Courant bracket as T-dual invariant extension of Lie bracket

We consider the symmetries of a closed bosonic string, starting with the general coordinate transformations. Their generator takes vector components $\xi^\mu$ as its parameter and its Poisson bracket algebra gives rise to the Lie bracket of its parameters. We are going to extend this generator in order for it to be invariant upon self T-duality, i.e. T-duality realized in the same phase space. The new generator is a function of a $2D$ double symmetry parameter $\Lambda$, that is a direct sum of vector components $\xi^\mu$, and 1-form components $\lambda_\mu$. The Poisson bracket algebra of a new generator produces the Courant bracket in a same way that the algebra of the general coordinate transformations produces Lie bracket. In that sense, the Courant bracket is T-dual invariant extension of the Lie bracket. When the Kalb-Ramond field is introduced to the model, the generator governing both general coordinate and local gauge symmetries is constructed. It is no longer self T-dual and its algebra gives rise to the $B$-twisted Courant bracket, while in its self T-dual description, the relevant bracket becomes the $\theta$-twisted Courant bracket.


Introduction
The Courant bracket [1,2] and various generalizations obtained by its twisting had been relevant to the string theory since its appearance in the algebra of generalized currents [3,4,5,6]. It represents the generalization of the Lie bracket on spaces of generalized vectors, understood as the direct sum of the elements of the tangent bundle and the elements of the cotangent bundle. Although the Lie bracket satisfies the Jacobi identity, the Courant bracket does not. Its Jacobiator is equal to the exterior derivative of the Nijenhuis operator.
It is well known that the commutator of two general coordinate transformations along two vector fields produces another general coordinate transformation along the vector field equal to their Lie bracket. Since the Courant bracket represents its generalization, it is worth considering how it is related to symmetries of the bosonic string σ-model. In [7], the field theory defined on the torus with restricted fields and its symmetries for restricted parameters were considered and the Courant bracket has been obtained as the T-dual invariant bracket governing the symmetry algebra. In this paper, we analyze the general classical bosonic string σ-model and algebra of its symmetries generators, without constraints on its parameters.
We firstly consider the closed bosonic string moving in the background characterized solely by the metric tensor. We extend the generator of the general coordinate transformations so that it becomes invariant upon self T-duality, understood as T-duality realized in the same phase space [6]. We obtain the Courant bracket in the Poisson bracket algebra of this extended generator. The Courant bracket is therefore a self T-dual invariant extension of the Lie bracket.
Furthermore, we consider the bosonic string σ-model that includes the antisymmetric Kalb-Ramond field too. The antisymmetric field is introduced by the action of Btransformation on the generalized metric. We construct the symmetry generator and recognize that it generates both the general coordinate and the local gauge transformations [8]. In this case, the symmetry generator is not invariant upon self T-duality and it gives rise to the twisted Courant bracket. The matrix that governs this twist is exactly the matrix of B-shifts. Lastly, we consider the self T-dual description of the theory, that we construct in the analogous manner, this time with the action of θ-transformation, T-dual to the Btransformation. We obtain the bracket governing the generator algebra that turns out to be the θ-twisted Courant bracket, also known as the Roytenberg bracket [4,9]. The twisted Courant and Roytenberg brackets had been shown to be related by self T-duality [6].
2 Bosonic string moving in the background characterized by the metric field Consider the closed bosonic string, moving in the background defined by the coordinate dependent metric field G µν (x), with the Kalb-Ramond field set to zero B µν = 0 and the constant dilaton field Φ = const. In the conformal gauge, the Lagrangian density is given by [10,11] where x µ (ξ), µ = 0, 1, ..., D − 1 are coordinates on the D-dimensional space-time, and η αβ , α, β = 0, 1 is the worldsheet metric, ǫ 01 = −1 is the Levi-Civita symbol, and κ = 1 2πα ′ with α ′ being the Regge slope parameter. The Legendre transformation of the Lagrangian gives the canonical Hamiltonian where π µ are canonical momenta conjugate to coordinates x µ , given by The Hamiltonian can be rewritten in the matrix notation where X M is a double canonical variable, given by and G M N is the so called generalized metric, that in the absence of the Kalb-Ramond field takes the diagonal form In this paper, we consider the T-duality realized without changing the phase space, which is called the self T-duality [6]. Two quantities are said to be self T-dual if they are invariant upon The first part of (2.7) corresponds to the T-duality interchanging the winding and momentum numbers, which are respectively obtained by integrating κx ′µ and π µ over the worldsheet space parameter σ [12]. The second part of (2.7) corresponds to swapping the background fields for the T-dual background fields. Our approach gives the same expression for the T-dual metric as the usual T-dualization procedure obtained by Buscher in the special case of zero Kalb-Ramond field [13,14,15].

Symmetry generator
Let us consider symmetries of the closed bosonic string. The canonical momenta π µ generate the general coordinate transformations. The generator is given by [8] with ξ µ being a symmetry parameter. The general coordinate transformations of the metric tensor are given by [7,8] where L ξ is the Lie derivative along the vector field ξ. Its action on the metric field is where D µ are covariant derivatives defined in a usual way It is easy to verify, using the standard Poisson bracket relations that the Poisson bracket of these generators can be written as where [ξ 1 , ξ 2 ] L is the Lie bracket. The Lie bracket is the commutator of two Lie derivatives which results in another Lie derivative along the vector ξ µ 3 , given by Let us now construct the symmetry generator that is related to the generator of general coordinate transformations by self T-duality (2.7) where λ µ is a gauge parameter.
The symmetry parameters ξ µ and λ µ are vector and 1-form components, respectively. They can be combined in a double gauge parameter, given by The double gauge parameter is a generalized vector, defined on the direct sum of elements of tangent and cotangent bundle. Combining (2.8) and (2.16), we obtain the symmetry generator that is self T-dual (2.7) where Ω M N is the O(D, D) invariant metric [16], given by The expression (Λ T ) M Ω M N X N can be recognized as the natural inner product on the space of generalized vectors We are interested in obtaining the algebra of this extended symmetry generator (2.18), analogous to (2.13). Using the Poisson bracket relations (2.12), we obtain In order to transform the anomalous part, we note that and Applying the previous two relations to the right hand side of (2.21), one obtains where the resulting gauge parameters are given by These relations define the Courant bracket [(ξ 1 , λ 1 ), (ξ 2 , λ 2 )] C = (ξ, λ) [1,2], allowing us to rewrite the generator algebra (2.24) (2.26) The Courant bracket represents the self T-dual invariant extension of the Lie bracket.
In the coordinate-free notation, the Courant bracket can be written as with i ξ being the interior product along the vector field ξ, and d being the exterior derivative. The Lie derivative L ξ can be written as their anticommutator The Courant bracket does not satisfy the Jacobi identity. Nevertheless, the Jacobiator of the Courant bracket is an exact 1-form [17] ( However, if one makes the following change of parameters λ µ → λ µ + ∂ µ ϕ, the generator (2.18) does not change since the total derivative integral vanishes for the closed string. Therefore, the deviation from Jacobi identity contributes to the trivial symmetry, and we say that the symmetry is reducible.
3 Bosonic string moving in the background characterized by the metric field and the Kalb-Ramond field In this chapter, we extend the Hamiltonian so that it includes the antisymmetric Kalb-Ramond field. It is possible to obtain this Hamiltonian from the transformation of the generalized metric G M N (2.6) under the so called B-transformations. The B-transformations (or B-shifts) [17] are realized by eB , wherê As a result ofB 2 = 0, the full transformation is easily obtained (3.2) Its transpose is given by from which it is easy to verify that where H M N is the generalized metric and G E µν is the effective metric perceived by the open strings, given by It is straightforward to write the canonical Hamiltonian as well as the Lagrangian in the canonical form On the equations of motion for π µ , we obtain Substituting (3.10) into (3.9) we find the well known expression for bosonic string Lagrangian [10,11] L(ẋ, It is possible to rewrite the canonical Hamiltonian (3.8) in terms of the generalized metric G M N , that characterizes background with the metric only tensor. Substituting (3.5) into (3.8), we obtain with i µ being the auxiliary current, given by 14) The algebra of auxiliary currents i µ gives rise to the H-flux [6] {i where the structural constants are the Kalb-Ramond field strength components, given by

Symmetry generator
Let us extend the symmetry transformations of the background fields for the theory with the non-trivial Kalb-Ramond field. The infinitesimal general coordinate transformations of the background fields are given by where the action of the Lie derivative L ξ (2.28) on the Kalb-Ramond field is given by [8] while its action on the metric field is the same as in (2.10). The local gauge transformations of the background fields are [8] Rewriting the symmetry generator G(ξ, λ) (2.18) in terms of the basis defined by components ofX M (3.13), one obtains where (3.4) was used in the last step, andΛ M is a new double gauge parameter, given bŷ We are going to mark the right hand side of (3.20) as a new generator

19).
Our goal is to obtain the algebra in the form due to (3.21). The Poisson bracket between canonical variables (2.12) remains the same after the introduction of the Kalb-Ramond field. Therefore the results from previous chapter, as well as mutual relations between coefficients in different bases can be used to obtain the algebra (3.23). Firstly, substituting (3.24) into the second equation in (2.25), one obtains Secondly, substituting the previous equation in (3.21), one obtains The above relations define the twisted Courant bracket [(ξ 1 ,λ 1 ), (ξ 2 ,λ 2 )] C B = (ξ,λ) [18]. This is the bracket of the symmetry transformations in the theory defined by both metric and Kalb-Ramond field.
In the coordinate free notation, the twisted Courant bracket is given by where H(ξ 1 , ξ 2 , .) represents the contraction of the H-flux H = dB (3.16) with two gauge parameters ξ 1 and ξ 2 . This term is the corollary of the non-commutativity of the auxiliary currents i µ (3.15), due to twisting of the Courant bracket with the Kalb-Ramond field. In special case when the Kalb-Ramond field B is a closed form dB = 0, the twisted Courant bracket (3.28) reduces to the Courant bracket (2.27). This can also be seen from the well known fact that B-shifts (3.2) are symmetries of the Courant bracket when B is a closed form [17].

Courant bracket twisted by θ µν
When both the metric and the Kalb-Ramond field are present in the theory, the expressions for T-dual fields are given by [13] where θ µν is the non-commutativity parameter for the string endpoints on a D-brane [19], given by We say that two quantities are self T-dual, if they are invariant under the interchange [6] When the Kalb-Ramond field is set to zero B µν = 0, (4.3) reduces to the self T-duality transformation laws in the background without the B field (2.7). From the relations (4.3), it is apparent that the introduction of Kalb-Ramond field breaks down the self T-duality invariance of the symmetry generator (3.22). To find a new self T-dual invariant generator, we will analogously to the prior construction start with the background containing only T-dual metric. The Hamiltonian in the metric only background, similar to (2.2), reads where ⋆ G M N is the T-dual generalized metric for the above Hamiltonian, given by (4.5) Note that the self T-duality is realized as the joint action of the permutation of the coordinate σ-derivatives with the canonical momenta and the swapping all the fields in (2.6) for their T-duals. This is equivalent to the Buscher's procedure [13,14,15], when it is done in the same phase space. In order to construct the Hamiltonian in the self T-dual description, we consider how the T-dual generalized metric (4.5) is transformed with respect to the so called θtransformations eθ, whereθ The full exponential eθ is given by and its transpose by Under (4.7), the T-dual generalized metric (4.5) transforms in the following way where which is exactly equal to the generalized metric (3.6). From it we can write the T-dual Hamiltonian The canonical Lagrangian is given by from which one easily obtains π µ = κG µνẋ ν − 2κB µν x ′ν . (4.14) We see that the canonical momentum remains the same, which is expected, since the self T-duality is realized in the same phase space. Substituting (4.14) into (4.13), one obtains It is obvious that both the Hamiltonian and the Lagrangian are invariant under the self T-duality.

Conclusion
In this paper, we firstly considered the bosonic string moving in the background defined solely by the metric tensor, in which the generalized metric G M N has a simple diagonal form (2.6). The general coordinate transformations are generated by canonical momenta π µ , parametrised with vector components ξ µ . We have extended this generator, so that it is self T-dual, adding the symmetry generated by coordinate σ-derivative x ′µ , that are T-dual to the canonical momenta π µ (2.7). The extended generator of both of these symmetries is a function of a double gauge parameter Λ M (2.17). The latter is a generalized vector, i.e. an element of a space obtained from a direct sum of vectors and 1-forms. The symmetry generator G(Λ) = G(ξ, λ) of both of aforementioned symmetries was expressed as the standard O(D, D) inner product of two generalized vectors (2.18). The Poisson bracket between the extended generators G(Λ 1 ) and G(Λ 2 ) resulted up to a sign in the generator G(Λ), with its argument being equal to the Courant bracket of the double gauge parameters Λ = [Λ 1 , Λ 2 ] C . As this is analogous to an appearance of the Lie bracket in the algebra of general coordinate transformations generators, we concluded that the Courant bracket is the self T-dual extension of the Lie bracket. Afterwards, we added the Kalb-Ramond field B µν to the background, transforming the diagonal generalized metric G M N acting by the B-transformation eB (3.2). The standard generalized metric for bosonic string H M N was obtained (3.6), as well as the well known expressions for the Hamiltonian (3.8) and the Lagrangian (3.11). We noted that it is possible to express the Hamiltonian in terms of the diagonal generalized metric G M N , on the expense of transforming the double canonical variable X M by the B-shift. This newly obtained canonical variableX was suitable for rewriting the symmetry generator G as GB, which is no longer self T-dual. This is the generator of both general coordinate, and local gauge transformations. The Poisson bracket algebra of this new generator was calculated and as an argument of the resulting generator the Courant bracket twisted by the Kalb-Ramond field was obtained. It deviates from the Courant bracket by the term related to the H-flux, which is the terms that breaks down the self T-duality invariance. Lastly, we considered the self T-dual description of the bosonic string σ-model. Analogously as in the first description, the complete Hamiltonian was constructed starting from the background characterized only by the T-dual metric ⋆ G µν = (G −1 E ) µν . We applied the θ-transformations eθ (4.7), T-dual to B-shifts, and obtained the same canonical Hamiltonian. Similarly to the previous case, the action of θ-transformation on the double canonical variable was chosen for an appropriate basis. In this basis, the symmetry generator dependent upon some new gauge parameters was constructed and its algebra gave rise to the θ-twisted Courant bracket. This bracket is characterized by the presence of terms related to non-geometric Q and R fluxes.
It would be interesting to obtain the bracket that includes all of the fluxes, while remaining invariant upon the self T-duality. The natural candidate for this is the Courant bracket twisted by both the Kalb-Ramond field and the non-commutativity parameter. This could be done by the matrix eB, wherȇ This transformation is not trivial, as the square of the matrixB is not zero. Nevertheless, the transformation is also an element of the O(D, D) group, and it remains an interesting idea for future research [23].