# Fluxes of Courant bracket twisted at the same time by $B$ and $\theta$
††thanks: Work supported in part by the Serbian Ministry of Science,
Technological Development and Innovation

Lj. Davidović , I. Ivanišević and B. Sazdović
Institute of Physics, University of Belgrade
Pregrevica 118, 11080 Belgrade, Serbia e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

This paper investigates the simultaneous twisting of the Courant bracket by a
2-form $B$ and a bi-vector $\theta$, exploring the generalized fluxes obtained
in Courant algebroid relations. We define the twisted Lie bracket and
demonstrate that the generalized $H$-flux can be expressed as the field
strength defined on this Lie algebroid. Similarly, we show that the $f$-flux
is a direct consequence of simultaneous twisting, which arises in the twisted
Lie bracket relations between holonomic partial derivatives. We obtain the
generalized $Q$ flux in terms of the twisted Koszul bracket, which is a quasi-
Lie algebroid bracket. The action of an exterior derivative related to the
twisted Koszul bracket on a bi-vector produces the generalized $R$-flux. We
show that the generalized $R$-flux is also the twisted Schouten-Nijenhuis
bracket, i.e. the natural graded bracket on multi-vectors defined with the
twisted Lie bracket.

##  1 Introduction

The unification of vectors and 1-forms into generalized vectors has led to the
development of generalized geometry, where various geometrical structures,
including Poisson, symplectic, and Calabi-Yau manifolds, naturally emerge. The
smooth section of the generalized tangent bundle is equipped with a Courant
bracket, which reduces to the Lie bracket on pure vector fields [1]. In
bosonic string theory, this skew-symmetric bracket emerges in the Poisson
bracket algebra of a generator governing general coordinate and local gauge
transformations [2, 3]. The two symmetries are generated by canonical
variables - the former by coordinate $\sigma$-derivatives, while the latter by
canonical momenta. The relation of these variables by T-duality allowed for
interpretation of the Courant bracket as the extension of Lie bracket
invariant under T-duality [3].

In addition, the framework of generalized geometry is suitable for description
of various fluxes. String fluxes [4] are important since they can create a
potential that stabilizes the vacuum expectation value and provides mass to
the massless non-physical particles called moduli [5, 6, 7]. They describe
geometry of different backgrounds connected by T-duality [8, 9, 10, 11].
Moreover, fluxes are closely linked to the non-commutativity and non-
associativity of string coordinates [12, 13]. In the context relevant for this
paper, fluxes can emerge as a structure functions in Poisson bracket relations
between non-canonical currents [14, 15]. The change of canonical variables
with currents in generator of symmetry results in various twisted Courant
brackets in Poisson bracket algebra of new generator [3]. Some of the well
studied examples include the $B$-twisted Courant bracket, in which $H$-flux
appears, the $\theta$-twisted Courant bracket, in which $Q$ and $R$-fluxes
appear, and the so called Roytenberg bracket, containing all generalized
fluxes [16, 17]. The generalized fluxes were also obtained from generalized
vielbeins [18]. Furthermore, generalized fluxes were shown to measure the
failure of twisted Courant brackets to satisfy Jacobi identity [19].

The twisting of the Courant bracket in general comes with the cost of losing
the invariance of that bracket under T-duality. For instance, the $B$-twisted
Courant bracket becomes the $\theta$-twisted Courant bracket when T-duality is
performed [15]. Moreover, there were some successful attempts to describe the
topological T-duality as an isomorphism between Courant algebroids with
different twisted Courant brackets as their underlining brackets [20].
Additionally, the T-duality between $H$ and $R$ fluxes via Courant algebroids
was also described in [21].

The construction of the extended Courant bracket containing all fluxes has
been done in multiple instances [18, 19, 16, 17]. All these approaches are
equivalent to twisting the Courant bracket firstly by $B$, and then by
$\theta$. The bracket obtained this way is however not invariant under the
T-duality, which is a consequence of the fact that $B$-shifts and
$\theta$-transformations do not commute. That is why in [22], the Courant
bracket that was simultaneously twisted by $B$ and $\theta$ was constructed.
The idea was that instead of twisting firstly by $B$ and then by $\theta$,
simultaneous twisting would a priori define currents that mutually transform
one into another under T-duality. In this paper, our goal is to obtain and
analyse all the fluxes appearing in the algebra relations of the Courant
bracket simultaneously twisted by both $B$ and $\theta$.

We begin the paper by outlining the basic mathematical notions of Lie
algebroids, including the bracket, exterior derivative and graded
generalization of Lie bracket on multi-vectors. We will introduce the Lie
bracket generalization on the space of differential forms, called the Koszul
bracket, as well as the Lie bracket generalization on the space of multi-
vectors, called the Schouten-Nijenhuis bracket. These brackets can easily be
generalized to arbitrary Lie algebroids, which will turn out to be useful for
understanding the nature of the fluxes in the rest of the paper. Additionally,
we will define the Courant bracket together with its twisted versions, and
show the most well known examples of the Courant brackets twisted by $B$ and
$\theta$, which are performed by $e^{\hat{B}}$ and $e^{\hat{\theta}}$
transformations, respectively.

In the third chapter, we will present the twist defining the Courant bracket
simultaneously twisted by both $B$ and $\theta$. The key in construction is
considering the matrix $\breve{B}$, which is a direct sum of matrices
$\hat{B}$ and $\hat{\theta}$. The resulting twisting matrix features
hyperbolic functions of the product of fields $\theta$ and $B$ and is very
complicated transformation to work with. That is why we will introduce another
$O(D,D)$ rotation, so that the currents have more recognizable form and are
easier to work with.

The expressions for generalized fluxes will be derived in the fourth chapter
from the Poisson bracket algebra of the mutually T-dual currents. It will be
demonstrated that the obtained generalized fluxes exhibit same relations as
the fluxes obtained in the algebra relations of Roytenberg bracket.

In the fifth chapter, we will analyse geometries suitable for writing fluxes
in the coordinate free notation. We introduce the twisted Lie bracket, as a
bracket of the Lie algebroid defined with the hyperbolic function ${\cal C}$
as its anchor, and show that its algebra between holonomic partial derivatives
gives rise to the $\breve{f}$-flux appearing in the algebra relations of
currents. As for the $\breve{H}$-flux, we will express it as the action of the
exterior derivative related to the twisted Lie bracket on the 2-form
$\hat{B}$.

In the following chapter, we also define the twisted Koszul bracket. We demand
that the relation between the twisted Koszul bracket and the twisted Lie
bracket mirrors the relation between their untwisted counterparts. This is a
bracket of the twisted Lie algebroid. We demonstrate that the twisted Koszul
bracket between two differential one-forms produces the $\breve{Q}$-flux.

Lastly, we introduce the twisted Schouten-Nijenhuis bracket as the graded
extension of the twisted Lie bracket. We show that the twisted Schouten-
Nijenhuis bracket of the appropriate bi-vector with itself produces exactly
the $\breve{R}$-flux.

##  2 Mathematical preliminaries: from Lie to Courant algebroids

This mathematical introduction we begin with the notion of Lie algebroid.
Broadly speaking, a Lie algebroid can be thought of as a structure in which
the traditional tangent bundle is substituted with a vector bundle that shares
similar properties. As a result, numerous geometrical concepts that
traditionally rely on the tangent bundle are easily generalized to the context
of Lie algebroids [23]. These geometrical concepts will appear to be
convenient to describe the fluxes that we are considering in this paper.

###  2.1 Lie algebroid

Lie algebroid [24, 25] is a triple $\Big{(}V,[,],\rho\Big{)}$ consisting of a
vector bundle $V$, the anchor $\rho:V\to T{\cal M}$, and the skew-symmetric
bracket $[,]$ on the space of smooth section of $V$, so that the following
compatibility conditions are satisfied:

$\displaystyle\rho[\xi_{1},\xi_{2}]=[\rho(\xi_{1}),\rho(\xi_{2})]_{L}\,,$
(2.1) $\displaystyle[\xi_{1},f\xi_{2}]=f[\xi_{1},\xi_{2}]+({\cal
L}_{\rho(\xi_{1})}f)\xi_{2}\,,$ (2.2)
$\displaystyle[\xi_{1},[\xi_{2},\xi_{3}]]+[\xi_{2},[\xi_{3},\xi_{1}]]+[\xi_{3},[\xi_{1},\xi_{2}]]=0\,,$
(2.3)

where $[,]_{L}$ denotes the usual Lie bracket. To gain some intuition, it is
worth considering the most simple example of the Lie algebroid, consisting of
the tangent bundle, the identity operator as the anchor $\rho=\text{Id}$ and
the usual Lie bracket. The first condition (2.1) is trivial, while the
remaining conditions (2.2) and (2.3) are respectively well known Leibniz rule
and Jacobi identity for the Lie bracket.

From the compatibility conditions in the Lie algebroid definition, we see that
on this structure we can define a Lie derivative. Its action on functions is
defined by

${\cal\hat{L}}_{\xi}f={\cal L}_{\rho(\xi)}f\,,$ (2.4)

and on vectors by the Lie algebroid bracket

${\cal\hat{L}}_{\xi_{1}}\xi_{2}=[\xi_{1},\xi_{2}]\,.$ (2.5)

Additionally, we can introduce the analog of the exterior derivative. Recall
that the exterior derivative extends the concept of the differential of a
function to differential forms. The analog of the exterior derivative is a
nilpotent operator given by [23]

$\displaystyle\hat{d}\lambda(\xi_{0},...,\xi_{p})$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{p}(-1)^{i}{\cal
L}_{\rho(\xi_{i})}\Big{(}\lambda(\xi_{0},...,\hat{\xi}_{i},...,\xi_{p})\Big{)}$
$\displaystyle+\sum_{i<j}(-1)^{i+j}\lambda([\xi_{i},\xi_{j}],\xi_{0},...,\hat{\xi}_{i},...,\hat{\xi}_{j},...,\xi_{p})\,.$

Here, $\lambda$ is a $p$-covariant tensor from the section of dual space to
the Lie algebroid bundle $V$. By
$\lambda(\xi_{0},...,\hat{\xi}_{i},...,\xi_{p})$ we marked this tensor
contracted with $p$ different vectors $\xi_{0}\,,...,\xi_{p}$, while the hat
above the vector $\hat{\xi}_{i}$ denotes omission of that specific element. In
case of a trivial Lie algebroid with identity anchor, the above relation
defines the action of usual exterior derivative on $p$-forms. However, we
should stress out that in case of general vector bundles, $\lambda$ should not
be understood as $p$-forms. For example, if a Lie algebroid is defined on
cotangent bundle, $\xi$ will be one-forms, while $\lambda$ will be
$p$-vectors.

####  2.1.1 Schouten-Nijenhuis bracket

The Schouten-Nijenhuis bracket [26] is a natural way to extend the
generalization of the Lie bracket to the multi-vectors. It is defined by

$[f,g]_{S}=0\,,\quad[\xi,f]_{S}={\cal
L}_{\xi}(f)\,,\quad[\xi_{1},\xi_{2}]_{S}=[\xi_{1},\xi_{2}]_{L}\,,$ (2.7)

where other brackets are obtained with the help of the graded Leibniz rule

$[\theta_{1},\theta_{2}\wedge\theta_{3}]_{S}=[\theta_{1},\theta_{2}]_{S}\wedge\theta_{3}+(-1)^{(p-1)q}\theta_{2}\wedge[\theta_{1},\theta_{3}]_{S}\,,$
(2.8)

and the graded commutative relation

$[\theta_{1},\theta_{2}]_{S}=-(-1)^{(p-1)(q-1)}[\theta_{2},\theta_{1}]_{S}\,,$
(2.9)

where $p$, and $q$ are orders of multi-vectors $\theta_{1}$ and $\theta_{2}$,
respectively.

It is also possible to obtain the coordinate expression for the Schouten-
Nijenhuis bracket. Let

$\displaystyle\theta_{1}=\frac{1}{p!}\theta_{1}^{\mu_{1},...\mu_{p}}\partial_{\mu_{1}}\wedge...\wedge\partial_{\mu_{p}}\,,\quad\theta_{2}=\frac{1}{q!}\theta_{2}^{\nu_{1},...\nu_{q}}\partial_{\nu_{1}}\wedge...\wedge\partial_{\nu_{q}}\,,$
(2.10)

then we have [26]

$\displaystyle[\theta_{1},\theta_{2}]_{S}$ $\displaystyle=$
$\displaystyle\frac{1}{(p+q-1)!}[\theta_{1},\theta_{2}]^{\mu_{1}...\mu_{p+q-1}}_{S}\partial_{\mu_{1}}\wedge...\wedge\partial_{\mu_{p+q-1}}\,,$
(2.11) $\displaystyle[\theta_{1},\theta_{2}]^{\mu_{1}...\mu_{p+q-1}}_{S}$
$\displaystyle=$
$\displaystyle\frac{1}{(p-1)!q!}\epsilon^{\mu_{1}...\mu_{p+q-1}}_{\nu_{1}...\nu_{p-1}\rho_{1}...\rho_{q}}\theta_{1}^{\sigma\nu_{1}...\nu_{p-1}}\partial_{\sigma}\theta_{2}^{\rho_{1}...\rho_{q}}$
$\displaystyle+\frac{(-1)^{p}}{p!(q-1)!}\epsilon^{\mu_{1}...\mu_{p+q-1}}_{\nu_{1}...\nu_{p}\rho_{1}...\rho_{q-1}}\theta_{2}^{\sigma\rho_{1}...\rho_{q-1}}\partial_{\sigma}\theta_{1}^{\nu_{1}...\nu_{p}}\,,$

where the antisymmetric Levi Civita symbol is defined by

$\displaystyle\epsilon^{\mu_{1}...\mu_{p}}_{\nu_{1}...\nu_{p}}=\begin{vmatrix}\delta^{\mu_{1}}_{\nu_{1}}&...&\delta^{\mu_{1}}_{\nu_{p}}\\\
.&&.\\\ .&&.\\\ .&&.\\\
\delta^{\mu_{p}}_{\nu_{1}}&...&\delta^{\mu_{p}}_{\nu_{p}}\end{vmatrix}\,.$
(2.12)

The Lie algebroid generalization of the Schouten-Nijenhuis bracket is defined
by

$[f,g]_{{\bf S}}=0\,,\quad[\xi,f]_{{\bf S}}={\cal
L}_{\rho(\xi)}(f)\,,\quad[\xi_{1},\xi_{2}]_{{\bf S}}=[\xi_{1},\xi_{2}]\,,$
(2.13)

with other brackets satisfying the graded identities (2.8) and (2.9). This is
a bracket on a smooth section of higher product of vector bundle on which the
Lie algebroid is defined. In case when the Lie algebroid bracket is defined on
the smooth section of tangent bundle, its corresponding Schouten-Nijenhuis
bracket will be defined on multi-vectors, but generally this is not the case.
As such, the expression (2.11) cannot be easily generalized for an arbitrary
Lie algebroid, rather it has to be derived in each different case.

####  2.1.2 Koszul bracket

Let us shortly illustrate the notion of Lie algebroid with a less trivial but
typical (and for our paper important) example, consisting of the cotangent
bundle $T^{\star}{\cal M}$ as its vector bundle. Elements of its smooth
section are differential 1-forms $\lambda$. Let $\theta$ be a Poisson bi-
vector, i.e. let $[\theta,\theta]_{S}=0$, where

$\left.[\theta,\theta]_{S}\right|^{\mu\nu\rho}=2(\theta^{\mu\sigma}\partial_{\sigma}\theta^{\nu\rho}+\theta^{\nu\sigma}\partial_{\sigma}\theta^{\rho\mu}+\theta^{\rho\sigma}\partial_{\sigma}\theta^{\mu\nu})=0\,.$
(2.14)

This bi-vector defines a morphism by

$\theta(\lambda_{1})\lambda_{2}=\theta(\lambda_{1},\lambda_{2})\,,\quad\Big{(}\theta(\lambda_{1})\Big{)}^{\\!\mu}=\lambda_{1\nu}\theta^{\nu\mu}\,,$
(2.15)

that satisfies

$\Big{(}\theta([\lambda_{1},\lambda_{2}]_{\theta})\Big{)}=[\theta(\lambda_{1}),\theta(\lambda_{2})]_{L}\,,$
(2.16)

where we introduced the Koszul bracket [27] which in the non-coordinate
notation is given by

$[\lambda_{1},\lambda_{2}]_{\theta}={\cal
L}_{\theta(\lambda_{1})}\lambda_{2}-{\cal
L}_{\theta(\lambda_{2})}\lambda_{1}-d(\theta(\lambda_{1},\lambda_{2}))\,,$
(2.17)

or in some local basis by

$\Big{(}[\lambda_{1},\lambda_{2}]_{\theta}\Big{)}_{\\!\\!\mu}=\theta^{\nu\rho}(\lambda_{1\nu}\partial_{\rho}\lambda_{2\mu}-\lambda_{2\nu}\partial_{\rho}\lambda_{1\mu})+\lambda_{1\rho}\lambda_{2\nu}\partial_{\mu}\theta^{\rho\nu}\,.$
(2.18)

The cotangent bundle, with Koszul bracket and Poisson bi-vector as an anchor
defines a Lie algebroid. The corresponding exterior derivative (2.1) acts on
multi-vector. Its action can be expressed in terms of Schouten-Nijenhuis
bracket by

$d_{\theta}=[\theta,.]_{S}\,.$ (2.19)

Its square is zero, if and only if the bi-vector is Poisson bi-vector.

Even in the case when a bi-vector is not Poisson bi-vector, i.e. the condition
(2.14) does not hold, it is still possible to define the Koszul bracket, even
the exterior derivative of the Koszul bracket. However, in this instance, the
Koszul bracket will not satisfy the Jacobi identity, and its corresponding
exterior derivative will not be nilpotent. In this instance, we say that we
have a quasi-Lie algebroid.

###  2.2 Generalized tangent bundle

The introduction of Lie algebroid allowed generalization of the geometric
terms to arbitrary vector bundles. This can be generalized further by
considering a direct sum of the tangent and cotangent bundle over a manifold
$T{\cal M}\oplus T^{\star}{\cal M}$, which is called the generalized tangent
bundle [28, 29]. The elements of its section are generalized vectors

$\Lambda^{M}=\xi^{\mu}\oplus\lambda_{\mu}=\begin{pmatrix}\xi^{\mu}\\\
\lambda_{\mu}\end{pmatrix}\,,$ (2.20)

where $\xi$ represents its vector components, and $\lambda$ represents its
1-form components. The interior product defines a natural way to combine
vectors and 1-forms into a scalar $i_{\xi}\lambda=\xi^{\mu}\lambda_{\mu}$. We
can use this to define an inner product on the smooth section of the
generalized tangent bundle by

$\langle\Lambda_{1},\Lambda_{2}\rangle=\langle\xi_{1}\oplus\lambda_{1},\xi_{2}\oplus\lambda_{2}\rangle=i_{\xi_{1}}\lambda_{2}+i_{\xi_{2}}\lambda_{1}\,.$
(2.21)

The group of transformations that keeps this inner product invariant is
$O(D,D)$ group [30, 31], which is closely related to the string theory
T-duality.

There is a natural extension of the Lie bracket on the smooth section of
generalized tangent bundle called Courant bracket. Its expression is given by

$\displaystyle[\Lambda_{1},\Lambda_{2}]_{\cal C}$ $\displaystyle=$
$\displaystyle\xi\oplus\lambda$ (2.22) $\displaystyle\xi$ $\displaystyle=$
$\displaystyle[\xi_{1},\xi_{2}]_{L}\,,$ $\displaystyle\lambda$
$\displaystyle=$ $\displaystyle{\cal L}_{\xi_{1}}\lambda_{2}-{\cal
L}_{\xi_{2}}\lambda_{1}-\frac{1}{2}d(i_{\xi_{1}}\lambda_{2}-i_{\xi_{2}}\lambda_{1})\,,$

or

$\displaystyle\xi^{\mu}$ $\displaystyle=$
$\displaystyle\xi_{1}^{\nu}\partial_{\nu}\xi_{2}^{\mu}-\xi_{2}^{\nu}\partial_{\nu}\xi_{1}^{\mu}\,,$
(2.23) $\displaystyle\lambda_{\mu}$ $\displaystyle=$
$\displaystyle\xi_{1}^{\nu}(\partial_{\nu}\lambda_{2\mu}-\partial_{\mu}\lambda_{2\nu})-\xi_{2}^{\nu}(\partial_{\nu}\lambda_{1\mu}-\partial_{\mu}\lambda_{1\nu})+\frac{1}{2}\partial_{\mu}(\xi_{1}\lambda_{2}-\xi_{2}\lambda_{1})\,.$

This bracket is known to appear in string theory context in the Poisson
bracket algebra of symmetry generators. Symmetries in string theory
$\sigma$-model are governed by generators that in classical theory act on
Hamiltonian via Poisson bracket. General coordinate transformations, or
diffeomorphisms, are generated by [32]

${\cal
G}_{\xi}=\int_{0}^{2\pi}d\sigma\xi^{\mu}(x(\sigma))\pi_{\mu}(\sigma)\,,$
(2.24)

that leads to string background field transformations

$\displaystyle\delta_{\xi}G_{\mu\nu}$ $\displaystyle=$ $\displaystyle{\cal
L}_{\xi}G_{\mu\nu}=\xi^{\rho}\partial_{\rho}G_{\mu\nu}+\partial_{\mu}\xi^{\rho}G_{\rho\nu}+\partial_{\nu}\xi^{\rho}G_{\rho\mu}\,,$
(2.25) $\displaystyle\delta_{\xi}B_{\mu\nu}$ $\displaystyle=$
$\displaystyle{\cal
L}_{\xi}B_{\mu\nu}=\xi^{\rho}\partial_{\rho}B_{\mu\nu}-\partial_{\mu}\xi^{\rho}B_{\rho\nu}+B_{\mu\rho}\partial_{\nu}\xi^{\rho}\,.$
(2.26)

Similarly, local gauge transformations are generated by [32]

${\cal G}_{\lambda}=\int_{0}^{2\pi}d\sigma\lambda_{\mu}(x(\sigma))\kappa
x^{\prime\mu}(\sigma)\,,$ (2.27)

which results in

$\displaystyle\delta_{\lambda}G_{\mu\nu}$ $\displaystyle=$ $\displaystyle
0\,,$ (2.28) $\displaystyle\delta_{\lambda}B_{\mu\nu}$ $\displaystyle=$
$\displaystyle(d\lambda)_{\mu\nu}=\partial_{\mu}\lambda_{\nu}-\partial_{\nu}\lambda_{\mu}\,.$

Taking the joint generator of diffeomorphisms and local gauge transformations,
we obtain

${\cal G}_{\Lambda}={\cal G}_{\xi}+{\cal G}_{\lambda}=\int
d\sigma\langle\Lambda,X\rangle\,,$ (2.29)

where

$X^{M}=\begin{pmatrix}\kappa x^{\prime\mu}\\\ \pi_{\mu}\end{pmatrix}\,,$
(2.30)

and

$\Lambda^{M}=\begin{pmatrix}\xi^{\mu}\\\ \lambda_{\mu}\end{pmatrix}\,.$ (2.31)

The Courant bracket appears in the Poisson bracket relation of this joint
generator

$\Big{\\{}{\cal G}_{\Lambda_{1}},\,{\cal G}_{\Lambda_{2}}\Big{\\}}=-{\cal
G}_{[\Lambda_{1},\Lambda_{2}]_{\cal C}}\,.$ (2.32)

###  2.3 Twisted Courant bracket

The Courant bracket is not the only choice of bracket on generalized tangent
bundle. Various twists of the bracket can be obtained, by which different
string fluxes appear. The simplest example is the $B$-twisted Courant bracket

$[\Lambda_{1},\Lambda_{2}]_{{\cal
C}_{B}}=e^{\hat{B}}[e^{-\hat{B}}\Lambda_{1},e^{-\hat{B}}\Lambda_{2}]_{\cal
C}\,,$ (2.33)

where $e^{\hat{B}}$ denotes $B$-transformation, given by

$e^{\hat{B}}=\begin{pmatrix}\delta^{\mu}_{\nu}&0\\\
2B_{\mu\nu}&\delta^{\nu}_{\mu}\end{pmatrix}\,,\ \ \hat{B}^{M}_{\
N}=\begin{pmatrix}0&0\\\ 2B_{\mu\nu}&0\\\ \end{pmatrix}\,.$ (2.34)

This bracket can be obtained by expressing the generator (2.29) in the basis

$\hat{X}^{M}=(e^{\hat{B}})^{M}_{\ N}\ X^{N}=\begin{pmatrix}\kappa
x^{\prime\mu}\\\ i_{\mu}\end{pmatrix}\,,$ (2.35)

where $i_{\mu}$ are currents given by

$i_{\mu}=\pi_{\mu}+2\kappa B_{\mu\nu}x^{\prime\nu}\,.$ (2.36)

The Poisson bracket between these currents gives rise to the relation [14, 3]

$\\{i_{\mu}(\sigma),i_{\nu}(\bar{\sigma})\\}=-2\kappa
B_{\mu\nu\rho}x^{\prime\rho}\delta(\sigma-\bar{\sigma})\,,$ (2.37)

where $B_{\mu\nu\rho}$ is the Kalb-Ramond field strength, or H-flux, given by

$B_{\mu\nu\rho}=\partial_{\mu}B_{\nu\rho}+\partial_{\nu}B_{\rho\mu}+\partial_{\rho}B_{\mu\nu}\,.$
(2.38)

Another example is the $\theta$-twisted Courant bracket

$[\hat{\Lambda}_{1},\hat{\Lambda}_{2}]_{{\cal
C}_{\theta}}=e^{\hat{\theta}}[e^{-\hat{\theta}}\Lambda_{1},e^{-\hat{\theta}}\Lambda_{2}]_{\cal
C}\,,$ (2.39)

where twisting is performed by the $\theta$-transformation, given by

$e^{\hat{\theta}}=\begin{pmatrix}\delta^{\mu}_{\nu}&\kappa\theta^{\mu\nu}\\\
0&\delta^{\nu}_{\mu}\end{pmatrix}\,,\quad\hat{\theta}^{M}_{\
N}=\begin{pmatrix}0&\kappa\theta^{\mu\nu}\\\ 0&0\end{pmatrix}\,.$ (2.40)

The Courant bracket twisted by $\theta$ can be obtained if the generator
(2.29) is expressed in basis defined by

$\hat{X}^{M}=(e^{\hat{\theta}})^{M}_{\ N}\ X^{N}=\begin{pmatrix}k^{\mu}\\\
\pi_{\mu}\end{pmatrix}\,,$ (2.41)

where $k^{\mu}$ are currents, given by [3]

$\displaystyle k^{\mu}=\kappa x^{\prime\mu}+\kappa\theta^{\mu\nu}\pi_{\nu}\,.$
(2.42)

The algebra of these currents closes on a different set of fluxes, i.e.

$\displaystyle\\{k^{\mu}(\sigma),k^{\nu}(\bar{\sigma})\\}=-\kappa Q_{\rho}^{\
\mu\nu}k^{\rho}\delta(\sigma-\bar{\sigma})-\kappa^{2}R^{\mu\nu\rho}\pi_{\rho}\delta(\sigma-\bar{\sigma})\,,$
(2.43)

where $Q$ and $R$ fluxes are given by

$Q_{\rho}^{\ \mu\nu}=\partial_{\rho}\theta^{\mu\nu},\ \ \ \ \
R^{\mu\nu\rho}=\theta^{\mu\sigma}\partial_{\sigma}\theta^{\nu\rho}+\theta^{\nu\sigma}\partial_{\sigma}\theta^{\rho\mu}+\theta^{\rho\sigma}\partial_{\sigma}\theta^{\mu\nu}\,.$
(2.44)

Interesting insights emerge when one takes into account T-duality, under which
momenta and winding around a compact dimension are swapped. The former are
integrals of canonical momenta $\pi_{\mu}$, while the latter are integrals of
the coordinate $\sigma$-derivatives $\kappa x^{\prime\mu}$ around the compact
dimension. Moreover, the Buscher T-dualization rules [33, 34] define the
T-dual of the Kalb-Ramond field $B$ to be the non-commutativity parameter
$\theta$. These T-duality relations we write as

$\kappa x^{\prime\mu}\cong\pi_{\mu}\,,\qquad
B_{\mu\nu}\cong\frac{\kappa}{2}\theta^{\mu\nu}\,.$ (2.45)

When these interchanges are made, the currents $i_{\mu}$ (2.36) transform into
currents $k^{\mu}$ (2.42), i.e.

$i_{\mu}\cong k^{\mu}\,.$ (2.46)

Given the relations (2.45) and (2.46), we can conclude that the $B$-twisted
Courant and $\theta$-twisted Courant bracket are in fact related by the
T-duality [15].

The T-dual relations between these twisted Courant brackets can also be
obtained from the double theory. In that instance, the generalization of the
generator (2.29) gives rise to the $C$-bracket [35, 36], which by the same
method can be twisted, giving rise to the twisted $C$-brackets [37]. When
these twisted $C$-brackets are projected to mutually T-dual phase spaces, the
$B$-twisted and $\theta$-twisted Courant brackets are respectively obtained.

##  3 Courant bracket twisted simultaneously by both B and $\theta$

The Courant bracket twisted simultaneously by both B and $\theta$ was
constructed in [22] with the goal of obtaining the twisted Courant bracket
containing all fluxes, while simultaneously being invariant under the
T-duality. In this chapter, we will outline the overview of its construction.
For details, reader is referred to the reference [22].

The twisting matrix is defined in the form $e^{\breve{B}}$, where

$\breve{B}=\hat{B}+\hat{\theta}=\begin{pmatrix}0&\kappa\theta^{\mu\nu}\\\
2B_{\mu\nu}&0\end{pmatrix}\,.$ (3.1)

Given that the Kalb-Ramond field $B$ is T-dual to the non-commutativity
parameter $\theta$, this transformation is invariant under T-duality by
design. The expression for $e^{\breve{B}}$ is obtained from its Taylor
expansion

$e^{\breve{B}}=\begin{pmatrix}{\cal C}^{\mu}_{\ \nu}&\kappa{\cal S}^{\mu}_{\
\rho}\theta^{\rho\nu}\\\ 2B_{\mu\rho}{\cal S}^{\rho}_{\ \nu}&({\cal C}^{T})^{\
\nu}_{\mu}\end{pmatrix}\,,$ (3.2)

where

${\cal C}^{\mu}_{\ \nu}=\cosh{\sqrt{\alpha}}^{\mu}_{\ \nu}\,,\quad{\cal
S}^{\mu}_{\
\nu}=\Big{(}\frac{\sinh{\sqrt{\alpha}}}{\sqrt{\alpha}}\Big{)}^{\mu}_{\
\nu}\,,$ (3.3)

and

$\alpha^{\mu}_{\ \nu}=2\kappa\theta^{\mu\rho}B_{\rho\nu}\,.$ (3.4)

Under T-duality we have $\alpha\cong\alpha^{T}$, which easily generalizes to
all analytic functions of $\alpha$

${\cal C}\cong{\cal C}^{T}\,,\ \ {\cal S}\cong{\cal S}^{T}\,.$ (3.5)

The Courant bracket twisted at the same time by $B$ and $\theta$ can be
obtained from the Poisson bracket algebra of the generator

${\cal G}_{\breve{\Lambda}}=\int
d\sigma\langle\breve{\Lambda},\breve{X}\rangle\,,$ (3.6)

which is expressed in the basis

$\breve{X}^{M}=(e^{\breve{B}})^{\\!M}_{\ N}\
X^{N}=\begin{pmatrix}\breve{k}^{\mu}\\\ \breve{\iota}_{\mu}\end{pmatrix}\,,$
(3.7)

where new currents are

$\displaystyle\breve{k}^{\mu}$ $\displaystyle=$ $\displaystyle\kappa{\cal
C}^{\mu}_{\ \nu}x^{\prime\nu}+\kappa({\cal S}\theta)^{\mu\nu}\pi_{\nu}\,,$
(3.8) $\displaystyle\breve{\iota}_{\mu}$ $\displaystyle=$ $\displaystyle
2(B{\cal S})_{\mu\nu}x^{\prime\nu}+({\cal C}^{T})^{\ \nu}_{\mu}\pi_{\nu}\,.$

Due to (3.5), these currents are mutually related by T-duality.

It is convenient to introduce an auxiliary basis defined by

$\mathring{k}^{\mu}={\cal C}^{\mu}_{\ \nu}\breve{k}^{\nu}=\kappa
x^{\prime\mu}+\kappa\mathring{\theta}^{\mu\nu}\mathring{\iota}_{\nu}\,,\quad\mathring{\iota}_{\mu}=\breve{\iota}_{\nu}({\cal
C}^{-1})^{\nu}_{\ \mu}=\pi_{\mu}+2\kappa\mathring{B}_{\mu\nu}x^{\prime\nu}\,,$
(3.9)

where

$\mathring{B}_{\mu\nu}=(B{\cal S}{\cal
C}^{-1})_{\mu\nu}\,,\quad\mathring{\theta}^{\mu\nu}=({\cal C}{\cal
S}\theta)^{\mu\nu}\,.$ (3.10)

The currents $\mathring{k}^{\mu}$ and $\mathring{\iota}_{\mu}$ have the same
form as the expressions that are often investigated when twisted Courant
brackets with all fluxes are considered [18, 19, 38, 16, 17]. The relevant
Poisson bracket relations between auxiliary currents are given by [22]

$\displaystyle\\{\mathring{\iota}_{\mu}(\sigma),\mathring{\iota}_{\nu}(\bar{\sigma})\\}=-2{\cal\mathring{B}}_{\mu\nu\rho}\mathring{k}^{\rho}\delta(\sigma-\bar{\sigma})-{\cal\mathring{F}}_{\mu\nu}^{\
\rho}\ \mathring{\iota}_{\rho}\delta(\sigma-\bar{\sigma})\,,$ (3.11)
$\displaystyle\\{\mathring{k}^{\mu}(\sigma),\mathring{k}^{\nu}(\bar{\sigma})\\}=-\kappa{\cal\mathring{Q}}_{\rho}^{\
\mu\nu}\mathring{k}^{\rho}\delta(\sigma-\bar{\sigma})-\kappa^{2}{\cal\mathring{R}}^{\mu\nu\rho}\mathring{\iota}_{\rho}\delta(\sigma-\bar{\sigma})\,,$
(3.12)
$\displaystyle\\{\mathring{\iota}_{\mu}(\sigma),\mathring{k}^{\nu}(\bar{\sigma})\\}=\kappa\delta_{\mu}^{\nu}\delta^{\prime}(\sigma-\bar{\sigma})+{\cal\mathring{F}}^{\
\nu}_{\mu\rho}\
\mathring{k}^{\rho}\delta(\sigma-\bar{\sigma})-\kappa{\cal\mathring{Q}}_{\mu}^{\
\nu\rho}\mathring{\iota}_{\rho}\delta(\sigma-\bar{\sigma})\,,$ (3.13)

where the auxiliary generalized fluxes are given by

$\displaystyle{\cal\mathring{B}}_{\mu\nu\rho}$ $\displaystyle=$
$\displaystyle\partial_{\mu}\mathring{B}_{\nu\rho}+\partial_{\nu}\mathring{B}_{\rho\mu}+\partial_{\rho}\mathring{B}_{\mu\nu}\,,$
(3.14) $\displaystyle{\cal\mathring{F}}_{\mu\nu}^{\ \rho}$ $\displaystyle=$
$\displaystyle-2\kappa{\cal\mathring{B}}_{\mu\nu\sigma}\mathring{\theta}^{\sigma\rho}\,,$
(3.15) $\displaystyle{\cal\mathring{Q}}^{\ \nu\rho}_{\mu}$ $\displaystyle=$
$\displaystyle\mathring{Q}^{\
\nu\rho}_{\mu}+2\kappa\mathring{\theta}^{\nu\sigma}\mathring{\theta}^{\rho\tau}{\cal\mathring{B}}_{\mu\sigma\tau}\,,\quad\mathring{Q}^{\
\nu\rho}_{\mu}=\partial_{\mu}\mathring{\theta}^{\nu\rho}$ (3.16)
$\displaystyle{\cal\mathring{R}}^{\mu\nu\rho}$ $\displaystyle=$
$\displaystyle\mathring{R}^{\mu\nu\rho}+2\kappa\mathring{\theta}^{\mu\lambda}\mathring{\theta}^{\nu\sigma}\mathring{\theta}^{\rho\tau}{\cal\mathring{B}}_{\lambda\sigma\tau}\,,$
(3.17) $\displaystyle\mathring{R}^{\mu\nu\rho}$ $\displaystyle=$
$\displaystyle\mathring{\theta}^{\mu\sigma}\partial_{\sigma}\mathring{\theta}^{\nu\rho}+\mathring{\theta}^{\nu\sigma}\partial_{\sigma}\mathring{\theta}^{\rho\mu}+\mathring{\theta}^{\rho\sigma}\partial_{\sigma}\mathring{\theta}^{\mu\nu}\,.$

The formulae (3.14)-(3.17) are the well known expressions for generalized
fluxes. These fluxes, as well as their Bianchi identities, were obtained from
the non-associative quasi-Poisson structures [19].

##  4 Generalized fluxes

The algebra of auxiliary currents closes with auxiliary fluxes as its
structure functions. We can now use the expression for currents $\breve{k}$
and $\breve{\iota}$ in terms of their auxiliary counterparts (3.9) to compute
the fluxes relevant for the Courant bracket twisted with both $B$ and
$\theta$. Besides the relations (3.14)-(3.17), we will require the algebra
relations in the form

$\\{{\cal C}^{\mu}_{\
\rho}(\sigma),\mathring{\iota}_{\nu}(\bar{\sigma})\\}=\partial_{\nu}{\cal
C}^{\mu}_{\ \rho}\delta(\sigma-\bar{\sigma})\,,\quad\\{{\cal C}^{\mu}_{\
\rho}(\sigma),\mathring{k}^{\nu}(\bar{\sigma})\\}=\kappa\mathring{\theta}^{\nu\sigma}\partial_{\sigma}{\cal
C}^{\mu}_{\ \rho}\delta(\sigma-\bar{\sigma})\,.$ (4.18)

Let us begin with writing the Poisson bracket relation between two sets of
currents $\breve{\iota}_{\mu}$

$\displaystyle\\{\breve{\iota}_{\mu}(\sigma),\breve{\iota}_{\nu}(\bar{\sigma})\\}$
$\displaystyle=$ $\displaystyle{\cal C}^{\rho}_{\ \mu}{\cal C}^{\sigma}_{\
\nu}\\{\mathring{\iota}_{\rho},\mathring{\iota}_{\sigma}\\}+{\cal C}^{\rho}_{\
\mu}\mathring{\iota}_{\sigma}\\{\mathring{\iota}_{\rho},{\cal C}^{\sigma}_{\
\nu}\\}+\mathring{\iota}_{\rho}{\cal C}^{\sigma}_{\ \nu}\\{{\cal C}^{\rho}_{\
\mu},\mathring{\iota}_{\sigma}\\}$ $\displaystyle=$ $\displaystyle-2{\cal
C}^{\rho}_{\ \mu}{\cal C}^{\sigma}_{\
\nu}\mathring{B}_{\rho\sigma\alpha}\mathring{k}^{\alpha}\delta(\sigma-\bar{\sigma})-{\cal
C}^{\rho}_{\ \mu}{\cal C}^{\sigma}_{\
\nu}\cal{\mathring{F}}_{\\!\\!\rho\sigma}^{\ \alpha}\
\mathring{\iota}_{\alpha}\delta(\sigma-\bar{\sigma})$
$\displaystyle-\Big{(}{\cal C}^{\rho}_{\ \mu}\partial_{\rho}{\cal
C}^{\sigma}_{\ \nu}-{\cal C}^{\rho}_{\ \nu}\partial_{\rho}{\cal C}^{\sigma}_{\
\mu}\Big{)}\mathring{\iota}_{\sigma}\delta(\sigma-\bar{\sigma})$
$\displaystyle=$ $\displaystyle-2\breve{\cal
B}_{\mu\nu\rho}\breve{k}^{\rho}\delta(\sigma-\bar{\sigma})-\breve{\cal
F}_{\\!\\!\mu\nu}^{\ \rho}\
\breve{\iota}_{\rho}\delta(\sigma-\bar{\sigma})\,.$

We firstly used (3.9), after which we substituted (3.11) and (4.18). The
resulting algebra we expressed in terms of $\breve{{\cal B}}$ flux, given by

$\breve{\cal B}_{\mu\nu\rho}={\cal C}^{\alpha}_{\ \mu}{\cal C}^{\beta}_{\
\nu}{\cal C}^{\gamma}_{\ \rho}\mathring{B}_{\alpha\beta\gamma}\,,$ (4.20)

and $\breve{{\cal F}}$ flux, given by

$\breve{\cal F}_{\\!\\!\mu\nu}^{\ \rho}=\mathring{\cal
F}_{\\!\\!\alpha\beta}^{\ \gamma}{\cal C}^{\alpha}_{\ \mu}{\cal C}^{\beta}_{\
\nu}({\cal C}^{-1})^{\rho}_{\ \gamma}+({\cal C}^{-1})^{\rho}_{\
\tau}\Big{(}{\cal C}^{\sigma}_{\ \mu}\partial_{\sigma}{\cal C}^{\tau}_{\
\nu}-{\cal C}^{\sigma}_{\ \nu}\partial_{\sigma}{\cal C}^{\tau}_{\
\mu}\Big{)}\,.$ (4.21)

In order to simplify the expression for $\breve{{\cal F}}$ flux, let us define

$\hat{\partial}_{\mu}=({\cal C}^{T})_{\mu}^{\ \nu}\ \partial_{\nu}\,.$ (4.22)

Recall that $({\cal C}^{T})_{\mu}^{\ \nu}$ is a hyperbolic cosine function
(3.3) of a matrix $\alpha$. It is non-zero, and there is a Taylor’s expansion
of its inverse that is defined everywhere. This means that we can invert the
relation (4.22), where we obtain

$\partial_{\mu}=({\cal C}^{-T})_{\mu}^{\ \nu}\ \hat{\partial}_{\nu}\,,$ (4.23)

where $({\cal C}^{-T})=({\cal C}^{T})^{-1}$. The set $\hat{\partial}_{\mu}$
constitute a basis in a chart where partial derivatives $\partial_{\mu}$ are
defined. Writing the Lie bracket between these new basis vectors, we obtain

$[\hat{\partial}_{\mu},\hat{\partial}_{\nu}]_{L}=\breve{f}_{\mu\nu}^{\ \rho}\
\hat{\partial}_{\rho}\,.$ (4.24)

Suppose the simplest case of constant background fields $B$ and $\theta$.
Then, the matrix ${\cal C}$ is constant, so the $\breve{f}$-flux is zero. In
this case, vectors $\hat{\partial}_{\mu}$ form a holonomic basis, and there is
some local coordinate system $\hat{x}^{\mu}$, such that

$\hat{\partial}_{\mu}=\frac{\partial}{\partial\hat{x}^{\mu}}\,.$ (4.25)

In fact, the matrix $\frac{\partial x^{\mu}}{\partial\hat{x}^{\nu}}=({\cal
C}^{T})_{\nu}^{\ \mu}$ would represent the Jacobian matrix, making it a simple
case of diffeomorphism.

For the general case of background fields $B$ and $\theta$, the
$\breve{f}$-flux is non-zero, and $\hat{\partial}_{\mu}$ constitute a non-
holonomic basis. According to the Frobenius theorem, we cannot integrate it;
that is, there is no coordinate system $\hat{x}^{\mu}$ such that
$\hat{\partial}_{\mu}$ are well-defined partial derivatives with respect to
coordinates $\hat{x}^{\mu}$.

We can rewrite the bi-vector $\mathring{\theta}$ in this new basis by

$\mathring{\theta}^{\mu\nu}\partial_{\mu}\partial_{\nu}=\breve{\theta}^{\mu\nu}\hat{\partial}_{\mu}\hat{\partial}_{\nu}\,,$
(4.26)

from which we obtain

$\breve{\theta}^{\mu\nu}=({\cal SC}^{-1})^{\mu}_{\
\rho}\theta^{\rho\nu}=(C^{-2})^{\mu}_{\ \rho}\mathring{\theta}^{\rho\nu}\,.$
(4.27)

After substituting (3.15) and (4.27) into (4.21), one obtains

$\breve{\cal F}_{\\!\\!\mu\nu}^{\ \rho}=\breve{f}_{\\!\mu\nu}^{\
\rho}-2\kappa\breve{{\cal B}}_{\mu\nu\sigma}\breve{\theta}^{\sigma\rho}\,,$
(4.28)

where

$\breve{f}_{\\!\mu\nu}^{\ \rho}=({\cal C}^{-1})^{\rho}_{\
\sigma}\Big{(}\hat{\partial}_{\mu}{\cal C}^{\sigma}_{\
\nu}-\hat{\partial}_{\nu}{\cal C}^{\sigma}_{\ \mu}\Big{)}\,.$ (4.29)

Now we can proceed with calculating the algebra of auxiliary currents
$\breve{k}$. Starting with (3.9), with the help of (3.12) and (4.18), we have

$\displaystyle\\{\breve{k}^{\mu}(\sigma),\breve{k}^{\nu}(\bar{\sigma})\\}$
$\displaystyle=$ $\displaystyle({\cal C}^{-1})^{\mu}_{\ \rho}({\cal
C}^{-1})^{\nu}_{\
\sigma}\\{\mathring{k}^{\rho},\mathring{k}^{\sigma}\\}+({\cal
C}^{-1})^{\mu}_{\ \rho}\mathring{k}^{\sigma}\\{\mathring{k}^{\rho},({\cal
C}^{-1})^{\nu}_{\ \sigma}\\}$ $\displaystyle+\mathring{k}^{\rho}\\{({\cal
C}^{-1})^{\mu}_{\ \rho},\mathring{k}^{\sigma}\\}({\cal C}^{-1})^{\nu}_{\
\sigma}$ $\displaystyle=$ $\displaystyle-\kappa({\cal C}^{-1})^{\mu}_{\
\rho}({\cal C}^{-1})^{\nu}_{\ \sigma}\mathring{Q}_{\tau}^{\
\rho\sigma}\mathring{k}^{\tau}\delta(\sigma-\bar{\sigma})-\kappa^{2}({\cal
C}^{-1})^{\mu}_{\ \rho}({\cal C}^{-1})^{\nu}_{\
\sigma}\mathring{R}^{\rho\sigma\tau}\mathring{\iota}_{\tau}\delta(\sigma-\bar{\sigma})$
$\displaystyle-\kappa\Big{(}\breve{\theta}^{\mu\alpha}\hat{\partial}_{\alpha}({\cal
C}^{-1})^{\nu}_{\
\rho}-\breve{\theta}^{\nu\alpha}\hat{\partial}_{\alpha}({\cal
C}^{-1})^{\mu}_{\ \rho}\Big{)}\mathring{k}^{\rho}\delta(\sigma-\bar{\sigma})$
$\displaystyle=$ $\displaystyle-\kappa\breve{\cal Q}_{\rho}^{\
\mu\nu}\breve{k}^{\rho}\delta(\sigma-\bar{\sigma})-\kappa^{2}\breve{{\cal
R}}^{\mu\nu\rho}\breve{\iota}_{\rho}\delta(\sigma-\bar{\sigma})\,,$

where we used the relations (4.22) and (4.27) to simplify result. The fluxes
that we obtained are

$\breve{\cal Q}_{\rho}^{\ \mu\nu}={\cal C}^{\alpha}_{\ \rho}({\cal
C}^{-1})^{\mu}_{\ \beta}({\cal C}^{-1})^{\nu}_{\ \gamma}\mathring{\cal
Q}_{\alpha}^{\ \beta\gamma}-{\cal C}^{\alpha}_{\
\rho}\Big{(}\breve{\theta}^{\nu\beta}\hat{\partial}_{\beta}({\cal
C}^{-1})^{\mu}_{\
\alpha}-\breve{\theta}^{\mu\beta}\hat{\partial}_{\beta}({\cal
C}^{-1})^{\nu}_{\ \alpha}\Big{)}\,,$ (4.31)

and

$\breve{\cal R}^{\mu\nu\rho}=\mathring{\cal R}^{\alpha\beta\gamma}({\cal
C}^{-1})^{\mu}_{\ \alpha}({\cal C}^{-1})^{\nu}_{\ \beta}({\cal
C}^{-1})^{\rho}_{\ \gamma}\,.$ (4.32)

These are expressions for analogs of $Q$ and $R$ fluxes (2.44). We will now
proceed to rewrite them in more recognizable forms. For
$\breve{\cal{Q}}$-flux, substituting (3.16) into (4.31), we obtain

$\breve{\cal Q}_{\rho}^{\ \mu\nu}=\breve{Q}_{\rho}^{\
\mu\nu}+2\kappa\breve{\theta}^{\mu\alpha}\breve{\theta}^{\nu\beta}{\cal\breve{B}}_{\rho\alpha\beta}\,,$
(4.33)

where

$\breve{Q}_{\rho}^{\ \mu\nu}={\cal C}^{\alpha}_{\ \rho}({\cal
C}^{-1})^{\mu}_{\ \beta}({\cal C}^{-1})^{\nu}_{\
\gamma}\mathring{Q}_{\alpha}^{\ \beta\gamma}-{\cal C}^{\alpha}_{\
\rho}\Big{(}\breve{\theta}^{\nu\beta}\hat{\partial}_{\beta}({\cal
C}^{-1})^{\mu}_{\
\alpha}-\breve{\theta}^{\mu\beta}\hat{\partial}_{\beta}({\cal
C}^{-1})^{\nu}_{\ \alpha}\Big{)}\,.$ (4.34)

After expressing the previous relation in terms of $\breve{\theta}$ (4.27), we
obtain

$\displaystyle\breve{Q}_{\rho}^{\ \mu\nu}$ $\displaystyle=$
$\displaystyle\partial_{\alpha}({\cal C}^{2}\breve{\theta})^{\beta\gamma}{\cal
C}^{\alpha}_{\ \rho}({\cal C}^{-1})^{\mu}_{\ \beta}({\cal C}^{-1})^{\nu}_{\
\gamma}-{\cal C}^{\alpha}_{\
\rho}\Big{(}\breve{\theta}^{\nu\beta}\hat{\partial}_{\beta}({\cal
C}^{-1})^{\mu}_{\
\alpha}-\breve{\theta}^{\mu\beta}\hat{\partial}_{\beta}({\cal
C}^{-1})^{\nu}_{\ \alpha}\Big{)}$ $\displaystyle=$
$\displaystyle\hat{\partial}_{\rho}\breve{\theta}^{\mu\nu}-\Big{(}({\cal
C}\breve{\theta})^{\beta\nu}\hat{\partial}_{\rho}({\cal C}^{-1})^{\mu}_{\
\beta}+({\cal C}\breve{\theta})^{\mu\gamma}\hat{\partial}_{\rho}({\cal
C}^{-1})^{\nu}_{\ \gamma})\Big{)}$ $\displaystyle-{\cal C}^{\alpha}_{\
\rho}\Big{(}\breve{\theta}^{\nu\beta}\hat{\partial}_{\beta}({\cal
C}^{-1})^{\mu}_{\
\alpha}-\breve{\theta}^{\mu\beta}\hat{\partial}_{\beta}({\cal
C}^{-1})^{\nu}_{\ \alpha}\Big{)}$ $\displaystyle=$
$\displaystyle\hat{\partial}_{\rho}\breve{\theta}^{\mu\nu}+{\cal
C}^{\alpha}_{\ \beta}{\breve{\theta}}^{\nu\beta}\hat{\partial}_{\rho}({\cal
C}^{-1})^{\mu}_{\ \alpha}-{\cal C}^{\alpha}_{\
\beta}{\breve{\theta}}^{\mu\beta}\hat{\partial}_{\rho}({\cal C}^{-1})^{\nu}_{\
\alpha}$ $\displaystyle-{\cal C}^{\alpha}_{\
\rho}\Big{(}\breve{\theta}^{\nu\beta}\hat{\partial}_{\beta}({\cal
C}^{-1})^{\mu}_{\
\alpha}-\breve{\theta}^{\mu\beta}\hat{\partial}_{\beta}({\cal
C}^{-1})^{\nu}_{\ \alpha}\Big{)}$ $\displaystyle=$
$\displaystyle\hat{\partial}_{\rho}\breve{\theta}^{\mu\nu}+{\breve{\theta}}^{\nu\beta}\Big{(}{\cal
C}^{\alpha}_{\ \beta}\hat{\partial}_{\rho}({\cal C}^{-1})^{\mu}_{\
\alpha}-{\cal C}^{\alpha}_{\ \rho}\hat{\partial}_{\beta}({\cal
C}^{-1})^{\mu}_{\ \alpha}\Big{)}$
$\displaystyle-{\breve{\theta}}^{\mu\beta}\Big{(}{\cal C}^{\alpha}_{\
\beta}\hat{\partial}_{\rho}({\cal C}^{-1})^{\nu}_{\ \alpha}-{\cal
C}^{\alpha}_{\ \rho}\hat{\partial}_{\beta}({\cal C}^{-1})^{\nu}_{\
\alpha}\Big{)}$ $\displaystyle=$
$\displaystyle\hat{\partial}_{\rho}\breve{\theta}^{\mu\nu}+\breve{f}^{\
\mu}_{\\!\rho\sigma}\ \breve{\theta}^{\sigma\nu}-\breve{f}^{\
\nu}_{\\!\rho\sigma}\ \breve{\theta}^{\sigma\mu}\,.$

Let us unpack what happened in the previous relation. In the first step, we
expressed the flux $\mathring{Q}$ using the non-commutative field
$\breve{\theta}$. Then, in the second step, we used partial integration on the
first term and rearranged the resulting expression using equation (4.22). In
subsequent steps, we recognized that $({\cal C}\breve{\theta})^{\beta\nu}$ can
be expressed as ${\cal C}^{\beta}_{\ \sigma}\breve{\theta}^{\sigma\nu}$ and
that ${\cal C}^{\mu}_{\ \beta}\partial_{\alpha}({\cal C}^{-1})^{\beta}_{\
\sigma}$ equals $-\partial_{\alpha}{\cal C}^{\mu}_{\ \beta}({\cal
C}^{-1})^{\beta}_{\ \sigma}$. By relabeling some indices and using equation
(4.29), we arrived at the final step of equation (4).

Similarly, substituting (3.17) into (4.32), we obtain

$\breve{\cal
R}^{\mu\nu\rho}=\breve{R}^{\mu\nu\rho}+2\kappa\breve{\theta}^{\mu\alpha}\breve{\theta}^{\nu\beta}\breve{\theta}^{\rho\gamma}{\cal\breve{B}}_{\alpha\beta\gamma}\,,$
(4.36)

where

$\breve{R}^{\mu\nu\rho}=\mathring{R}^{\alpha\beta\gamma}({\cal
C}^{-1})^{\mu}_{\ \alpha}({\cal C}^{-1})^{\nu}_{\ \beta}({\cal
C}^{-1})^{\rho}_{\ \gamma}$ (4.37)

The $\breve{R}$-flux can further be rewritten by

$\displaystyle\breve{R}^{\mu\nu\rho}$ $\displaystyle=$ $\displaystyle({\cal
C}^{2}\breve{\theta})^{\alpha\sigma}\partial_{\sigma}({\cal
C}^{2}\breve{\theta})^{\beta\gamma}({\cal C}^{-1})^{\mu}_{\ \alpha}({\cal
C}^{-1})^{\nu}_{\ \beta}({\cal C}^{-1})^{\rho}_{\ \gamma}+cyclic$
$\displaystyle=$ $\displaystyle({\cal
C}\breve{\theta})^{\mu\sigma}\partial_{\sigma}\breve{\theta}^{\nu\rho}-({\cal
C}\breve{\theta})^{\mu\sigma}({\cal
C}\breve{\theta})^{\beta\gamma}(\partial_{\sigma}({\cal C}^{-1})^{\nu}_{\
\beta}({\cal C}^{-1})^{\rho}_{\ \gamma}+({\cal C}^{-1})^{\nu}_{\
\beta}\partial_{\sigma}({\cal C}^{-1})^{\rho}_{\ \gamma})+cyclic$
$\displaystyle=$
$\displaystyle\breve{\theta}^{\mu\sigma}\hat{\partial}_{\sigma}\breve{\theta}^{\nu\rho}+\breve{\theta}^{\mu\alpha}\breve{\theta}^{\rho\beta}\hat{\partial_{\alpha}}({\cal
C}^{-1})^{\nu}_{\ \tau}{\cal C}^{\tau}_{\
\beta}-\breve{\theta}^{\mu\beta}\breve{\theta}^{\nu\alpha}\hat{\partial}_{\beta}({\cal
C}^{-1})^{\rho}_{\ \tau}{\cal C}^{\tau}_{\ \alpha}+cyclic$ $\displaystyle=$
$\displaystyle\breve{\theta}^{\mu\sigma}\hat{\partial}_{\sigma}\breve{\theta}^{\nu\rho}+\breve{\theta}^{\nu\sigma}\hat{\partial}_{\sigma}\breve{\theta}^{\rho\mu}+\breve{\theta}^{\rho\sigma}\hat{\partial}_{\sigma}\breve{\theta}^{\mu\nu}$
$\displaystyle-\breve{\theta}^{\mu\alpha}\breve{\theta}^{\rho\beta}({\cal
C}^{-1})^{\nu}_{\ \tau}\hat{\partial_{\alpha}}{\cal C}^{\tau}_{\
\beta}+\breve{\theta}^{\mu\beta}\breve{\theta}^{\nu\alpha}({\cal
C}^{-1})^{\rho}_{\ \tau}\hat{\partial}_{\beta}{\cal C}^{\tau}_{\
\alpha}-\breve{\theta}^{\nu\alpha}\breve{\theta}^{\mu\beta}({\cal
C}^{-1})^{\rho}_{\ \tau}\hat{\partial_{\alpha}}{\cal C}^{\tau}_{\ \beta}$
$\displaystyle+\breve{\theta}^{\nu\beta}\breve{\theta}^{\rho\alpha}({\cal
C}^{-1})^{\mu}_{\ \tau}\hat{\partial}_{\beta}{\cal C}^{\tau}_{\
\alpha}-\breve{\theta}^{\rho\alpha}\breve{\theta}^{\nu\beta}({\cal
C}^{-1})^{\mu}_{\ \tau}\hat{\partial_{\alpha}}{\cal C}^{\tau}_{\
\beta}+\breve{\theta}^{\rho\beta}\breve{\theta}^{\mu\alpha}({\cal
C}^{-1})^{\nu}_{\ \tau}\hat{\partial}_{\beta}{\cal C}^{\tau}_{\ \alpha}\,$
$\displaystyle=$
$\displaystyle\breve{\theta}^{\mu\sigma}\hat{\partial}_{\sigma}\breve{\theta}^{\nu\rho}+\breve{\theta}^{\nu\sigma}\hat{\partial}_{\sigma}\breve{\theta}^{\rho\mu}+\breve{\theta}^{\rho\sigma}\hat{\partial}_{\sigma}\breve{\theta}^{\mu\nu}-(\breve{\theta}^{\mu\alpha}\breve{\theta}^{\rho\beta}\breve{f}_{\\!\alpha\beta}^{\
\nu}+\breve{\theta}^{\nu\alpha}\breve{\theta}^{\mu\beta}\breve{f}_{\\!\alpha\beta}^{\
\rho}+\breve{\theta}^{\rho\alpha}\breve{\theta}^{\nu\beta}\breve{f}_{\\!\alpha\beta}^{\
\mu})\,.$

First, we expressed the flux as a function of $\breve{\theta}$ according to
equation (4.27). Then, we applied the chain rule and the derivative
$\hat{\partial}$ (4.22). Finally, we applied the chain rule again to
hyperbolic functions and used equation (4.29) to obtain the final expression.

Lastly, the remaining algebra between currents is obtained from (3.13) and
(4.18) in a following way

$\displaystyle\\{\breve{\iota}_{\mu}(\sigma),\breve{k}^{\nu}(\bar{\sigma})\\}$
$\displaystyle=$ $\displaystyle{\cal C}^{\sigma}_{\ \mu}({\cal
C}^{-1})^{\nu}_{\
\rho}\\{\mathring{\iota}_{\sigma},\mathring{k}^{\rho}(\bar{\sigma})\\}+{\cal
C}^{\sigma}_{\ \mu}\\{\mathring{\iota}_{\sigma},({\cal C}^{-1})^{\nu}_{\
\rho}\\}\mathring{k}^{\rho}+\mathring{\iota}_{\sigma}\\{{\cal C}^{\sigma}_{\
\mu},\mathring{k}^{\rho}\\}({\cal C}^{-1})^{\nu}_{\ \rho}$ $\displaystyle=$
$\displaystyle\kappa{\cal C}^{\sigma}_{\ \mu}(\sigma)({\cal C}^{-1})^{\nu}_{\
\sigma}(\bar{\sigma})\delta^{\prime}(\sigma-\bar{\sigma})$
$\displaystyle+\Big{(}{\cal\mathring{F}}^{\ \sigma}_{\\!\\!\rho\tau}\ {\cal
C}^{\rho}_{\ \mu}({\cal C}^{-1})^{\nu}_{\ \sigma}-\hat{\partial}_{\mu}({\cal
C}^{-1})^{\nu}_{\ \tau}\Big{)}\mathring{k}^{\tau}\delta(\sigma-\bar{\sigma})$
$\displaystyle+\Big{(}-\kappa{\cal\mathring{Q}}_{\rho}^{\ \sigma\tau}{\cal
C}^{\rho}_{\ \mu}({\cal C}^{-1})^{\nu}_{\ \sigma}+\kappa({\cal
C}^{-1})^{\nu}_{\ \sigma}\mathring{\theta}^{\sigma\rho}\partial_{\rho}{\cal
C}^{\tau}_{\ \mu}\Big{)}\mathring{\iota}_{\tau}\delta(\sigma-\bar{\sigma})\,.$

The anomalous term becomes

$\displaystyle\kappa{\cal C}^{\sigma}_{\ \mu}(\sigma)({\cal C}^{-1})^{\nu}_{\
\sigma}(\bar{\sigma})\delta^{\prime}(\sigma-\bar{\sigma})$ $\displaystyle=$
$\displaystyle\kappa\delta^{\nu}_{\mu}\delta^{\prime}(\sigma-\bar{\sigma})+\kappa{\cal
C}^{\sigma}_{\ \mu}\partial_{\rho}({\cal C}^{-1})^{\nu}_{\
\sigma}x^{\prime\rho}\delta(\sigma-\bar{\sigma})$ $\displaystyle=$
$\displaystyle\kappa\delta^{\nu}_{\mu}\delta^{\prime}(\sigma-\bar{\sigma})+{\cal
C}^{\rho}_{\ \mu}\partial_{\sigma}({\cal C}^{-1})^{\nu}_{\
\rho}\mathring{k}^{\sigma}\delta(\sigma-\bar{\sigma})$
$\displaystyle-\kappa{\cal C}^{\rho}_{\ \mu}\partial_{\sigma}({\cal
C}^{-1})^{\nu}_{\
\rho}\mathring{\theta}^{\sigma\tau}\mathring{\iota}_{\tau}\delta(\sigma-\bar{\sigma})\,,$

where we used the well-known delta function identity

$f(\bar{\sigma})\delta^{\prime}(\sigma-\bar{\sigma})=f^{\prime}(\sigma)\delta(\sigma-\bar{\sigma})+f(\sigma)\delta^{\prime}(\sigma-\bar{\sigma})\,.$
(4.41)

Substituting (4) into (4), we obtain

$\displaystyle\\{\breve{\iota}_{\mu}(\sigma),\breve{k}^{\nu}(\bar{\sigma})\\}$
$\displaystyle=$
$\displaystyle\kappa\delta^{\nu}_{\mu}\delta^{\prime}(\sigma-\bar{\sigma})$
$\displaystyle+\Big{(}{\cal C}^{\rho}_{\ \mu}(\partial_{\sigma}({\cal
C}^{-1})^{\nu}_{\ \rho}-\partial_{\rho}({\cal C}^{-1})^{\nu}_{\ \sigma})+{\cal
C}^{\rho}_{\ \mu}({\cal C}^{-1})^{\nu}_{\ \tau}\mathring{{\cal
F}}_{\\!\\!\rho\sigma}^{\
\tau}\Big{)}\mathring{k}^{\sigma}\delta(\sigma-\bar{\sigma})$
$\displaystyle+\kappa\Big{(}({\cal C}^{-1})^{\nu}_{\
\sigma}\partial_{\rho}{\cal C}^{\tau}_{\
\mu}\mathring{\theta}^{\sigma\rho}-{\cal C}^{\rho}_{\
\mu}\partial_{\sigma}({\cal C}^{-1})^{\nu}_{\
\rho}\mathring{\theta}^{\sigma\tau}$ $\displaystyle-{\cal C}^{\rho}_{\
\mu}({\cal C}^{-1})^{\nu}_{\ \sigma}{\cal\mathring{Q}}_{\rho}^{\
\sigma\tau}\Big{)}\mathring{\iota}_{\tau}\delta(\sigma-\bar{\sigma})\,.$

Substituting relations between currents (3.9) in the previous expression, we
obtain

$\displaystyle\\{\breve{\iota}_{\mu}(\sigma),\breve{k}^{\nu}(\bar{\sigma})\\}=\kappa\delta^{\nu}_{\mu}\delta^{\prime}(\sigma-\bar{\sigma})+\breve{\cal
F}^{\ \nu}_{\\!\\!\mu\rho}\
\breve{k}^{\rho}\delta(\sigma-\bar{\sigma})-\kappa\breve{\cal Q}_{\mu}^{\
\nu\rho}\breve{\iota}_{\rho}\delta(\sigma-\bar{\sigma})\,,$ (4.43)

All Poisson bracket relations between currents (4), (4), (4.43) can be
summarized by

$\\{\breve{X}^{M},\breve{X}^{N}\\}=-\breve{F}^{MN}_{\ \ \ \ P}\
\breve{X}^{P}\delta(\sigma-\bar{\sigma})+\kappa\eta^{MN}\delta^{\prime}(\sigma-\bar{\sigma})\,,$
(4.44)

where

$\displaystyle\breve{F}^{MN\rho}=\begin{pmatrix}\kappa^{2}\breve{{\cal
R}}^{\mu\nu\rho}&-\kappa\breve{{\cal Q}}_{\nu}^{\ \mu\rho}\\\
\kappa\breve{{\cal Q}}_{\mu}^{\ \nu\rho}&\breve{{\cal F}}^{\
\rho}_{\\!\\!\mu\nu}\\\
\end{pmatrix}\,,\qquad\breve{F}^{MN}{}_{\rho}=\begin{pmatrix}\kappa\breve{{\cal
Q}}_{\rho}^{\ \mu\nu}&\breve{{\cal F}}^{\ \mu}_{\\!\\!\nu\rho}\\\
-\breve{{\cal F}}^{\ \nu}_{\\!\\!\mu\rho}&2{\cal\breve{B}}_{\mu\nu\rho}\\\
\end{pmatrix}\,.$ (4.45)

We aim to derive a coordinate-independent expressions for the generalized
fluxes associated with the simultaneous twisting of the Courant bracket by
both $B$ and $\theta$. To accomplish this, we will introduce several new
mathematical structures, including some Lie and quasi-Lie algebroids.

##  5 Twisted Lie bracket

We will begin by searching for the Lie algebroid with ${\cal C}$ serving as
its anchor. The bracket of this Lie algebroid should be related to the Lie
bracket by

$\displaystyle\Big{(}{\cal C}[\xi_{1},\xi_{2}]_{\hat{L}}\Big{)}^{\mu}$
$\displaystyle=$ $\displaystyle\Big{(}[{\cal C}\xi_{1},{\cal
C}\xi_{2}]_{L}\Big{)}^{\mu}={\cal C}^{\nu}_{\
\rho}\xi_{1}^{\rho}\partial_{\nu}({\cal C}^{\mu}_{\
\sigma}\xi_{2}^{\sigma})-{\cal C}^{\nu}_{\
\rho}\xi_{2}^{\rho}\partial_{\nu}({\cal C}^{\mu}_{\ \sigma}\xi_{1}^{\sigma})$
$\displaystyle=$ $\displaystyle{\cal C}^{\nu}_{\ \rho}{\cal C}^{\mu}_{\
\sigma}\Big{(}\xi_{1}^{\rho}\partial_{\nu}\xi_{2}^{\sigma}-\xi_{2}^{\rho}\partial_{\nu}\xi_{1}^{\sigma}\Big{)}+\xi_{1}^{\rho}\xi_{2}^{\sigma}\Big{(}{\cal
C}^{\nu}_{\ \rho}\partial_{\nu}{\cal C}^{\mu}_{\ \sigma}-{\cal C}^{\nu}_{\
\sigma}\partial_{\nu}{\cal C}^{\mu}_{\ \rho}\Big{)}$ $\displaystyle=$
$\displaystyle{\cal C}^{\mu}_{\
\sigma}\Big{(}\xi_{1}^{\rho}\hat{\partial}_{\rho}\xi_{2}^{\sigma}-\xi_{2}^{\rho}\hat{\partial}_{\rho}\xi_{1}^{\sigma}\Big{)}+\xi_{1}^{\rho}\xi_{2}^{\sigma}\Big{(}\hat{\partial}_{\rho}{\cal
C}^{\mu}_{\ \sigma}-\hat{\partial}_{\sigma}{\cal C}^{\mu}_{\ \rho}\Big{)}\,,$

where we used (4.22) and relabeled some indices. Multiplying the previous
relation with ${\cal C}^{-1}$ and taking into the account (4.29), we obtain

$\Big{(}[\xi_{1},\xi_{2}]_{\hat{L}}\Big{)}^{\mu}=\xi_{1}^{\nu}\hat{\partial}_{\nu}\xi_{2}^{\mu}-\xi_{2}^{\nu}\hat{\partial}_{\nu}\xi_{1}^{\mu}+\breve{f}^{\
\mu}_{\\!\nu\rho}\ \xi_{1}^{\nu}\xi_{2}^{\rho}\,.$ (5.47)

Analogous to our notation for twisted Courant brackets, we will denote this
bracket as the twisted Lie bracket, since

$[\xi_{1},\xi_{2}]_{\hat{L}}={\cal C}^{-1}[{\cal C}\xi_{1},{\cal
C}\xi_{2}]_{L}\,.$ (5.48)

We can now see the interpretation of $\breve{f}$-flux - it is a direct
consequence of non-commutativity of partial derivatives under the twisted Lie
bracket. Taking the twisted Lie bracket between two partial derivatives, one
obtains

$[\partial_{\mu},\partial_{\nu}]_{\hat{L}}=\breve{f}_{\\!\mu\nu}^{\ \rho}\
\partial_{\rho}\,,$ (5.49)

and therefore, the $\breve{f}$-flux represent the structure functions of the
twisted Lie bracket between two holonomic partial derivatives.

What does the twisted Lie bracket represent? Again, consider a case of
constant background fields $B$ and $\theta$, resulting in constant matrix
${\cal C}$. In this instance, recall that we can define proper coordinate
system $\hat{x}^{\mu}$, such that
$\hat{\partial}_{\mu}=\frac{\partial}{\partial\hat{x}^{\mu}}$, and the matrix
${\cal C}$ is the Jacobiator. Then, this is just a push-forward
$\Phi_{\star,p}:T_{p}{\cal M}\to T_{\Phi(p)}{\cal M}$ of a transformation
$\Phi:{\cal M}\to{\cal M}$ where $\partial_{\mu}\Phi^{\nu}=({\cal
C}^{T})_{\mu}^{\ \nu}$, which can be integrated. As such, relation (5.48) can
be rewritten as

$\left.\Phi_{\star}[\xi_{1},\xi_{2}]_{L}\right|_{p}=\left.[\Phi_{\star}\xi_{1},\Phi_{\star}\xi_{2}]_{L}\right|_{\Phi(p)}\,,$
(5.50)

which is just the well known property that the push-forward of Lie bracket is
the Lie bracket of push-forward. In a more general case, the twisted Lie
bracket cannot be seen as the push-forward of the (ordinary) Lie bracket, and
we have a completely new bracket.

One can be tempted to think of the twisted Lie bracket as the Lie bracket
rewritten in a non-holonomic basis spanned by $\hat{\partial}_{\mu}$. This is
however not the case, as can be easily verified

$\displaystyle[\xi_{1},\xi_{2}]_{L}$ $\displaystyle=$
$\displaystyle\hat{\xi}_{1}^{\mu}\hat{\partial}_{\mu}(\hat{\xi}_{2}^{\nu}\hat{\partial}_{\nu})-\hat{\xi}_{1}^{\mu}\hat{\partial}_{\mu}(\hat{\xi}_{2}^{\nu}\hat{\partial}_{\nu})$
$\displaystyle=$
$\displaystyle(\hat{\xi}_{1}^{\nu}\hat{\partial}_{\nu}\hat{\xi}_{2}^{\mu}-\hat{\xi}_{2}^{\nu}\hat{\partial}_{\nu}\hat{\xi}_{1}^{\mu}+\breve{f}^{\
\mu}_{\\!\nu\rho}\
\hat{\xi}_{1}^{\nu}\hat{\xi}_{2}^{\rho})\hat{\partial}_{\mu}\,.$

We obtain the same form as in (5.47), but the components in the expression (5)
are expressed in basis $\hat{\partial}_{\mu}$, while in (5.47) the basis is
$\partial_{\mu}$. We can interpret this as that there is a new geometry
defined by matrix ${\cal C}$.

In order for ${\cal C}$ to be a proper anchor of a Lie algebroid, we have to
demonstrate that the Leibniz rule and Jacobi identity are satisfied. Firstly,
using definition of the twisted Lie bracket (5.48) together with Leibniz rule
for Lie bracket, we obtain

$\displaystyle[\xi_{1},f\xi_{2}]_{\hat{L}}$ $\displaystyle=$
$\displaystyle{\cal C}^{-1}[{\cal C}\xi_{1},f{\cal C}\xi_{2}]_{L}={\cal
C}^{-1}\Big{(}({\cal L}_{{\cal C}\xi_{1}}f)\ \xi_{2}+f[{\cal C}\xi_{1},{\cal
C}\xi_{2}]_{L}\Big{)}$ $\displaystyle=$ $\displaystyle({\cal L}_{{\cal
C}\xi_{1}}f)\ \xi_{2}+f[\xi_{1},\xi_{2}]_{\hat{L}}\,,$

and therefore we proved (2.2). The Jacobi identity (2.3) is also satisfied,
since

$[\xi_{1},[\xi_{2},\xi_{3}]_{\hat{L}}]_{\hat{L}}+cyclic={\cal C}^{-1}[{\cal
C}\xi_{1},[{\cal C}\xi_{2},{\cal C}\xi_{3}]_{L}]_{L}+cyclic={\cal
C}^{-1}[\xi_{1},[\xi_{2},\xi_{3}]_{L}]_{L}+cyclic=0\,,$ (5.53)

and therefore the tangent bundle equipped with the twisted Lie bracket indeed
defines a Lie algebroid with ${\cal C}$ as its anchor.

From the Leibniz rule for twisted Lie bracket, we can derive the action of
corresponding Lie derivative on functions

${\cal\hat{L}}_{\xi}\ f={\cal L}_{{\cal C}\xi}\
f=\xi^{\mu}\hat{\partial}_{\mu}f\,.$ (5.54)

Its action on vectors is simply given by the twisted Lie bracket. To write the
action of Lie derivative ${\cal L}_{\hat{\xi}}$ on 1-forms, we firstly apply
the Leibniz rule (5) on 1-form-vector contraction

$\displaystyle{\cal\hat{L}}_{\xi_{1}}(\xi_{2}^{\mu}\lambda_{2\mu})$
$\displaystyle=$
$\displaystyle({\cal\hat{L}}_{\xi_{1}}\xi_{2})^{\mu}\lambda_{2\mu}+\xi_{2}^{\mu}({\cal\hat{L}}_{\xi_{1}}\lambda_{2})_{\mu}$
$\displaystyle=$
$\displaystyle(\xi_{1}^{\nu}\hat{\partial}_{\nu}\xi_{2}^{\mu}-\xi_{2}^{\nu}\hat{\partial}_{\nu}\xi_{1}^{\mu})\lambda_{2\mu}+\breve{f}^{\
\mu}_{\\!\nu\rho}\
\xi_{1}^{\nu}\xi_{2}^{\rho}\lambda_{2\mu}+\xi_{2}^{\mu}({\cal\hat{L}}_{\xi_{1}}\lambda_{2})_{\mu}\,,$

and then (5.54) on that contraction, since it is effectively a scalar

$\displaystyle{\cal\hat{L}}_{\xi_{1}}(\xi_{2}^{\mu}\lambda_{2\mu})=\xi_{1}^{\nu}\hat{\partial}_{\nu}(\xi_{2}^{\mu}\lambda_{2\mu})=\xi_{1}^{\nu}\hat{\partial}_{\nu}\xi_{2}^{\mu}\lambda_{2\mu}+\xi_{1}^{\nu}\xi_{2}^{\mu}\hat{\partial}_{\nu}\lambda_{2\mu}\,.$
(5.56)

When we equate right-hand sides of previous two relations, we obtain

$({\cal\hat{L}}_{\xi_{1}}\lambda_{2})_{\mu}=\hat{\partial}_{\mu}\xi_{1}^{\nu}\lambda_{2\nu}+\xi_{1}^{\nu}\hat{\partial}_{\nu}\lambda_{2\mu}+\breve{f}^{\
\rho}_{\\!\mu\nu}\
\xi_{1}^{\nu}\lambda_{2\rho}=\xi_{1}^{\nu}(\hat{\partial}_{\nu}\lambda_{2\mu}-\hat{\partial}_{\mu}\lambda_{2\nu})+\hat{\partial}_{\mu}(\xi_{1}^{\nu}\lambda_{2\nu})+\breve{f}^{\
\rho}_{\\!\mu\nu}\ \xi_{1}^{\nu}\lambda_{2\rho}\,.$ (5.57)

The exterior algebra is easily derived from the relation (2.1). Let us
explicitly obtain the action of exterior derivative on functions and 1-forms.
Functions correspond to the case $p=0$, so we have

$\hat{d}f(\xi)={\cal
C}\xi(f)=\xi^{\mu}\hat{\partial}_{\mu}f\,,\quad(\hat{d}f)_{\mu}=\hat{\partial}_{\mu}f\,.$
(5.58)

From (5.54), we see that the usual relation for the action of Lie derivatives
on function ${\cal\hat{L}}_{\xi}f=i_{\xi}\hat{d}f$ still holds in the twisted
case. On the other hand, 1-forms correspond to the case of $p=1$ in (2.1),
from which we obtain

$\displaystyle\hat{d}\lambda(\xi_{1},\xi_{2})$ $\displaystyle=$
$\displaystyle{\cal C}\xi_{1}(\lambda(\xi_{2}))-{\cal
C}\xi_{2}(\lambda(\xi_{1}))-\lambda([\xi_{1},\xi_{2}]_{\hat{L}})$
$\displaystyle=$
$\displaystyle\xi_{1}^{\mu}\xi_{2}^{\nu}\Big{(}\hat{\partial}_{\mu}\lambda_{\nu}-\hat{\partial}_{\nu}\lambda_{\mu}-\hat{f}^{\
\rho}_{\\!\mu\nu}\lambda_{\rho}\Big{)}\,,$
$\displaystyle(\hat{d}\lambda)_{\mu\nu}$ $\displaystyle=$
$\displaystyle\hat{\partial}_{\mu}\lambda_{\nu}-\hat{\partial}_{\nu}\lambda_{\mu}-\hat{f}^{\
\rho}_{\\!\mu\nu}\ \lambda_{\rho}\,.$

The Cartan formula
${\cal\hat{L}}_{\xi}\lambda=i_{\xi}\hat{d}\lambda+\hat{d}i_{\xi}\lambda$ can
be easily demonstrated using (5.57) and (5), and holds true for any $p$-form
$\lambda$.

###  5.1 Generalized H-flux

Let us now reconsider the generalized $\breve{H}$-flux (4.20) in terms of new
Lie algebroid that we obtained. It has a structure of a 3-form, that when
contracted with three vectors can be written by

$\breve{{\cal
B}}_{\mu\nu\rho}\xi_{1}^{\mu}\xi_{2}^{\nu}\xi_{3}^{\rho}=\mathring{B}_{\alpha\beta\gamma}{\cal
C}^{\alpha}_{\ \mu}\xi_{1}^{\mu}{\cal C}^{\beta}_{\ \nu}\xi_{2}^{\nu}{\cal
C}^{\gamma}_{\ \rho}\xi_{3}^{\rho}=d\mathring{B}({\cal C}\xi_{1},{\cal
C}\xi_{2},{\cal C}\xi_{3})\,.$ (5.60)

where $\mathring{B}_{\mu\nu\rho}$ is defined in (3.14). The right-hand side of
the previous relation is expressed in a non coordinate notation, which using
the definition for exterior derivative can be further transformed by

$\displaystyle(d\mathring{B})({\cal C}\xi_{1},{\cal C}\xi_{2},{\cal
C}\xi_{3})$ $\displaystyle=$ $\displaystyle{\cal
C}\xi_{1}\Big{(}\mathring{B}({\cal C}\xi_{2},{\cal C}\xi_{3})\Big{)}-{\cal
C}\xi_{2}\Big{(}\mathring{B}({\cal C}\xi_{1},{\cal C}\xi_{3})\Big{)}+{\cal
C}\xi_{3}\Big{(}\mathring{B}({\cal C}\xi_{1},{\cal C}\xi_{2})\Big{)}$
$\displaystyle-\mathring{B}\Big{(}[{\cal C}\xi_{1},{\cal C}\xi_{2}]_{L},{\cal
C}\xi_{3}\Big{)}+\mathring{B}\Big{(}[{\cal C}\xi_{1},{\cal
C}\xi_{3}]_{L},{\cal C}\xi_{2}\Big{)}-\mathring{B}\Big{(}[{\cal
C}\xi_{2},{\cal C}\xi_{3}]_{L},{\cal C}\xi_{1}\Big{)}$ $\displaystyle=$
$\displaystyle{\cal C}\xi_{1}\Big{(}\hat{B}(\xi_{2},\xi_{3})\Big{)}-{\cal
C}\xi_{2}\Big{(}\hat{B}(\xi_{1},\xi_{3})\Big{)}+{\cal
C}\xi_{3}\Big{(}\hat{B}(\xi_{1},\xi_{2})\Big{)}$
$\displaystyle-\hat{B}\Big{(}[\xi_{1},\xi_{2}]_{\hat{L}},\xi_{3}\Big{)}+\hat{B}\Big{(}[\xi_{1},\xi_{3}]_{\hat{L}},\xi_{2}\Big{)}-\hat{B}\Big{(}[\xi_{2},\xi_{3}]_{\hat{L}},\xi_{1}\Big{)}$
$\displaystyle=$ $\displaystyle\hat{d}\hat{B}(\xi_{1},\xi_{2},\xi_{3})\,,$

where $\hat{B}$ is a new field that we defined by

$\hat{B}_{\mu\nu}=\mathring{B}_{\alpha\beta}{\cal C}^{\alpha}_{\ \mu}{\cal
C}^{\beta}_{\ \nu}=(B{\cal SC})_{\mu\nu}\,,$ (5.62)

and in the last step recognized the expression for twisted exterior derivative
acting on a 2-form.

##  6 Twisted Koszul bracket

We define the twisted Koszul bracket by

$[\lambda_{1},\lambda_{2}]_{\breve{\theta}}={\cal\hat{L}}_{\breve{\theta}(\lambda_{1})}\lambda_{2}-{\cal\hat{L}}_{\breve{\theta}(\lambda_{2})}\lambda_{1}-\hat{d}(\breve{\theta}(\lambda_{1},\lambda_{2}))\,,$
(6.1)

where
$\breve{\theta}(\lambda_{1})^{\mu}=\lambda_{1\nu}\breve{\theta}^{\nu\mu}$.
This is an analogous definition to the one for the (non-twisted) Koszul
bracket (2.17). In some local basis, its components are given by

$\displaystyle\Big{(}[\lambda_{1},\lambda_{2}]_{\breve{\theta}}\Big{)}_{\mu}$
$\displaystyle=$
$\displaystyle\breve{\theta}^{\nu\rho}(\lambda_{1\nu}\hat{\partial}_{\rho}\lambda_{2\mu}-\lambda_{2\nu}\hat{\partial}_{\rho}\lambda_{1\mu})+(\hat{\partial}_{\mu}\breve{\theta}^{\nu\rho}+\breve{f}^{\
\nu}_{\\!\mu\sigma}\ \breve{\theta}^{\sigma\rho}-\breve{f}^{\
\rho}_{\\!\mu\sigma}\
\breve{\theta}^{\sigma\nu})\lambda_{1\nu}\lambda_{2\rho}$ (6.2)
$\displaystyle=$
$\displaystyle\breve{\theta}^{\nu\rho}(\lambda_{1\nu}\hat{\partial}_{\rho}\lambda_{2\mu}-\lambda_{2\nu}\hat{\partial}_{\rho}\lambda_{1\mu})+\breve{Q}_{\mu}^{\
\nu\rho}\ \lambda_{1\nu}\lambda_{2\rho}\,.$

Taking the twisted Koszul bracket between one-forms $dx^{\mu}$, one obtains

$[dx^{\mu},dx^{\nu}]_{\breve{\theta}}=\breve{Q}_{\rho}^{\ \mu\nu}\
dx^{\rho}\,,$ (6.3)

where we used (6.2) and (4). In an analogous way that the $\breve{f}$-flux was
defined in terms of twisted Lie bracket (5.49), the $\breve{Q}$-flux is
defined as the structure functions of the twisted Koszul bracket of the
covectors $dx^{\mu}$.

The twisted Koszul bracket can be related to the twisted Lie bracket. In order
to do that, we firstly obtain

$\displaystyle\breve{\theta}\Big{(}[\lambda_{1},\lambda_{2}]_{\breve{\theta}}\Big{)}$
$\displaystyle=$
$\displaystyle\breve{\theta}^{\nu\mu}\Big{(}\breve{\theta}^{\sigma\rho}(\lambda_{1\sigma}\hat{\partial}_{\rho}\lambda_{2\nu}-\lambda_{2\sigma}\hat{\partial}_{\rho}\lambda_{1\nu})+\hat{\partial}_{\nu}\breve{\theta}^{\sigma\rho}\lambda_{1\sigma}\lambda_{2\rho}$
$\displaystyle+(\breve{f}^{\ \rho}_{\\!\nu\tau}\
\breve{\theta}^{\tau\sigma}-\breve{f}^{\ \sigma}_{\\!\nu\tau}\
\breve{\theta}^{\tau\rho})\lambda_{1\rho}\lambda_{2\sigma}\Big{)}\,.$

Next, we obtain

$\displaystyle[\breve{\theta}(\lambda_{1}),\breve{\theta}(\lambda_{2})]_{\hat{L}}$
$\displaystyle=$
$\displaystyle\breve{\theta}^{\nu\rho}\lambda_{1\rho}\hat{\partial}_{\nu}(\breve{\theta}^{\mu\sigma}\lambda_{2\sigma})-\breve{\theta}^{\nu\rho}\lambda_{2\rho}\hat{\partial}_{\nu}(\breve{\theta}^{\mu\sigma}\lambda_{1\sigma})+\breve{f}^{\
\mu}_{\\!\nu\rho}\
\breve{\theta}^{\nu\sigma}\breve{\theta}^{\rho\tau}\lambda_{1\sigma}\lambda_{2\tau}$
$\displaystyle=$
$\displaystyle\breve{\theta}^{\nu\rho}\breve{\theta}^{\mu\sigma}\Big{(}\lambda_{1\rho}\hat{\partial}_{\nu}\lambda_{2\sigma}-\lambda_{2\rho}\hat{\partial}_{\nu}\lambda_{1\sigma}\Big{)}+\breve{f}^{\
\mu}_{\\!\nu\rho}\
\breve{\theta}^{\nu\sigma}\breve{\theta}^{\rho\tau}\lambda_{1\sigma}\lambda_{2\tau}$
$\displaystyle+(\breve{\theta}^{\rho\nu}\hat{\partial}_{\nu}\breve{\theta}^{\sigma\mu}+\breve{\theta}^{\sigma\nu}\hat{\partial}_{\nu}\breve{\theta}^{\mu\rho})\lambda_{1\rho}\lambda_{2\sigma}\,.$

After relabeling of some dummy indices, we obtain the relation

$\Big{[}\breve{\theta}\Big{(}[\lambda_{1},\lambda_{2}]_{\breve{\theta}})\Big{)}\Big{]}^{\mu}=\Big{(}[\breve{\theta}(\lambda_{1}),\breve{\theta}(\lambda_{2})]_{\hat{L}}\Big{)}^{\mu}+\breve{R}^{\mu\nu\rho}\lambda_{1\nu}\lambda_{2\rho}\,,$
(6.6)

where $\breve{R}$ is defined in (4). We can use the definition of the twisted
Lie bracket (5.48), to rewrite the previous relation as

$\Big{[}{\cal
C}\breve{\theta}\Big{(}[\lambda_{1},\lambda_{2}]_{\breve{\theta}}\Big{)}\Big{]}^{\mu}=\Big{(}[{\cal
C}\breve{\theta}(\lambda_{1}),{\cal
C}\breve{\theta}(\lambda_{2})]_{L}\Big{)}^{\mu}+{\cal C}^{\mu}_{\
\sigma}\breve{R}^{\sigma\nu\rho}\lambda_{1\nu}\lambda_{2\rho}\,.$ (6.7)

The twisted Koszul bracket defines a quasi-Lie algebroid with anchor
$\hat{\rho}_{\breve{\theta}}={\cal C}\breve{\theta}$, where the
$\breve{R}$-flux is a deformation from the Lie algebroid structure.

We can still define the exterior derivative associated with the quasi-Lie
algebroid defined with the twisted Koszul bracket. On functions, its action is
obtained from (2.1)

$\hat{d}_{\breve{\theta}}f(\lambda)=\breve{\theta}^{\mu\nu}\hat{\partial}_{\nu}f\lambda_{\mu}\,,\qquad(\hat{d}_{\breve{\theta}}f)^{\mu}=\breve{\theta}^{\mu\nu}\hat{\partial}_{\nu}f\,.$
(6.8)

Similarly, on vectors it becomes

$\displaystyle\hat{d}_{\breve{\theta}}\xi(\lambda_{1},\lambda_{2})$
$\displaystyle=$
$\displaystyle(\lambda_{1\rho}\breve{\theta}^{\rho\nu})\hat{\partial}_{\nu}(\xi^{\mu}\lambda_{2\mu})-(\lambda_{2\rho}\breve{\theta}^{\rho\nu})\hat{\partial}_{\nu}(\xi^{\mu}\lambda_{1\mu})-\xi^{\mu}\Big{(}[\lambda_{1},\lambda_{2}]_{\breve{\theta}}\Big{)}_{\mu}$
$\displaystyle=$
$\displaystyle\Big{(}\breve{\theta}^{\mu\rho}\hat{\partial}_{\rho}\xi^{\nu}-\breve{\theta}^{\nu\rho}\hat{\partial}_{\rho}\xi^{\mu}-\xi^{\rho}(\hat{\partial}_{\rho}\breve{\theta}^{\mu\nu}+\breve{f}^{\
\mu}_{\\!\rho\sigma}\ \breve{\theta}^{\sigma\nu}-\breve{f}^{\
\nu}_{\\!\rho\sigma}\
\breve{\theta}^{\sigma\mu})\Big{)}\lambda_{1\mu}\lambda_{2\nu}\,.$

The exterior derivative $\hat{d}_{\breve{\theta}}$ satisfies Leibniz rule, but
is not idempotent, unless the $\breve{R}$-flux is zero.

##  7 Twisted Schouten-Nijenhuis bracket

Lastly, let us define the twisted Schouten-Nijenhuis bracket as the
generalization of the twisted Lie bracket on the space of multi-vectors

$\displaystyle[f,g]_{\hat{S}}=0\,,\quad[\xi,f]_{\hat{S}}={\cal\hat{L}}_{\xi}(f)\,,\quad[\xi_{1},\xi_{2}]_{\hat{S}}=[\xi_{1},\xi_{2}]_{\hat{L}}\,,$
(7.1)

with other brackets obtained via graded relations (2.8) and (2.9). The
important feature is that we can express the $\breve{R}$-flux (4) in terms of
the twisted Schouten-Nijenhuis bracket via

$\breve{R}=\frac{1}{2}[\breve{\theta},\breve{\theta}]_{\hat{S}}\,,$ (7.2)

analogous to their non-twisted counterparts.

The expression (7.2) can be proven using the principle of mathematical
induction. We begin by showing the relation (7.2) holds for a simple bi-vector
$\breve{\theta}$. Next, assuming that the relation (7.2) holds for a composite
bi-vector consisting of $n$ simple bi-vectors, we proceed to demonstrate that
it also holds for the composite bi-vector consisting of $n+1$ simple bi-
vectors. While this problem is straightforward to resolve, it involves several
technical steps in computations. Detailed derivations for this process are
included in the paper’s appendix.

###  7.1 Relation to the twisted Koszul bracket

In the previous chapters, we saw that the Schouten-Nijenhuis bracket can be
written as $d_{\theta}\theta=[\theta,\theta]_{S}$. This motivates us to
consider the action of exterior derivative $\hat{d}_{\breve{\theta}}$ on the
bi-vector $\breve{\theta}$. From definition (2.1), we have

$\displaystyle\hat{d}_{\breve{\theta}}\breve{\theta}(\lambda_{1},\lambda_{2},\lambda_{3})$
$\displaystyle=$ $\displaystyle{\cal
C}\breve{\theta}(\lambda_{1})\Big{(}[\lambda_{2},\lambda_{3}]_{\breve{\theta}}\Big{)}-{\cal
C}\breve{\theta}(\lambda_{2})\Big{(}[\lambda_{1},\lambda_{3}]_{\breve{\theta}}\Big{)}+{\cal
C}\breve{\theta}(\lambda_{3})\Big{(}[\lambda_{1},\lambda_{2}]_{\breve{\theta}}\Big{)}$
(7.3)
$\displaystyle-\breve{\theta}\Big{(}[\lambda_{1},\lambda_{2}]_{\breve{\theta}},\lambda_{3}\Big{)}+\breve{\theta}\Big{(}[\lambda_{1},\lambda_{3}]_{\breve{\theta}},\lambda_{2}\Big{)}-\breve{\theta}\Big{(}[\lambda_{2},\lambda_{3}]_{\breve{\theta}},\lambda_{1}\Big{)}\,.$

There are two types of terms, so let us calculate the components of a
representative of each type. Firstly, we have

$\displaystyle{\cal
C}\breve{\theta}(\lambda_{1})\Big{(}[\lambda_{2},\lambda_{3}]_{\breve{\theta}}\Big{)}$
$\displaystyle=$
$\displaystyle\lambda_{1\mu}\breve{\theta}^{\mu\sigma}\hat{\partial}_{\sigma}\Big{(}\breve{\theta}^{\nu\rho}\lambda_{2\nu}\lambda_{3\rho}\Big{)}$
$\displaystyle=$
$\displaystyle\breve{\theta}^{\mu\sigma}\hat{\partial}_{\sigma}\breve{\theta}^{\nu\rho}\lambda_{1\mu}\lambda_{2\nu}\lambda_{3\rho}+\breve{\theta}^{\mu\sigma}\breve{\theta}^{\nu\rho}\lambda_{1\mu}(\hat{\partial}_{\sigma}\lambda_{2\nu}\lambda_{3\rho}+\lambda_{2\nu}\hat{\partial}_{\sigma}\lambda_{3\rho})\,,$

and secondly

$\displaystyle-\breve{\theta}\Big{(}[\lambda_{1},\lambda_{2}]_{\breve{\theta}},\lambda_{3}\Big{)}$
$\displaystyle=$
$\displaystyle\lambda_{1\mu}\lambda_{2\nu}\lambda_{3\rho}\Big{(}\breve{\theta}^{\rho\sigma}\hat{\partial}_{\sigma}\breve{\theta}^{\mu\nu}+\breve{f}^{\
\mu}_{\\!\sigma\tau}\
\breve{\theta}^{\nu\sigma}\breve{\theta}^{\rho\tau}-\breve{f}^{\
\nu}_{\\!\sigma\tau}\
\breve{\theta}^{\mu\sigma}\breve{\theta}^{\rho\tau}\Big{)}$
$\displaystyle-\breve{\theta}^{\mu\rho}\breve{\theta}^{\nu\sigma}(\lambda_{1\nu}\hat{\partial}_{\sigma}\lambda_{2\mu}-\lambda_{2\nu}\hat{\partial}_{\sigma}\lambda_{1\mu})\lambda_{3\rho}\,.$

When (7.1) and (7.1) are substituted in (7.3), we obtain

$\displaystyle\hat{d}_{\breve{\theta}}\breve{\theta}(\lambda_{1},\lambda_{2},\lambda_{3})=2\breve{R}^{\mu\nu\rho}\lambda_{1\mu}\lambda_{2\nu}\lambda_{3\rho}\,,$
(7.6)

which is exactly what we expected. We see that equivalently, the twisted
Schouten-Nijenhuis bracket, and consequentially $\breve{R}$-flux, can be
defined as the action of the twisted exterior derivative related to the quasi
Lie algebroid defined with the twisted Koszul bracket:

$[\breve{\theta},\breve{\theta}]_{\hat{S}}=\hat{d}_{\breve{\theta}}\breve{\theta}\,.$
(7.7)

##  8 Conclusions

In this paper, we investigated the generalized fluxes that appear as structure
functions in the Poisson bracket algebra of non-canonical currents, the linear
combination of which produces the Courant bracket simultaneously twisted by
$B$ and $\theta$. These currents are mutually related by T-duality, which
effectively makes the Courant bracket simultaneously twisted by $B$ and
$\theta$ into the T-dual invariant extension of the Courant bracket containing
all fluxes. In order to obtain these fluxes, we used the relation between this
twisted Courant bracket and another twisted Courant bracket that also contains
all fluxes [22]. These brackets are related by the hyperbolic function ${\cal
C}$ that depends on the product of $B$ and $\theta$ fields. It turned out that
in the description of underlining geometries in which fluxes appear this
hyperbolic function plays an important role.

Firstly, we considered the Lie algebroid defined on the smooth sections of the
tangent bundle, with the requirement that the hyperbolic ${\cal C}$ matrix
serves as its anchor. The resulting compatible bracket is the twisted Lie
bracket. The algebra of the twisted Lie bracket between two partial
derivatives $\partial_{\mu}$ closes with $\breve{f}$-flux as the structure
functions. Generally, the $f$-flux is seen as the so called geometric flux,
and appears in the Lie algebra of some non-holonomic basis in which metric is
diagonal. In our case, we do not have diagonal metric on the manifold, and
moreover, the $f$-flux depends on both $B$ and $\theta$ background fields.
Given that a bi-vector is not always defined globally, we cannot talk about
the $\breve{f}$-flux as a ”geometric” flux that is defined globally. Instead,
the $\breve{f}$-flux is a direct consequence of simultaneous twisting.

Secondly, we defined the exterior derivative related to the twisted Lie
bracket. We demonstrated that the $\breve{H}$ flux can be seen as the twisted
exterior derivative of the new background field $\hat{B}$. This flux is also
dependent on both fields $B$ and $\theta$, which differs from analogous fluxes
in previous works. We can think of simultaneous twisting of the Courant
bracket resulting in the twisting of the $H$-flux as well.

Thirdly, we defined the twisted Koszul bracket by replacing the standard
invariant terms in the expression for (untwisted) Koszul bracket with their
twisted counterparts. We showed that when two one-forms $dx^{\mu}$ are
considered, their twisted Koszul bracket produces the $\breve{Q}$-flux. This
way, we obtained the analogous relation between the $\breve{f}$-flux and
$\breve{Q}$-flux. The former are structure functions in the twisted Lie
bracket algebra between holonomic partial derivatives, while the latter are
structure functions in the twisted Koszul bracket algebra between their
mathematical duals. In addition, we obtained the analogous relation between
the usual $Q$-flux and Koszul bracket on one side, and the $\breve{Q}$-flux
and the twisted Koszul bracket on the other side.

Lastly, in order to write the $\breve{R}$-flux in coordinate-free form, we
introduced the twisted Schouten-Nijenhuis bracket as the graded twisted Lie
bracket. The $\breve{R}$-flux can also be obtained from the action of the
exterior derivative associated with the twisted Koszul bracket on the bi-
vector $\breve{\theta}$.

It is well known that the Koszul bracket is a Lie algebroid bracket only when
the bi-vector is of the Poisson type, while in the other case it is the
bracket of a quasi-Lie algebroid. In case of the twisted Koszul bracket, the
Lie algebroid conditions will be satisfied if the twisted Schouten-Nijenhuis
bracket of the bi-vector $\breve{\theta}$ with itself is zero.

In a trivial case of constant fields $B$ and $\theta$, the matrix ${\cal C}$
is also constant and represent the diffeomorphisms. Obviously, all fluxes are
zero in this instance, but it is interesting to see the relations between the
newly obtained twisted brackets. The compatibility condition between the
anchor and the twisted Lie bracket becomes the push-forward invariance of the
Lie bracket.

The research of Courant algebroids found many applications relevant for
physicists. For instance, to each Courant algebroid it is possible to
construct a corresponding Courant $\sigma$-model, characterized by topological
action of AKSZ type that satisfies classical master equation [39]. Physically,
this is a theory of a topological brane. This can be used to analyze bosonic
string in different flux backgrounds. Moreover, the complete BV-BRST
formulation of the Courant $\sigma$-model can be described in terms of
$L_{\infty}$ algebras [40]. It would be interesting to explore whether we can
gain insights from the membrane described by this Courant algebroid, which
encompasses all fluxes and is manifestly invariant under T-duality.

Additionally, backgrounds were considered in which algebra of generalized
vielbeins is described by Courant algebroids. For example, this way torus with
fluxes obtained by application of T-duality along different directions can be
described. We hope that the Courant algebroid with generalized fluxes that we
obtained in this paper could also describe some background relevant for string
theory, which is a work that remains to be done.

##  Appendix A Derivation of the relation between the twisted Schouten-
Nijenhuis bracket and $\breve{R}$-flux

In this appendix, we will show that the relation (7.2) is indeed correct for
all bi-vectors

$\breve{\theta}=\sum_{i=1}^{n}\hat{\xi}_{i1}\wedge\hat{\xi}_{i2}\,,$ (A.1)

using the principle of mathematical induction. For the base of our induction,
we will consider the case of a simple bi-vector (for which $n=1$). Then, we
will show that if one assumes that the relation (7.2) hold for $n$, it will
also hold for $n+1$.

###  A.1 Simple bi-vector

Let us assume that the bi-vector $\breve{\theta}$ is decomposable bi-vector,
meaning that it can be expressed as a wedge product between some two vectors
$\hat{\xi}_{1}$ and $\hat{\xi}_{2}$

$\breve{\theta}=\hat{\xi}_{1}\wedge\hat{\xi}_{2}\,,\qquad\breve{\theta}^{\mu\nu}=\hat{\xi}_{1}^{\mu}\hat{\xi}_{2}^{\nu}-\hat{\xi}_{1}^{\nu}\hat{\xi}_{2}^{\mu}\,.$
(A.2)

Using the graded Leibniz identity (2.8), we obtain

$\displaystyle[\hat{\xi}_{1},\breve{\theta}]_{\hat{S}}=[\hat{\xi}_{1},\hat{\xi}_{1}\wedge\hat{\xi}_{2}]_{\hat{S}}=[\hat{\xi}_{1},\hat{\xi}_{1}]_{\hat{S}}\wedge\hat{\xi}_{2}+\hat{\xi}_{1}\wedge[\hat{\xi}_{2},\hat{\xi}_{1}]_{\hat{S}}=\hat{\xi}_{1}\wedge[\hat{\xi}_{2},\hat{\xi}_{1}]_{\hat{L}}\,,$
(A.3)

where we also used the third relation in (2.7). Similarly, we have

$\displaystyle[\hat{\xi}_{2},\breve{\theta}]_{\hat{S}}=-[\hat{\xi}_{1},\hat{\xi}_{2}]_{\hat{L}}\wedge\hat{\xi}_{2}\,.$
(A.4)

Now, the application of relations (2.8), (2.9), (A.3) and (A.4), we obtain

$\displaystyle[\breve{\theta},\breve{\theta}]_{\hat{S}}=[\breve{\theta},\hat{\xi}_{1}]_{\hat{S}}\wedge\hat{\xi}_{2}-\hat{\xi}_{1}\wedge[\breve{\theta},\hat{\xi}_{2}]_{\hat{S}}=-2\hat{\xi}_{1}\wedge[\hat{\xi}_{1},\hat{\xi}_{2}]_{\hat{L}}\wedge\hat{\xi}_{2}\,.$
(A.5)

In local basis spanned by $\partial_{\mu}$, the components of the twisted
Schouten-Nijenhuis bracket are

$\displaystyle\frac{1}{2}\Big{(}[\breve{\theta},\breve{\theta}]_{\hat{S}}\Big{)}^{\mu\nu\rho}$
$\displaystyle=$
$\displaystyle-\sum_{\tiny{odd}}\hat{\xi}_{1}^{\mu}([\hat{\xi}_{1},\hat{\xi}_{2}]_{\hat{L}})^{\nu}\hat{\xi}_{2}^{\rho}$
$\displaystyle=$
$\displaystyle-\hat{\xi}_{1}^{\mu}(\hat{\xi}_{1}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{2}^{\nu}-\hat{\xi}_{2}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{1}^{\nu}+\breve{f}_{\\!\alpha\beta}^{\
\nu}\ \hat{\xi}_{1}^{\alpha}\hat{\xi}_{2}^{\beta})\hat{\xi}_{2}^{\rho}$
$\displaystyle+\hat{\xi}_{1}^{\rho}(\hat{\xi}_{1}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{2}^{\nu}-\hat{\xi}_{2}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{1}^{\nu}+\breve{f}_{\\!\alpha\beta}^{\
\nu}\ \hat{\xi}_{1}^{\alpha}\hat{\xi}_{2}^{\beta})\hat{\xi}_{2}^{\mu}$
$\displaystyle-\hat{\xi}_{1}^{\nu}(\hat{\xi}_{1}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{2}^{\rho}-\hat{\xi}_{2}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{1}^{\rho}+\breve{f}_{\\!\alpha\beta}^{\
\rho}\ \hat{\xi}_{1}^{\alpha}\hat{\xi}_{2}^{\beta})\hat{\xi}_{2}^{\mu}$
$\displaystyle+\hat{\xi}_{1}^{\mu}(\hat{\xi}_{1}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{2}^{\rho}-\hat{\xi}_{2}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{1}^{\rho}+\breve{f}_{\\!\alpha\beta}^{\
\rho}\ \hat{\xi}_{1}^{\alpha}\hat{\xi}_{2}^{\beta})\hat{\xi}_{2}^{\nu}$
$\displaystyle-\hat{\xi}_{1}^{\rho}(\hat{\xi}_{1}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{2}^{\mu}-\hat{\xi}_{2}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{1}^{\mu}+\breve{f}_{\\!\alpha\beta}^{\
\mu}\ \hat{\xi}_{1}^{\alpha}\hat{\xi}_{2}^{\beta})\hat{\xi}_{2}^{\nu}$
$\displaystyle+\hat{\xi}_{1}^{\nu}(\hat{\xi}_{1}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{2}^{\mu}-\hat{\xi}_{2}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{1}^{\mu}+\breve{f}_{\\!\alpha\beta}^{\
\mu}\ \hat{\xi}_{1}^{\alpha}\hat{\xi}_{2}^{\beta})\hat{\xi}_{2}^{\rho}$
$\displaystyle=$
$\displaystyle-2\hat{\xi}_{1}^{\mu}(\hat{\xi}_{1}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{2}^{\nu}-\hat{\xi}_{2}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{1}^{\nu}+\breve{f}_{\\!\alpha\beta}^{\
\nu}\ \hat{\xi}_{1}^{\alpha}\hat{\xi}_{2}^{\beta})\hat{\xi}_{2}^{\rho}$
$\displaystyle-2\hat{\xi}_{1}^{\nu}(\hat{\xi}_{1}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{2}^{\rho}-\hat{\xi}_{2}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{1}^{\rho}+\breve{f}_{\\!\alpha\beta}^{\
\rho}\ \hat{\xi}_{1}^{\alpha}\hat{\xi}_{2}^{\beta})\hat{\xi}_{2}^{\mu}$
$\displaystyle-2\hat{\xi}_{1}^{\rho}(\hat{\xi}_{1}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{2}^{\mu}-\hat{\xi}_{2}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{1}^{\mu}+\breve{f}_{\\!\alpha\beta}^{\
\mu}\ \hat{\xi}_{1}^{\alpha}\hat{\xi}_{2}^{\beta})\hat{\xi}_{2}^{\nu}\,,$

where we summed over all odd permutations of indices $\mu$, $\nu$, and $\rho$,
and used the fact that expressions of the type
$\hat{\xi}_{1}^{\mu}(\hat{\xi}_{1}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{2}^{\nu}-\hat{\xi}_{2}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{1}^{\nu}+\breve{f}_{\\!\alpha\beta}^{\
\nu}\ \hat{\xi}_{1}^{\alpha}\hat{\xi}_{2}^{\beta})\hat{\xi}_{2}^{\rho}$ are
antisymmetric with respect to interchange of $\hat{\xi}_{1}$ and
$\hat{\xi}_{2}$.

In order to show that this is in fact the expression for $\breve{R}$-flux (4),
we firstly write

$\displaystyle\breve{\theta}^{\mu\sigma}\hat{\partial}_{\sigma}\breve{\theta}^{\nu\rho}+cyclic$
$\displaystyle=$
$\displaystyle\hat{\xi}_{1}^{\mu}\hat{\xi}_{2}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{1}^{\nu}\hat{\xi}_{2}^{\rho}+\hat{\xi}_{1}^{\mu}\hat{\xi}_{2}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{2}^{\rho}\hat{\xi}_{1}^{\nu}-\hat{\xi}_{1}^{\mu}\hat{\xi}_{2}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{2}^{\nu}\hat{\xi}_{1}^{\rho}-\hat{\xi}_{1}^{\mu}\hat{\xi}_{2}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{1}^{\rho}\hat{\xi}_{2}^{\nu}$
$\displaystyle+\hat{\xi}_{1}^{\nu}\hat{\xi}_{2}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{1}^{\rho}\hat{\xi}_{2}^{\mu}+\hat{\xi}_{1}^{\nu}\hat{\xi}_{2}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{2}^{\mu}\hat{\xi}_{1}^{\rho}-\hat{\xi}_{1}^{\nu}\hat{\xi}_{2}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{2}^{\rho}\hat{\xi}_{1}^{\mu}-\hat{\xi}_{1}^{\nu}\hat{\xi}_{2}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{1}^{\mu}\hat{\xi}_{2}^{\rho}$
$\displaystyle+\hat{\xi}_{1}^{\rho}\hat{\xi}_{2}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{1}^{\mu}\hat{\xi}_{2}^{\nu}+\hat{\xi}_{1}^{\rho}\hat{\xi}_{2}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{2}^{\nu}\hat{\xi}_{1}^{\mu}-\hat{\xi}_{1}^{\rho}\hat{\xi}_{2}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{2}^{\mu}\hat{\xi}_{1}^{\nu}-\hat{\xi}_{1}^{\rho}\hat{\xi}_{2}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{1}^{\nu}\hat{\xi}_{2}^{\mu}$
$\displaystyle-\hat{\xi}_{2}^{\mu}\hat{\xi}_{1}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{1}^{\nu}\hat{\xi}_{2}^{\rho}-\hat{\xi}_{2}^{\mu}\hat{\xi}_{1}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{2}^{\rho}\hat{\xi}_{1}^{\nu}+\hat{\xi}_{2}^{\mu}\hat{\xi}_{1}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{2}^{\nu}\hat{\xi}_{1}^{\rho}+\hat{\xi}_{2}^{\mu}\hat{\xi}_{1}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{1}^{\rho}\hat{\xi}_{2}^{\nu}$
$\displaystyle-\hat{\xi}_{2}^{\nu}\hat{\xi}_{1}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{1}^{\rho}\hat{\xi}_{2}^{\mu}-\hat{\xi}_{2}^{\nu}\hat{\xi}_{1}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{2}^{\mu}\hat{\xi}_{1}^{\rho}+\hat{\xi}_{2}^{\nu}\hat{\xi}_{1}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{2}^{\rho}\hat{\xi}_{1}^{\mu}+\hat{\xi}_{2}^{\nu}\hat{\xi}_{1}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{1}^{\mu}\hat{\xi}_{2}^{\rho}$
$\displaystyle-\hat{\xi}_{2}^{\rho}\hat{\xi}_{1}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{1}^{\mu}\hat{\xi}_{2}^{\nu}-\hat{\xi}_{2}^{\rho}\hat{\xi}_{1}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{2}^{\nu}\hat{\xi}_{1}^{\mu}+\hat{\xi}_{2}^{\rho}\hat{\xi}_{1}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{2}^{\mu}\hat{\xi}_{1}^{\nu}+\hat{\xi}_{2}^{\rho}\hat{\xi}_{1}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{1}^{\nu}\hat{\xi}_{2}^{\mu}$
$\displaystyle=$
$\displaystyle\hat{\xi}_{1}^{\mu}(\hat{\xi}_{2}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{1}^{\nu}-\hat{\xi}_{1}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{2}^{\nu})\hat{\xi}_{2}^{\rho}-\hat{\xi}_{1}^{\mu}(\hat{\xi}_{2}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{1}^{\rho}-\hat{\xi}_{1}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{2}^{\rho})\hat{\xi}_{2}^{\nu}$
$\displaystyle+\hat{\xi}_{1}^{\nu}(\hat{\xi}_{2}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{1}^{\rho}-\hat{\xi}_{1}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{2}^{\rho})\hat{\xi}_{2}^{\mu}-\hat{\xi}_{1}^{\nu}(\hat{\xi}_{2}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{1}^{\mu}-\hat{\xi}_{1}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{2}^{\mu})\hat{\xi}_{2}^{\rho}$
$\displaystyle+\hat{\xi}_{1}^{\rho}(\hat{\xi}_{2}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{1}^{\mu}-\hat{\xi}_{1}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{2}^{\mu})\hat{\xi}_{2}^{\nu}-\hat{\xi}_{1}^{\rho}(\hat{\xi}_{2}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{1}^{\nu}-\hat{\xi}_{1}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{2}^{\nu})\hat{\xi}_{2}^{\mu}$
$\displaystyle=$
$\displaystyle-2\hat{\xi}_{1}^{\mu}(\hat{\xi}_{1}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{2}^{\nu}-\hat{\xi}_{2}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{1}^{\nu})\hat{\xi}_{2}^{\rho}-2\hat{\xi}_{1}^{\nu}(\hat{\xi}_{1}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{2}^{\rho}-\hat{\xi}_{2}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{1}^{\rho})\hat{\xi}_{2}^{\mu}$
$\displaystyle-2\hat{\xi}_{1}^{\rho}(\hat{\xi}_{1}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{2}^{\mu}-\hat{\xi}_{2}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{1}^{\mu})\hat{\xi}_{2}^{\nu}\,.$

Secondarily, we have

$\displaystyle-\breve{\theta}^{\mu\alpha}\breve{\theta}^{\rho\beta}\breve{f}_{\\!\alpha\beta}^{\
\nu}+cyclic$ $\displaystyle=$
$\displaystyle-(\hat{\xi}_{1}^{\mu}\hat{\xi}_{2}^{\alpha}-\hat{\xi}_{2}^{\mu}\hat{\xi}_{1}^{\alpha})(\hat{\xi}_{1}^{\rho}\hat{\xi}_{2}^{\beta}-\hat{\xi}_{2}^{\rho}\hat{\xi}_{1}^{\beta})\breve{f}_{\\!\alpha\beta}^{\
\nu}+cyclic$ $\displaystyle=$
$\displaystyle-\hat{\xi}_{1}^{\mu}\hat{\xi}_{1}^{\rho}\breve{f}_{\\!\alpha\beta}^{\
\nu}\
\hat{\xi}_{2}^{\alpha}\hat{\xi}_{2}^{\beta}+\hat{\xi}_{1}^{\mu}\hat{\xi}_{2}^{\rho}\breve{f}_{\\!\alpha\beta}^{\
\nu}\ \hat{\xi}_{2}^{\alpha}\hat{\xi}_{1}^{\beta}$
$\displaystyle+\hat{\xi}_{2}^{\mu}\hat{\xi}_{1}^{\rho}\breve{f}_{\\!\alpha\beta}^{\
\nu}\
\hat{\xi}_{1}^{\alpha}\hat{\xi}_{2}^{\beta}-\hat{\xi}_{2}^{\mu}\hat{\xi}_{2}^{\rho}\breve{f}_{\\!\alpha\beta}^{\
\nu}\ \hat{\xi}_{1}^{\alpha}\hat{\xi}_{1}^{\beta}+cyclic$ $\displaystyle=$
$\displaystyle-2\hat{\xi}_{1}^{\mu}\hat{\xi}_{2}^{\rho}\breve{f}_{\\!\alpha\beta}^{\
\nu}\
\hat{\xi}_{1}^{\alpha}\hat{\xi}_{2}^{\beta}-2\hat{\xi}_{1}^{\nu}\hat{\xi}_{2}^{\mu}\breve{f}_{\\!\alpha\beta}^{\
\rho}\
\hat{\xi}_{1}^{\alpha}\hat{\xi}_{2}^{\beta}-2\hat{\xi}_{1}^{\rho}\hat{\xi}_{2}^{\nu}\breve{f}_{\\!\alpha\beta}^{\
\mu}\ \hat{\xi}_{1}^{\alpha}\hat{\xi}_{2}^{\beta}\,,$

where we used the antisymmetric properties of the $\breve{f}$-flux, such as

$\breve{f}_{\\!\alpha\beta}^{\ \nu}\
\hat{\xi}_{1}^{\alpha}\hat{\xi}_{1}^{\beta}=0\,,\quad\breve{f}_{\\!\alpha\beta}^{\
\nu}\
\hat{\xi}_{1}^{\alpha}\hat{\xi}_{2}^{\beta}=-\breve{f}_{\\!\alpha\beta}^{\
\nu}\ \hat{\xi}_{2}^{\alpha}\hat{\xi}_{1}^{\beta}\,.$ (A.9)

Substituting (A.1) and (A.1) into (4), we obtain exactly (A.1). As such, we
showed that (7.2) stands for any simple bi-vector $\breve{\theta}$.

###  A.2 Composite bi-vector

Let us assume that for some number $n$, the twisted Schouten-Nijenhuis bracket
of a bi-vector that is composite of $n$ simple bi-vectors with itself is given
by the $\breve{R}$-flux formula. If we can prove that then for $n+1$ the
relation holds, according to the principle of mathematical induction this is
sufficient to prove for all $n$. Let

$\breve{\theta}_{n}=\sum_{i=1}^{n}\hat{\xi}_{i1}\wedge\hat{\xi}_{i2}\,,\qquad\breve{\theta}_{n}^{\mu\nu}=\sum_{i=1}^{n}(\hat{\xi}_{i1}^{\mu}\hat{\xi}_{i2}^{\nu}-\hat{\xi}_{i1}^{\nu}\hat{\xi}_{i2}^{\mu})\,,$
(A.10)

where $\hat{\xi}_{i}$ and $\hat{\zeta}_{i}$ are vectors. Assume that for that
number $n$ we have

$\frac{1}{2}[\breve{\theta}_{n},\breve{\theta}_{n}]_{\hat{S}}=\breve{\theta}_{n}^{\mu\alpha}\hat{\partial}_{\alpha}\breve{\theta}_{n}^{\nu\rho}-\breve{\theta}_{n}^{\mu\alpha}\breve{\theta}_{n}^{\rho\beta}\breve{f}_{\\!\alpha\beta}^{\
\nu}+cyclic\,.$ (A.11)

Now it is necessary to prove that for some

$\breve{\theta}_{n+1}=\breve{\theta}_{n}+\breve{\beta}\,,\qquad\breve{\beta}=\hat{\eta}_{1}\wedge\hat{\eta}_{2}\,,$
(A.12)

the relation (A.11) still holds.

We have

$\displaystyle[\breve{\theta}_{n+1},\breve{\theta}_{n+1}]_{\hat{S}}=[\breve{\theta}_{n},\breve{\theta}_{n}]_{\hat{S}}+[\breve{\theta}_{n},\breve{\beta}]_{\hat{S}}+[\breve{\beta},\breve{\theta}_{n}]_{\hat{S}}+[\breve{\beta},\breve{\beta}]_{\hat{S}}\,.$
(A.13)

Using the graded Leibniz identity (2.8), we obtain

$\displaystyle[\breve{\theta}_{n},\breve{\beta}]_{\hat{S}}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}[\hat{\xi}_{i1}\wedge\hat{\xi}_{i2},\hat{\eta}_{1}\wedge\hat{\eta}_{2}]_{\hat{S}}=\sum_{i=1}^{n}\Big{(}[\hat{\xi}_{i1}\wedge\hat{\xi}_{i2},\hat{\eta}_{1}]_{\hat{S}}\wedge\hat{\eta}_{2}-\hat{\eta}_{1}\wedge[\hat{\xi}_{i1}\wedge\hat{\xi}_{i2},\hat{\eta}_{2}]_{\hat{S}}\Big{)}$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\Big{(}-[\hat{\eta}_{1},\hat{\xi}_{i1}]_{\hat{L}}\wedge\hat{\xi}_{i2}\wedge\hat{\eta}_{2}-\hat{\xi}_{i1}\wedge[\hat{\eta}_{1},\hat{\xi}_{i2}]_{\hat{L}}\wedge\hat{\eta}_{2}$
$\displaystyle\qquad+\hat{\eta}_{1}\wedge[\hat{\eta}_{2},\hat{\xi}_{i1}]_{\hat{L}}\wedge\hat{\xi}_{i2}+\hat{\eta}_{1}\wedge\hat{\xi}_{i1}\wedge[\hat{\eta}_{2},\hat{\xi}_{i2}]_{\hat{L}}\Big{)}\,,$

Furthermore, with the help of (5.47), we firstly note that

$\displaystyle-\Big{(}[\hat{\eta}_{1},\hat{\xi}_{i1}]_{\hat{L}}\wedge\hat{\xi}_{i2}\wedge\hat{\eta}_{2}\Big{)}^{\mu\nu\rho}$
$\displaystyle=$
$\displaystyle(\hat{\xi}_{i1}^{\sigma}\hat{\partial}_{\sigma}\hat{\eta}_{1}^{\mu}-\hat{\eta}_{1}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{i1}^{\mu}-\breve{f}_{\\!\alpha\beta}^{\
\mu}\
\hat{\eta}_{1}^{\alpha}\hat{\xi}_{i1}^{\beta})\hat{\xi}_{i2}^{\nu}\hat{\eta}_{2}^{\rho}$
$\displaystyle-(\hat{\xi}_{i1}^{\sigma}\hat{\partial}_{\sigma}\hat{\eta}_{1}^{\mu}-\hat{\eta}_{1}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{i1}^{\mu}-\breve{f}_{\\!\alpha\beta}^{\
\mu}\
\hat{\eta}_{1}^{\alpha}\hat{\xi}_{i1}^{\beta})\hat{\xi}_{i2}^{\rho}\hat{\eta}_{2}^{\nu}+cyclic$
$\displaystyle=$
$\displaystyle(\hat{\xi}_{i1}^{\sigma}\hat{\partial}_{\sigma}\hat{\eta}_{1}^{\rho}-\hat{\eta}_{1}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{i1}^{\rho}+\breve{f}_{\\!\alpha\beta}^{\
\rho}\
\hat{\xi}_{i1}^{\alpha}\hat{\eta}_{1}^{\beta})\hat{\xi}_{i2}^{\mu}\hat{\eta}_{2}^{\nu}$
$\displaystyle-(\hat{\xi}_{i1}^{\sigma}\hat{\partial}_{\sigma}\hat{\eta}_{1}^{\nu}-\hat{\eta}_{1}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{i1}^{\nu}+\breve{f}_{\\!\alpha\beta}^{\
\nu}\
\hat{\xi}_{i1}^{\alpha}\hat{\eta}_{1}^{\beta})\hat{\xi}_{i2}^{\mu}\hat{\eta}_{2}^{\rho}+cyclic\,,$

where we chose the index permutation that will be convenient for us later.
Secondly, we note

$\displaystyle-\Big{(}\hat{\xi}_{i1}\wedge[\hat{\eta}_{1},\hat{\xi}_{i2}]_{\hat{L}}\wedge\hat{\eta}_{2}\Big{)}^{\mu\nu\rho}$
$\displaystyle=$
$\displaystyle(\hat{\xi}_{i2}^{\sigma}\hat{\partial}_{\sigma}\hat{\eta}_{1}^{\nu}-\hat{\eta}_{1}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{i2}^{\nu}+\breve{f}_{\\!\alpha\beta}^{\
\nu}\
\hat{\xi}_{i2}^{\alpha}\hat{\eta}_{1}^{\beta})\hat{\xi}_{i1}^{\mu}\hat{\eta}_{2}^{\rho}$
$\displaystyle-(\hat{\xi}_{i2}^{\sigma}\hat{\partial}_{\sigma}\hat{\eta}_{1}^{\rho}-\hat{\eta}_{1}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{i2}^{\rho}+\breve{f}_{\\!\alpha\beta}^{\
\rho}\
\hat{\xi}_{i2}^{\alpha}\hat{\eta}_{1}^{\beta})\hat{\xi}_{i1}^{\mu}\hat{\eta}_{2}^{\nu}+cyclic.$

In addition, we have

$\displaystyle\Big{(}\hat{\eta}_{1}\wedge[\hat{\eta}_{2},\hat{\xi}_{i1}]_{\hat{L}}\wedge\hat{\xi}_{i2}\Big{)}^{\mu\nu\rho}$
$\displaystyle=$
$\displaystyle(\hat{\eta}_{2}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{i1}^{\rho}-\hat{\xi}_{i1}^{\sigma}\hat{\partial}_{\sigma}\hat{\eta}_{2}^{\rho}+\breve{f}_{\\!\alpha\beta}^{\
\rho}\
\hat{\eta}_{2}^{\alpha}\hat{\xi}_{i1}^{\beta})\hat{\xi}_{i2}^{\mu}\hat{\eta}_{1}^{\nu}$
$\displaystyle-(\hat{\eta}_{2}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{i1}^{\nu}-\hat{\xi}_{i1}^{\sigma}\hat{\partial}_{\sigma}\hat{\eta}_{2}^{\nu}+\breve{f}_{\\!\alpha\beta}^{\
\nu}\
\hat{\eta}_{2}^{\alpha}\hat{\xi}_{i1}^{\beta})\hat{\xi}_{i2}^{\mu}\hat{\eta}_{1}^{\rho}+cyclic\,,$

and

$\displaystyle\Big{(}\hat{\xi}_{i1}\wedge\hat{\eta}_{1}\wedge[\hat{\xi}_{i2},\hat{\eta}_{2}]_{\hat{L}}\Big{)}^{\mu\nu\rho}$
$\displaystyle=$
$\displaystyle(\hat{\xi}_{i2}^{\sigma}\hat{\partial}_{\sigma}\hat{\eta}_{2}^{\rho}-\hat{\eta}_{2}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{i2}^{\rho}+\breve{f}_{\\!\alpha\beta}^{\
\rho}\
\hat{\xi}_{i2}^{\alpha}\hat{\eta}_{2}^{\beta})\hat{\xi}_{i1}^{\mu}\hat{\eta}_{1}^{\nu}$
$\displaystyle-(\hat{\xi}_{i2}^{\sigma}\hat{\partial}_{\sigma}\hat{\eta}_{2}^{\nu}-\hat{\eta}_{2}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{i2}^{\nu}+\breve{f}_{\\!\alpha\beta}^{\
\nu}\
\hat{\xi}_{i2}^{\alpha}\hat{\eta}_{2}^{\beta})\hat{\xi}_{i1}^{\mu}\hat{\eta}_{1}^{\rho}+cyclic.$

Lastly, using graded commutative identity (2.9), we obtain

$[\breve{\beta},\breve{\theta}_{n}]_{\hat{S}}=-(-1)^{1}[\breve{\theta}_{n},\breve{\beta}]_{\hat{S}}=[\breve{\theta}_{n},\breve{\beta}]_{\hat{S}}\,.$
(A.19)

Combining (A.2), (A.2), (A.2), (A.2) and (A.19), as well as induction
hypothesis (A.11) into (A.13), we obtain

$\displaystyle\frac{1}{2}\Big{(}[\breve{\theta}_{n+1},\breve{\theta}_{n+1}]_{\hat{S}}\Big{)}^{\mu\nu\rho}$
$\displaystyle=$
$\displaystyle\breve{\theta}_{n}^{\mu\alpha}\hat{\partial}_{\alpha}\breve{\theta}_{n}^{\nu\rho}-\breve{\theta}_{n}^{\mu\alpha}\breve{\theta}_{n}^{\rho\beta}\breve{f}_{\\!\alpha\beta}^{\
\nu}+\breve{\beta}^{\mu\alpha}\hat{\partial}_{\alpha}\breve{\beta}^{\nu\rho}-\breve{\beta}^{\mu\alpha}\breve{\beta}^{\rho\beta}\breve{f}_{\\!\alpha\beta}^{\
\nu}$
$\displaystyle+\sum_{i=1}^{n}\Big{(}(\hat{\xi}_{i1}^{\sigma}\hat{\partial}_{\sigma}\hat{\eta}_{1}^{\rho}-\hat{\eta}_{1}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{i1}^{\rho}+\breve{f}_{\\!\alpha\beta}^{\
\rho}\
\hat{\xi}_{i1}^{\alpha}\hat{\eta}_{1}^{\beta})\hat{\xi}_{i2}^{\mu}\hat{\eta}_{2}^{\nu}$
$\displaystyle\qquad-(\hat{\xi}_{i1}^{\sigma}\hat{\partial}_{\sigma}\hat{\eta}_{1}^{\nu}-\hat{\eta}_{1}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{i1}^{\nu}+\breve{f}_{\\!\alpha\beta}^{\
\nu}\
\hat{\xi}_{i1}^{\alpha}\hat{\eta}_{1}^{\beta})\hat{\xi}_{i2}^{\mu}\hat{\eta}_{2}^{\rho}$
$\displaystyle\qquad+(\hat{\xi}_{i2}^{\sigma}\hat{\partial}_{\sigma}\hat{\eta}_{1}^{\nu}-\hat{\eta}_{1}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{i2}^{\nu}+\breve{f}_{\\!\alpha\beta}^{\
\nu}\
\hat{\xi}_{i2}^{\alpha}\hat{\eta}_{1}^{\beta})\hat{\xi}_{i1}^{\mu}\hat{\eta}_{2}^{\rho}$
$\displaystyle\qquad-(\hat{\xi}_{i2}^{\sigma}\hat{\partial}_{\sigma}\hat{\eta}_{1}^{\rho}-\hat{\eta}_{1}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{i2}^{\rho}+\breve{f}_{\\!\alpha\beta}^{\
\rho}\
\hat{\xi}_{i2}^{\alpha}\hat{\eta}_{1}^{\beta})\hat{\xi}_{i1}^{\mu}\hat{\eta}_{2}^{\nu}$
$\displaystyle\qquad+(\hat{\eta}_{2}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{i1}^{\rho}-\hat{\xi}_{i1}^{\sigma}\hat{\partial}_{\sigma}\hat{\eta}_{2}^{\rho}+\breve{f}_{\\!\alpha\beta}^{\
\rho}\
\hat{\eta}_{2}^{\alpha}\hat{\xi}_{i1}^{\beta})\hat{\xi}_{i2}^{\mu}\hat{\eta}_{1}^{\nu}$
$\displaystyle\qquad-(\hat{\eta}_{2}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{i1}^{\nu}-\hat{\xi}_{i1}^{\sigma}\hat{\partial}_{\sigma}\hat{\eta}_{2}^{\nu}+\breve{f}_{\\!\alpha\beta}^{\
\nu}\
\hat{\eta}_{2}^{\alpha}\hat{\xi}_{i1}^{\beta})\hat{\xi}_{i2}^{\mu}\hat{\eta}_{1}^{\rho}$
$\displaystyle\qquad+(\hat{\xi}_{i2}^{\sigma}\hat{\partial}_{\sigma}\hat{\eta}_{2}^{\rho}-\hat{\eta}_{2}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{i2}^{\rho}+\breve{f}_{\\!\alpha\beta}^{\
\rho}\
\hat{\xi}_{i2}^{\alpha}\hat{\eta}_{2}^{\beta})\hat{\xi}_{i1}^{\mu}\hat{\eta}_{1}^{\nu}$
$\displaystyle\qquad-(\hat{\xi}_{i2}^{\sigma}\hat{\partial}_{\sigma}\hat{\eta}_{2}^{\nu}-\hat{\eta}_{2}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{i2}^{\nu}+\breve{f}_{\\!\alpha\beta}^{\
\nu}\
\hat{\xi}_{i2}^{\alpha}\hat{\eta}_{2}^{\beta})\hat{\xi}_{i1}^{\mu}\hat{\eta}_{1}^{\rho}+cyclic.$

On the other hand, the $\breve{R}$-flux for this composite bi-vector can be
expressed as

$\displaystyle\breve{R}^{\mu\nu\rho}$ $\displaystyle=$
$\displaystyle(\breve{\theta}_{n}^{\mu\alpha}+\breve{\beta}^{\mu\alpha})\hat{\partial}_{\alpha}(\breve{\theta}_{n}^{\nu\rho}+\breve{\beta}^{\nu\rho})-(\breve{\theta}_{n}^{\mu\alpha}+\breve{\beta}^{\mu\alpha})(\breve{\theta}_{n}^{\rho\beta}+\breve{\beta}^{\rho\beta})\breve{f}_{\\!\alpha\beta}^{\
\nu}+cyclic$ $\displaystyle=$
$\displaystyle\breve{\theta}_{n}^{\mu\alpha}\hat{\partial}_{\alpha}\breve{\theta}_{n}^{\nu\rho}-\breve{\theta}_{n}^{\mu\alpha}\breve{\theta}_{n}^{\rho\beta}\breve{f}_{\\!\alpha\beta}^{\
\nu}+\breve{\beta}^{\mu\alpha}\hat{\partial}_{\alpha}\breve{\beta}^{\nu\rho}-\breve{\beta}^{\mu\alpha}\breve{\beta}^{\rho\beta}\breve{f}_{\\!\alpha\beta}^{\
\nu}$
$\displaystyle+\breve{\beta}^{\mu\alpha}\hat{\partial}_{\alpha}\breve{\theta}_{n}^{\nu\rho}+\breve{\theta}_{n}^{\mu\alpha}\hat{\partial}_{\alpha}\breve{\beta}^{\nu\rho}-\breve{\theta}_{n}^{\mu\alpha}\breve{\beta}^{\rho\beta}\breve{f}_{\\!\alpha\beta}^{\
\nu}-\breve{\beta}^{\mu\alpha}\breve{\theta}_{n}^{\rho\beta}\breve{f}_{\\!\alpha\beta}^{\
\nu}+cyclic$

We have

$\displaystyle\breve{\theta}_{n}^{\mu\sigma}\hat{\partial}_{\sigma}\breve{\beta}^{\nu\rho}+cyclic$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n}(\hat{\xi}_{i1}^{\mu}\hat{\xi}_{i2}^{\sigma}-\hat{\xi}_{i2}^{\mu}\hat{\xi}_{i1}^{\sigma})\hat{\partial}_{\sigma}(\hat{\eta}_{1}^{\nu}\hat{\eta}_{2}^{\rho}-\hat{\eta}_{2}^{\nu}\hat{\eta}_{1}^{\rho})+cyclic$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\Big{(}\hat{\xi}_{i1}^{\mu}\hat{\xi}_{i2}^{\sigma}(\hat{\partial}_{\sigma}\hat{\eta}_{1}^{\nu}\hat{\eta}_{2}^{\rho}+\hat{\partial}_{\sigma}\hat{\eta}_{2}^{\rho}\hat{\eta}_{1}^{\nu}-\hat{\partial}_{\sigma}\hat{\eta}_{2}^{\nu}\hat{\eta}_{1}^{\rho}-\hat{\partial}_{\sigma}\hat{\eta}_{1}^{\rho}\hat{\eta}_{2}^{\nu})$
$\displaystyle-\hat{\xi}_{i2}^{\mu}\hat{\xi}_{i1}^{\sigma}(\hat{\partial}_{\sigma}\hat{\eta}_{1}^{\nu}\hat{\eta}_{2}^{\rho}+\hat{\partial}_{\sigma}\hat{\eta}_{2}^{\rho}\hat{\eta}_{1}^{\nu}-\hat{\partial}_{\sigma}\hat{\eta}_{2}^{\nu}\hat{\eta}_{1}^{\rho}-\hat{\partial}_{\sigma}\hat{\eta}_{1}^{\rho}\hat{\eta}_{2}^{\nu})+cyclic\Big{)}$

Similarly, we have

$\displaystyle\breve{\beta}^{\mu\sigma}\hat{\partial}_{\sigma}\breve{\theta}_{n}^{\nu\rho}+cyclic$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n}(\hat{\eta}_{1}^{\mu}\hat{\eta}_{2}^{\sigma}-\hat{\eta}_{2}^{\mu}\hat{\eta}_{1}^{\sigma})\hat{\partial}_{\sigma}(\hat{\xi}_{i1}^{\nu}\hat{\xi}_{i2}^{\rho}-\hat{\xi}_{i2}^{\nu}\hat{\xi}_{i1}^{\rho})+cyclic$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\Big{(}\hat{\eta}_{1}^{\mu}\hat{\eta}_{2}^{\sigma}(\hat{\partial}_{\sigma}\hat{\xi}_{i1}^{\nu}\hat{\xi}_{i2}^{\rho}+\hat{\partial}_{\sigma}\hat{\xi}_{i2}^{\rho}\hat{\xi}_{i1}^{\nu}-\hat{\partial}_{\sigma}\hat{\xi}_{i2}^{\nu}\hat{\xi}_{i1}^{\rho}-\hat{\partial}_{\sigma}\hat{\xi}_{i1}^{\rho}\hat{\xi}_{i2}^{\nu})$
$\displaystyle-\hat{\eta}_{2}^{\mu}\hat{\eta}_{1}^{\sigma}(\hat{\partial}_{\sigma}\hat{\xi}_{i1}^{\nu}\hat{\xi}_{i2}^{\rho}+\hat{\partial}_{\sigma}\hat{\xi}_{i2}^{\rho}\hat{\xi}_{i1}^{\nu}-\hat{\partial}_{\sigma}\hat{\xi}_{i2}^{\nu}\hat{\xi}_{i1}^{\rho}-\hat{\partial}_{\sigma}\hat{\xi}_{i1}^{\rho}\hat{\xi}_{i2}^{\nu})+cyclic\Big{)}$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\Big{(}\hat{\eta}_{1}^{\nu}\hat{\eta}_{2}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{i1}^{\rho}\hat{\xi}_{i2}^{\mu}+\hat{\eta}_{1}^{\rho}\hat{\eta}_{2}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{i2}^{\nu}\hat{\xi}_{i1}^{\mu}-\hat{\eta}_{1}^{\nu}\hat{\eta}_{2}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{i2}^{\rho}\hat{\xi}_{i1}^{\mu}$
$\displaystyle-\hat{\eta}_{1}^{\rho}\hat{\eta}_{2}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{i1}^{\nu}\hat{\xi}_{i2}^{\mu}-\hat{\eta}_{2}^{\nu}\hat{\eta}_{1}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{i1}^{\rho}\hat{\xi}_{i2}^{\mu}-\hat{\eta}_{2}^{\rho}\hat{\eta}_{1}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{i2}^{\nu}\hat{\xi}_{i1}^{\mu}$
$\displaystyle+\hat{\eta}_{2}^{\nu}\hat{\eta}_{1}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{i2}^{\rho}\hat{\xi}_{i1}^{\mu}+\hat{\eta}_{2}^{\rho}\hat{\eta}_{1}^{\sigma}\hat{\partial}_{\sigma}\hat{\xi}_{i1}^{\nu}\hat{\xi}_{i2}^{\mu}+cyclic\Big{)}$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\Big{[}\hat{\xi}_{i2}^{\mu}\Big{(}\hat{\eta}_{2}^{\sigma}(\hat{\eta}_{1}^{\nu}\hat{\partial}_{\sigma}\hat{\xi}_{i1}^{\rho}-\hat{\eta}_{1}^{\rho}\hat{\partial}_{\sigma}\hat{\xi}_{i1}^{\nu})-\hat{\eta}_{1}^{\sigma}(\hat{\eta}_{2}^{\nu}\hat{\partial}_{\sigma}\hat{\xi}_{i1}^{\rho}-\hat{\eta}_{2}^{\rho}\hat{\partial}_{\sigma}\hat{\xi}_{i1}^{\nu})\Big{)}$
$\displaystyle+\hat{\xi}_{i1}^{\mu}\Big{(}\hat{\eta}_{2}^{\sigma}(\hat{\eta}_{1}^{\rho}\hat{\partial}_{\sigma}\hat{\xi}_{i2}^{\nu}-\hat{\eta}_{1}^{\nu}\hat{\partial}_{\sigma}\hat{\xi}_{i2}^{\rho})-\hat{\eta}_{1}^{\sigma}(\hat{\eta}_{2}^{\rho}\hat{\partial}_{\sigma}\hat{\xi}_{i2}^{\nu}-\hat{\eta}_{2}^{\nu}\hat{\partial}_{\sigma}\hat{\xi}_{i2}^{\rho})\Big{)}\Big{]}+cyclic\,,$

where we firstly manipulated with cyclic indices and then just rewrote the
expression for future convenience. Lastly, we have

$\displaystyle-\breve{\theta}_{n}^{\mu\alpha}\breve{\beta}^{\rho\beta}\breve{f}_{\\!\alpha\beta}^{\
\nu}+cyclic$ $\displaystyle=$
$\displaystyle-\sum_{i=1}^{n}(\hat{\xi}_{i1}^{\mu}\hat{\xi}_{i2}^{\alpha}-\hat{\xi}_{i1}^{\alpha}\hat{\xi}_{i2}^{\mu})(\hat{\eta}_{1}^{\rho}\hat{\eta}_{2}^{\beta}-\hat{\eta}_{1}^{\beta}\hat{\eta}_{2}^{\rho})\breve{f}_{\alpha\beta}^{\
\nu}+cyclic\,,$ (A.24)

and

$\displaystyle-\breve{\beta}^{\mu\alpha}\breve{\theta}_{n}^{\rho\beta}\breve{f}_{\\!\alpha\beta}^{\
\nu}+cyclic$ $\displaystyle=$
$\displaystyle-\sum_{i=1}^{n}(\hat{\eta}_{1}^{\mu}\hat{\eta}_{2}^{\alpha}-\hat{\eta}_{1}^{\alpha}\hat{\eta}_{2}^{\mu})(\hat{\xi}_{i1}^{\rho}\hat{\xi}_{i2}^{\beta}-\hat{\xi}_{i1}^{\beta}\hat{\xi}_{i2}^{\rho})\breve{f}_{\alpha\beta}^{\
\nu}+cyclic$ $\displaystyle=$
$\displaystyle-\sum_{i=1}^{n}(\hat{\eta}_{1}^{\nu}\hat{\eta}_{2}^{\alpha}-\hat{\eta}_{1}^{\alpha}\hat{\eta}_{2}^{\nu})(\hat{\xi}_{i1}^{\mu}\hat{\xi}_{i2}^{\beta}-\hat{\xi}_{i1}^{\beta}\hat{\xi}_{i2}^{\mu})\breve{f}_{\alpha\beta}^{\
\rho}+cyclic\,.$

Substituting relations (A.2)-(A.2) into (A.2), one obtains exactly the
relation (A.2). Therefore, we have demonstrated via principle of mathematical
induction, that the twisted Schouten-Nijenhuis bracket of the arbitrary bi-
vector with itself produces exactly the $\breve{R}$-flux.

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