The degree goes along
with the intrinsic parity of the structure sheaf in the following sense:
coordinates of even degree commute with themselves, while those of odd degree
anticommute. Denoting with $\upsilon(x),U(x)$ and $V(x)$ linear functions of
$p,\chi,\theta$ respectively, and $f(x)\in C^{\infty}(M)$, the non-trivial
Poisson brackets are given by:

$\begin{matrix}[l]\\{p_{j},x^{i}\\}=\delta_{j}{}^{i},&\,&\\{\upsilon(x),f(x)\\}=\upsilon^{i}(x)\partial_{i}f(x),\\\
\\{\chi_{i},\theta^{j}\\}=\delta_{i}{}^{j}=\\{\theta^{j},\chi_{i}\\},&\,&\\{U(x),V(x)\\}=\eta(U(x),V(x)),\end{matrix}$
(1)

where $\eta:=\delta_{i}{}^{j}\mathrm{d}\theta^{i}\wedge\mathrm{d}\chi_{j}$.
Canonical transformations for this graded phase space represented by
degree-$2$ maps preserve the total degree. Canonical transformations in which
the degree is higher or lower than $2$ modify the grading. The most general
degree 2 generators are

$h=\upsilon^{i}(x)p_{i}+\frac{1}{2}N^{\alpha\beta}(x)\xi_{\alpha}\xi_{\beta},\quad
N^{\alpha\beta}(x)\in\Gamma(\Lambda^{2}\left(T^{*}[1]M\oplus T[1]M\right)).$
(2)

$h$ acts on functions of the graded Poisson space via Poisson brackets to
produce infinitesimal canonical transformations. The first term yields
infinitesimal diffeomorphisms of $M$ while the second term induces $O(d,d)$
rotations on the elements of the vector space $T^{*}[1]M\oplus T[1]M$. A
general excursion on infinitesimal canonical transformations is postponed to
section 2.3.

_Remark_ : Focusing on the space of the degree-$1$ coordinates, a degree-
preserving canonical transformation introduces what is commonly known in
Generalized Geometry as the generalized metric $\mathcal{H}$ on
$T^{*}[1]M\oplus T[1]M$. As a symmetric tensor on the fibers, in every
coordinate patch it depends on a Riemannian metric $g$ on the base and on a
$2$-form $B\in\Lambda^{2}T^{*}M\equiv\Omega^{2}(M)$. Globally the generalized
metric characterizes the bundle with the pair $(g,H)$ for $g$ as before and
$H$ a closed $3$-form. In the realm of Generalized Geometry, $\mathcal{H}$ is
defined from $\eta$ and an involution $\tau\in\text{End}(T^{*}[1]M\oplus
T[1]M$), $\tau^{2}=\text{id}$. In the graded symplectic setting the generating
function (of type 1) for the canonical transformation
$F(\tilde{\theta},\theta)$ must be antisymmetric,
$F(\tilde{\theta},\theta)=-F(\theta,\tilde{\theta})$ in order to define an
involution. One can easily verify that

$F(\tilde{\theta},\theta)=\theta
g(x)\tilde{\theta}+\frac{1}{2}\tilde{\theta}B(x)\tilde{\theta}-\frac{1}{2}\theta
B(x)\theta,\quad g(x)\in\text{Sym}(T^{*}M\otimes T^{*}M),B(x)\in\Omega^{2}(M)$

is the most general degree $2$ function with the required properties. In fact
given the expression for the conjugate new coordinates, denoted with a tilde
on top, and the old coordinates as functions of the known coordinates:

$\chi=\dfrac{\partial F}{\partial\theta},\quad\tilde{\chi}=-\dfrac{\partial
F}{\partial\tilde{\theta}},$

we can derive the following formula:

$\begin{pmatrix}\tilde{\chi},&\tilde{\theta}\end{pmatrix}=\begin{pmatrix}\chi,&\theta\end{pmatrix}\begin{pmatrix}g^{-1}B&&g^{-1}\\\
g-Bg^{-1}B&&-Bg^{-1}\end{pmatrix}=\begin{pmatrix}\chi,&\theta\end{pmatrix}\mathcal{H},$

and the canonical Poisson brackets
$\\{\tilde{\theta}^{j},\tilde{\chi}_{i}\\}=\delta^{j}{}_{i}$ while
$\\{\tilde{\theta},\tilde{\theta}\\}=0=\\{\tilde{\chi},\tilde{\chi}\\}$.

Our graded Poisson structure can be equipped with a further object: a
homological vector field Q compatible with the above Poisson structure,
$\mathcal{L}_{\text{Q}}\omega=0$. Then $(T^{*}[2]T[1]M,\omega,\text{Q})$
becomes what is technically called a QP-manifold. Contrary to the ungraded
case, in graded symplectic geometry every such vector field is Hamiltonian,
under the mild condition that the degree of Q and $\omega$ do not sum to zero.
If Q has degree $1$, then by degree counting it is determined by a degree $3$
Hamiltonian $\Theta$ as

$\text{Q}=\\{\cdot,\Theta\\}.$ (3)

The most general instance of $\Theta$ is

$\Theta=\xi_{\alpha}\tilde{\rho}^{\alpha
i}(x)p_{i}+\frac{1}{3!}C^{\alpha\beta\gamma}(x)\xi_{\alpha}\xi_{\beta}\xi_{\gamma}\,.$
(4)

In the expression for the Hamiltonian we use the shorthand notation
$\xi_{\alpha}:=\left(\chi_{i},\theta^{i}\right)$, where $\alpha$, and all
Greek indices from now on, are understood to take values from $1$ to $2d$;
$C^{\alpha\beta\gamma}(x)$ is a section of the tensor space
$\Lambda^{3}\left(T[1]M\oplus T^{*}[1]M\right)$. Then $\tilde{\rho}$ is a
smooth map from the dual of $T[1]M\oplus T^{*}[1]M$, taken with respect to the
bilinear and symmetric form there, to $T^{*}[2]M$.

$\tilde{\rho}^{\alpha i}:=\left(\rho\circ\eta\right)^{\alpha
i},\quad\text{for}\;\rho:T^{*}[1]M\oplus T[1]M\mapsto T^{*}[2]M.$

The homological vector field associated with (4) can be computed explicitly
with the Poisson brackets (1):

$\text{Q}=\xi_{\alpha}\partial_{k}\tilde{\rho}^{\alpha
i}(x)p_{i}\dfrac{\partial}{\partial p_{k}}-\xi_{\alpha}\tilde{\rho}^{\alpha
i}(x)\dfrac{\partial}{\partial x^{i}}+\tilde{\rho}^{\alpha
i}(x)p_{i}\dfrac{\partial}{\partial\tilde{\xi}^{\alpha}}+\frac{1}{2}C^{\alpha\beta\gamma}(x)\xi_{\beta}\xi_{\gamma}\dfrac{\partial}{\partial\tilde{\xi}^{\alpha}}+\frac{1}{6}\partial_{k}C^{\alpha\beta\gamma}(x)\xi_{\alpha}\xi_{\beta}\xi_{\gamma}\dfrac{\partial}{\partial
p_{k}},$

where $\tilde{\xi}^{\alpha}=\left(\theta^{i},\chi_{i}\right)$ is the dual
coordinate to $\xi_{\alpha}=\left(\chi_{i},\theta^{i}\right)$. The dual is
taken with the help of the metric $\eta$.

The (co)homological condition $\text{Q}^{2}=0$ can be rearranged using the
definition (3) and the graded Jacobi identity into:

$\\{\Theta,\Theta\\}=0,$ (5)

known as the classical master equation, which is highly non-trivial because of
the even degree of the expression. This differential equation imposes
restrictions on the possible functions $\tilde{\rho}(x)$ and $C(x)$ in the
Hamiltonian for a given graded symplectic manifold. It is known Heller:2016abk
that (5) gives rise to a system of three algebraic and differential equations:
one constrains the image of $\tilde{\rho}$, a subset of $T^{*}[1]M\oplus
T[1]M$, to be the kernel of $\rho$. Another one involves the tensors
collectively incorporated in $C(x)$, which can be unpacked in
$H_{ijk}\theta^{i}\theta^{j}\theta^{k}$,
$f^{i}{}_{jk}\chi_{i}\theta^{j}\theta^{k}$,
$Q^{ij}{}_{k}\chi_{i}\chi_{j}\theta^{k}$ and
$R^{ijk}\chi_{i}\chi_{j}\chi_{k}$, in analogy to the stringy fluxes. The
equation is hence untied in the Bianchi identities for $H$, $f$, $Q$ and $R$,
which were firstly derived in the context of Generalized Geometry and
algebroids in reference Blumenhagen:2012pc , and were originally obtained by
applying a T-duality on the Bianchi identity for $H$ in the seminal paper
Shelton:2005cf . The fluxes are locally given in terms of a $2$-form $B$ and a
non-Poisson bivector $\Pi$, their most general coordinate expressions in a
flat background being:

$\begin{matrix}[l]H_{ijk}=\partial_{[i}B_{jk]}\\\
f^{i}{}_{jk}=[e_{j},e_{k}]^{i}+\Pi^{im}H_{mjk}\\\
Q^{ij}{}_{k}=\partial_{k}\Pi^{ij}+\Pi^{il}\Pi^{jm}H_{lmk}\\\
R^{ijk}=3\Pi^{[i|m}\partial_{m}\Pi^{|jk]}+\Pi^{il}\Pi^{jm}\Pi^{kn}H_{lmn}.\end{matrix}$
(6)

In the second row, $f$ is based on the Lie bracket of gauge fields
$e=e_{j}{}^{m}(x)\partial_{m}$. For our purposes we will resort to a less
general definition in due course, namely $f$, $Q$ and $R$ will not depend on
the $H$ flux.

The remaining equation retrieved from (5) shows that, depending on $\rho$,
some of the fluxes are identically zero. The canonical choice for $\Theta$ is
met with the functions

$\rho(\chi_{j})=p_{j},\quad\rho(\theta^{j})=0,\quad\quad
C^{\alpha\beta\gamma}\xi_{\alpha}\xi_{\beta}\xi_{\gamma}=H_{ijk}\theta^{i}\theta^{j}\theta^{k}.$
(7)

###  2.2 Non-canonical $T^{*}[2]T[1]M$ via deformation

Now we want to consider a different closed non-degenerate $2$-form
$\omega^{\prime}$ on the same graded manifold. In the case of a regular non-
graded symplectic manifold, Moser lemma Moser1965  guides us in the choice.
For a given $\omega$, it allows us to take a whole $1$-parameter family of
symplectic forms $\omega_{t}$ such that $\omega_{t}=\omega+F_{t}$, where
$F_{0}=0$. Moser lemma then states that there exists a $1$-parameter family of
diffeomorphisms $\varphi_{t}$ relating $\omega_{t}$ to $\omega$ in the usual
way:

$\varphi_{t}^{*}\omega_{t}=\omega.$ (8)

The proof goes as follow. Although the isotopy
$\varphi:\mathcal{M}\times[0,1]\mapsto\mathcal{M}$ could cease to be a flow
because the composition law ($\varphi_{t}\circ\varphi_{s}=\varphi_{t+s}$)
might not hold, it can still be regarded as a flow generated by a vector field
$X_{t}$. Differentiating (8) with respect to $t$, the candidate generating
vector $X_{t}$ must satisfy the differential equation:

$\iota_{X_{t}}\omega_{t}=-\dfrac{\mathrm{d}}{\mathrm{d}t}A_{t},$

where $A_{t}$ is a local gauge field satisfying $F_{t}=\mathrm{d}A_{t}$. The
above expression can be solved for $X_{t}$, since the symplectic form is non-
degenerate. Moser lemma generalizes to the case of graded symplectic
manifolds. Diffeomorphisms will be vector bundles isomorphisms covering
morphism of the base manifold.

Note that the result of the lemma holds globally. Despite this, in this
article the deformation will be characterized locally, as we are considering a
patch for the neighborhood $U_{\text{q}}$ of a point q $\in M$, in which the
symplectic form $\omega$ is written in Darboux coordinates. In due course we
will point out that our statements are global.

Given the above premises, let us deform $\omega$ by means of a linear,
differentiable and invertible local map

$\mathcal{E}(x):T^{*}[1]U_{\text{q}}\oplus T[1]U_{\text{q}}\rightarrow
T^{*}[1]U_{\text{q}}\oplus T[1]U_{\text{q}}.$ (9)

Then in the same neighborhood $U_{\text{q}}$ the new symplectic form
$\omega^{\prime}\equiv\omega_{1}$ coincides with the canonical form when
pulled back by $\mathcal{E}(x)$ (9):

$\displaystyle\omega^{\prime}$
$\displaystyle=\mathrm{d}x^{i}\wedge\mathrm{d}p_{i}+\mathrm{d}\left(\xi_{\gamma}\left(\mathcal{E}(x)^{-1}\right)^{\gamma}{}_{\alpha}\right)\eta^{\alpha\beta}\wedge\mathrm{d}\left(\xi_{\delta}\left(\mathcal{E}(x)^{-1}\right)^{\delta}{}_{\beta}\right),$
$\displaystyle=\mathrm{d}x^{i}\wedge\mathrm{d}p_{i}+\mathrm{d}\xi_{\gamma}\mathcal{G}^{\gamma\delta}(x)\wedge\mathrm{d}\xi_{\delta}-\mathrm{d}x^{i}\left[\xi_{\gamma}\mathit{\Gamma}_{i\alpha}{}^{\gamma}\right]\mathcal{G}^{\alpha\delta}\wedge\mathrm{d}\xi_{\delta}+\mathrm{d}x^{k}\left[\xi_{\gamma}\mathit{\Gamma}_{k\mu}{}^{\gamma}\mathcal{G}^{\mu\nu}\xi_{\delta}\mathit{\Gamma}_{l\nu}{}^{\delta}\right]\wedge\mathrm{d}x^{l}.$
(10)

We define
$\mathcal{G}^{\gamma\delta}\equiv\left(\mathcal{G}^{-1}\right)^{\gamma\delta}$
to be the inverse of a pseudo-Riemannian metric:

$\mathcal{G}^{-1}=\mathcal{E}^{-1}\eta\mathcal{E}^{-T}.$ (11)

Furthermore $\mathcal{E}$ (9) is a local section for the frame bundle, where
the local frames over q are $O(d,d)$ symmetric. Equivalently, $\mathcal{E}$ is
a family of $1$-forms with values in $\mathfrak{o}(d,d)$. We also define
$\mathit{\Gamma}_{i\beta}{}^{\alpha}$ to be

$\mathit{\Gamma}_{i\beta}{}^{\alpha}:=\left(\mathcal{E}^{-1}\right)_{\beta}{}^{\delta}\partial_{i}\mathcal{E}_{\delta}{}^{\alpha}.$
(12)

In the notation of (1), the local deformed graded Poisson brackets are now:

$\begin{matrix}[l]\\{p_{i},x^{j}\\}=\delta_{i}{}^{j},&\,&\\{\upsilon(x),f(x)\\}=\upsilon(x)^{i}\partial_{i}f(x),\\\
\\{\xi_{\alpha},\xi_{\beta}\\}=\mathcal{G}_{\alpha\beta},&\,&\\{U(x),V(x)\\}=\mathcal{G}(U,V),\\\
\\{p_{i},\xi_{\alpha}\\}=\mathit{\Gamma}_{i\alpha}{}^{\beta}\xi_{\beta},&\,&\\{\upsilon(x),U(x)\\}=\nabla_{\upsilon}U(x),\\\
\\{p_{i},p_{j}\\}=\text{R}^{\alpha}{}_{\beta
ij}\xi_{\alpha}\mathcal{G}^{\beta\gamma}\xi_{\gamma},&\,&\\{\upsilon(x),\kappa(x)\\}=[\upsilon(x),\kappa(x)]_{\text{Lie}}+\text{R}(\upsilon(x),\kappa(x)),\end{matrix}$
(13)

where $\text{R}^{\alpha}{}_{\beta ij}$ is the curvature tensor of $\nabla$:

$\text{R}^{\alpha}{}_{\beta
ij}=\partial_{[i}\mathit{\Gamma}_{j]\beta}{}^{\alpha}+\mathit{\Gamma}_{[i|\delta}{}^{\alpha}\mathit{\Gamma}_{|j]\beta}{}^{\delta}.$
(14)

Metric, connection and curvature are $x$-dependent. However with the
connection coefficients in (12) the curvature tensor is identically zero:

$\text{R}^{\alpha}{}_{\beta
ji}=\partial_{[j|}\mathcal{E}^{\alpha}{}_{\delta}\partial_{|i]}\left(\mathcal{E}^{-1}\right)^{\delta}{}_{\beta}+\mathcal{E}^{\alpha}{}_{\delta}\partial_{[j|}\left(\mathcal{E}^{-1}\right)^{\delta}{}_{\gamma}\,\mathcal{E}^{\gamma}{}_{\mu}\partial_{|i]}\left(\mathcal{E}^{-1}\right)^{\mu}{}_{\beta}=0.$

This aspect is due to the particular choice of the diffeomorphism, which is a
redefinition of the degree-$1$ coordinates leaving the base invariant. One
could have sought more general types of transformations, yielding additional
$\mathrm{d}x\wedge\mathrm{d}x$ $2$-forms which by definition have the right
properties of a curvature tensor (see (17)). The current symplectic form in
the chart discussed here is enough for our purposes. For the rest of the
subsection $\text{R}^{\alpha}{}_{\beta ij}$ will nevertheless be kept explicit
to stress the features of the construction.

Some considerations are in order: the connection is metric w.r.t.
$\mathcal{G}$. This feature is implied by Jacobi identity of the Poisson
bracket, for a pair of linear functions in $\chi,\theta$ and one linear
function of $p$.

$\displaystyle\\{\upsilon(x),\\{U(x),V(x)\\}\\}$
$\displaystyle=\\{\\{\upsilon(x),U(x)\\},V(x)\\}+\\{U(x),\\{\upsilon(x),V(x)\\}\\},$
$\displaystyle\Rightarrow v(x).\mathcal{G}(U,W)$
$\displaystyle=\mathcal{G}(\nabla_{\upsilon(x)}U,W)+\mathcal{G}(U,\nabla_{\upsilon(x)}W).$
(15)

The dot refers to the action of the derivation $\upsilon(x)$ on the
$C^{\infty}(M)$-element that follows. (15) is the condition for the metric to
be covariantly constant, i.e. $\nabla_{\upsilon(x)}\mathcal{G}=0$.

The differential Bianchi identity (or second Bianchi identity) for the
curvature is itself an outcome of Jacobi identity. Let us call $\upsilon(x)$,
$\kappa(x)$ and $\varepsilon(x)$ a triplet of linear functions in the momenta:

$\displaystyle 0$
$\displaystyle=\left\\{\upsilon(x),\left\\{\kappa(x),\varepsilon(x)\right\\}\right\\}-\left\\{\left\\{\upsilon(x),\kappa(x)\right\\},\varepsilon(x)\right\\}-\left\\{\kappa(x),\left\\{\upsilon(x),\varepsilon(x)\right\\}\right\\},$
$\displaystyle\Rightarrow$
$\displaystyle[\upsilon,[\kappa,\varepsilon]]-[[\upsilon,\kappa],\varepsilon]-[\kappa,[\upsilon,\varepsilon]]=0,\;\;\;\;\;\;\;\text{Jacobi
identity for the Lie bracket}$
$\displaystyle\nabla_{\upsilon}\text{R}^{\alpha}{}_{\beta}(\kappa,\varepsilon)+\nabla_{\varepsilon}\text{R}^{\alpha}{}_{\beta}(\upsilon,\kappa)+\nabla_{\kappa}\text{R}^{\alpha}{}_{\beta}(\upsilon,\varepsilon)+\text{R}^{\alpha}{}_{\beta}(\upsilon,[\kappa,\varepsilon])-\text{R}^{\alpha}{}_{\beta}([\upsilon,\kappa],\varepsilon)-\text{R}^{\alpha}{}_{\beta}(\kappa,[\upsilon,\varepsilon])=0.$
(16)

In the last row the second Bianchi identity is reproduced in the form

$\left(\nabla_{\upsilon}\text{R}\right)(\kappa,\epsilon)+\text{R}(\text{T}(\upsilon,\kappa),\epsilon)+\text{cyclic}=0.\quad\text{T}(\upsilon,\kappa):=\nabla_{\upsilon}\kappa-\nabla_{\kappa}\upsilon-[\upsilon,\kappa].$

Degree counting constrains the possible combinations of functions in the
Poisson brackets. The only possibility we are left to check is
$\\{V,\\{\upsilon,\kappa\\}\\}$. The Jacobi identity is verified if R is bound
to be the curvature tensor.

$\displaystyle\\{V,\\{\upsilon,\kappa\\}\\}-\\{\\{V,\upsilon\\},\kappa\\}-\\{\upsilon,\\{V,\kappa\\}\\}=0,$
$\displaystyle\Rightarrow\nabla_{\upsilon}\nabla_{\kappa}U-\nabla_{\kappa}\nabla_{\upsilon}U-\nabla_{[\upsilon,\kappa]}U-\text{R}(\upsilon,\kappa)U=0.$
(17)

A compatible differentiable structure (Q-structure) on
$(T^{*}[2]T[1]M,\omega^{\prime})$ is still formally given by the same
Hamiltonian function and vector field of the canonical case. To verify the
classical master equation with the new Poisson structure (13), the maps
$\tilde{\rho}$ and $C$ shall subsequently respect different conditions than in
the previous case. Let us uncover a bit more what this means in the completely
general setup, when $\text{R}^{\alpha}{}_{\beta ij}$ is not set to its null
value:

$\begin{matrix}[l]\tilde{\rho}^{\gamma
j}\mathcal{G}_{\gamma\alpha}\tilde{\rho}^{\alpha i}=0,\\\ \tilde{\rho}^{\gamma
i}\mathcal{G}_{\gamma\sigma}C^{\sigma\alpha\beta}+2\tilde{\rho}^{\beta
j}\left(\mathit{\Gamma}_{j\gamma}{}^{\alpha}\tilde{\rho}^{\gamma
i}+\partial_{j}\tilde{\rho}^{\alpha i}\right)=0,\\\
\dfrac{1}{3}\tilde{\rho}^{\alpha
i}\left(\partial_{i}C^{\beta\gamma\delta}+3\mathit{\Gamma}_{i\sigma}{}^{\beta}C^{\sigma\gamma\delta}\right)+\dfrac{1}{4}C^{\alpha\beta\sigma}\mathcal{G}_{\sigma\mu}C^{\mu\gamma\delta}&=-\text{R}^{\gamma\delta}{}_{ij}\tilde{\rho}^{\alpha
i}\tilde{\rho}^{\beta j}.\end{matrix}$ (18)

All the non-contracted Greek indices are understood to be antisymmetrized. The
first algebraic equation, if $\mathcal{G}$ is locally diagonalized by some
vielbeins $\mathcal{F}$, is a rewriting of the claim that the image of
$\mathcal{F}^{*}\circ\tilde{\rho}^{*}$, a subset of $T^{*}[1]M\oplus T[1]M$,
is the kernel of $\tilde{\rho}\circ\mathcal{F}$, i.e.
$\eta(\mathcal{F}^{*}\circ\tilde{\rho}^{*},\mathcal{F}^{*}\circ\tilde{\rho}^{*})=0$;
as we will see in the next section, this appoints the underlying Courant
algebroid to be exact (see around (32)). The other equations are covariant
versions of what discussed below (5). The second row tells us how the given
metric and its connection $\mathit{\Gamma}$ must be related to the tensors $C$
and the linear map $\tilde{\rho}$. Then the third line displays the
generalization of the Bianchi identities for the fluxes $C$: usually in
differential form they will be written with a twisted de Rham differential,
but here they are expressed with the covariant derivative and the inner
product with $C$ taken with the $G$ metric. The non-zero curvature R of the
connection $\nabla$ obstructs the cochain complex.

In the present situation, the curvature is locally zero, and the setup is
comparable to having Riemann normal coordinates in standard differential
geometry. Thus the curved metric can be “off-diagonalized” into $\eta$, by
means of the frame fields $\mathcal{F}$, and it is possible to seek as metric
connection an antisymmetric connection (i.e. with non-zero torsion). If
$\mathcal{G}$ and $\mathcal{F}$ are fixed in the first algebraic equation, its
meaning is thus clear, and it can be explicitly solved for $\tilde{\rho}$ with
the aforementioned properties. Then an available solution to the third
equation can always be found in $C\equiv 0$222Focusing on a Hamiltonian (4)
that consists only of the first summand leads to the same conclusion: the
third equation in (18) is $0\equiv 0$ and in the second equation the first
term drops out.. Then the second differential equation in (18) reduces to
$\nabla^{\alpha}\tilde{\rho}^{\beta i}=0$ and determines $\mathit{\Gamma}$ in
terms of derivatives of the vielbein and of $\tilde{\rho}$, and viceversa. For
(12) this equation is immediately true.

Notice that we can anyway consider generalized closed $3$-forms built from the
local antisymmetric $2$-tensor gauge fields. These gauge fields are
respectively the Kalb-Ramond field $B$, arbitrary and non-constant, and for
the other tensors in the T- duality chain the bivector $\Pi$. They can
actually be incorporated through the deformation (in the vielbein and
ultimately the connection). Instead if we were deforming the Poisson bracket
in other ways which yielded non-zero curvature in the Poisson bracket of two
functions of $p$, the components of $C$ could not be set all simultaneously to
zero. Focusing on $B$ and $\Pi$ and their field strengths alone would not be
possible, as cohomologically non-trivial $H$, $f$, $Q$ or $R$ can be present
as well.

In the case where the local differentiable and invertible map that pulls back
$\omega^{\prime}$ into $\omega$, expressed in the Darboux chart, is known (it
can be a vielbein $\mathcal{E}(x)$ like here, but also something else), an
effortless expedient to obtain the solutions $\tilde{\rho}$ and $C$ to the
classical master equation is at hand. It consists in exploiting the local map
to pull back or push forward an already known solution for the canonical case.
Hence in our case the following transformations applied to $\tilde{\rho}$ and
$C$ give consistent tensors that can be used in the Hamiltonian $\Theta$:

$\tilde{\rho}^{\alpha
i}\mapsto\left(\mathcal{E}^{-1}\right)^{\alpha}{}_{\beta}\tilde{\rho}^{\beta
i}=:\underline{\tilde{\rho}}^{\alpha i},\quad
C^{\alpha\beta\gamma}\mapsto\left(\Lambda^{3}\mathcal{E}^{*}C\right)^{\alpha\beta\gamma}=:\underline{C}^{\alpha\beta\gamma}.$
(19)

We will hence use the canonical maps in (7) and transform them by means of the
vielbein map $\mathcal{E}$. We will omit the subscript from now on, as this
shall not cause confusion. Moreover we will take $C=0$ as already announced.

###  2.3 Gauge symmetries

In this part we pause from the main exposition to spend a few words on the
symmetries of our model, expanding the discussion around formula (2). A graded
Poisson algebra of any degree and a corresponding Hamiltonian can be invariant
under some transformations with infinitesimal generators $\delta_{\alpha}$,
acting on the coordinates via Poisson brackets. Since $\delta_{\alpha}$ is a
graded derivation for the Poisson algebra, the infinitesimal transformations
it induces are canonical: the Poisson brackets remain invariant especially
because their associativity property is preserved. A first distinction between
these transformations can be made following the degree: we can distinguish
between degree-preserving transformations and degree changing ones. In the
case under investigation here of an odd Hamiltonian $\Theta$ of degree $3$,
and even Poisson brackets $|\\{,\\}|=-2$, the transformed Hamiltonian
$\Theta+\delta_{\alpha}\Theta$ will also solve the classical master equation
to first order in $\delta_{\alpha}$ (higher orders are neglected because the
transformation is infinitesimal).

Infinitesimal canonical transformations of a Poisson algebra of degree $2$
were classified by Roytenberg in Roytenberg:2002nu  for the degree-preserving
case. He found that they generate diffeomorphisms and
$O(d,d)$-transformations, (2). In particular $B\in\Lambda^{2}T[1]M$ and
$\Pi\in\Lambda^{2}T^{*}[1]M$ belong to the Lie algebra $\mathfrak{o}(d,d)$.

When a Q-structure is present, one may wonder whether the algebraic objects
obtained via derived brackets can share the same symmetry with the original
Poisson algebra. This boils down to ask if the Hamiltonian can be invariant
under the symmetry, $\delta_{\alpha}\Theta=0$, or equivalently if $\alpha$ can
be closed with respect to the differential Q, $\text{Q}\alpha=0$. This implies
that in a locally contractible open set the function $\alpha$ must be Q-exact
and hence given by

$\alpha=\\{\varrho,\Theta\\},\quad\quad|\varrho|+|\Theta|+|\\{\cdot,\cdot\\}|=|\alpha|,$

since then
$\delta_{\alpha}\Theta=\\{\alpha,\Theta\\}=\\{\\{\varrho,\Theta\\},\Theta\\}=\dfrac{1}{2}\\{\varrho,\\{\Theta,\Theta\\}\\}=0$,
irregardless of the even or odd degree of the functions involved. For the sake
of our purposes, we are interested in a degree-$2$ function $\alpha$ of the
structure sheaf of the degree $2$ dg-symplectic manifold $T^{*}[2]T[1]M$.
Therefore we want to let $\Theta$ act on a degree $1$ function,
$\varrho=\varrho^{\mu}(x)\xi_{\mu}$. If the Hamiltonian is given by

$\Theta=\xi_{\alpha}\tilde{\rho}^{\alpha i}p_{i},$

and using the deformed Poisson brackets (13) we obtain:

$\displaystyle\\{\varrho,\Theta\\}=\varrho^{\mu}(x)\mathcal{G}_{\mu\alpha}\tilde{\rho}^{\alpha
i}p_{i}+\xi_{\alpha}\tilde{\rho}^{\alpha
i}\nabla_{i}\left(\varrho^{\mu}(x)\xi_{\mu}\right).$

The first term generates “Q-exact” diffeomorphisms, while the second is
responsible of “Q-exact” $O(d,d)$ transformations. Anyway, we would like to
restrict our considerations to the latter ones. In this way we are
representing the gauge symmetries of the endomorphisms of $T^{*}[1]M\oplus
T[1]M$, of $B\in\Lambda^{2}T[1]M$ and $\Pi\in\Lambda^{2}T^{*}[1]M$ in the
graded geometrical setting. We impose that only degree $2$ functions quadratic
in the $\xi$ can appear: this occurs when $\varrho$ is mapped to the null
vector field by the anchor:

$\rho(\varrho)=0.$ (20)

In conclusion, for future reference we are going to need this particular
$\alpha$:

$\alpha^{\beta\gamma}=\tilde{\rho}^{\beta i}\nabla_{i}\varrho^{\gamma}(x),$
(21)

with $\varrho$ subjected to the above (20). In particular, in the next
sections we are going to check invariance under the gauge symmetries of the
$\Pi$ field.

###  2.4 Deformation and open-closed string relation

For our purposes we will assign to the local invertible differentiable map for
the degree $1$ coordinates $\mathcal{E}(x)$, the block matrix expression333The
inverse of the local vielbein $\mathcal{E}(x)$ (23) will also be essential:
$\mathcal{E}^{-1}=\begin{pmatrix}\frac{g}{2}^{-1}\left(g-B\right)&-\frac{g}{2}^{-1}\\\
+\frac{G}{2}&\left(g-B\right)\frac{g}{2}^{-1}\end{pmatrix}.$ (22) :

$\mathcal{E}=\begin{pmatrix}\mathbb{1}&\left(g-B\right)^{-1}\\\
-\left(g+B\right)&\mathbb{1}\end{pmatrix}.$ (23)

The metric $g$ is Riemannian, while the $2$-form $B$, seen as a smooth linear
map $B:\Gamma(TM)\mapsto\Gamma(T^{*}M)$, is invertible. In the upper right
$d$-dimensional block we invoke the open-closed string relation

$\left(g-B\right)^{-1}=:G^{-1}-\Pi,$ (24)

where $G^{-1}\in\vee^{2}T^{*}[1]M$ and $\Pi\in\Lambda^{2}T^{*}[1]M$ are non-
degenerate tensors that can be separately written in terms of $g$ and $B$ as:

$G^{-1}:=\left(g+B\right)^{-1}g\left(g-B\right)^{-1},\quad\Pi:=-\left(g+B\right)^{-1}B\left(g-B\right)^{-1}.$
(25)

In particular $G^{-1}$ is the inverse of a Riemannian metric. Equation (24) is
the closed-open string relation Seiberg:1999vs  (in units where
$2\pi\alpha^{\prime}\equiv 1$): in that reference it was obtained comparing
the open string propagator at the boundary of the cylinder, with the closed
string propagator extrapolated for points in the interior of the cylinder. $G$
is the metric as seen by the open string, opposed to the metric $g$ seen by
the closed string. In this context we do not need a deeper understanding of
closed and open string theory but the relation between the quadruplet of
tensors ($g,B,G,\Pi$) stated by (24). In our related work Boffo:2020vqx  we
analyzed a similar generalized vielbein, with the upper right block put to
zero. The vielbein used here returns a class of block diagonal metrics, i.e.

$\mathcal{G}=\begin{pmatrix}-2g&0\\\ 0&2G^{-1}\end{pmatrix}.$ (26)

The group $O(d)\times O(d)$ admits an action on $\mathcal{G}$; in the
canonical setting $\eta$ is invariant under the larger group $O(d,d)$. Notice
that, due to the opposite sign, $\mathcal{G}$ is Lorentzian of signature
$(-d,d)$, as the $O(d,d)$-invariant pairing is.

Subsequently the symbols for the connection are as well functions of the
fields $g,B,G^{-1}$ and $\Pi$, and they satisfy (15). $\mathcal{E}^{-1}$ is
given in the footnote, (22).

$\begin{matrix}[l]\mathit{\Gamma}_{ij}{}^{k}=\frac{1}{2}g^{km}\partial_{i}\left(g+B\right)_{mj},&\;&\mathit{\Gamma}_{i}{}^{jk}=\frac{1}{2}g^{kl}\left(g-B\right)_{lm}\partial_{i}\left(g-B\right)^{mj},\\\
\;&\;&\;\\\
\mathit{\Gamma}_{ijk}=-\frac{1}{2}\left(g-B\right)_{kl}g^{lm}\partial_{i}\left(g+B\right)_{mj},&\;&\mathit{\Gamma}_{i\;\;k}^{\;j}=\frac{1}{2}G_{km}\partial_{i}\left(g-B\right)^{mj},\end{matrix}$
(27)

Upper indices denote the (component of) the inverse of the full expression,
i.e. $g^{ij}:=\left(g^{-1}\right)^{ij}$ and
$\left(g-B\right)^{ij}:=\left(\left(g-B\right)^{-1}\right)^{ij}$.

The new coordinates obtained composing with (23) on the right deserve further
attention. Introducing $B(x)\in\Lambda^{2}T[1]M$ and modifying the basis as
$\chi_{j}\mapsto\chi_{j}-\theta^{i}\left(g_{ij}(x)+B_{ij}(x)\right)$
introduces the gauge potential for the stringy $H$-flux, and a metric $g(x)$.
Remarkably, a regular metric for tangent vectors is already directly present
in the generalized vielbein itself, rather than being implemented via
$d$-dimensional vielbeins, as sometimes happens. The relative sign between $g$
and $B$ is only a matter of convenience; most importantly, these two tensors
are demanded to appear on the same footing (without different prefactors).
Eventually this configuration is summed with its inverse (with reverted sign):
$\theta^{j}\mapsto\chi_{i}\left(g-B\right)^{ij}+\theta^{j}$. The change of
sign assures that $\mathcal{E}$ is invertible and hence an isomorphism of the
Poisson algebra. The bivector $\Pi\in\Lambda^{2}T^{*}[1]M$ will source
$Q\in\Lambda^{2}TM\otimes T^{*}M$ and $R\in\mathfrak{X}^{3}(M)$, in the same
way as $B$ is the gauge field for a $3$-form $H$. Notice that the deformation
of $\omega$, carried by the vielbein (23) based on $g$ and $B$, hints at the
abelian $1$-gerbe structure of the related Courant algebroid (treated in
section 3). That is also the reason why we do not look at the most general
Hamiltonian function (i.e. the $C^{\alpha\beta\gamma}(x)$ tensors are set to
zero): the $H$-flux of the T-duality chain will already appear directly in the
derived Courant algebroid bracket thanks to the new graded form
$\omega^{\prime}$.

Let us replicate for future reference the deformed Poisson brackets we are
going to analyze in the course of the article:

$\begin{matrix}[l]\\{p_{j},x^{i}\\}^{\prime}=\delta_{j}^{i},&\\{\upsilon(x),f(x)\\}^{\prime}=\upsilon(x).f(x),\\\
\\{\xi_{\alpha},\xi_{\beta}\\}^{\prime}=\begin{pmatrix}-2g&0\\\
0&2G^{-1}\end{pmatrix},&\\{U(x),V(x)\\}^{\prime}=-2g(X,Y)+2G^{-1}(\gamma,\sigma),\\\
\\{p_{i},\xi_{\alpha}\\}^{\prime}=\mathit{\Gamma}_{i\alpha}{}^{\beta}\xi_{\beta}\;\eqref{matrix-
conn},&\\{\upsilon(x),U(x)\\}^{\prime}=\nabla_{\upsilon}U(x),\\\
\\{p_{i},p_{j}\\}^{\prime}=0,&\\{\upsilon(x),\varkappa(x)\\}^{\prime}=[\upsilon,\varkappa]_{\text{Lie}}.\end{matrix}$
(28)

For what concerns the explicit shape of the Hamiltonian
$\Theta=\xi_{\alpha}\tilde{\rho}^{\alpha i}p_{i}$, substitution of (23) into
(19), where for the canonical graded Poisson algebra we adopt the standard
projector $\rho_{\beta}{}^{i}=\left(\delta_{k}{}^{i},0\right)$, returns

$\rho_{\alpha}{}^{i}=\left(\delta_{k}{}^{i},\left(g+B\right)^{ki}\right),\quad\tilde{\rho}^{\alpha
i}\equiv\mathcal{G}^{\alpha\beta}\rho_{\beta}{}^{i}=\left(-\dfrac{g}{2}^{ki},\left[\left(g-B\right)\dfrac{g}{2}^{-1}\right]_{k}^{i}\right).$
(29)

On the linear function $U=X+\sigma\in\Gamma(T^{*}[1]M\oplus T[1]M)$ it acts in
this way:

$\rho(U)=X+\left(g+B\right)^{-1}(\sigma).$ (30)

The Hamiltonian eventually becomes

$\Theta=-\chi_{l}\frac{g}{2}^{li}p_{i}+\theta^{l}\left(g-B\right)_{lm}\frac{g}{2}^{mi}p_{i}\,.$
(31)

Checking whether the Poisson bracket of $\Theta$ (31) with itself closes to
zero is easy once the Hamiltonian is constructed with (30). In conclusion, the
graded Poisson algebra defined by (13), where the metric is (26), the
connection coefficients are given in (27) and the curvature is zero, together
with $\Theta$ (31) yields a consistent QP-manifold.

##  3 Courant algebroids and graded Poisson structures

In this section we switch to Courant algebroids and generalized tangent
bundles, since we would like to exploit tools of generalized differential
geometry in our investigation. Again, we will try to present these
mathematical concepts in a user-friendly way. Standard Courant algebroids were
shown to be in direct relation with the $2$-graded QP-manifolds just
illustrated. We are now going to first unveil the necessary information about
Courant algebroids and then review the correspondence.

###  3.1 Correspondence between $2$-graded QP-manifolds and exact Courant
algebroids

It was shown in Roytenberg:2002nu  that a graded QP-manifold of degree $2$
with canonical symplectic structure corresponds to a Courant algebroid on the
bundle $T^{*}[1]M\oplus T[1]M\cong TM\oplus T^{*}M$. A Courant algebroid (in
short CA) is a higher algebraic structure which consists of a vector or
principal bundle endowed with a bracket, a surjective map from the total space
to the tangent bundle and a symmetric bilinear form, so that some
integrability properties are ensured. A CA on the tangent bundle is equivalent
to a Lie algebroid, and over each point of the base space to a Lie algebra.
Although the cases in which the construction of a Courant algebroid applies
are much more general, for our purposes we will consider just _exact_ CAs on
$E$. A CA on a vector bundle $E$ is exact when the short exact sequence

$0\rightarrow T^{*}M\xrightarrow{j}E\xrightarrow{\rho}TM\rightarrow 0$ (32)

holds. Every $E$ satisfying (32) is hence isomorphic to $TM\oplus T^{*}M$ by
means of a splitting of the sequence, and their algebraic structures
intertwine.

######  Definition 3.1.

A _Courant algebroid_ is a quadruple
$(E,\rho,[\cdot,\cdot],\langle\cdot,\cdot\rangle)$, where $\rho:E\rightarrow
TM$ is the _anchor_ map,
$[\cdot,\cdot]:\Gamma(E)\times\Gamma(E)\rightarrow\Gamma(E)$ is a bracket on
sections and $\langle\cdot,\cdot\rangle:\Gamma(E)\vee\Gamma(E)\rightarrow
C^{\infty}(M)$ is a fiber-wise symmetric bilinear form. The following minimal
set of conditions must hold:

  1. 1.

$[U,[V,W]]=[[U,V],W]+[V,[U,W]]\quad\quad\quad\text{(left Leibniz identity for
\,}[\cdot,\cdot]$),

  2. 2.

$\frac{1}{2}\rho(U)\langle
V,V\rangle=\left\langle[U,V],V\right\rangle\quad\quad\quad\text{(the pairing
is invariant under the adjoint action),}$

  3. 3.

$\frac{1}{2}\rho(U)\langle
V,V\rangle=\left\langle[V,V],U\right\rangle\quad\quad\quad\text{(behaviour of
the symmetric part of the bracket),}$

for $U,V,W\in\Gamma(E)$. Notice furthermore that the anchor map is subjected
to be actually $\rho\in\text{Hom}(E,TM)$ from the axioms.

It is worth pointing out that the extended tangent bundle $TM\oplus T^{*}M$
has an abelian $1$-gerbe structure. The open sets are patched together through
transition functions depending on a $2$-form $B$. In the non-empty
intersection of open sets $U_{i}\cap U_{j}$,

$B_{(i)}=B_{(j)}-\mathrm{d}\lambda_{(ij)}.$

The exact $2$-form $\mathrm{d}\lambda$ must fulfill the cocycle condition:

$\lambda_{(ij)}+\lambda_{(jk)}+\lambda_{(ki)}=c^{-1}_{(ijk)}\mathrm{d}c_{(ijk)}$

in the three overlapping patches $U_{i}\cap U_{j}\cap U_{k}\neq\\{0\\}$, with
$c_{(ijk)}$ holomorphic function with values in $U(1)$. $B$ is also invariant
under $\lambda\mapsto\lambda+\mathrm{d}f$, $f\in C^{\infty}(M)$. Exact CAs are
classified by the third cohomology class $H^{3}(M,\mathbb{R})$ Severa:2017oew
.

To be convinced that a QP-manifold $T^{*}[2]T[1]M$ encodes the same data as a
CA on $TM\oplus T^{*}M$, the clue is to compute the master equation
$\\{\Theta,\Theta\\}=0$ for the Hamiltonian given in (4). It yields the set of
defining conditions for a Courant algebroid of 3.1. If $U(x),V(x)\in\Gamma(E)$
are given by
$U^{\alpha}(x)\xi_{\alpha}=X^{i}(x)\chi_{i}+\sigma_{i}(x)\theta^{i},V^{\alpha}(x)\xi_{\alpha}=Y^{j}(x)\chi_{j}+\kappa_{j}(x)\theta^{j}$,
from the equivalence with the canonical QP-manifold ($T^{*}[2]T[1]M,\omega$
(1), $\Theta$ (4) with $\rho_{\alpha}{}^{i}=(\delta_{k}{}^{i},0)$ and
$C=H_{ijk}\theta^{i}\theta^{j}\theta^{k}$) one retains as natural bracket for
the CA the so called Dorfman bracket,

$[U,V]_{\text{D}}:=[X,Y]+\mathcal{L}_{X}\kappa-\iota_{Y}\mathrm{d}\sigma,$

twisted by $H$:

$\\{\\{U,\Theta\\},V\\}=[U,V]_{H}:=[U,V]_{\text{D}}+H(X,Y),\quad
H(X,Y)\in\Gamma(T^{*}M).$

This expression shows that the twisted Dorfman bracket is the derived bracket
for the canonical graded Poisson algebra of degree-$1$ functions.

###  3.2 Courant algebroid for a non-canonical QP-manifold

So far we commented upon the relation between the Courant algebroid (CA)
$(TM\oplus T^{*}M,\rho,[\cdot,\cdot],\langle\cdot,\cdot\rangle)$ and the
associated canonical QP-manifold $(T^{*}[2]T[1]M,\omega,\Theta)$. Now we want
to unveil the expressions of the CA objects for the non-canonical QP-manifold
of our model instead. As already announced, in the Hamiltonian for the
canonical case we resort to the identity map for $\rho$, while the tensors $C$
are set to zero. We will also present the results for the example of
subsection 2.4 along with those for a generic vielbein $\mathcal{E}$. This
should not create confusion but rather help to focus more immediately on the
main outcomes.

The pairing $\langle\cdot,\cdot\rangle^{\prime}$ for the Courant algebroid is
easily retrieved directly from the Poisson bracket between $U(x)$ and $V(x)$
linear functions in $\xi$:

$\\{U,V\\}=\begin{pmatrix}-2g(X,Y)&0\\\
0&2G^{-1}(\sigma,\kappa)\end{pmatrix}=:\langle U,V\rangle^{\prime}.$

The Dorfman bracket is generalized via derived bracket with the Hamiltonian in
(31), $\\{\\{\\{U,\Theta\\},V\\},\\\
W\\}=\left\langle[U,V]^{\prime}_{\text{D}},W\right\rangle^{\prime}$, with
$W^{\alpha}(x)\xi_{\alpha}=Z^{k}(x)\chi_{k}+\zeta_{k}(x)\theta^{k}$ and $U,V$
as before, to the following:

$\displaystyle\langle[U,V]^{\prime}_{\text{D}},W\rangle^{\prime}$
$\displaystyle=\left\langle\nabla_{\rho(U)}V-\nabla_{\rho(V)}U,W\right\rangle^{\prime}+\left\langle\nabla_{\rho(W)}U,V\right\rangle^{\prime}$
(33)
$\displaystyle=\iota_{[\rho(U),\rho(V)]}\left(\zeta-\left(g-B\right)(Z)\right)+\iota_{[\rho(V),\rho(W)]}\left(\sigma-\left(g-B\right)(X)\right)$
$\displaystyle\;\;\;-\iota_{[\rho(U),\rho(W)]}\left(\kappa-\left(g-B\right)(Y)\right)+\rho(U)\left\langle\big{(}\kappa-\left(g-B\right)\left(Y\right)\big{)},\rho(W)\right\rangle$
$\displaystyle\;\;\;-\rho(V)\left\langle\big{(}\sigma-\left(g-B\right)\left(X\right)\big{)},\rho(W)\right\rangle+\rho(V)^{i}\rho(W).\big{(}\sigma-\left(g-B\right)\left(X\right)\big{)}_{i},$
(34)

where, in accordance with (30), the anchored vectors are

$\rho(U)=X+\left(g+B\right)^{-1}(\sigma)$

and (34) displays the particular deformed graded Poisson structure under
consideration from the general formula (33). It might not be evident from the
expression (34), but the bracket relies just on global objects: the anchor
map, the Riemannian metric $g$ and $\mathrm{d}_{\rho}B\equiv
H_{\rho}\in\Lambda^{3}(T^{*}M)$. The anchor-dependent differential
$\mathrm{d}_{\rho}$ will be discussed in the next section 4.2.

Notice also that part of expression (34) for the bracket can be rearranged so
to form again the fully contracted standard Dorfman bracket, i.e.
$\langle[X,Y],\zeta\rangle+\langle\mathcal{L}_{X}\kappa-\iota_{Y}d\sigma,Z\rangle$,
as one would expect when setting $g,B=0$ in the deformation $\mathcal{E}$,
since in that limit it reduces to the identity.

As a further consistency check, it is straightforward to directly prove that
the new bracket $[\cdot,\cdot]^{\prime}_{\text{D}}$, the new block diagonal
pairing and the new anchor $\rho$ respect the axioms in 3.1. They are
retrieved thanks to associativity of the deformed Poisson brackets, the master
equation and Leibniz rule for the derivation Q. See also Heller:2016abk  for
details.

Let us point out that in the Generalized Geometry description the generalized
vielbeins $\mathcal{E}(x)$ are local sections of the $O(d,d)$ frame bundle.
They can be extended on the overlapping patches: the transition functions are
orthogonal transformations (which comprise diffeomorphisms, $B$ transforms and
transformations with a bivector). These are canonical transformations of the
underlying graded Poisson algebra, see the overview of section 2.1 and
equation (2). Every two set of vielbeins will differ by an exact $2$-form:
this is a feature of the underlying $1$-gerbe structure of the generalized
tangent bundle. More importantly though, to get a CA on $TM\oplus T^{*}M$ one
needs also a Hamiltonian function that can solve the classical master
equation: different anchor maps and tensors $C$ can be good solutions of (18).
In this scenario, we will end up with a different representative of the third
cohomology class of the Courant algebroid, or even with other tensors or
multivectors involved. Nevertheless our results in the next sections will be
affected just by addition of the possibly non-zero $C$-tensors to the derived
connection (42).

##  4 Differential geometry of $TM\oplus T^{*}M$

In the upcoming section we review a bit of differential geometry of $TM\oplus
T^{*}M\cong T^{*}[1]M\oplus T[1]M$. The material we cover is well established
in the literature, but sometimes we complement it with some less standard
notions. For example we will suggest a generalized version of the Lie bracket.
Then, inspired by the expression of the deformed Dorfman bracket (33) and
(34), and by subtraction of this generalization of a Lie bracket, we will be
able to extract another metric connection (w.r.t. $\mathcal{G}$) on
$E$-sections, with non-zero curvature in some sense yet to be defined. This
fact is rather crucial. Indeed in the way we incorporated the gauge fields $B$
and $\Pi$ in the beginning we could just get the algebra of functions for a
particle in a flat background, coupled to $B$ and $\Pi$ with the connection
symbols (27). These fields were not dynamical themselves, since the curvature
was identically zero.

With the new connection, and by suggesting an invariant scalar of the
curvature tensor fully projected into some subbundles, we will eventually
obtain an action for a theory of gravity. Comments on its relation to the
action of compactified string theory with $Q$ and $R$ fluxes, also known as
the NS-NS action in a non-geometric frame Blumenhagen:2013aia , will be
postponed to the discussion 6. We already mentioned that the fluxes in the
T-dual chain have found their description within the ordinary QP-manifold
$(T^{*}[2]T[1]M,\omega,\Theta)$ as the $C^{\alpha\beta\gamma}(x)$ tensors in
the Hamiltonian, that twist the CA bracket. In fact their nature is clear from
the following piece of the derived bracket:

$\displaystyle\left\\{\left\\{\left\\{U,\frac{1}{3!}C^{\alpha\beta\gamma}(x)\xi_{\alpha}\xi_{\beta}\xi_{\gamma}\right\\},V\right\\},W\right\\}=$
$\displaystyle\,H(X,Y,Z)+f(X,Y,\zeta)+f(X,\kappa,Z)+f(\sigma,Y,Z)$
$\displaystyle+Q(X,\kappa,\zeta)+Q(\sigma,\kappa,Z)+Q(\sigma,Y,\zeta)+R(\sigma,\kappa,\zeta).$

It was already pointed out in the previous section that they satisfy the
Bianchi identities, in accordance with the results from flux compactification
of IIB string theory, as an outcome of the master equation. In the present
situation we interpret them as local field strengths of their respective gauge
potentials and more importantly they are treated on the same footing as the
respective symmetric part ($g$ or $G^{-1}$, the open and closed strings
metrics), according to the open-closed strings relation. We believe that this
rather uncommon perspective is equally interesting. Notice moreover that we do
not resort to a doubling of the space coordinates with the coordinates
corresponding to the winding modes, as in Double Field Theory Hohm:2010pp . We
are rather exploiting solely the doubling of the tangent bundle with the
cotangent and the ansatz (23).

###  4.1 Generalized Lie bracket, connection and torsion tensor

In this part we resume and expand the notions of generalized Lie bracket,
connection and torsion tensor on sections of a vector bundle
$E\xrightarrow{\pi}M$ as initially presented in our previous paper
Boffo:2019zus . The definitions for the torsion tensor and for the skew-
symmetric Lie-like bracket are original. Our motivation relies on the
possibility to obtain a new connection
$\widetilde{\nabla}:\Gamma(E)\times\Gamma(E)\rightarrow\Gamma(E)$ from the
covariant version of the Dorfman bracket, requiring that an antisymmetric
bracket is subtracted from it. On the other hand, the existence of a
connection whenever we are given (a symmetric bilinear form, a surjective
$\rho:\Gamma(E)\mapsto\Gamma(TM)$ and) a bracket on $E$-sections which
respects all the axioms of a CA bracket but Jacobi identity, and a skew-
symmetric bracket (for which Jacobi identity holds instead) is also assured by
the proposition enunciated in Boffo:2019zus . The new connection
$\widetilde{\nabla}$ would then be the previous one $\nabla$ summed to its
torsion tensor, which is defined with the help of the Lie-like bracket. Let us
begin with the introduction of an antisymmetric bracket on the sections of a
vector bundle $E$.

######  Definition 4.1.

A _generalized Lie bracket_ (or Lie-like bracket)
$\llbracket\cdot,\cdot\rrbracket$ is a bilinear operation
$\llbracket\cdot,\cdot\rrbracket:\Gamma(E)\times\Gamma(E)\mapsto\Gamma(E)$
which is antisymmetric, $\llbracket U,V\rrbracket=-\llbracket V,U\rrbracket$,
and respects the Leibniz rule:

$\llbracket U,fV\rrbracket=\rho(U)f\,V+f\llbracket U,V\rrbracket,\quad f\in
C^{\infty}(M),$

$\rho$ being the surjective application $\rho:\Gamma(E)\mapsto\Gamma(TM)$.

Any two generalized Lie brackets differ by an antisymmetric $2$-tensor. In
analogy to the Lie brackets of vector fields, if the basis
$\\{\xi_{\alpha}\\}$ for $E$ is holonomic, it is reasonable to set
$\llbracket\xi_{\alpha},\xi_{\beta}\rrbracket=0$. Notice that the basis
induced by the Darboux chart of the minimal symplectic realization of $E[1]$,
$T^{*}[2]E[1]$, is holonomic. In this basis, being $U,V\in\Gamma(E)$,
$U=U^{\alpha}(x)\xi_{\alpha}$ and $V=V^{\beta}(x)\xi_{\beta}$, a minimal
example of a Lie-like bracket can be

$\llbracket
U,V\rrbracket=\left(\rho(U)V^{\alpha}(x)-\rho(V)U^{\alpha}\right)\xi_{\alpha}.$
(35)

The example is coordinate-dependent and thus local, but it can be made global
by extending it to every patch covering the base manifold, forcing the
definition (35) in the overlaps and applying the Leibniz property, that the
image of $\rho$ must be a well-posed derivation, to the transition functions.
Remind that the transition functions when $E$ is a $O(d,d)$-principal bundle
comprise also $B$-field transformations. The whole procedure is consistent if
$\rho$ is globally defined. Furthermore (35) respects Jacobi identity when
$\rho$ is a homomorphism with the Lie bracket of vector fields. Our
generalized Lie bracket in 4.1 is related to a bracket dubbed _dull bracket_
which was proposed in lean2012dorfman . By definition, it fulfills the Leibniz
property on both the entries and

$\rho([U,V]_{dull})=[\rho(U),\rho(V)]_{\text{Lie}}.$

On the contrary, antisymmetry is not demanded.

It would be interesting if the generalized Lie bracket could be a derived
bracket. As seen in the previous section, the CA bracket is the derived
bracket for the degree-$2$ symplectic structure on $T^{*}[2]T[1]M$ with
homological vector field Q, on the algebra of degree-$1$ functions. The
generalized Lie bracket (35) can be interpreted as the antisymmetrization of a
derived bracket, for the canonical Poisson structure of degree $2$ equipped
with an odd vector field Q, which fails to be Hamiltonian. For this purpose,
the only non-zero Poisson brackets are:

$\\{p_{i},x^{j}\\}=\delta_{i}^{j}=\\{\chi_{i},\theta^{j}\\}.$ (36)

Together with this structure we would like to consider the following Q̌:

$\text{\v{Q}}:=\tilde{\rho}^{\alpha
i}p_{i}\dfrac{\partial}{\partial\xi_{\alpha}}.$ (37)

It is easy to see that $\text{\v{Q}}^{2}=0$ although it is not Hamiltonian for
the symplectic form $\omega_{0}$ associated with (36):
$\iota_{\text{\v{Q}}}\omega_{0}\neq\mathrm{d}\vartheta$. Then on the algebra
of degree-$1$ functions the antisymmetrization of the derived bracket
corresponds to the generalized Lie bracket (35):

$\\{\text{\v{Q}}U,V\\}-\\{\text{\v{Q}}V,U\\}=\llbracket U,V\rrbracket.$ (38)

This perspective will be helpful when we will have to determine some more or
less hidden invariances pertaining the Courant algebroid.

The concept of a differential depending on $\rho$ is also quite important. It
is first of all defined from its action on functions.

######  Definition 4.2.

A differential $\mathrm{d}_{\rho}:\Omega^{0}(E)\cong
C^{\infty}(M)\mapsto\Omega^{1}(E)$ can be given in coordinates
$\\{\xi_{\alpha}\\}$ for the fiber of $E\xrightarrow{\pi}M$ by

$\mathrm{d}_{\rho}:=\tilde{\xi}^{\alpha}\rho(\xi_{\alpha}),$ (39)

where
$\tilde{\xi}_{\alpha}:=g_{E}^{\alpha\beta}\xi_{\beta},\,g_{E}\equiv\langle\cdot,\cdot\rangle\in\vee^{2}E^{*}$.
Then it acts on the function $f$ according to the following fully contracted
formula:

$\langle\mathrm{d}_{\rho}f,e\rangle=\rho(e)f.$

The action of $\mathrm{d}_{\rho}$ on the whole exterior algebra
$\Omega^{\bullet}(E)$ is allowed if an antisymmetric bracket
$\llbracket\cdot,\cdot\rrbracket$ as in definition 4.1 is used.

$\displaystyle\mathrm{d}_{\rho}\psi(U_{1},\dots,U_{k+1})=$
$\displaystyle\sum_{i}(-)^{i}\rho(U_{i})\psi(U_{1},\dots,\hat{U}_{i},\dots,U_{k+1})$
$\displaystyle+\sum_{i<j}(-)^{i+j}\psi(\llbracket
U_{i},U_{j}\rrbracket,\dots,\hat{U}_{i},\dots,\hat{U}_{j},\dots U_{k+1}).$

The differential $\mathrm{d}_{\rho}$ squares to zero if
$\text{Im}\left(g^{-1}_{E}\circ\rho^{*}\right)=\text{Ker}\rho$ and
$\llbracket\cdot,\cdot\rrbracket$ respects the Jacobi identity.

If the existence of $\mathrm{d}_{\rho}$ is disjoint from that of
$\llbracket\cdot,\cdot\rrbracket$, in the sense that a differential is given
from the beginning, then it is possible to express (35) in terms of
$\mathrm{d}_{\rho}$. Let $\varsigma$ be the tautological $1$-form (otherwise
called soldering form) $\varsigma:E\overset{\sim}{\longrightarrow}E^{\prime}$.
Then the generalized Lie bracket can be obtained from:

$\varsigma\left(\llbracket
U,V\rrbracket\right)=\rho(U)\varsigma(V)-\rho(V)\varsigma(U)-d_{\rho}\varsigma(U,V)$
(40)

In the holonomic coordinate basis
$\\{\xi_{\alpha}\\}=\\{\partial_{i},\mathrm{d}x^{i}\\}$ for $E=TM\oplus
T^{*}M$, where $\\{x^{i}\\}$ are coordinates on $M$ (associated with the
Darboux chart on our degree-$2$ dg-symplectic manifold), the soldering form
has the expression

$\varsigma=\partial_{i}\otimes\mathrm{d}x^{i}\oplus\mathrm{d}x^{i}\otimes\partial_{i},$

thus in (40) the $\rho$-differential on $\varsigma$ is zero, and in the
coordinate basis the generalized Lie bracket is (35):

$\llbracket U,V\rrbracket^{i}\partial_{i}\oplus\llbracket
U,V\rrbracket_{i}\mathrm{d}x^{i}=\left(\rho(U)Y^{i}-\rho(V)X^{i}\right)\partial_{i}\oplus\left(\rho(U)\kappa_{i}-\rho(V)\sigma_{i}\right)\mathrm{d}x^{i}.$

(40) prescribes also how the expression transforms under a change of basis. In
our companion work Boffo:2019zus  we relegated to a footnote the description
of the differential $\mathrm{d}_{\rho}$, but the brackets here and there are
indeed the same.

At this point, with the help of a Lie-like bracket
$\llbracket\cdot,\cdot\rrbracket$ many definitions of standard differential
geometry for vector fields can be reproduced identically for $E$-sections too.
The torsion tensor is among these.

######  Definition 4.3.

A _torsion tensor_
$\text{T}:\Gamma\left(\Lambda^{2}E^{*}\right)\rightarrow\Gamma(E)$ is defined
by:

$\text{T}(U,V):=\nabla_{U}V-\nabla_{V}U-\llbracket U,V\rrbracket.$ (41)

Because of the generalized Lie bracket (4.1) the above defined T is clearly
antisymmetric and $C^{\infty}(M)$-linear.

Let us now state a useful result. The proof is contained in Boffo:2019zus .

######  Proposition 4.4.

For any bracket on $E$-sections respecting at least axioms 2 and 3 in 3.1 for
a given pairing $\langle\cdot,\cdot\rangle$ and anchor map
$\rho:\Gamma(E)\mapsto\Gamma(TM)$, while Jacobi identity is not required, any
generalized Lie commutator as in 4.1 and any affine connection on generalized
vector fields, metric with respect to the pairing, and with fully
antisymmetric torsion $\text{T}\in\Gamma\left(\Lambda^{3}E^{*}\right)$ (41),
the following relation holds:

$\langle[U,V]-\llbracket U,V\rrbracket,W\rangle=\langle\nabla_{W}U,V\rangle.$
(42)

Notice that ultimately the above relation says that the Dorfman bracket for a
CA $[\cdot,\cdot]_{\text{D}}$, a generalized Lie bracket
$\llbracket\cdot,\cdot\rrbracket$ and a connection $\nabla$ are dependent on
each other.

We will be interested exclusively in extracting a covariant derivative from
given $[\cdot,\cdot]$ and $\llbracket\cdot,\cdot\rrbracket$, using the
proposition. It might hence be helpful to go through the evidences that the
derivative operator $\nabla$ (42) has indeed the correct properties:

$\begin{matrix}[l]\left\langle\nabla_{fW}U,V\right\rangle=\left\langle\nabla_{W}U,fV\right\rangle=f\left\langle\nabla_{W}U,V\right\rangle,&\left\langle\nabla_{W}fU,V\right\rangle=\rho(W)f\langle
U,V\rangle+f\left\langle\nabla_{W}U,V\right\rangle,\\\ \rho(W)\langle
U,V\rangle=\langle\nabla_{W}U,V\rangle+\langle
U,\nabla_{W}V\rangle,&\langle\text{T}(U,V),V\rangle=0.\end{matrix}$ (43)

Tensoriality in $W$ is obvious. Tensoriality in $V$ is due to the fact that
$[\cdot,\cdot]$ and $\llbracket\cdot,\cdot\rrbracket$ respect the Leibniz
property in their respective second slot. $\mathbb{R}$-linearity in $U$ is due
to:

$[fU,V]=-\rho(V)f\,U+f[U,V]+\mathrm{d}_{\rho}f\,\langle U,V\rangle,$

which stems from the axiom about the symmetric part of the bracket and Leibniz
rule for $\rho$. Then, metricity of the connection is an outcome of the third
axiom for a CA bracket together with the antisymmetry of the generalized Lie
bracket. The torsion is completely skew-symmetric due to both the conditions
(2 and 3) for the bracket $[\cdot,\cdot]$ in 3.1, and because
$\llbracket\cdot,\cdot\rrbracket$ is antisymmetric.

Now we want to employ some of the brackets previously discussed in the
proposition to straightforwardly yield a new connection $\widetilde{\nabla}$.
As for the bracket respecting two of the CA axioms we pick up the deformed
Dorfman bracket (34). Recall that it comes from the derived Poisson brackets
with undeformed Hamiltonian $\Theta$. Then we are led to express the Lie-like
bracket w.r.t. the same holonomic coordinate basis, while employing the new
anchor. Hence $\llbracket\cdot,\cdot\rrbracket$ in (35) does the job. With
these elements, the connection
$\widetilde{\nabla}:\Gamma(E)\rightarrow\Gamma(E^{*})\otimes\Gamma(E)$
corresponds to

$\left\langle\widetilde{\nabla}_{W}U,V\right\rangle^{\prime}=\left\langle\text{T}(U,V),W\right\rangle^{\prime}+\left\langle\nabla_{\rho(W)}U,V\right\rangle^{\prime}.$
(44)

As displayed here, the torsion tensor for the connection $\nabla$ we started
with, emerges from the fully general contracted expression of the Dorfman
bracket (33) upon subtraction of $\langle\llbracket
U,V\rrbracket,W\rangle^{\prime}$. We now have a perfectly consistent Courant
algebroid connection on $E$. However for our scope the relevant connections
are obtained by restriction of the “big” connection to tangent or cotangent
space exclusively. In both the resulting connections the direction of
derivation will be anchored to tangent space by means of the anchor.

###  4.2 Connections on tangent space and on cotangent space for the given
deformation

In this part we are going to halve the dimension of the fiber of our bundle by
restricting our considerations to $TM$ or $T^{*}M$ respectively. The case of
$TM$, analyzed extensively in many other works e.g. Jurco:2015ywk , is
presented just for reference and is regarded as dual to the case of $T^{*}M$,
which is the relevant one for the description of a gravitational field paired
to the local expressions of the stringy non-geometric fluxes. As subspaces of
$E$, $TM$ and $T^{*}M$ are isotropic with respect to the natural pairing
$\eta$ of vector fields with forms but not with respect to the metric
$\mathcal{G}\equiv\langle\cdot,\cdot\rangle^{\prime}$. They are involutive
under the standard Dorfman bracket but not under the non-canonical one.

A clever way to project onto tangent or cotangent space utilizes a splitting
$s:\Gamma(TM)\rightarrow\Gamma(E)$, $\rho\circ s=\text{id}_{TM}$, or an
embedding $r:\Gamma(T^{*}M)\rightarrow\Gamma(E)$. $r$ itself can be considered
a splitting of the dual short exact sequence

$0\rightarrow TM\xrightarrow{\Delta}E\xrightarrow{\Delta^{*}}T^{*}M\rightarrow
0,$

if it fulfills the condition $\Delta^{*}\circ r=\text{id}_{T^{*}M}$. If $\rho$
is given by (30), a consistent choice for $\Delta^{*}$ can be444We are
requiring that $\rho\circ\Delta=\text{id}$.

$\Delta^{*}(U)=-\left(g-B\right)(X)+\sigma.$ (45)

Of all the possible splittings, we are going to use only the trivial
embeddings, $s:TM\mapsto E$:

$s(X)=\begin{pmatrix}X\\\ 0\end{pmatrix}\in E\cong TM\oplus T^{*}M;$ (46)

and correspondingly, $r:T^{*}M\mapsto E$:

$r(\sigma)=\begin{pmatrix}0\\\ \sigma\end{pmatrix}\in E\cong TM\oplus T^{*}M.$
(47)

As announced, the induced metrics on $TM$ and $T^{*}M$ are non-degenerate and
corresponds to:

$\langle s(X),s(Y)\rangle=-2g(x,Y),\quad\langle
r(\sigma),r(\zeta)\rangle=2G^{-1}(\sigma,\zeta).$

Eventually we can force our connection (44) on tangent (cotangent) space by
_defining_ it in the following way:

$\langle\widetilde{\nabla}_{s(Z)}s(X),s(Y)\rangle^{\prime}=:-2g\left(\widetilde{\nabla}_{Z}X,Y\right),\quad\langle\widetilde{\nabla}_{r(\zeta)}r(\sigma),r(\kappa)\rangle^{\prime}=:2G^{-1}\left(\widetilde{\nabla}_{\zeta}\sigma,\kappa\right).$

Let us now briefly comment upon the regular connection on vector fields first.
Our list of prescriptions (the new Dorfman bracket homomorphic to the
canonical one via $\mathcal{E}$, the anchor $\rho$, the generalized Lie
bracket and the splitting $s$ which leads to a non-degenerate induced metric
on $TM$) yield:

$2g\left(\widetilde{\nabla}_{Z}X,Y\right)=\left(g+B\right)\left([X,Y],Z\right)+\iota_{Z}\left(\mathcal{L}_{X}\left(g-B\right)(Y)-\iota_{Y}d\left(g-B\right)(X)\right)-2g([X,Y],Z),\,$
(48)

where the last term is due to the generalized Lie bracket (fully contracted to
a $C^{\infty}(M)$-function). It is hence immediate to show that the connection
for tangent space is the Levi-Civita one (metric w.r.t. $g$ and torsion-free)
plus an antisymmetric term based on the exterior derivative of $B$, the
$3$-form $H=\mathrm{d}B$:

$\widetilde{\nabla}_{Z}X=\nabla^{\text{L.C.}}_{Z}X-\dfrac{g}{2}^{-1}\circ
H(Z,X,\cdot),$ (49)

Its torsion is the fully antisymmetric $H$ itself. The reader should be aware
that thanks to our systematic method the connection is a globally defined
object, although the starting point (graded symplectic structure with a
homological vector field) was not.

We can now turn our attention to cotangent space, and study this instance of
derived connection which, following reference Asakawa:2015jza , we would like
to call “contravariant” connection. Let us first remind the reader that the
anchor $\rho$ provides a natural derivative operator in the direction of
forms,

$\rho(r(\sigma))=\left(G^{-1}+\Pi\right)(\sigma):C^{\infty}(M)\rightarrow\Gamma(TM).$

Hence using the splitting $r$ of the dual sequence we obtain the connection
$\widetilde{\nabla}:\Gamma(T^{*}M)\mapsto\Gamma(TM)\otimes\Gamma(T^{*}M)$:

$\displaystyle 2G^{-1}\left(\widetilde{\nabla}_{\zeta}\sigma,\kappa\right)=$
$\displaystyle\iota_{\left[\rho(r(\sigma)),\rho(r(\kappa))\right]}\zeta+\iota_{\rho(r(\zeta))}\left(\mathcal{L}_{\rho(r(\sigma))}\kappa-\iota_{\rho(r(\kappa))}d\sigma\right)$
$\displaystyle-2G^{-1}\left(\rho(r(\sigma))r(\kappa)-\rho(r(\kappa))r(\sigma),r(\zeta)\right),$
(50)

where the last row is due to the generalized Lie bracket. The expression might
look unwieldy, however the right properties of an affine connection (43) are
checked. Notice that the direction of derivation, given by $\zeta$, is always
anchored to $TM$, via $\rho\circ r:\Gamma(T^{*}M)\mapsto\Gamma(TM)$. In the
generalized Lie bracket this happens thanks to an easy manipulation

$2G^{-1}\left(\dots,r(\zeta)\right)=2g\left(G^{-1}+\Pi\right)\left(\dots,\rho(r(\zeta))\right),$

while the first addend in (50) can be massaged to

$\iota_{\left[\rho(r(\sigma)),\rho(r(\kappa))\right]}\zeta\equiv
2G^{-1}\left(\dfrac{G}{2}\left(\left[\rho(r(\sigma)),\rho(r(\kappa))\right]\right),\zeta\right)=G\left(\left[\rho(r(\sigma)),\rho(r(\kappa))\right],G^{-1}(\zeta)\right).$

In the coordinate basis the connection becomes definitely more familiar.
Assigning indices $i,j,k$ to $\sigma,\kappa$ and $\zeta$ respectively, and
using $G^{-1}$ to raise or lower indices,

$\mathit{\tilde{\Gamma}}^{ki}{}_{j}=\dfrac{1}{2}G_{jl}\left(2\left(G^{-1}+\Pi\right)^{[i|m}\partial_{m}\left(G^{-1}+\Pi\right)^{|l]k}+\left(G^{-1}+\Pi\right)^{km}\partial_{m}\left(G^{-1}+\Pi\right)^{il}\right).$
(51)

The expression in components now involves the $R$555$R$ is as well $R\equiv
3[\Pi,\Pi]_{\text{S}}$, where $[,]_{\text{S}}$ is the Schouten bracket of
multivector fields. and $Q$ fluxes, which according to Andriot:2012wx  in the
supergravity frame are

$\displaystyle R^{ijk}:=$ $\displaystyle 3\Pi^{[i|l}\partial_{l}\Pi^{|jk]},$
(52) $\displaystyle Q^{ij}{}_{k}:=$ $\displaystyle\partial_{k}\Pi^{ij}.$ (53)

The connection coefficient (51) consists of a symmetric part, equivalent to
the Christoffel symbol $\Gamma_{G}$ for $G^{-1}$ and to another triplet of
terms. We will dub $\mathcal{Y}$ the following:

$\mathcal{Y}^{ki}{}_{j}:=\frac{1}{2}G_{jl}\left(\Pi^{km}\partial_{m}G^{il}+2\Pi^{[i|m}\partial_{m}G^{|l]k}\right),\quad\mathcal{Y}^{ki}{}_{j}=\mathcal{Y}^{ik}{}_{j},\quad\mathcal{Y}^{ki}{}_{k}=\frac{1}{2}G_{lm}\Pi^{ip}\partial_{p}G^{lm}.$

Together, $\Gamma_{G}$ and $\mathcal{Y}$ build up a metric and torsionless
connection for the metric $2G^{-1}$. It can be convenient to rewrite them so
that the anchor is prominent:

$\Gamma_{G}^{ki}{}_{j}+\mathcal{Y}^{ki}{}_{j}=\dfrac{1}{2}G_{mj}\left(\rho^{kl}\partial_{l}G^{im}+\rho^{il}\partial_{l}G^{mk}-\rho^{jl}\partial_{l}G^{ik}\right).$

The torsion of the connection is made up by derivatives hitting $\Pi$, that
lead to the $R$ tensor and to a triplet of $Q$ terms, which for later
convenience we denote in the following notation:

$\mathcal{X}^{ki}{}_{j}:=-\frac{1}{2}Q^{ki}{}_{j}+G_{jl}G^{[k|m}Q^{|i]l}{}_{m},\quad\mathcal{X}^{ki}{}_{j}=-\mathcal{X}^{ik}{}_{j},\quad\mathcal{X}^{ki}{}_{k}=0.$
(54)

Later, when needed, we will untie $\mathcal{X}$ in its $Q\sim\partial\Pi$
constituents. Eventually the connection symbols (51) are given by the
combination

$\mathit{\tilde{\Gamma}}^{ki}{}_{j}=\Gamma_{G\;\;\;j}^{\;\;\;ki}+\mathcal{Y}^{ki}{}_{j}+\frac{1}{2}G_{jl}R^{kil}+\mathcal{X}^{ki}{}_{j}.$
(55)

The torsion tensor of $\widetilde{\nabla}$ is

$\text{T}^{ijk}=2R^{ijk}+4\mathcal{X}^{ijk},$

and it is easy to see that it is fully antisymmetric, as predicted in
proposition (4.4).

###  4.3 Curvature tensor

In this part we would like to present our working definitions for the analogue
of the Riemann tensor and of Ricci tensor, specialized to the relevant
subbundles for this article. First we will sketch how to get a genuine Riemann
curvature tensor for the $E$-connection $\widetilde{\nabla}$ on the
generalized tangent bundle $E$. Usually in Generalized Geometry (see for
example Coimbra:2011nw ) a curvature tensor, built with the non-skewsymmetric
bracket for the CA, is defined in the special circumstance that the
generalized metric $\mathcal{H}$ is introduced, and its expression needs to be
evaluated on a combination of the subbundles associated with $\mathcal{H}$.
Here instead it can mimic the Riemann tensor of standard differential geometry
and be simply the commutator of the covariant derivative $\widetilde{\nabla}$
(44) minus the covariant derivative of the commutator, where the latter is
constructed with $\llbracket\cdot,\cdot\rrbracket$.
$\text{Riem}\in\Gamma(\overset{3}{\bigotimes}E^{*}\otimes E)$ is thus

$\left[\widetilde{\nabla}_{U},\widetilde{\nabla}_{V}\right]W-\widetilde{\nabla}_{\llbracket
U,V\rrbracket}W=:\text{Riem}(U,V,W)\in\Gamma(E).$ (56)

The definition is well-posed, i.e. Riem is a true rank-$4$ tensor in $E$. This
claim is easily checked from the behaviour under function multiplication of
the commutator of covariant derivatives and of the generalized Lie bracket.
This definition is definitely much more straightforward than the most common
one illustrated before.

In the end we are interested in specializing Riem in the obvious way to the
regular connection on tangent vectors only, and to that on covectors. Their
curvatures stem from evaluating Riem with $s$-vectors only or $r$-forms only,
and project the result on the corresponding subbundle. For example, in the
$TM$ case the generalized Lie bracket when the splitting $s$ is given by (46),
matches the Lie bracket of vector fields

$\text{Riem}_{TM}(X,Y,Z):=\rho\left(\text{Riem}(s(X),s(Y),s(Z))\right)\overset{\eqref{non-
iso-s}}{=}\left[\widetilde{\nabla}_{X},\widetilde{\nabla}_{Y}\right]Z-\widetilde{\nabla}_{[X,Y]}Z.$
(57)

The Riemann curvature tensor
$\text{Riem}_{T^{*}M}\in\Gamma(\otimes^{3}TM\otimes T^{*}M)$ corresponds to:

$\text{Riem}_{T^{*}M}(\zeta,\kappa,\sigma):=\Delta^{*}\left(\text{Riem}(r(\zeta),r(\kappa),r(\sigma))\right)\overset{\eqref{r-splitting}}{=}\left[\widetilde{\nabla}_{\zeta},\widetilde{\nabla}_{\kappa}\right]\sigma-\widetilde{\nabla}_{\llbracket
r(\zeta),r(\kappa)\rrbracket}\sigma.$ (58)

The Ricci tensor $\text{Ric}_{TM}\in\Gamma(T^{*}M\otimes T^{*}M)$ (resp.
$\text{Ric}_{T^{*}M}\in\Gamma(TM\otimes TM)$) are obtained from the partial
trace of $\text{Riem}_{TM}$ (resp. $\text{Riem}_{T^{*}M}$):

$\text{Ric}_{TM}(Y,Z)=\sum_{i}^{d}\langle\text{Riem}_{TM}\left(\partial_{i},Y,Z\right),\mathrm{d}x^{i}\rangle,\quad\langle\cdot,\cdot\rangle\;\text{canonical
pairing.}$

In the next section we can present a pair of action functionals for a peculiar
contraction of $\text{Ric}_{TM}$ and of $\text{Ric}_{T^{*}M}$.

##  5 Gravitational actions with fluxes

This section addresses the application to physics of the algebraic and
geometric methods developed in the previous sections. Before showing and
discussing the gravitational action involving $G^{-1}$ and the $Q$ and
$R$-fluxes, let us briefly present the result for the connection on tangent
space (49). The Ricci tensor of that connection, integrated against
$g^{-1}\circ\left(g-B\right)\circ g^{-1}$, and by means of Stokes’ theorem in
the integration, is known to reproduce the effective action functional of the
low-energy closed strings, i.e. the Ricci scalar and the square of the field
strength $H$ Jurco:2015ywk , Boffo:2019zus ,

$S_{NS}[g,B]=\int_{M}\text{vol}_{g}\,R(g)-\dfrac{1}{12}H^{2}.$ (59)

The above formula is the Hilbert-Einstein action of the connection (49),
formed with the non-symmetric combination $g-B$. It could be derived deploying
elements of differential geometry of $TM\oplus T^{*}M$, as well as the
$\mathcal{E}(x)$-deformed graded Poisson algebra with its Q structure.

A similar result can now be unveiled for the contravariant connection (51). It
concerns a gravitational field together with a kinetic term for the field
strength of $\Pi$, where the exterior derivative is taken with the anchor and
hence on the $1$-forms corresponds to $R$ and $\mathcal{X}$ as in our formulas
(52) and (54). These are the local expressions of the stringy $R$ and $Q$
fluxes. The invariant Lagrangian is given by a scalar built from the Ricci
curvature for the $T^{*}M$-subbundle. In components the Ricci tensor
$\text{Ric}_{T^{*}M}\in\Gamma(TM\otimes TM)$ from (58) is

$\text{Ric}^{jk}_{T^{*}M}=\left(G^{-1}+\Pi\right)^{[l|m}\partial_{m}\mathit{\tilde{\Gamma}}^{|j]k}{}_{l}+\mathit{\tilde{\Gamma}}^{[l|m}{}_{l}\mathit{\tilde{\Gamma}}^{|j]k}{}_{m},$
(60)

where indices are contracted by means of the identity matrix of
$\text{End}(T^{*}M)$. Notice also that the antisymmetrization is performed
here without factors of $\frac{1}{2}$. Let us abbreviate the symmetric part of
the connection with the covariant derivative $\tilde{D}_{G}$:

$\tilde{D}_{G}^{i}\varsigma:=\left(G^{-1}+\Pi\right)^{im}\partial_{m}\varsigma+\varsigma_{k}\left(\Gamma_{G\;\;\;j}^{\;\;\;ik}+\mathcal{Y}^{ik}{}_{j}\right)dx^{j}.$

Then the Ricci tensor is:

$\text{Ric}^{jk}_{T^{*}M}=\,\text{Ric}_{G}^{jk}-\frac{1}{4}R^{jm}{}_{l}R^{lk}{}_{m}-\frac{1}{2}R^{(j|m}{}_{i}\mathcal{X}^{i|k)}{}_{m}-\mathcal{X}^{jm}{}_{i}\mathcal{X}^{ik}{}_{m}+\frac{1}{2}\tilde{D}_{G}^{i}R^{jk}{}_{i}+\tilde{D}_{G}^{i}\mathcal{X}^{jk}{}_{i}.$
(61)

In this expression, the first four terms are completely symmetric, while the
last two are purely antisymmetric. $\text{Ric}_{G}$ is the Ricci tensor, on
the cotangent bundle, for the symmetric part of the connection, i.e. it is the
commutator of a pair of $\tilde{D}_{G}$. It is worth to stress again that it
is the curvature tensor for a connection on forms, metric with respect to
$G^{-1}$. The explicit expression for $\text{Ric}_{G}^{jk}$ is:

$\text{Ric}_{G}^{jk}=\left(G^{-1}+\Pi\right)^{[l|m}\partial_{m}\left(\Gamma_{G\;\;\;l}^{\;\;|j]k}+\mathcal{Y}^{|j]k}{}_{l}\right)+\left(\Gamma_{G\;\;\;\;\;\;l}^{\;\;[l|m}+\mathcal{Y}^{[l|m}{}_{l}\right)\left(\Gamma_{G\;\;\;\,m}^{\;\;|j]k}+\mathcal{Y}^{|j]k}{}_{m}\right).$

To retain the antisymmetric terms of the Ricci curvature tensor we can
contract $\text{Ric}_{T^{*}M}$ with the volume form and the non-symmetric
$\left(G^{-1}-\Pi\right)$, in this combination:

$\sqrt{\det G^{-1}}\left(G\circ\left(G^{-1}-\Pi\right)\circ G\right)_{jk}.$
(62)

In fact the following relation holds:

$\left(G^{-1}+\Pi\right)^{jl}\partial_{l}\left(\sqrt{\det
G^{-1}}\,w\right)=\sqrt{\det
G^{-1}}\,\left(\Gamma_{G\;\;\;k}^{\;\;\;kj}+\mathcal{Y}_{\;\;\;k}^{kj}\right)w=\sqrt{\det
G^{-1}}\,\tilde{D}_{G}^{j}w\equiv\sqrt{\det G^{-1}}\,\widetilde{\nabla}^{j}w.$

This says that the total derivative of the volume form times a scalar is equal
to the divergence, accompanied with the scalar half-density $\sqrt{\det
G^{-1}}$. Hence we can perform integration by parts on the contracted action,
use Poincaré lemma for the boundary term and implement the following
observation:

$\tilde{D}^{l}_{G}\Pi_{jk}\equiv
G_{jn}G_{rk}\left(G^{-1}+\Pi\right)^{lm}\partial_{m}\Pi^{nr}=G_{jn}\left(\frac{1}{3}R^{lnr}+G^{lm}Q^{nr}{}_{m}\right)G_{rk}.$

Eventually we end up with the action functional666Despite the possible
confusion, we deploy $\mathrm{d}x$ to allude to a $d$-dimensional top form.:

$\displaystyle S[G^{-1},\Pi]=\int_{M}\mathrm{d}x\,\sqrt{\det G^{-1}}$
$\displaystyle\bigg{[}\text{R}_{G}-\frac{1}{12}R^{2}+\dfrac{3}{4}\left(Q^{jn}{}_{m}\left(G^{pm}Q^{lk}{}_{p}-2Q^{ml}{}_{p}G^{pk}\right)\right)G_{jl}G_{nk}$
$\displaystyle-\left(\dfrac{1}{6}Q^{jn}{}_{s}+\dfrac{2}{3}G_{rs}G^{jp}Q^{nr}{}_{p}\right)R^{slk}G_{jl}G_{nk}\bigg{]}.$
(63)

It comprehends therefore the relevant tensors that are T-dual to the $H$-flux
in compactifications (e.g. on a $\mathbb{T}_{3}$ torus) of string theory (NS-
NS sector, with zero dilaton). It is manifestly invariant under
diffeomorphisms. The action (63) is the Einstein-Hilbert term for a connection
living in the cotangent space solely, and built with the open string metric
and bivector $\Pi$. It is obtained through the same mechanism with which we
managed to rebuild the NS-NS sector of closed string effective action for zero
dilaton (59), from the Levi-Civita connection with $H$ torsion
$\nabla^{\text{L.C.}}-\frac{1}{2}g^{-1}\circ H$. Also, the final contraction
retained the antisymmetric part as well.

###  5.1 Invariance under gauge symmetries of $\Pi$

The gravitational actions that we constructed are manifestly invariant under
diffeomorphisms, but this is not the end the story. The first action (59) is
also invariant under the gauge symmetry of the $B$ field, $B\mapsto
B+\mathrm{d}\lambda$. Therefore one could imagine that the second action (63)
might be invariant under the gauge symmetry of $\Pi$.

As anticipated in section 2.3, the generators of gauge symmetries are Q-exact
degree $2$ functions $\alpha$ quadratic in the $\xi$ coordinates. Though for
the non-symmetric theory of gravity (59) we can guess them easily by knowing
that $H=\mathrm{d}B$, for $S[G^{-1},\Pi]$ (63) we must crank the machinery of
section 2.3, and find $\alpha$ from (21) subjected to the condition (20). The
kernel of our anchor map comprises the sections:

$\Upsilon:=\begin{pmatrix}\varrho\\\
-\left(G^{-1}+\Pi\right)^{-1}(\varrho)\end{pmatrix},\quad\varrho\in\Gamma(TM).$
(64)

These sections are the graph of $-\left(G^{-1}+\Pi\right)^{-1}=-(g+B)$.
However, also the graph of the inverse map could be taken to characterize the
kernel of the anchor. The result is independent of the choice, but the
computation can gain some feasibility. Expression (64) is very efficient to
derive the gauge symmetries of $B\in\Omega^{2}(M)$ in the graded geometric
setting, where they can be easily recognized as the exterior derivative of a
$1$-form. To better tackle the case of $\Pi$, we can resort to the following
expression for the sections in the kernel of $\rho$:

$\Upsilon:=\begin{pmatrix}-\left(G^{-1}+\Pi\right)(\varsigma)\\\
\varsigma\end{pmatrix},\quad\varsigma\in\Gamma(T^{*}M).$ (65)

Now we must carefully select the Q-exact sections $\alpha$,

$\alpha^{\beta\gamma}=\tilde{\rho}^{\beta i}\nabla_{i}\Upsilon^{\gamma},$

that belong to the subspace of quadratic functions in $\theta$. In fact in the
deformed Poisson algebra, dualization occurs by means of $G^{-1}$. Using the
connection symbols in the second row of (27) and sending $\theta$ into
$G^{-1}\chi$ we end up with the sought-after $\delta\Pi^{nm}=\alpha^{nm}$:

$\displaystyle\delta\Pi^{jk}=\dfrac{1}{2}\left(G^{-1}+\Pi\right)^{jl}\bigg{(}$
$\displaystyle\varsigma_{i}\partial_{l}\left(G^{-1}+\Pi\right)^{ik}-\left(G^{-1}+\Pi\right)(\varsigma)^{i}\left(G^{-1}-\Pi\right)^{-1}_{im}\partial_{l}\left(G^{-1}-\Pi\right)^{mk}$
$\displaystyle+2G^{km}\partial_{l}\varsigma_{m}(x)\bigg{)}.$ (66)

The gauge symmetry of $\Pi$ (66) looks a bit contrived, but it would be plain
wrong to guess $\Pi\mapsto\Pi+\mathrm{d}_{\rho}\varsigma$, because the
bivector sources both the $Q$-flux and the $R$-flux in our model.

We have thus represented the gauge symmetry of $\Pi$ (66) in the graded
symplectic setting. But then the Dorfman bracket, being the derived bracket of
the graded Poisson bracket, preserves this very same symmetry. Now to be able
to conclude that the whole construction, which relies on proposition 4.4, does
not spoil the symmetry, we must verify that the generalized Lie bracket is
invariant under the same symmetry of the Dorfman bracket. For this to happen,
the derived bracket construction is again useful: according to the derived
bracket formula for the generalized Lie bracket (38),
$\varsigma_{i}(x)\left(-\left(G^{-1}+\Pi\right)^{ij}\chi_{j}+\theta^{i}\right)$
must be Q̌-closed (37). With our anchor map, Q̌ is:

$\text{\v{Q}}=p_{i}\dfrac{\partial}{\partial\chi_{i}}+\left(G^{-1}+\Pi\right)^{ji}p_{i}\dfrac{\partial}{\partial\theta^{j}}.$

The conclusion is immediate: $\Upsilon$ is closed under the homological vector
field Q̌ which fails to be Hamiltonian. Hence, requiring that also every $\Pi$
involved with the anchor map is left unchanged, $R$ (52) and $\mathcal{X}$
(54) are invariant under the transformation $\Pi\mapsto\Pi+\delta\Pi$.
Invariance of the action (63) follows.

A similar action with non-geometric fluxes, that is also invariant under the
gauge symmmetry of the bivector $\Pi$, can be found in Andriot:2013xca  and
Blumenhagen:2013aia . Under different conditions, the work Asakawa:2015jza
proposed a gravitational theory for a metric in the background of the
$R$-flux, exhibiting invariance under the gauge symmetries of a Poisson
bivector. The authors used a connection on a Poisson Courant algebroid (i.e. a
CA twisted with a Poisson bivector). Its curvature scalar consists of the
Ricci scalar for what in our notation is $\Gamma_{G}$ and of the square of a
$R$ flux given by a Poisson bivector $\theta$ and our $\Pi$ as
$R:=[\theta,\Pi]_{\text{S}}$. They proved the invariance of the gravity action
under $\Pi\mapsto\Pi+\mathrm{d}_{\theta}\pi$, where the differential is the
same as the $\rho$-differential used here and $\theta$ serves as an anchor
map.

##  6 Discussion and outlook

Let us conclude the article with some comments on the achievements, some
remarks on the important details and an outlook on future research.

The purpose of the work was twofold: On one hand, Moser lemma allowed us to
select a non-canonical graded symplectic structure, involving the geometric
data of a metric and a $2$-form on the space of degree-$1$ (Grassmann)
variables. The other coordinates of the graded manifold were not transformed.
The degree-$1$ objects therefore ceased to be $O(d,d)$-vectors but they rather
belonged to a $O(d)\times O(d)$ representation. On the other hand, the derived
bracket of the Poisson brackets with Hamiltonian
$\Theta=\xi_{\alpha}\tilde{\rho}^{\alpha i}p_{i}$, $\tilde{\rho}^{\alpha i}$
in (29) and $C^{\alpha\beta\gamma}=0$, alas the Dorfman bracket of the exact
CA $E\cong TM\oplus T^{*}M$, had a simple coordinate expression in which
covariant derivatives replaced the partial ones. The connection giving rise to
a covariant derivative is due to the modified Poisson bracket between degree
$1$ functions and degree $2$ linear functions of $p$ (see (28)). The CA
bracket is a derivation on sections, but when the antisymmetric Lie-like
bracket on the generalized vector fields (35) is subtracted it becomes an
affine connection $\widetilde{\nabla}$. This new connection turns out to be
given by the previous covariant derivative added to its torsion tensor
(defined with the antisymmetric $\llbracket\cdot,\cdot\rrbracket$). We studied
$\widetilde{\nabla}$ in two “limits”: when all of its arguments were projected
onto $TM$, and when they were projected onto $T^{*}M$. Since the induced
metrics on $TM$ and $T^{*}M$ are non-degenerate, we obtained the connections
$\widetilde{\nabla}:\Gamma(TM)\mapsto\Gamma(T^{*}M\otimes TM)$ (49) and
$\widetilde{\nabla}:\Gamma(T^{*}M)\mapsto\Gamma\left(TM\otimes T^{*}M\right)$
(51). The curvature invariants for these were shown to yield Lagrangians of
physical interest. In the former case, we have recovered the NS-NS sector of
low-energy closed string theory (without dilaton), from a curvature scalar
that retained the antisymmetric contribution. Something similar happened in
the latter case, upon integrating the Ricci curvature against a non-symmetric
tensor. The action obtained in that way consisted of the Ricci scalar for the
Levi-Civita connection of the contravariant metric $G^{-1}$ (where however the
anchor comprises both $G^{-1}$ and $\Pi$) and other invariant terms built with
the local expressions of the $Q$ and $R$ fields of string compactification.
Moreover $S[g,B]$ (59) and $S\left[G^{-1},\Pi\right]$ (63) hold for an
arbitrary number of dimensions.

The findings presented here share some similarities with other works on the
subjects, and they differ in some aspects. We focused on a very minimal
deformation of the graded symplectic structure, as only the algebra of degree
$1$ functions was reshaped. In this way, and by the chosen particular
deformation with a $2$-form $B$ and a bivector $\Pi$, dependent on $B$, we
could interpret the stringy 3-form $H$ and the connection coefficients $Q$ and
$R$ (where the first has 1 contravariant and 2 covariant indices, and the
latter is a trivector) as formal “field strengths” of $B$ and $\Pi$. In our
companion work Boffo:2020vqx  we considered a deformation based only the open
string quantities $G^{-1}$ and $\Pi$, and obtained the same action functional.
Usually in dg-symplectic geometry the stringy fluxes appear in the
Hamiltonian, and are responsible for a twisting of the Dorfman bracket of the
associated CA. Our approach via deformation yields global objects
(connections, actions) and does not leave the realm of (more or less) standard
geometry in favor of what is conventionally known as non-geometry. Instead
this happens for backgrounds with $R$-flux: local patchings cannot be glued
together. Instead our $R$ is merely the dual exterior derivative of the
bivector $\Pi$ and therefore a standard trivector, and that explains why it
was possible to suggest an action for a contravariant symmetric tensor in the
presence of $\Pi$.

Deforming the $2$-graded Poisson algebra gives a technique to include
interactions with gauge potentials, which is most commonly deployed for the
phase space algebra of a point particle in a given electromagnetic field.
Locally the (graded) Poisson algebra has the facets of a gauge theory, i.e. a
principal bundle with some structure group, and in this perspective some
problematic aspects can be tackled in an easier manner: For example, the new
graded Poisson algebra (13) can be quantized. Its quantization can be of Weyl
type or Clifford type, depending on the parity of the algebra of functions.
Anyway this quantization concerns the particle in a non-flat background having
interactions with $B$ and $\Pi$, not the gravity theories themselves.

Our construction differs from the T-dual field redefinition of the NS-NS
bosonic closed string action Andriot:2013xca , its Lie algebroid description
Blumenhagen:2013aia , Blumenhagen:2012nk  and their respective generalized
Ricci scalars, for example because of our different choice of the basis of the
fiber, that treats equally the metric and the antisymmetric field, and also
because of the introduction of the generalized Lie bracket. Another big
difference is due to the simultaneous presence in the ansatz for the vielbein
of $g-B$ and $\left(g+B\right)^{-1}$, so that the anchor map is forced to be
$(G^{-1}+\Pi)(\sigma)$, while so far just $\Pi(\sigma)$ was employed as anchor
in the above references. Clearly if we had to choose just $\iota_{e^{a}}\Pi$
as new basis for the $1$-forms we would not get a block diagonal generalized
metric on the CA $E$. Notice that our systematic method returns a well-defined
connection on generalized vectors, provided that a generalized Lie bracket is
given, and it is simple to define a curvature tensor in our framework. This is
not so obvious in Generalized Geometry. We did not double the dimension of the
base manifold. A further analysis on the relations with DFT is beyond the
scopes of this work: however it is rather reasonable to expect our action
functional to be some limit of the DFT invariant action of Hohm, Hull and
Zwiebach Hohm:2010pp  when the section condition is applied. A similar
setting, namely a graded Poisson algebra with the trivector $R$, was related
to the section condition of DFT in Heller:2017mwz .

We currently lack a construction that takes the R-R fields into account. We
nevertheless expect these fields, which are tensors of the spin bundle, to be
treatable within the realm of a graded Poisson structure of degree $2$; they
have been already described in Generalized Geometry for example in
Severa:2018pag  and Coimbra:2011nw .

Acknowledgements: E.B. and P.S. are grateful to the RTG 1620 “Models of
Gravity” for funding. E.B. also thanks GAČR Grant EXPRO 19-28628X for support.
The authors want to thank A. Chatzistavrakidis and M. Pinkwart-Walker for
discussion and suggestions on the work. Thanks to M. Jotz Lean for bringing
the dull bracket to our attention.

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