Topological Membranes, Current Algebras and H-flux - R-flux Duality based on Courant Algebroids

We construct a topological sigma model and a current algebra based on a Courant algebroid structure on a Poisson manifold. In order to construct models, we reformulate the Poisson Courant algebroid by supergeometric construction on a QP-manifold. A new duality of Courant algebroids which transforms H-flux and R-flux is proposed, where the transformation is interpreted as a canonical transformation of a graded symplectic manifold.


Introduction
There exist various dualities in string theory. Among them, T-duality is directly connected with the geometry of the target space and thus has to be a characteristic property of stringy geometry.
One of the formulations to analyze T-duality is the approach of doubled geometry, which has manifest O(d, d) invariance, and there, the existence of so-called nongeometric fluxes has been proposed [1]. On the other hand, the fluxes H, F , Q and R and their transformations have also been conjectured from T-duality analysis in supergravity compactification scenario [2,3]. It has been proposed that T-duality converts H-, F -, Q-and R-fluxes into each other.
Recently, there are further developments related to T-duality. Double field theory [4] is a manifestly O(d, d) covariant field theory which allows also for T-duality along non-isometry directions. Examples for other developments are the branes as sources for Q-and R-fluxes [5,6] and the β-supergravity [7]. The topological T-duality [8,9] is also proposed to analyze T-duality with flux. However, the background geometric structures for nongeometric fluxes are not well understood.
A background geometry in string theory with NS H-flux [10] is known to be a Courant algebroid [11,12], and the standard Courant algebroid of the generalized tangent bundle T M ⊕ T * M is of particular interest in the framework of generalized geometry [13,14]. The T-duality on the H-flux is well understood as an automorphism on the standard Courant algebroid if ι X ι Y H = 0 [15]. However, we cannot simultaneously introduce all degrees of freedom of H-, F -, Q-, R-fluxes as deformation of the Courant algebroid. The only independent deformation in the exact Courant algebroid is a 3-form (H-flux) degree of freedom [16].
Recently, the Courant algebroid on a Poisson manifold, i.e. the Poisson Courant algebroid, has been introduced in [17] as a geometric object for a background with R-flux. It is shown that the nontrivial flux R of a 3-vector can be introduced consistently on a Poisson manifold as a deformation of the Courant algebroid. It is the 'contravariant object' [18] with respect to the standard Courant algebroid, which is the exchange of T * M with T M and H-flux with R-flux. The T-duality on the R-flux has also been analyzed and it has been shown that the duality of R-flux with Q-flux is also understood as an automorphism on the Poisson Courant algebroid [19].
In this paper, we analyze the geometric structure of the Poisson Courant algebroid and a duality between H-flux and R-flux, which we call flux duality, in detail. We also construct the corresponding worldvolume theories, a topological sigma model and a current algebra with the structure of this Poisson Courant algebroid.
We first discuss the mathematical features and some structural correspondences between the standard Courant algebroid with H-flux and the Poisson Courant algebroid with R-flux.
By analyzing both coboundary operators, we generalize the duality between the de Rham cohomology and the Poisson cohomology as the background algebraic structure. Moreover, we have a new interpretation of this duality as a canonical transformation on a graded symplectic manifold 5 , and we formulate a flux duality, a duality between the H-flux and R-flux.
Then, we discuss field theoretic realizations of the Poisson Courant algebroid as a symmetry; we construct a topological sigma model and a current algebra. To this end, first we reformulate the Courant algebroid in terms of supergeometry. The construction of the Courant algebroid by using supergeometry and the derived brackets are introduced in [22]. This formulation uses a so-called QP-manifold, a differential graded symplectic manifold [23,24]. The advantage of the use of supergeometry is that the topological sigma model and the current algebra are constructed straightforwardly from the supergeometric data. The general theories are known as the AKSZ construction of topological sigma models [25] and the supergeometric BFV formulation of current algebras [26].
It is known that the AKSZ sigma model in three dimensions generally has the structure of a Courant algebroid [27,28,29]. Physically, this is a theory of a topological membrane. Following general arguments, we construct a topological sigma model from the Poisson Courant algebroid in three dimensions. When the three-dimensional world volume has a boundary, i.e. when we consider the open membrane, we obtain a two-dimensional boundary sigma model a la WZW. This is the Poisson sigma model with R-flux on the Dirac structure of a Poisson Courant algebroid. From the point of view of the sigma model, T-duality is changing the boundary conditions of the topological membranes. There is an approach with a similar concept proposed in [30]. The difference is that our formalism is based on the Poisson Courant algebroid.
We also construct a current algebra of the Poisson Courant algebroid on loop space, coming 5 T-duality has been formulated as a canonical transformation on the string phase space in [20,21].
Canonical transformations in this paper are defined on a graded target manifold.
from the canonical formulation of the theories on (1 + 1)-dimensional spacetime S 1 × R. In the H-flux case, this is the Alekseev-Strobl current algebra [31], which has the structure of the standard Courant algebroid with H-flux as underlying symmetry. This type of current algebra can also be reformulated by using the supergeometric construction [32,26]. Following these general formulations, we construct a corresponding current algebra with R-flux. This paper is organized as follows. In section 2, we review the supergeometric construction of Courant algebroids and, in section 3, we apply it to the Poisson Courant algebroid. In section 4, we discuss the mathematical structure of the duality of the standard Courant algebroid and the Poisson Courant algebroid. In section 5, we discuss the meaning of R-flux of the Poisson Courant algebroid from the perspective of double field theory. In section 6, we review the AKSZ sigma model. Then, we construct a topological sigma model of the Poisson Courant algebroid and analyze its boundary theories. In section 7, we construct the current algebra of the Poisson Courant algebroid. Finally, section 8 is devoted to conclusion and discussion. In the Appendix, our notation of supergeometry in this paper is summarized.

Courant algebroids and supergeometry
In this section, we briefly review supergeometry, its definition and related terms which are necessary to construct the topological sigma models from the standard Courant algebroids using the AKSZ construction in section 5. Here, we review definitions of Courant algebroids in the first subsection. Courant algebroids provide the background geometry of T-duality.
The second subsection then reviews a differential graded symplectic manifold, which is called a QP-manifold. In the AKSZ formulation, a QP-manifold is used to construct a topological sigma model. Finally, in the third subsection, the supergeometric construction of the Courant algebroids from QP-manifolds of degree 2 is explained. The formulation is based on the fundamental theorem that general Courant algebroids are equivalent to QP-manifolds of degree 2. This short review of the techniques involved provides the foundation to flow into the definition of Poisson Courant algebroids and their realization through supergeometric construction.

Courant algebroids
Let us start with recalling the definition of the Courant algebroid. where e 1 , e 2 , e 3 ∈ Γ(E).
Regarding its application to string theory, Courant algebroids appeared in the context of generalized geometry [13,14]. In this case, the vector bundle is the direct sum of tangent The operations of the Courant algebroid are as follows: The Courant algebroids on T M ⊕ T * M, more precisely, exact Courant algebroids are classified by H 3 (M, R) [16]. This means that we can only introduce the H-flux deformation as independent degree of freedom among all fluxes in the standard Courant algebroid.

Supergeometric construction
In the following, we review the supergeometric formulation of the Courant algebroids based on a so-called QP-manifold. Here, the structures of a Courant algebroid in the previous section is reconstructed by the supergeometric method.
First, we give a definition of a graded manifold. A graded manifold M is a ringed space, whose structure sheaf is a Z-graded commutative algebra over an ordinary smooth manifold M. The grading is compatible with the supermanifold grading, that is, a variable of even degree is commutative and a variable of odd degree is anticommutative. By definition, the V is a graded vector space, and S • (V ) is a free graded commutative ring on V . For rigorous mathematical definitions, we refer to [34,35].
In this paper, we only consider nonnegatively graded manifolds. An N-manifold (i.e., a nonnegatively graded manifold) M equipped with a graded symplectic structure ω of degree n is called P-manifold of degree n and denoted by (M, ω). ω is also called P -structure. We where the Hamiltonian vector field X f is defined by Definition 2.4 A QP-manifold is a P -manifold (M, ω) endowed with a degree 1 homological vector field Q such that L Q ω = 0 [23].
We call the homological vector field Q a Q-structure and the corresponding triple (M, ω, Q) We consider the canonical embedding map of the vector bundle E into M: The embedding map j can be written using local coordinates by For a section e ∈ Γ(E) the pushforward is a function, j * e ∈ C ∞ (E [1]). We use the same symbol for E and jE, if there is no risk of confusion.
We decompose the structure sheaf by degree, i.e., the space of functions on M as is the space of smooth functions of degree i. We have the following equivalences by the map j: The next step is to introduce a graded symplectic form of degree 2 on M. We take the following symplectic structure, where η a , η b = k ab is the fiber metric. This defines a P-structure and leads to the corresponding graded Poisson bracket {x i , ξ j } = δ i j and {η a , η b } = k ab . Finally, a Q-structure on M is defined by a homological function Θ of degree 3 as in (2.2).
Using local coordinates, the general form of a degree 3 function is given by where ρ i a (x) and C abc (x) are arbitrary local functions of x. The homological function satisfies the classical master equation, {Θ, Θ} = 0. This gives a set of relations among the degree zero local functions ρ i a (x) and C abc (x). The above triple (M, ω, Q) defines a QP-manifold of degree 2.
The operations on the Courant algebroid are defined using a graded Poisson bracket and derived brackets. The pseudo-metric, Dorfman bracket and anchor map are then reconstructed via the following expressions: where f is a function on M and e, e 1 , e 2 ∈ Γ(E). As a consequence of the classical master equation, these three operations satisfy the defining relations of a Courant algebroid.
In the case of E = T M ⊕ T * M, we take local coordinates (x i , q i , p i , ξ i ) with degree If we take the Q-structure function as We can regard the Poisson Courant algebroid as a contravariant object associated to the standard Courant algebroid. Contravariant geometry is a differential calculus in which the roles of T M and T * M are exchanged [18,36]. Therefore, we call [−, −] π R the contravariant Dorfman bracket and we can call this structure the contravariant Courant algebroid.
After giving the definition of the Poisson Courant algebroid, we reconstruct this algebroid by supergeometric methods. For this, we use the same graded manifold M = T * [2]T * [1]M as in the case of the standard Courant algebroid. We also take the same symbols for the local coordinates (x i , q i , p i , ξ i ) and the canonical graded symplectic form (2.7).
Then the homological function defining the Q-structure for the Poisson Courant algebroid We can confirm the above relations using local coordinates. The first equation gives (3.13) and thus this is the anchor map ρ = 0 ⊕ π ♯ : T M ⊕ T * M → T M. The second equation gives The classical master equation then imposes structural restrictions onto the expansion coefficients. One of the conditions for τ and σ is τ i k σ jk + σ ik τ j k = 0. The two simplest solutions are τ = 0, σ = 0 or τ = 0, σ = 0. In the standard Courant algebroid case, τ i j = δ i j and σ = F = Q = R = 0, and in the Poisson Courant algebroid case, σ = π, Q i jk (x) = − ∂π jk ∂x i (x) and τ = H = F = 0.

Duality between H-flux and R-flux
In this section, we study the meaning of H-flux geometry and R-flux geometry. We analyze the 'duality' transformation between two Courant algebroids in terms of supergeometry and the homological algebra. The key operation is a canonical transformation on the graded symplectic manifold (the P-manifold). This duality is a generalization of the correspondence between de Rham cohomology on differential forms and Poisson cohomology on multivector fields.
In this section, we denote the homological function Θ of the standard Courant algebroid in (2.8) as Θ H and the one of the Poisson Courant algebroid in (3.9) as Θ R .

Flux duality transformations as canonical transformations
Suppose the Poisson structure π is nondegenerate. We construct the duality transformation between the standard Courant algebroid and the Poisson Courant algebroid, which is derived from the transformation between the two homological functions Θ H and Θ R . This leads to a duality between the standard Courant algebroid cohomology and the Poisson Courant algebroid cohomology.
First, we define a canonical transformation on a P-manifold. Let α ∈ C ∞ (M). e δα is the exponential adjoint operation, for any f ∈ C ∞ (M). If α is of degree n, this transformation preserves degree and satisfies Note that α q is a trivial transformation on Θ H , and {Θ R , α p } = 0. By direct computation, we get the relation where R = ∧ 3 π ♯ H. On the basis of the QP-manifold, this canonical transformation acts as The Liouville 1-form is transformed as Note that j * π ♯ (α) = e δα p e −δα q e δα p j * α for a 1-form α.
γ is mapped to a differential form by pullback, The first ∂ ∂x i is the basis of the tangent bundle in the Courant algebroid and the third ∂ ∂x i is the basis of the tangent bundle of the image of the anchor map.

Therefore, the restriction of the standard Courant cohomology H
Next, we consider the Poisson Courant algebroid case. Note that T * [1]M is isomorphic to T M. Let us consider the subspace C ∞ (T * [1]M), whose elements can be written as The pullback maps u to a multivector field, and The cohomology defined by the coboundary operator d π is the Poisson cohomology, H k P (M, d π ). Therefore, the restriction of the Poisson Courant cohomology (4.37)

Duality of cohomologies
Since we are considering an even dimensional manifold and a nondegenerate Poisson structure, . This is a generalization of the well known result that if M is symplectic, the de Rham cohomology and the Poisson induces a homomorphism between de Rham cohomology and Poisson cohomology, If π −1 is symplectic, the de Rham cohomology on Ω • (M) is isomorphic to the Poisson cohomology: Therefore, we obtain maps from elements of the Q H -complex to elements of the Q R -complex, The flux duality map of complexes T : . We obtain the following theorem.
Theorem 4.2 Let π be a nondegenerate Poisson structure, that is, π −1 is symplectic, and R = ∧ 3 π ♯ H. Then, the standard Courant algebroid cohomology is isomorphic to the Poisson Courant algebroid cohomology,

Poisson Courant algebroids from double field theory
In this section, we show that the Poisson Courant algebroid is a solution of the section condition (the strong constraint) in double field theory. This shows that the Poisson Courant algebroid is directly connected to the geometry of double field theory.

Supergeometric formulation, Poisson structure and double field theory
We start with the supergeometric formulation of the geometry of double field theory [37].
We take a doubled configuration space M in 2d dimensions with local coordinates (y i ,ỹ i ) and a QP-manifold of degree 2, On this P-manifold, the geometry of double field theory is formulated using the Q-structure homological function, The classical master equation, {Θ C , Θ C } = 0, gives rise to the section condition, The C-bracket is constructed by the derived bracket, where e 1 , e 2 are sections of T M ⊕ T * M.
We choose a nontrivial physical configuration space, a d-dimensional submanifold M ⊂ M with local coordinate x i under the assumption that M has a Poisson structure π. We can consider a local coordinate transformation with following Jacobian, Alternatively, this local coordinate transformation can be realized as a twist of the original Θ C by a canonical function α p = 1 2 π ij (x)p i p j . Here, we denote the original homological function The canonical transformation deforms the homological function, This corresponds to the change of variables, We need the second term in (5.46) for consistency of the Poisson structure with the local coordinate transformation (5.44). The section condition is deformed tõ Finally, we takeξ i = 0 corresponding to the submanifold defined byx i = 0, and obtain homological functions of both standard and Poisson Courant algebroids, Note that Θ C defines a double complex, since {Θ H=0 , Θ R=0 } = 0. In this paper, we analyze these two Courant algebroids. In fact, Θ H=0 + Θ R=0 defines a Lie bialgebroid on T M ⊕ T * M. 8 Since we change the section conditionη i = 0 toξ i = 0, in general, the configuration space M is not embedded as a direct product M ×M in the doubled space, but is a nontrivial submanifold of the doubled configuration space.  ∂π ij ∂x k q k p i p j in (3.9) is a so-called Poisson connection with vanishing curvature. Therefore, it must be distinguished from a Q-flux term in double field theory.

Poisson Courant algebroid R-flux in double field theory
In the second case, if the Poisson Courant algebroid is seen as a stand-alone object, we can make contact to the double field theory R-flux. Through identification of (5.50) to (3.9) we find which leads to the identification π ij (x)ξ i =η j or π ij (x) ∂ ∂x i = ∂ ∂ỹ j and we can read off how the section condition is solved. Integration of this equation leads tõ Since there is no H-flux coefficient in the Poisson Courant algebroid, we obtain the relation B ij = 0, which leads to ∂ ∂y i = 0 due to (5.52). The term − 1 2 ∂π ij ∂x k q k p i p j in (3.9) is not sourced by the potentials B or β, but is a Poisson connection arising from the underlying space, and its origin is different from the Q-flux in double field theory. Finally, the local description of R-flux is then given in terms of the β-potential via To summarize, the Poisson Courant algebroid can be interpreted in two different ways, depending on the frame chosen in double field theory. In order to analyze the property of R-flux, we can use this correspondence, and on spacetime with a Poisson structure, some parts of R-flux geometry can be analyzed as H-flux geometry.

Topological sigma models
We want to consider field theoretical models with Poisson Courant algebroid symmetry. Here, we construct a 3-dimensional AKSZ sigma model, i.e., a theory of a topological membrane with 3-vector flux R, following the construction of a topological membrane theory based on the standard Courant algebroid [41,42]. For this purpose, first we shortly review the concept of AKSZ sigma models [25,43,29].

AKSZ sigma models
Let (X , D, µ) be a differential graded manifold X with a D-invariant nondegenerate measure A P-structure ω on Map(X , M) is defined by Note that ω is nondegenerate and closed since the operation µ * ev * preserves these properties.  If ∂X = ∅, the boundary conditions are deformed by α, so that S ′ satisfies the classical master equation. In this case, using Stokes' theorem, a straightforward computation gives

AKSZ sigma models with boundary
Therefore, twisting of S by α introduces a boundary term induced by transgression of α,

Contravariant Courant sigma models
We construct the AKSZ sigma model induced from the Poisson Courant algebroid.
Let us take a 3-dimensional manifold X with boundary ∂X. The worldvolume is a supermanifold X = T If α = 0, the Q-structure function has the following form: We call (6.61) the Poisson Courant sigma model or the contravariant Courant sigma model.
We take the variation of S, The equations of motion for ξ and q are obtained by integration by parts. Since We can take boundary conditions ξ i | ∂X = 0 and p i | ∂X = 0, such that (6.63) and (6.64) vanish.
Therefore, in terms of the target space, the structure is ξ i = p i = 0. This corresponds to the Dirac structure [12] T * M of the Poisson Courant algebroid T M ⊕ T * M.
Next, we consider a more nontrivial case with boundary term, modifying the Q-structure by a canonical function α. As an example, we take α where we have used the expression S ′′ = e −δα S ′ in (6.59). The boundary term deforms the boundary conditions. The variation δS ′′ restricted to the boundary is Since these terms must vanish, consistent boundary conditions are as follows: The master equation, {S ′′ , S ′′ } BV = 0, requires another consistency condition, i.e., the integrand of S 1 is zero on the boundary, Similarly, (6.66) and (6.67) can be converted to a condition on the target space, such that holds on the Lagrangian submanifold L ′ , defined by Substituting (6.69) into (6.68), we obtain the following geometric structure on L ′ : The commutator of a 2-form B with respect to the Koszul bracket is twisted by a 3-vector field R. If B = π −1 , (6.70) becomes Next, we construct the boundary action on T [1]∂X by integrating out the superfield ξ i and using the Stokes' theorem. Suppose π is nondegenerate, i.e., π −1 is symplectic. Integrating out ξ i from the action (6.65), we obtain the equations of motion p i = −π −1 ij dx j . By substituting this equation to (6.65), the action becomes the boundary (twisted) AKSZ sigma model with WZ term in two dimensions, This is the Poisson sigma model [45,46,47] deformed by a WZ term [48].
The action without ghosts can be obtained as follows. We expand the superfields in components as If we integrate over θ µ and drop ghost fields with nonzero degrees, we get the physical action: If we add the kinetic term, we obtain a string sigma model action with R-flux,

Duality of Courant sigma models
We have discussed duality transformations of the standard and the Poisson Courant algebroids in section 4. In this subsection, we derive the same result from the analysis of the corresponding sigma models.
We perform the duality transformation on the level of sigma models. The AKSZ construction on a three-dimensional manifold X with boundary gives rise to two Courant sigma models, one with the Poisson Courant algebroid structure with R-flux, constructed in the previous subsection, and one with the standard Courant algebroid structure with H-flux.
The BV action of the Poisson Courant sigma model is (6.61). On the other hand, from the AKSZ construction, the BV action of the standard Courant sigma model is We consider the following twisting of the standard Courant sigma model by applying the twist (4.22) as This is equivalent to where α p and α q are understood as µ ∂X * ev * α p and µ ∂X * ev * α q . We have Therefore, by this twist, the action becomes the Courant sigma model with boundary term, From (4.23)-(4.26), redefining the superfields as we can simplify the total action as The resulting action is the same as the BV action of the Poisson (contravariant) Courant sigma model (6.61), and we obtain the relation between H and R, R = ∧ 3 π ♯ H, again. Therefore, in the theory of the topological membrane, the duality transformation between H-flux and R-flux is a change of boundary conditions.

Current algebras
In this section, we consider a current algebraà la Alekseev and Strobl [31,51,32,26] corresponding to the Poisson Courant algebroid in two-dimensional spacetime.

Poisson brackets with fluxes from target QP-structures
In this subsection, we briefly review the Hamiltonian method to construct a Poisson bracket of canonical variables with fluxes from general target QP-structures [26].
Let (M, ω) be a P-manifold of degree n − 1. We take a worldvolume X = Σ × R in n dimensions and a space supermanifold X = T [1]Σ, since we consider the Hamiltonian formalism.
The simplest method to determine a Poisson bracket is to construct it from ω = µ * ev * ω.
Since Σ is in n − 1 dimensions and ω is of degree n − 1, the graded symplectic structure ω is of degree zero due to the integration µ * . Then, we obtain a graded Poisson bracket of degree zero, that is, an ordinary Poisson bracket {−, −} P B . However, the Poisson brackets obtained in this way cannot include fluxes, since the target space geometric datum Θ is not used. In [31], the symplectic form was deformed by b-transformation to include H-flux.
Here, we use another method to incorporate the geometric datum Θ. We can easily prove that {f, g} L is antisymmetric and satisfies both the Leibniz rule and the Jacobi identity using {pr * f , pr * g} = 0.
Simple candidates for L are canonical Lagrangian submanifolds, which we denote by L 0 .
For instance, in the case of the Courant algebroids, two simple Lagrangian submanifolds in Generally, we cannot obtain the twisted Poisson bracket with fluxes by simple restriction to these canonical Lagrangian submanifolds. We do a special twisting of functions on the mapping space, before restricting the space to a canonical Lagrangian submanifold, where the twisting does not depend on fluxes. This procedure derives a Poisson bracket with flux.
Here, we have twisted the space of functions on the mapping space and restricted it to the canonical Lagrangian submanifold. We could also obtain the same Poisson bracket by twisting the canonical Lagrangian submanifold with a canonical transformation e −δα and restricting the derived bracket to the twisted Lagrangian submanifold.
We demonstrate the procedure in the H-flux case.  1, 1, 2). The canonical graded symplectic structure is expressed by ω = δx i ∧ δξ i + δp i ∧ δq i . We take a canonical Lagrangian submanifold with respect to ω, Since the Q-structure function of the standard Courant algebroid is Θ = ξ i q i + 1 3! H ijk (x)q i q j q k , the derived brackets for the canonical quantities on L 0 are ) of degree (0, 1) and canonical conju- ). The transgression of (7.91)-(7.93) induces the derived bracket on superfields. The concrete expression is The Liouville 1-form on the Lagrangian submanifold is α 0 = ιDµ * ev * ϑ L = − µ p i dx i .
Twisting by the Liouville 1-form α 0 gives rise to the transformation q k → q k − dx k . If we reduce to the canonical Lagrangian submanifold L 0 defined by ξ i = q i = 0, we obtain We expand the superfields by the local coordinate θ on T [1]S 1 , The degree zero component in the expansion is the physical field (and degree nonzero components are ghost fields). In this example, physical fields are x i (σ) = x (0)i (σ) and p i (σ) = p (1) i (σ). The Poisson brackets of the physical canonical quantities are degree zero components of (7.97)-(7.99): These are the Poisson brackets of the canonical quantities with H-flux in [31]. The symplectic form of Alekseev-Strobl type, which induces (7.101)-(7.103), is For the H-flux case, we demonstrate the construction in the following example.

Current algebras from target QP-Structures
The derived brackets of functions are easily computed, {{j 0(f ) , Θ}, j 0(g) } = 0, Currents are identified with twisted functions on the Lagrangian submanifold of the mapping space. In order to construct currents, we apply the transgression map to j 0 and j 1 . Then, we twist them by α 0 and finally restrict the resulting functions to the Lagrangian submanifold defined by ξ i = q i = 0. The corresponding currents are which are the correct AS currents.

Poisson bracket twisted by R-flux
By the same method as in the previous section, we derive a current algebra with R-flux from the supergeometric data of the Poisson Courant algebroid.
Let us consider a two-dimensional worldsheet Σ = S 1 × R and take the tangent bundle We would like to construct Poisson brackets between x i (σ) and q i (σ). For that, we use the QP-manifold of the Poisson Courant algebroid, which is The transgression of (7.115)-(7.117) gives the derived brackets on the superfields. If we restrict them without twisting to the canonical Lagrangian submanifold parametrized by ξ i = p i = 0, we obtain the Poisson bracket without R-flux, In order to introduce R-flux, we would like to consider a nontrivial restriction with twisting.
For simplicity, we assume that π is nondegenerate. In order to obtain an AS type current algebra, we take the Liouville 1-form induced by the symplectic form ω L 0 defined by the Poisson bracket (7.118)-(7.120). This is α 0 = − X µ q i (π −1 ) ij dx j + · · · , where · · · contains terms without q i .
Twisting by α 0 induces the twist p i → p i − (π −1 ) ij dx j . After the restriction ξ i = p i = 0 to the canonical Lagrangian submanifold, we get the Poisson brackets with R-flux, Physical Poisson brackets are the degree zero components of these equations. Here we denote physical fields by x i (σ) = x (0)i (σ) and q i (σ) = q (1)i (σ). Then, the Poisson brackets on the physical canonical quantities are The relations (7.124)-(7.126) can also be derived by β-transformation. If R = 0, the relations (7.124)-(7.126) are obtained by pullback of the Poisson structure on T M, lifted from the Poisson structure π on M [52], to the mapping space. In this case, the R-term is is a bivector field such that [π, β] S = R.

Contravariant current algebras with R-flux
Here, currents are constructed from functions of degree equal to or less than one on the target , which is the same space as in the case of the AS current algebra. Take a function of degree zero j 0 = f (x) and a function of degree one j 1 = X i (x)p i + α i (x)q i . By transgression of j 0 and j 1 to the mapping space, twisting by α 0 such that p i → p i − (π −1 ) ij dx j , and restricting them to the canonical Lagrangian submanifold, we obtain the supergeometric currents, If we take the degree zero components of the superfields, we obtain AS type currents, The algebra of these supergeometric currents is computed from the Poisson brackets of the canonical quantities (7.121)-(7.123): is the contravariant Dorfman bracket with R-flux on T M ⊕ T * M and ρ(X + α) = π ♯ (α) is the anchor map. Component expansions give rise to physical current algebras: {J 0(f ) (σ), J 0(g) (σ ′ )} P B = 0, (7.131) This formula (7.131)-(7.133) is consistent, even if the Poisson structure π is degenerate. Therefore, we do not need to impose a nondegeneracy condition for π in the current algebra.

Conclusions and discussion
The Poisson Courant algebroid, which is a contravariant object of the standard Courant algebroid, has been formulated by supergeometric construction. The duality between these two specific Courant algebroids has been analyzed in detail. As a result, the duality transformation is a canonical transformation on the graded symplectic manifold and the transformation between the 3-form H-flux in the standard Courant algebroid and the trivector R-flux in the Poisson Courant algebroid has been derived. In [28,48,53], twisting of a bivector field by a 3-form H, a so-called twisted Poisson structure, has been discussed. From the above duality, we have obtained its contravariant geometric structure in (6.71), twisting of a 2-form by a trivector field R.
Moreover, we have shown that this duality is, from the mathematical viewpoint, the generalization of the correspondence between the de Rham cohomology and the Poisson cohomology. We also discussed that the same duality can be derived on the sigma model level. We have also constructed a current algebra with R-flux on the tangent space of the loop space from the target space QP-manifold data. The resulting current algebra is the contravariant counterpart of the current algebra with H-flux of Alekseev-Strobl type.
The R-flux has also been discussed in [54,55] using double field theory. There, the nongeometric R-flux is characterized as a Jacobiator, the quantity corresponding to the anomaly of the Jacobi identity, i.e. R ijk ∼ β l[i ∂β jk] ∂x l . In section 5, we have discussed the Poisson Courant algebroid and its trivector field R from the point of view of double field theory. If we take the special solution of the section condition defined by the Poisson structure π, the resulting spacetime has the Poisson Courant algebroid structure. Therefore, we have found that our formalism describes the R-flux in the frame specified by this particular solution.
In our formulation, β is independent of the Poisson bivector π, thus we can consider the special case [π + β, π + β] S = 0. It means that π + β is again a Poisson structure. Note that it does not mean a deformation of the Poisson Courant algebroid. This is a Maurer-Cartan condition of β, d π β + 1 2 [β, β] S = 0, and we obtain the R-flux as the Jacobiator [56,57], This is the same formula in the definition inspired by the double field theory. The meaning of this observation will be discussed in future work.

A Formulas in graded differential calculus
We summarize formulas of graded symplectic geometry.

A.1 Basic definitions
Let z be a local coordinate on a graded manifold M. A differential on a function is defined by A vector field X is expanded using local coordinates by The interior product is defined by the differentiation by the following graded vector field on For a graded differential form α, we denote |α| as total degree (form degree plus degree by grading) of α. Note that |d| = 1, |dz a | = |z a | + 1 and We obtain the following formula, Therefore,

A.2 Cartan formulas
The Lie derivative is defined by Its degree is |L X | = |X|.
Let α and β be graded differential forms. We can show the following graded Cartan formulas,

A.4 Graded symplectic form and Poisson bracket
Let ω be a symplectic form of degree n. Since ω is a 2-form, its total degree is |ω| = n + 2.

B.2 Graded symplectic geometry
In this subsection, we map the structures on M to structures on the target space Map(X , M) by the transgression map µ * ev * .
Let z i (σ, θ) be a local basis superfield of the mapping space Map(X , M), corresponding to a local coordinate z i on M. We write a vector field on the mapping space for X = X i (z) .
The differential on a function f is . and therefore |ι X f | = |f |.