Courant Bracket Twisted Both by a 2-Form B and by a Bi-Vector Θ

Eur. Phys. J. C (2021) 81:685 https://doi.org/10.1140/epjc/s10052-021-09447-4

Regular Article - Theoretical Physics

Courant bracket twisted both by a 2-form B and by a bi-vector θ

Ljubica Davidovića, Ilija Ivaniševićb, Branislav Sazdovićc Institute of Physics, University of Belgrade, Pregrevica 118, 11080 Belgrade, Serbia

Received: 6 May 2021 / Accepted: 12 July 2021 © The Author(s) 2021

Abstract We obtain the Courant bracket twisted simulta- obtained from the action of the appropriate O(D, D) trans- neously by a 2-form B and a bi-vector θ by calculating the formation [8]. The Courant bracket is usually twisted by a Poisson bracket algebra of the symmetry generator in the 2-form B, giving rise to what is known as the twisted Courant basis obtained acting with the relevant twisting matrix. It is bracket [9], and by a bi-vector θ, giving rise to the θ-twisted the extension of the Courant bracket that contains well known Courant bracket [10]. In [3,8,11,12], the former bracket was Schouten–Nijenhuis and Koszul bracket, as well as some new obtained in the generalized currents algebra, and it was shown star brackets. We give interpretation to the star brackets as to be related to the latter by self T-duality [13], when the T- projections on isotropic subspaces. dual of the B field is the bi-vector θ. The B-twisted Courant bracket contains H flux, while the θ-twisted Courant bracket contains non-geometric Q and R 1 Introduction fluxes. The fluxes are known to play a crucial role in the compactification of additional dimensions in string theory The Courant bracket [1,2] represents the generalization of [14]. Non-geometric fluxes can be used to stabilize moduli. In the Lie bracket on spaces of generalized vectors, understood this paper, we are interested in obtaining the Poisson bracket as the direct sum of the elements of the tangent bundle and representation of the twisted Courant brackets that contain the elements of the cotangent bundle. It was obtained in the all fluxes from the generators algebra. Though it is possible algebra of generalized currents firstly in [3]. Generalized cur- to obtain various twists of the C-bracket as well [15], we do rents are arbitrary functionals of the fields, parametrized by not deal with them in this paper. a pair of vector field and covector field on the target space. The realization of all fluxes using the generalized geom- Although the Lie bracket satisfies the Jacobi identity, the etry was already considered, see [16] for a comprehensive Courant bracket does not. review. In [17], one considers the generalized tetrads origi- In bosonic string theory, the Courant bracket is govern- nating from the generalized metric of the string Hamiltonian. ing both local gauge and general coordinate transformations, As the Lie algebra of tetrads originating from the initial met- invariant upon T-duality [4,5]. It is a special case of the more ric defines the geometric flux, it is suggested that all the general C-bracket [6,7]. The C-bracket is obtained as the other fluxes can be extracted from the Courant bracket of T-dual invariant bracket of the symmetry generator algebra, the generalized tetrads. Different examples of O(D, D) and when the symmetry parameters depend both on the initial and O(D) × O(D) transformations of generalized tetrads lead T-dual coordinates. It reduces to the Courant bracket once to the Courant bracket algebras with different fluxes as its when parameters depend solely on the coordinates from the structure constants. initial theory. In [18], one considers the standard Lie algebroid defined It is possible to obtain the twisted Courant bracket, when with the Lie bracket and the identity map as an anchor on the the self T-dual generator algebra is considered in the basis tangent bundle, as well as the Lie algebroid with the Koszul bracket and the bi-vector θ as an anchor on the cotangent Work supported in part by the Serbian Ministry of Education and bundle. The tetrad basis in these Lie algebroids is suitable Science, under Contract no. 171031. for defining the geometric f and non-geometric Q fluxes. a e-mail: [email protected] It was shown that by twisting both of these Lie algebroids b e-mail: [email protected] (corresponding author) by H-flux one can construct the Courant algebroid, which c e-mail: [email protected] gives rise to all of the fluxes in the Courant bracket algebra.

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Unlike previous approaches where generalized fluxes were Lastly, we return to the previous basis and obtain the full defined using the Courant bracket algebra, in a current paper expression for the Courant bracket twisted by both B and θ. we obtain them in the Poisson bracket algebra of the sym- It has a similar form as C˚ -twisted Courant bracket, but in this metry generator. case the other brackets contained within it are also twisted. Firstly, we consider the symmetry generator of local gauge The Courant bracket twisted by both B and θ and the one and global coordinate transformations, defined as a standard twisted by C˚ are directly related by a O(D, D) transforma- inner product in the generalized tangent bundle of a dou- tion represented with the block diagonal matrix. ble gauge parameter and a double canonical variable. The O(D, D) group transforms the double canonical variable into some other basis, in terms of which the symmetry gen- 2 The bosonic string essentials erator can be expressed. We demonstrate how the Poisson bracket algebra of this generator can be used to obtain twist The canonical Hamiltonian for closed bosonic string, moving of the Courant bracket by any such transformation. We give a in the D-dimensional space-time with background charac- Bˆ θˆ brief summary of how e and e produce respectively the B- terized by the metric field Gμν and the antisymmetric Kalb– θ twisted and -twisted Courant bracket in the Poisson bracket Ramond field Bμν is given by [19,20] algebra of generators [8]. B˘ κ Secondly, we consider the matrix e used for twisting the 1 −1 μν μ E ν HC = πμ(G ) πν + x Gμν x Courant bracket simultaneously by a 2-form and a bi-vector. 2κ 2 ˘ μ −1 ρν The argument B is defined simply as a sum of the arguments −2x Bμρ (G ) πν, (2.1) Bˆ and θˆ. Unlike Bˆ or θˆ, the square of B˘ is not zero. The full Taylor series gives rise to the hyperbolic functions of the where πμ are canonical momenta conjugate to coordinates parameter depending on the contraction of the 2-form with xμ, and μ μρ the bi-vector α ν = 2κθ Bρν. We represent the symmetry generator in the basis obtained acting with the twisting E = − ( −1 ) ˘ Gμν Gμν 4 BG B μν (2.2) matrix eB on the double canonical variable. This generator is manifestly self T-dual and its algebra closes on the Courant is the effective metric. The Hamiltonian can be rewritten in bracket twisted by both B and θ. the matrix notation Instead of computing the B − θ twisted Courant bracket directly, we introduce the change of basis in which we define 1 some auxiliary generators, in order to simplify the calcula- H = (X T )M H X N , C κ MN (2.3) tions. This change of basis is also realized by the action of 2 an element of the O(D, D) group. The structure constants M appearing in the Poisson bracket algebra have exactly the where X is a double canonical variable given by same form as the generalized fluxes obtained in other papers μ [16–18]. The expressions for fluxes is given in terms of new M κx μ X = , (2.4) auxiliary fields B˚ and θ˚, both being the function of α . πμ The algebra of these new auxiliary generators closes on another bracket, that we call C˚ -twisted Courant bracket. We and HMN is the so called generalized metric, given by obtain its full Poisson bracket representation, and express it in terms of generalized fluxes. We proceed with rewrit- E − ( −1)ρν ing it in the coordinate free notation, where many terms are = Gμν 2Bμρ G , HMN ( −1)μρ ( −1)μν (2.5) recognized as the well known brackets, such as the Koszul 2 G Bρν G or Schouten–Nijenhuis bracket, but some new brackets, that we call star brackets, also appear. These star brackets as a with M, N ∈{0, 1}. In the context of generalized geometry domain take the direct sum of tangent and cotangent bun- [21], the double canonical variable X M represents the gen- dle, and as a result give the graph of the bi-vector θ˚ in the eralized vector. The generalized vectors are 2D structures cotangent bundle, i.e. the sub-bundle for which the vector that combine both vector and 1-form components in a single μ μν and 1-form components are related as ξ = κθ˚ λν.We entity. show that they can be defined in terms of the projections on The standard T-duality [22,23] laws for background fields isotropic subspaces acting on different twists of the Courant have been obtained by Buscher [24] bracket. μν − μν μν κ μν G = (G 1) , B = θ , (2.6) E 2 123 Eur. Phys. J. C (2021) 81:685 Page 3 of 15 685

( −1)μν ξ where G E is the inverse of the effective metric (2.2), where iξ is the interior product along the vector ﬁeld , and μν and θ is the non-commutativity parameter, given by ηMN is O(D, D) metric, given by θ μν =−2 ( −1)μρ ( −1)σν. 01 G E Bρσ G (2.7) η = . (2.13) κ MN 10 The T-duality can be realized without changing the phase space, which is called the self T-duality [13]. It has the Now it is possible to rewrite the generator (2.9)as same transformation rules for the background ﬁelds like T- duality (2.6), with additionally interchanging the coordinate G() = dσ , X. (2.14) μ σ -derivatives κx with canonical momenta πμ

μ ∼ κx = πμ. (2.8) In [8], the Poisson bracket algebra of generator (2.9)was obtained in the form Since momenta and winding numbers correspond to σ inte- μ gral of respectively πμ and κx , we see that the self T- G(1), G(2) =−G [1,2]C , (2.15) duality, just like the standard T-duality, swaps momenta and winding numbers. where the standard Poisson bracket relations between coordinates and canonical momenta were assumed 2.1 Symmetry generator μ μ {x (σ), πν(σ)¯ }=δ νδ(σ −¯σ). (2.16) We consider the symmetry generator that at the same time governs the general coordinate transformations, parametrized The bracket [ , ]C is the Courant bracket [1], deﬁned by by ξ μ, and the local gauge transformations, parametrized by 1 2 λμ. The generator is given by [25] [ , ] = ⇔[(ξ ,λ ), (ξ ,λ )] = (ξ, λ), 2π 2π 1 2 C 1 1 2 2 C (2.17) μ μ G(ξ, λ) = dσ G(ξ, λ) = dσ ξ πμ + λμκx . 0 0 where (2.9) ξ μ = ξ ν∂ ξ μ − ξ ν∂ ξ μ, It has been shown that the general coordinate transformations 1 ν 2 2 ν 1 and the local gauge transformations are related by self T- duality [25], meaning that this generator is self T-dual. If one and makes the following change of parameters λμ → λμ + ∂μϕ, ν ν λμ = ξ (∂νλ μ − ∂μλ ν) − ξ (∂νλ μ − ∂μλ ν) the generator (2.9) does not change 1 2 2 2 1 1 1 + ∂μ(ξ1λ2 − ξ2λ1). (2.18) 2π 2 G(ξ, λ + ∂ϕ) = G(ξ, λ) + κ ϕ dσ = G(ξ, λ), (2.10) 0 It is the generalization of the Lie bracket on spaces of generalized vectors. since the total derivative integral vanishes for the closed string. Therefore, the symmetry is reducible. M Let us introduce the double gauge parameter ,asthe 3 O(D, D) group generalized vector, given by O ξ μ Consider the orthogonal transformation , i.e. the transfor- M = , (2.11) mation that preserves the inner product (2.12) λμ

ξ μ λ O , O = , ⇔(O )T η(O ) = T η , where represent the vector components, and μ represent 1 2 1 2 1 2 1 2 the 1-form components. The space of generalized vectors is (3.19) endowed with the natural inner product which is satisﬁed for the condition , =(T )M η N ⇔(ξ ,λ ), (ξ ,λ ) 1 2 1 MN 2 1 1 2 2 μ μ = ξ λ + ξ λ = ξ λ μ + ξ λ μ, T i 1 2 i 2 1 1 2 2 1 (2.12) O η O = η. (3.20) 123 685 Page 4 of 15 Eur. Phys. J. C (2021) 81:685

ˆ There is a solution for the above equation in the form O = eT , where eB is the twisting matrix, given by see Sec. 2.1 of [21], where μ ˆ δ B = ν 0 , ˆ M = 00. A θ e δν B N (3.28) T = , (3.21) 2Bμν μ 2Bμν 0 B −AT This bracket has been obtained in the algebra of generalized θ : T M → TM B : TM → T M with and being anti- currents [11,13]. A : TM → TM symmetric, and being the endomorphism. In case of A = 0, B = 0, the bracket (3.26) becomes the B θ O In general case, and can be independent for to satisfy Courant bracket twisted by a bi-vector θ condition (3.20). Consider now the action of some element of O(D, D) on [ , ] = θˆ[ −θˆ , −θˆ ] , the double coordinate X (2.4) and the double gauge parameter 1 2 Cθ e e 1 e 2 C (3.29) (2.11) ˆ where eθ is the twisting matrix, given by Xˆ M = OM X N , ˆ M = OM N , (3.22) N N δμ κθμν κθμν θˆ = ν , θˆM = 0 . e ν N (3.30) and note that the relation (2.15) can be written as 0 δμ 00 B θ dσ 1, X, 2, X =− dσ[1,2]C, X, The -twisted Courant bracket (3.27) and -twisted Courant bracket (3.29) are related by self T-duality [13]. It is easy to ˆ ˆ (3.23) demonstrate that both eB and eθ satisfy the condition (3.20). We can now deduce a simple algorithm for ﬁnding the and using (3.19) and (3.22)as Courant bracket twisted by an arbitrary O(D, D) transformation. One rewrites the double symmetry generator G(ξ, λ) ˆ ˆ ˆ ˆ T dσ 1, X, 2, X =− dσ[1,2]C, X in the basis obtained by the action of the matrix e on the double coordinate (2.4). Then, the Poisson bracket algebra =− σ[ˆ , ˆ ] , ˆ , between these generators gives rise to the appropriate twist d 1 2 CT X of the Courant bracket. In this paper, we apply this algorithm (3.24) to obtain the Courant bracket twisted by both B and θ. where we expressed the right hand side in terms of some new bracket [ˆ , ˆ ]C . Moreover, using (3.19) and (3.22), the 1 2 T 4 Twisting matrix right hand side of (3.23) can be written as

Bˆ θˆ −1 ˆ −1 ˆ −1 ˆ The transformations e and e do not commute. That is why [1,2]C, X=[O 1, O 2]C, O X we define the transformations that simultaneously twists the =O[O−1ˆ , O−1ˆ ] , ˆ . ˘ 1 2 C X (3.25) Courant bracket by B and θ as eB, where Using (3.24) and (3.25), one obtains μν 0 κθ B˘ = Bˆ + θˆ = . (4.1) 2Bμν 0 [ˆ , ˆ ] =O[O−1ˆ , O−1ˆ ] = T [ −T ˆ , −T ˆ ] . 1 2 CT 1 2 C e e 1 e 2 C (3.26) The Courant bracket twisted at the same time both by a 2- form B and by a bi-vector θ is given by This is a definition of a T -twisted Courant bracket. Through- out this paper, we use the notation where [, ]C is the Courant [ , ] = B˘ [ −B˘ , −B˘ ] . 1 2 C θ e e 1 e 2 C (4.2) bracket, while when C has an additional index, it represents B the twist of the Courant bracket by the indexed field, e.g. ˘ [, ] The full expression for eB can be obtained from the well CB is the Courant bracket twisted by B. In a special case, when A = 0, θ = 0, the bracket (3.26) known Taylor series expansion of exponential function becomes the Courant bracket twisted by a 2-form B [9] ∞ ˘ B˘ n eB = . Bˆ −Bˆ −Bˆ (4.3) [ , ]C = [ , ]C, n! 1 2 B e e 1 e 2 (3.27) n=0 123 Eur. Phys. J. C (2021) 81:685 Page 5 of 15 685

The square of the matrix B˘ is easily obtained One easily obtains the relation

μ κ(θ ) ˘ ˘ ˘ 2 B ν 0 ( B)T η B = η, B = 2 ν , (4.4) e e (4.14) 0 κ(Bθ)μ as well as its cube therefore the transformation (4.11) is indeed an element of O(D, D). μ κ2(θ θ)μν α ˘ 3 0 B It is worth pointing out characteristics of the matrix ν. B = 2 . (4.5) μ ρν μρ T ν ρ 2κ(Bθ B)μν 0 It is easy to show that α ρθ = θ (α )ρ and Bμρ α ν = ρ (αT )μ Bρν, which is further generalized to ˘ The higher degree of B are given by μ ρν μρ T ν ρ ( f (α)) ρθ = θ ( f (α ))ρ , Bμρ ( f (α)) ν μ (αn) T ρ ˘ 2n = ν 0 , = ( f (α ))μ Bρν, (4.15) B T n ν (4.6) 0 ((α ) )μ for any analytical function f (α). Moreover, the well known for even degrees, and for odd degrees by hyperbolic identity cosh(x)2 − sinh(x)2 = 1 can also be expressed in terms of newly deﬁned tensors n μν + 0 κ(α θ) B˘ 2n 1 = , (4.7) ( αn) 2 μ μ 2 ρ μ 2 B μν 0 (C ) ν − α ρ(S ) ν = δν . (4.16) where we have marked Lastly, the self T-duality relates the matrix α to its transpose ∼ T μ μρ α = α , due to (2.6). Consequently, we write the following α = 2κθ Bρν. (4.8) ν self T-duality relations Substituting (4.6) and (4.7)into(4.3), we obtain the twisting C =∼ CT , S =∼ ST . matrix (4.17) ⎛ ⎞ μ μ ∞ αn ∞ αn ρν = κ = θ ˘ ⎜ n 0 (2n)! ν n 0 (2n+1)! ρ ⎟ B = ⎝ ρ ν ⎠ . 5 Symmetry generator in an appropriate basis e ∞ αn ∞ (αT )n 2Bμρ = = n 0 (2n+1)! ν n 0 (2n)! μ The direct computation of the bracket (4.2) would be difﬁcult, (4.9) ˘ given the form of the matrix eB . Therefore, we use the indi- rect computation of the bracket, by computing the Poisson Taking into the account the Taylor’s expansion of hyperbolic bracket algebra of the symmetry generator (2.9), rewritten in functions the appropriate basis. As elaborated at the end of the Chapter ∞ ∞ x2k x2k+1 3, this basis is obtained by the action of the matrix (4.11)on cosh(x) = , sinh(x) = , (4.10) the double coordinate (2.4) (2k)! (2k + 1)! n=0 n=0 μ ˘ k˘ the twisting matrix (4.9) can be rewritten as X˘ M = (eB)M X N = , (5.18) N ι˘μ Cμ κSμ θ ρν B˘ = ν ρ , e ρ T ν (4.11) 2Bμρ S ν (C )μ where μ μ ν μν √ ˘ = κC + κ(Sθ) π , μ α μ μ √ μ k ν x ν sinh√ with S ν = and C ν = cosh α . Its deter- ν T ν α ν ν ι˘μ = 2(BS)μν x + (C )μ πν, (5.19) minant is given by are new currents. Applying (2.6), (2.8) and (4.17) to currents ˘ ( ˘ ) ˘μ ˘μ det(eB) = eTr B = 1, (4.12) k and ι˘μ we obtain ι˘μ and k respectively, meaning that these currents are directly related by self T-duality. Multi- and the straightforward calculations show that its inverse is plying the Eq. (5.18) with the matrix (4.13), we obtain the given by relations inverse to (5.19)

μ μ μ μ ˘ν μν C −κS θ ρν κx = C νk − κ(Sθ) ι˘ν, −B˘ = ν ρ . e ρ T ν (4.13) ˘ν T ν −2Bμρ S ν (C )μ πμ =−2(BS)μνk + (C )μ ι˘ν. (5.20) 123 685 Page 6 of 15 Eur. Phys. J. C (2021) 81:685

Applying the transformation (4.11) to a double gauge param- Substituting (4.16) in the previous equation, and keeping in eter (2.11), we obtain new gauge parameters mind that C, S and θ commute (4.15), we obtain μ μ ν μν μ ˘ν μ ρν −1 σ ξ˘ ˘ C ξ + κ(Sθ) λ C k = κx + κ(CSθ) (πν + 2κ(BSC )νσ x ). (5.28) ˘ M = = ( B)M N = ν ν . ν ˘ e N ν T ν λμ 2(BS)μνξ + (C )μ λν (5.21) Using (5.25), the results are marked as a new auxiliary current

The symmetry generator (2.9) rewritten in a new basis G(Cξ+ μ μ μν κSθλ,2(BS)ξ + CT λ) ≡ G˘(ξ,˘ λ)˘ is given by k˚ = κx + κθ˚ ι˚ν, (5.29) θ˚ ˘ ˘ ˘ μ ˘μ where is given by G()˘ = dσ ,˘ X⇔G(ξ,˘ λ)˘ = dσ ξ˘ ι˘μ + λ˘ μk . μν μ ρ σν (5.22) θ˚ = C ρS σ θ . (5.30)

Substituting (5.18) and (5.21)into(5.22), the symmetry gen- There is no explicit dependence on either C nor S in rede- erator in the initial canonical basis (2.9) is obtained. Due fined auxiliary currents, rather only on canonical variables to mutual self T-duality between basis currents (5.19), this and new background fields. From (5.29), it is easy to express generator is invariant upon self T-duality. the coordinate σ -derivative in the basis of new auxiliary cur- Rewriting the Eq. (2.15) in terms of new gauge parameters rents (5.21) in the basis of auxiliary currents (5.19), the Courant μν μ μ μν bracket twisted by both a 2-form Bμν and by a bi-vector θ κx = k˚ − κθ˚ ι˚ν. (5.31) is obtained in the new generator (5.22) algebra The first equation of (5.19) could have been multiplied ˘ (˘ ), ˘ (˘ ) =−˘ [˘ , ˘ ] . C C−1 G 1 G 2 G 1 2 CBθ (5.23) with , instead of , given that the latter would also produce a current that would not explicitly depend on C.How- μ 5.1 Auxiliary generator ever, the expression for coordinate σ -derivative κx would explicitly depend on C2 in that case, while with our choice Let us define a new auxiliary basis, so that both the matri- of basis it does not (5.31). ces C and S are absorbed in some new fields, giving rise to Substituting (5.24) and (5.28) in the expression for the the generator algebra that is much more readable. When the generator (5.22), we obtain algebra in this basis is obtained, simple change of variables ˘ ˘ ˘ ˘ −1 μ ν ˘ μ T ν back to the initial ones will provide us with the bracket in G(ξ,λ) = dσ λμ(C ) νk˚ + ξ (C )μ ι˚ν , (5.32) need. Multiplying the second equation of (5.19) with the matrix from which it is easily seen that the generator (5.22) is equal C−1, we obtain to an auxiliary generator −1 ν −1 ν ι˘ν(C ) μ = πμ + 2κ(BSC )μν x , (5.24) μ μ G˚ ()˚ = dσ X˚ , ˚ ⇔G˚ (ξ,˚ λ)˚ = dσ λ˚ μk˚ + ξ˚ ι˚μ , −1 ν −1 where we have used (BS)νρ (C ) μ =−(BSC )ρμ = − (5.33) (BSC 1)μρ , due to tensor BS being antisymmetric, and properties (4.15). We will mark the result as a new auxiliary current, given by provided that ν ξ˚ μ ι˚μ = πμ + 2κ B˚μν x , (5.25) M ˘ −1 ν μ μ ˘ ν ˚ = , λ˚ μ = λν(C ) μ, ξ˚ = C νξ , (5.34) λ˚ μ where B˚ is an auxiliary B-field, given by and ρ −1 σ B˚μν = Bμρ S σ (C ) ν. (5.26) k˚μ X˚ M = . (5.35) On the other hand, multiplying the first equation of (5.19) ι˚μ with the matrix C, we obtain Once that the algebra of (5.33) is known, the algebra of gen- μ ˘ν 2 μ ν μν C νk = (C ) νκx + κ(CSθ) πν. (5.27) erator (5.22) can be easily obtained using (5.34). 123 Eur. Phys. J. C (2021) 81:685 Page 7 of 15 685

The change of basis to the one suitable for the auxiliary 6 Courant bracket twisted by C˚ from the generator generator (5.33) corresponds to the transformation algebra

μ In order to obtain the Poisson bracket algebra for the genera- (C) ν 0 AM = , ˚ M = AM ˘ N , X˚ M = AM X˘ N , tor (5.33), let us ﬁrstly calculate the algebra of basis vectors, N 0 ((C−1)T ) ν N N μ using the standard Poisson bracket relations (2.16). The aux- (5.36) iliary currentsι ˚μ algebra is that can be rewritten as ρ ρ {ι˚μ(σ), ι˚ν(σ)¯ }=−2B˚μνρk˚ δ(σ −¯σ)− F˚μν ι˚ρδ(σ −¯σ), ˚ M = ( B˘ )M N , ˚ M = ( B˘ )M N , (6.1) X Ae N X Ae N (5.37) ˚ where (5.18) and (5.21) were used. It is easy to show that where Bμνρ is the generalized H-ﬂux, given by M ( B˘ )M the transformation A N , and consequentially Ae N ,isthe element of O(D, D) group B˚μνρ = ∂μ B˚νρ + ∂ν B˚ρμ + ∂ρ B˚μν, (6.2)

˘ ˘ ρ AT η A = η, (AeB)T η(AeB) = η, (5.38) and F˚μν is the generalized f-ﬂux, given by

ρ σρ which means that there is C˚ ,forwhich[21] F˚μν =−2κ B˚μνσ θ˚ . (6.3)

˚ ˘ eC = AeB. (5.39) The algebra of currents k˚μ is given by

The generator (5.33) gives rise to algebra that closes on C˚ - μ ν μν ρ 2 μνρ {k˚ (σ ), k˚ (σ)¯ }=−κQ˚ k˚ δ(σ −¯σ)−κ R˚ ι˚ρδ(σ −¯σ), twisted Courant bracket ρ (6.4) G˚ (˚ 1), G˚ (˚ 1) =−G˚ [˚ 1, ˚ 2]C , (5.40) C˚ where

˚ νρ νρ νσ ρτ νρ νρ where the C-twisted Courant bracket is deﬁned by Q˚ μ = Q˚ μ + 2κθ˚ θ˚ B˚μστ , Q˚ μ = ∂μθ˚ (6.5)

C˚ −C˚ −C˚ [˚ , ˚ ]C = e [e ˚ , e ˚ ]C. (5.41) and 1 2 C˚ 1 2

μνρ μνρ μλ νσ ρτ In the next chapter, we will obtain this bracket by direct com- R˚ = R˚ + 2κθ˚ θ˚ θ˚ B˚λστ , putation of the generators Poisson bracket algebra. μνρ μσ νρ νσ ρμ ρσ μν R˚ = θ˚ ∂σ θ˚ + θ˚ ∂σ θ˚ + θ˚ ∂σ θ˚ . (6.6) Lastly, let us briefly comment on reducibility conditions for the C˚ -twisted Courant bracket. Since we are working with the closed strings, the total derivatives vanishes when The terms in (6.4) containing both θ˚ and B˚ are the conse- integrated out over the worldsheet. Using (5.31), we obtain quence of non-commutativity of auxiliary currentsι ˚μ.The μ remaining algebra of currents k˚ andι ˚μ can be as easily obtained μ μ μν dσκϕ = dσκx ∂μϕ = dσ k˚ ∂μϕ + κι˚μθ˚ ∂νϕ = 0, {ι (σ), ˚ν(σ)¯ }=κδν δ(σ −¯σ) (5.42) ˚μ k μ ν ρ νρ + F˚μρ k˚ δ(σ −¯σ)− κQ˚ μ ι˚ρδ(σ −¯σ). for any parameter λ. Hence, the generator (5.33) remains (6.7) invariant under the following change of parameters The basic algebra relations can be summarized in a sin- μ μ μν ξ˚ → ξ˚ + κθ˚ ∂νϕ, λ˚ μ → λ˚ μ + ∂μϕ. (5.43) gle algebra relation where the structure constants contain all generalized fluxes These are reducibility conditions (2.10) in the basis spanned ˚μ ι { ˚ M , ˚ N }=−˚ MN ˚ P δ(σ −¯σ)+κηMNδ(σ −¯σ), by k and ˚μ. X X F P X (6.8) 123 685 Page 8 of 15 Eur. Phys. J. C (2021) 81:685 with only the first term in the last line of (6.10) μ ν dσ dσ¯ ξ˚ (σ)ι˚μ(σ), λ˚ ν(σ)¯ k˚ (σ)¯ 2 ˚ μνρ ˚ μρ 1 2 MNρ κ R −κQν F = νρ ρ , κQ˚ μ F˚μν = σ ˚μ − ξ˚ ν∂ λ˚ − F˚ ν ξ˚ ρλ˚ d k 1 ν 2μ μρ 1 2ν κQ˚ μν F˚ μ MN ρ νρ νρ μ νμ ρ F ρ = . (6.9) +ι κ(λ˚ θ˚ )∂ ξ˚ − κQ˚ ξ˚ λ˚ ˚ ν ˚μ 2ν ρ 1 ρ 1 2ν −Fμρ 2B˚μνρ + σ σκ¯ ξ˚ ν(σ)λ˚ (σ)∂¯ δ(σ −¯σ). d d 1 2ν σ (6.13) The form of the generalized fluxes is the same as the ones already obtained using the tetrad formalism [16–18]. In our In order to transform the anomalous part, we note that approach, the generalized fluxes are obtained in the Pois- 1 1 son bracket algebra, only from the fact that the generalized ∂σ δ(σ −¯σ) = ∂σ δ(σ −¯σ)− ∂σ¯ δ(σ −¯σ), (6.14) 2 2 canonical variable X M is transformed with an element of the O(D, D) group that twists the Courant bracket both by B and and θ at the same time. Consequentially, the fluxes obtained ˚ in this paper are functions of some new effective fields, Bμν f (σ)∂¯ σ δ(σ −¯σ) = f (σ)∂σ δ(σ −¯σ)+ f (σ)δ(σ −¯σ). (5.26) and θ˚μν (5.30). (6.15) We now proceed to obtain the full bracket. Let us rewrite the generator (5.33) algebra Applying (6.14) and (6.15) to the last row of (6.13), we obtain σ σκ¯ ξ˚ ν(σ)λ˚ (σ)∂¯ δ(σ −¯σ) d d 1 2ν σ G˚ (ξ˚1, λ˚ 1)(σ), G˚ (ξ˚2, λ˚ 2)(σ)¯ 1 μ ν = σκ μ ξ˚ ν∂ λ˚ − ∂ ξ˚ νλ˚ = dσ dσ¯ ξ˚ (σ)ι˚μ(σ), ξ˚ (σ)¯ ι˚ν(σ)¯ d x 1 μ 2ν μ 1 2ν 1 2 2 κ μ ν + σ σ¯ ξ˚ ν(σ)λ˚ (σ)∂ δ(σ −¯σ) + λ˚ 1μ(σ)k˚ (σ), λ˚ 2ν(σ)¯ k˚ (σ)¯ d d 1 2ν σ 2 + ξ˚ μ(σ)ι (σ), λ˚ (σ)¯ ˚ν(σ)¯ −ξ˚ ν(σ)¯ λ˚ (σ)∂¯ δ(σ −¯σ) 1 ˚μ 2ν k 1 2ν σ¯ μ ν 1 μ ν ν + λ˚ 1μ(σ)k˚ (σ), ξ˚ (σ)¯ ι˚ν(σ)¯ . (6.10) = σ ˚ ξ˚ ∂ λ˚ − ∂ ξ˚ λ˚ 2 d k 1 μ 2ν μ 1 2ν 2 μρ ν ν +ι˚μκθ˚ ξ˚ ∂ρλ˚ ν − ∂ρξ˚ λ˚ ν , (6.16) The first term of (6.10) is obtained, using (6.1) 1 2 1 2 where (5.31) was used, as well as antisymmetry of θ˚. Sub- stituting (6.16)to(6.13), we obtain σ σ¯ ξ˚ μ(σ)ι (σ), ξ˚ ν(σ)¯ ι (σ)¯ d d 1 ˚μ 2 ˚ν μ ν dσ dσ¯ ξ˚ (σ)ι˚μ(σ), λ˚ 2ν(σ)¯ k˚ (σ)¯ = σ ι ξ˚ ν∂ ξ˚ μ − ξ˚ ν∂ ξ˚ μ − F˚ μ ξ˚ νξ˚ ρ 1 d ˚μ 2 ν 1 1 ν 2 νρ 1 2 μ ν = dσ k˚ ξ˚ (∂μλ˚ ν − ∂νλ˚ μ) − ˚ ˚μξ˚ νξ˚ ρ . 1 2 2 2Bμνρk 1 2 (6.11) 1 ν ρ − ∂μ(ξ˚ λ˚ ) − F˚μρ ξ˚ λ˚ ν 2 1 2 1 2 The second term is obtained, using (6.4) +ι κ(λ˚ θ˚νρ )∂ ξ˚ μ + κθ˚μρ ξ˚ ν∂ λ˚ − 1∂ (ξ˚ λ˚ ) ˚μ 2ν ρ 1 1 ρ 2ν ρ 1 2 2 νμ ρ μ ν −κQ˚ ρ ξ˚ λ˚ 2ν . (6.17) dσ dσ¯ λ˚ 1μ(σ)k˚ (σ), λ˚ 2ν(σ)¯ k˚ (σ)¯ 1 Substituting (6.11), (6.12) and (6.17)into(6.10), we write μ νρ νρ = dσ k˚ κθ˚ (λ˚ 2ν∂ρλ˚ 1μ − λ˚ 1ν∂ρλ˚ 2μ) − κQ˚ μ the full algebra of generator in the form 2 μνρ λ˚ 1νλ˚ 2ρ − ι˚μκ R˚ λ˚ 1νλ˚ 2ρ . (6.12) G˚ (˚ 1), G˚ (˚ 2) =−G˚ ()˚ ⇔ G˚ (ξ˚1, λ˚ 1), G˚ (ξ˚2, λ˚ 2) =−G˚ (ξ,˚ λ),˚ The remaining terms are antisymmetric with respect to 1 ↔ 2,σ↔¯σ interchange. Therefore, it is sufficient to calculate (6.18) 123 Eur. Phys. J. C (2021) 81:685 Page 9 of 15 685 where that gives the same bracket as (5.41). Both (6.19)–(6.20) ˚ μ ν μ ν μ μρ and (6.22)–(6.23) are the products of C-twisted Courant ξ˚ = ξ˚ ∂νξ˚ − ξ˚ ∂νξ˚ − κθ˚ 1 2 2 1 bracket. The former shows explicitly how the gauge parame- ν ν 1 ters depend on the generalized fluxes. In the latter, similarities × ξ˚ ∂ρλ˚ 2ν − ξ˚ ∂ρλ˚ 1ν − ∂ρ(ξ˚1λ˚ 2 − ξ˚2λ˚ 1) 1 2 2 between the expressions for two parameters is easier to see. +κθ˚νρ (λ˚ ∂ ξ˚ μ − λ˚ ∂ ξ˚ μ) 1ν ρ 2 2ν ρ 1 +κ2R˚ μνρλ˚ λ˚ + F˚ μ ξ˚ ρξ˚ σ 6.1 Special cases and relations to other brackets 1ν 2ρ ρσ 1 2 νμ ρ ρ +κQ˚ (ξ˚ λ˚ ν − ξ˚ λ˚ ν), (6.19) ρ 1 2 2 1 Even though the non-commutativity parameter θ and the and Kalb Ramond field B are not mutually independent, while obtaining the bracket (6.24) the relation between these fields λ˚ = ξ˚ ν(∂ λ˚ − ∂ λ˚ ) − ξ˚ ν(∂ λ˚ − ∂ λ˚ ) μ 1 ν 2μ μ 2ν 2 ν 1μ μ 1ν (2.7) was not used. Therefore, the results stand even if a 1 bi-vector and a 2-form used for twisting are mutually inde- + ∂μ(ξ˚ λ˚ − ξ˚ λ˚ ) 2 1 2 2 1 pendent. This will turn out to be convenient to analyze the νρ origin of terms appearing in the Courant bracket twisted by +κθ˚ (λ˚ 1ν∂ρλ˚ 2μ − λ˚ 2ν∂ρλ˚ 1μ) ν ρ νρ ν C˚ . + B˚μνρξ˚ ξ˚ + κQ˚ λ˚ νλ˚ ρ + F˚ μν 2 1 2 μ 1 2 μσ Primarily, consider the case of zero bi-vector θ = 0 ×(ξ˚ σ λ˚ − ξ˚ σ λ˚ ). 1 2ν 2 1ν (6.20) with the 2-form Bμν arbitrary. Consequently, the parameter α (4.8) is zero, while the hyperbolic functions C and S are It is possible to rewrite the previous two equations, if we identity matrices. Therefore, the auxiliary fields (5.26) and note the relations between the generalized fluxes (5.30) simplify in a following way

μνρ μνρ μσ ντ ρ νρ νρ νσ ρ μν R˚ = R˚ + θ˚ θ˚ F˚στ, Q˚ μ = Q˚ μ + θ˚ F˚μσ . B˚μν → Bμν θ˚ → 0, (6.25) (6.21) ˘ ˆ and the twisting matrix eB (4.11) becomes the matrix eB Now we have (3.28). The expressions (6.19) and (6.20) respectively reduce to ξ˚ μ = ξ˚ ν∂ ξ˚ μ − ξ˚ ν∂ ξ˚ μ 1 ν 2 2 ν 1 μρ ν ν μ ν μ ν μ +κθ˚ ξ˚ (∂ λ˚ − ∂ λ˚ ) − ξ˚ (∂ λ˚ − ∂ λ˚ ) ξ˚ = ξ˚ ∂νξ˚ − ξ˚ ∂νξ˚ , 1 ν 2ρ ρ 2ν 2 ν 1ρ ρ 1ν 1 2 2 1 (6.26)

1 + ∂ρ(ξ˚1λ˚ 2 − ξ˚2λ˚ 1) and 2 +κξ˚ ρ∂ (λ˚ θ˚νμ) − κ(λ˚ θ˚νρ )∂ ξ˚ μ − κξ˚ ρ∂ (λ˚ θ˚νμ) 1 ρ 2ν 2ν ρ 1 2 ρ 1ν ν ν λ˚ μ = ξ˚ (∂νλ˚ μ − ∂μλ˚ ν) − ξ˚ (∂νλ˚ μ − ∂μλ˚ ν) νρ μ 2 μνρ 1 2 2 2 1 1 +κ(λ˚ 1νθ˚ )∂ρξ˚ + κ R˚ λ˚ 1νλ˚ 2ρ 2 1 ν ρ μ ρ σ μσ ν ρ ρ + ∂ (ξ˚ λ˚ − ξ˚ λ˚ ) + ξ˚ ξ˚ , +F˚ ξ˚ ξ˚ + κθ˚ F˚ (ξ˚ λ˚ − ξ˚ λ˚ ) μ 1 2 2 1 2Bμνρ 1 2 (6.27) ρσ 1 2 σρ 1 2ν 2 1ν 2 2 μσ ντ ρ +κ θ˚ θ˚ F˚στ λ˚ 1νλ˚ 2ρ, (6.22) where Bμνρ is the Kalb–Ramond field strength, given by and Bμνρ = ∂μ Bνρ + ∂ν Bρμ + ∂ρ Bμν. (6.28) λ˚ = ξ˚ ν(∂ λ˚ − ∂ λ˚ ) − ξ˚ ν(∂ λ˚ − ∂ λ˚ ) μ 1 ν 2μ μ 2ν 2 ν 1μ μ 1ν 1 B + ∂μ(ξ˚1λ˚ 2 − ξ˚2λ˚ 1) The equations (6.26) and (6.27) define exactly the -twisted 2 Courant bracket (3.27)[9]. νρ νρ +κθ˚ (λ˚ 1ν∂ρλ˚ 2μ − λ˚ 2ν∂ρλ˚ 1μ) + κ Q˚ μ λ˚ 1νλ˚ 2ρ Secondarily, consider the case of zero 2-form Bμν = 0 μν + ˚ ξ˚ νξ˚ ρ + F˚ ν and the bi-vector θ arbitrary. Similarly, α = 0 and C and 2Bμνρ 1 2 μσ μν S are identity matrices. The auxiliary fields B˚μν and θ˚ are ×(ξ˚ σ λ˚ − ξ˚ σ λ˚ ) + κθ˚νσ F˚ ρ λ˚ λ˚ , 1 2ν 2 1ν μσ 1ν 2ρ (6.23) given by where the partial integration was used in the equation (6.22). ˚ → θ˚μν → θ μν. Therelation(6.18) defines the C˚ -twisted Courant bracket Bμν 0 (6.29)

˘ ThetwistingmatrixeB becomes the matrix of θ-transformations θˆ [˚ , ˚ ]C = ˚ ⇔[(ξ˚ , λ˚ ), (ξ˚ , λ˚ )]C = (ξ,˚ λ),˚ (6.24) e (3.30). The gauge parameters (6.19) and (6.20) are respec- 1 2 C˚ 1 1 2 2 C˚ 123 685 Page 10 of 15 Eur. Phys. J. C (2021) 81:685 tively given by The Lie derivative along the vector ﬁeld ξ is given by

μ ν μ ν μ ξ˚ = ξ˚ ∂νξ˚ − ξ˚ ∂νξ˚ L = i d + di , (6.33) 1 2 2 1 ξ˚ ξ˚ ξ˚ μρ ν ν + κθ ξ˚ (∂νλ˚ ρ − ∂ρλ˚ ν) − ξ˚ (∂νλ˚ ρ − ∂ρλ˚ ν) 1 2 2 2 1 1 ξ˚ with iξ˚ being the interior product along the vector field and 1 d being the exterior derivative. Using the Lie derivative one + ∂ρ(ξ˚1λ˚ 2 − ξ˚2λ˚ 1) 2 easily defines the Lie bracket + κξ˚ ν∂ (λ˚ θ ρμ) − κξ˚ ν∂ (λ˚ θ ρμ) 1 ν 2ρ 2 ν 1ρ μ μ [ξ˚1, ξ˚2]L = Lξ˚ ξ˚2 − Lξ˚ ξ˚1. (6.34) + κ(λ˚ θ νρ )∂ ξ˚ − κ(λ˚ θ νρ )∂ ξ˚ 1 2 1ν ρ 2 2ν ρ 1 2 μνρ The generalization of the Lie bracket on a space of 1-forms + κ R λ˚ νλ˚ ρ, (6.30) 1 2 is a well known Koszul bracket [26] and [λ˚ , λ˚ ]θ = L λ˚ − L λ˚ + d(θ(˚ λ˚ , λ˚ )). (6.35) 1 2 θ˚λ˚ 1 2 θ˚λ˚ 2 1 1 2 ν ν λ˚ μ = ξ˚ (∂νλ˚ μ − ∂μλ˚ ν) − ξ˚ (∂νλ˚ μ − ∂μλ˚ ν) 1 2 2 2 1 1 The expressions (6.19) and (6.20) in the coordinate free 1 + ∂μ(ξ˚ λ˚ − ξ˚ λ˚ ) notation are given by 2 1 2 2 1 νρ ρν ξ˚ =[ξ˚1, ξ˚2]L −[ξ˚2, λ˚ 1κθ˚]L +[ξ˚1, λ˚ 2κθ˚]L + κθ (λ˚ 1ν∂ρλ˚ 2μ − λ˚ 2ν∂ρλ˚ 1μ) + κλ˚ 1ρλ˚ 2ν Qμ , 1 (6.31) − Lξ˚ λ˚ 2 − Lξ˚ λ˚ 1 − d(iξ˚ λ˚ 2 − iξ˚ λ˚ 1) κθ˚ 1 2 2 1 2 νρ μνρ +F˚ (ξ˚ , ξ˚ ,.)− κθ˚F˚ (λ˚ ,.,ξ˚ ) + κθ˚F˚ (λ˚ ,.,ξ˚ ) where by Qμ and R we have marked the non-geometric 1 2 1 2 2 1 fluxes, given by +R˚ (λ˚ 1, λ˚ 2,.), (6.36)

νρ νρ μνρ μσ νρ νσ ρμ ρσ μν and Qμ = ∂μθ , R = θ ∂σ θ +θ ∂σ θ +θ ∂σ θ . 1 (6.32) λ˚ = Lξ˚ λ˚ 2 − Lξ˚ λ˚ 1 − d(iξ˚ λ˚ 2 − iξ˚ λ˚ 1) −[λ˚ 1, λ˚ 2]κθ˚ 1 2 2 1 2 ˚ ˚ The bracket defined by these relations is θ-twisted Courant + H˚ (ξ˚1, ξ˚2,.)− F(λ˚ 1,.,ξ˚2) + F(λ˚ 2,.,ξ˚1) bracket (3.29)[8] and it features the non-geometric fluxes + κθ˚F˚ (λ˚ 1, λ˚ 2,.), (6.37) only. Let us comment on terms in the obtained expressions for where gauge parameters (6.22) and (6.23). The first line of (6.22) appears in the Courant bracket and in all brackets that can be H˚ = 2d B˚ . (6.38) obtained from its twisting by either a 2-form or a bi-vector. The next two lines correspond to the terms appearing in the We have marked the geometric H flux as H˚ , so that it is θ -twisted Courant bracket (6.30). The other terms do not distinguished from the 2-form B˚ . In the local basis, the full θ appear in either B-or -twisted Courant bracket. term containing H-flux is given by Similarly, the first line of (6.20) appears in the Courant bracket (2.18) and in all other brackets obtained from its ν ρ H˚ (ξ˚ , ξ˚ ,.) = 2B˚μνρ ξ˚ ξ˚ . (6.39) twisting, while the terms in the second line appear exclusively 1 2 μ 1 2 in the θ twisted Courant bracket (6.27). The first term in the last line appear in the B-twisted Courant bracket (6.31), while Similarly are defined the terms containing F˚ flux the rest are some new terms. We see that all the terms that do μ not appear in neither of two brackets are the terms containing F˚ (ξ˚ , ξ˚ ,.) = F˚ μ ξ˚ νξ˚ ρ, 1 2 νρ 1 2 (6.40) F˚ flux. and the non-geometric R˚ flux 6.2 Coordinate free notation μ R˚ (λ˚ , λ˚ ,.) = R˚ μνρλ˚ λ˚ , In order to obtain the formulation of the C˚ -twisted Courant 1 2 1ν 2ρ (6.41) bracket in the coordinate free notation, independent of the local coordinate system that is used on the manifold, let us as well as firstly provide definitions for a couple of well know brackets μ θ˚F˚ (λ˚ ,.,ξ˚ ) = θ˚νσ F˚ μ λ˚ ξ˚ ρ. and derivatives. 1 2 σρ 1ν 2 (6.42) 123 Eur. Phys. J. C (2021) 81:685 Page 11 of 15 685

It is possible to rewrite the coordinate free notation in and the vector ξ˚2 terms of the H˚ -flux and θ˚ bi-vector only. The geometric F˚ μ flux is just the contraction of the H˚ -flux with a bi-vector ∧2 κθ˚ ˚ (λ˚ ,.,ξ˚ ) = κ2θ˚νσ θ˚μρ ˚ λ˚ ξ˚ τ , H 1 2 2 Bσρτ 1ν 2 (6.51) F˚ = κθ˚ ˚ . H (6.43) and similarly when contraction is done with two forms ˚ The non-geometric R flux can be rewritten as 2 2 τρ νσ ∧ κθ˚H˚ (λ˚ , λ˚ ,.) = 2κ θ˚ θ˚ B˚ρσμλ˚ τ λ˚ ν. (6.52) 1 2 μ 1 2 1 R˚ = [θ,˚ θ˚] +∧3(κθ)˚ H˚ , (6.44) S 3 2 Lastly, the term ∧ κθ˚H˚ (λ˚ 1, λ˚ 2,.) is obtained by taking a wedge product of three bi-vectors with a 3-form and than ∧ [θ,˚ θ˚] where is the wedge product, and by S we have marked contracting it with two 1-forms. It is given by the Schouten–Nijenhuis bracket [27], given by μ μνρ 3 3 νσ ρτ μλ μνρ σα βγ μνρ ∧ κθ˚H˚ (λ˚ 1, λ˚ 2,.) = 2κ θ˚ θ˚ θ˚ B˚στλλ˚ 1νλ˚ 2ρ, [θ,˚ θ˚]S = αβγ θ˚ ∂σ θ˚ = 2R˚ , (6.45) (6.53) where μ ν ρ δα δβ δγ 7 Star brackets μνρ ν ρ μ αβγ = δα δβ δγ . (6.46) ρ μ ν δα δβ δγ The expressions for gauge parameters (6.36) and (6.37)produce some well known bracket, such as Lie bracket and Koszul bracket. The remaining terms can be combined so that Expressing both F˚ and R˚ fluxes in terms of the bi-vector θ˚ they are expressed by some new brackets, acting on pairs of and 3-form H˚ , we obtain generalized vectors. It turns out that these brackets produce a μ ξ˚ =[ξ˚1, ξ˚2]L −[ξ˚2, λ˚ 1κθ˚]L +[ξ˚1, λ˚ 2κθ˚]L generalized vector, where the vector part ξ˚ and the 1-form μ μν 1 part λ˚ μ are related by ξ˚ = κθ˚ λ˚ ν, effectively resulting − Lξ˚ λ˚ 2 − Lξ˚ λ˚ 1 − d(iξ˚ λ˚ 2 − iξ˚ λ˚ 1) κθ˚ 1 2 2 1 2 in the graphs in the generalized cotangent bundle T M of κ2 the bi-vector θ˚, i.e. ξ = κθ(.,λ). The star brackets can be + [θ,˚ θ˚]S(λ˚ , λ˚ ,.) 2 1 2 interpreted in terms of projections on isotropic subspaces. +κθ˚H˚ (., ξ˚ , ξ˚ ) −∧2κθ˚H˚ (λ˚ ,.,ξ˚ ) 1 2 1 2 θ 2 3 7.1 -star bracket +∧ κθ˚H˚ (λ˚ 2,.,ξ˚1) +∧ κθ˚H˚ (λ˚ 1, λ˚ 2,.), (6.47) and Let us firstly consider the second line of (6.22) and the first line of (6.23). When combined, they define a bracket acting λ˚ = L λ˚ − L λ˚ − 1 ( λ˚ − λ˚ ) −[λ˚ , λ˚ ] ξ˚ 2 ξ˚ 1 d iξ˚ 2 iξ˚ 1 1 2 κθ˚ on a pair of generalized vectors 1 2 2 1 2 +H˚ (ξ˚1, ξ˚2,.)− κθ˚H˚ (λ˚ 1,.,ξ˚2) [˚ , ˚ ] = ˚ ⇔[(ξ˚ , λ˚ ), (ξ˚ , λ˚ )] = (ξ˚ , λ˚ ), (7.1) 2 1 2 θ˚ 1 1 2 2 θ˚ +κθ˚H˚ (λ˚ 1,.,ξ˚2) +∧ κθ˚H˚ (λ˚ 1, λ˚ 2,.). (6.48) where The term κθ˚H˚ (., ξ˚1, ξ˚2) is the wedge product of a bi- vector with a 3-form, contracted with two vectors, given by ξ˚ μ = κθ˚μρ ξ˚ ν(∂ λ˚ − ∂ λ˚ ) − ξ˚ ν(∂ λ˚ − ∂ λ˚ ) 1 ν 2ρ ρ 2ν 2 ν 1ρ ρ 1ν

μ ρ 1 κθ˚ ˚ (., ξ˚ , ξ˚ ) = κθ˚μν ˚ ξ˚ ξ˚ σ , + ∂ρ(ξ˚1λ˚ 2 − ξ˚2λ˚ 1) , (7.2) H 1 2 2 Bνρσ 1 2 (6.49) 2

and and κθ˚H˚ (λ˚ 1,.,ξ˚2) is similarly deﬁned, with the 1-form con- λ˚ = ξ˚ ν(∂ λ˚ − ∂ λ˚ ) − ξ˚ ν(∂ λ˚ − ∂ λ˚ ) tracted instead of one vector ﬁeld μ 1 ν 2μ μ 2ν 2 ν 1μ μ 1ν 1 νρ σ + ∂ (ξ˚ λ˚ − ξ˚ λ˚ ), κθ˚H˚ (λ˚ ,.,ξ˚ ) = 2κθ˚ B˚ρμσ λ˚ νξ˚ . (6.50) μ 1 2 2 1 (7.3) 1 2 μ 1 2 2 from which one easily reads the relation 2 The terms like ∧ κθ˚H˚ (λ˚ 1,.,ξ˚2) are the wedge product of μ μρ two bi-vectors with a 3-form, contracted with the 1-form λ˚ 1 ξ˚ = κθ˚ λ˚ ρ. (7.4) 123 685 Page 12 of 15 Eur. Phys. J. C (2021) 81:685

In a coordinate free notation, this bracket can be written as for any bi-vector θ, and

μ λ μ = 2Bμνξ .(i = 1, 2) Bμν =−Bνμ, (7.12) [˚ 1, ˚ 2] =[(ξ˚1, λ˚ 1), (ξ˚2, λ˚ 2)] i i θ˚ θ˚ = κθ˚ ., L λ˚ − L λ˚ , L λ˚ − L λ˚ . for any 2-form B satisfy the condition (7.10). ξ˚1 2 ξ˚2 1 ξ˚1 2 ξ˚2 1 Furthermore, it is straightforward to introduce projections (7.5) on these isotropic subspaces by

7.2 Bθ-star bracket θ M θ μ μν I ( ) = I (ξ ,λμ) = (κ θ λν,λμ), (7.13)

The remaining terms contain geometric H˚ and F˚ ﬂuxes. Note and that they are the only terms that depend on the new effective M μ μ ν Kalb–Ramond ﬁeld B˚ . Firstly, we mark the last line of (6.23) IB( ) = IB(ξ ,λμ) = (ξ , 2Bμνξ ). (7.14) as Now it is easy to give an interpretation to star brackets. The

∗ ν ρ ν σ σ νσ ρ θ-star bracket (7.1) can be deﬁned as the projection of the λ˚ = 2B˚μνρξ˚ ξ˚ +F˚ ξ˚ λ˚ ν−ξ˚ λ˚ ν)+κθ˚ F˚ λ˚ νλ˚ ρ. μ 1 2 μσ 1 2 2 1 μσ 1 2 Courant bracket (3.29) on the isotropic subspace (7.13) (7.6)

θ˚ [˚ 1, ˚ 2] = I [˚ 1, ˚ 2]C . (7.15) Secondly, using the deﬁnition of F˚ (6.3) and the fact that θ˚ θ˚ is antisymmetric, the last line of (6.22) can be rewritten as Similarly, note that all the terms in (6.37) that do not appear ξ˚ μ = κθ˚μν ˚ ξ˚ ρξ˚ σ + κθ˚μσ F˚ ν (ξ˚ ρλ˚ − ξ˚ ρλ˚ ) ∗ 2 Bνρσ 1 2 σρ 1 2ν 2 1ν in the θ-twisted Courant bracket, contribute exactly to the 2 μν τσ ρ Bθ-star bracket. From that, it is easy to obtain the deﬁnition +κ θ˚ θ˚ F˚νσ λ˚ 1τ λ˚ 2ρ μν ∗ of the Bθ-star bracket (7.8) = κθ˚ λ˚ ν. (7.7)

Now relations (7.6) and (7.7) deﬁne the Bθ-star bracket ∗ θ˚ θ˚ [˚ 1, ˚ 2] = I [˚ 1, ˚ 2]C − I [˚ 1, ˚ 2]C . (7.16) by B˚ θ˚ C˚ θ˚

∗ ∗ ∗ ∗ [˚ , ˚ ] = ˚ ⇔[(ξ˚ , λ˚ ), (ξ˚ , λ˚ )] = (ξ˚∗, λ˚ ), 1 2 B˚ θ˚ 1 1 2 2 B˚ θ˚ 8 Courant bracket twisted by B and θ (7.8) Now it is possible to write down the expression for the We can write the full bracket (6.24)as Courant bracket twisted by B and θ (4.2), using the expression for C˚ -twisted Courant bracket [(ξ˚1, λ˚ 1), (ξ˚2, λ˚ 2)]C C˚ [˘ , ˘ ] = −1[ ˘ , ˘ ] , 1 2 C θ A A 1 A 2 C (8.1) = [ξ˚1, ξ˚2]L −[ξ˚2, λ˚ 1κθ˚]L +[ξ˚1, λ˚ 2κθ˚]L B C˚ κ2 + [θ,˚ θ˚]S(λ˚ 1, λ˚ 2,.),−[λ˚ 1, λ˚ 2]κθ˚ where A is defined in (5.36). Substituting (8.1)into(6.36), 2 we obtain +[(ξ˚ , λ˚ ), (ξ˚ , λ˚ )]∗ +[(ξ˚ , λ˚ ), (ξ˚ , λ˚ )] . 1 1 2 2 ˚ ,θ˚ 1 1 2 2 θ˚ (7.9) B ˘ −1 ˘ ˘ ξ = C [Cξ1, Cξ2]L −C−1[Cξ˘ , λ˘ κC−1θ˚] + C−1[Cξ˘ , λ˘ κC−1θ˚] 7.3 Isotropic subspaces 2 1 L 1 2 L ˘ −1 ˘ −1 − L ˘ (λ C ) − L ˘ (λ C ) Cξ1 2 Cξ2 1 In order to give an interpretation to newly obtained starred brackets, it is convenient to consider isotropic subspaces. A 1 ˘ ˘ −1 − d(iξ˘ λ2 − iξ˘ λ1) κθ˚C subspace L is isotropic if the inner product (2.12)ofanytwo 2 1 2 κ2 generalized vectors from that sub-bundle is zero −1 ˘ −1 ˘ −1 + C [θ,˚ θ˚]S(λ C , λ C ,.) 2 1 2 −1 ˘ ˘ 1,2=0,1,2 ∈ L. (7.10) +κC θ˚H˚ (., Cξ1, Cξ2) −1 2 ˘ −1 ˘ −C ∧ κθ˚H˚ (λ1C ,.,Cξ2) From (2.12), one easily finds that −1 2 ˘ −1 ˘ +C ∧ κθ˚H˚ (λ2C ,.,Cξ1) ξ μ = κθμνλ .(= , )θμν =−θ νμ, +C−1 ∧3 κθ˚ ˚ (λ˘ C−1, λ˘ C−1,.), i iν i 1 2 (7.11) H 1 2 (8.2) 123 Eur. Phys. J. C (2021) 81:685 Page 13 of 15 685 and similarly, substituting (8.1)into(6.37), we obtain The expressions for the Courant bracket twisted by both B and θ can be written in a form ˘ ˘ −1 ˘ −1 1 ˘ ˘ λ = LCξ˘ (λ2C ) − LCξ˘ (λ1C ) − d(iξ˘ λ2 − iξ˘ λ1) C 1 2 2 1 2 [(ξ˘ , λ˘ ), (ξ˘ , λ˘ )] ˘ ˘ 1 1 2 2 C θ +H˚ (Cξ1, Cξ2,.)C B ˘ −1 ˘ −1 ˘ −1 ˘ = [ξ˘ , ξ˘ ] −[ξ˘ , λ˘ κC−1θ˘] +[ξ˘ , λ˘ κC−1θ˘] −[λ1C , λ2C ]κθ˚C − κθ˚H˚ (λ1C ,.,Cξ2)C 1 2 LC 2 1 LC 1 2 LC ˘ −1 ˘ 2 +κθ˚H˚ (λ2C ,.,Cξ1)C κ + [θ,˘ θ˘] (λ˘ , λ˘ ,.),−[λ˘ , λ˘ ]θ 2 ˘ −1 ˘ −1 SC 1 2 1 2 C +∧ κθ˚H˚ (λ1C , λ2C ,.)C, (8.3) 2 ˘ ˘ −1 ˘ ˘ −1 √ μ +[(Cξ1, λ1C ), (Cξ2, λ2C )] − ˘ C Cμ = α ˘ = (ξ,˘ λ)˘ C 1θ where ν cosh and (5.21). This is ∗ ν +[(ξ˘ , λ˘ ), (ξ˘ , λ˘ ] . (8.12) somewhat a cumbersome expression, making it difficult to 1 1 2 2 B˘ ,C−1θ˘ work with. To simplify it, with the accordance of our con- θ vention, we define the twisted Lie bracket by When the Courant bracket is twisted by both B and ,it results in a bracket similar to C˚ -twisted Courant bracket, ˘ ˘ −1 ˘ ˘ where Lie brackets, Schouten Nijenhuis bracket and Koszul [ξ1, ξ2]LC = C [Cξ1, Cξ2]L , (8.4) bracket are all twisted as well. as well as the twisted Schouten–Nijenhuis bracket

μνρ σλτ 9 Conclusion ˘ ˘ −1 μ −1 ν −1 ρ ˘ ˘ [θ,θ]SC = (C ) σ (C ) λ(C ) τ [Cθ,Cθ]S , (8.5) We examined various twists of the Courant bracket, that appear in the Poisson bracket algebra of symmetry gener- and twisted Koszul bracket ators written in a suitable basis, obtained acting on the double canonical variable (2.4) by the appropriate elements of ˘ ˘ T −1 T ˘ T ˘ O(D, D) group. In this paper, we considered the transfor- [λ1, λ2]θC = (C ) [C λ1, C λ2]θC, (8.6) mations that twists the Courant bracket simultaneously by a2-formB and a bi-vector θ. When these fields are mutu- where the transpose of the matrix is necessary because the ally T-dual, the generator obtained by this transformation is Koszul bracket acts on 1-forms. Now, the first three terms of invariant upon self T-duality. (8.2) can be written as We obtained the matrix elements of this transformation, B˘ − − that we denoted e (4.11), expressed in terms of the hyper- [ξ˘ , ξ˘ ] −[ξ˘ , λ˘ κC 1θ˘] +[ξ˘ , λ˘ κC 1θ˘] , (8.7) 1 2 LC 2 1 LC 1 2 LC bolic functions of a parameter α (4.8). In order to avoid working with such a complicated expression, we considered where another O(D, D) transformation A (5.36) and introduced a new generator, written in a basis of auxiliary currentsι ˚μ θ˘μν = (C−1)μ θ˚ρν = Sμ θ ρν. ρ ρ (8.8) and k˚μ. The Poisson bracket algebra of a new generator was obtained and it gave rise to the C˚ -twisted Courant bracket, The second line of (8.2) and the first line of (8.3) originating which contains all of the fluxes. θ˚ from star bracket (7.1) can be easily combined into The generalized fluxes were obtained using different methods [10–12,16–18]. In our approach, we started by [(Cξ˘ , λ˘ C−1), (Cξ˘ , λ˘ C−1)] C. 1 1 2 2 C−1θ˘ (8.9) an O(D, D) transformation that twists the Courant bracket simultaneously by a 2-form B and bi-vector θ, making it The terms originating from B˚ θ˚ star bracket (7.8) are com- manifestly self T-dual. We obtained the expressions for all bined into fluxes, written in terms of the effective fields √ ˘ ˘ ˘ ˘ ∗ κθ ρ [(ξ1, λ1), (ξ2, λ2] ˘ − ˘, (8.10) tanh 2 B B,C 1θ B˚μν = Bμρ √ ν, κθ √ 2 B μ where μν sinh 2 2κθB σν θ˚ = √ σ θ . (9.1) κθ ˘ α β γ −1 2 2 B Bμνρ = B˚αβγ C μC νC ρ = ∂α(BSC )βγ

−1 −1 α β γ The ﬂuxes, as a function of these effective ﬁelds, appear nat- + ∂β (BSC )γα + ∂γ (BSC )αβ C C C . (8.11) μ ν ρ urally in the Poisson bracket algebra of such generators. 123 685 Page 14 of 15 Eur. Phys. J. C (2021) 81:685

Similar bracket was obtained in the algebra of general- bracket that includes all fluxes. It has been already shown ized currents in [11,12] and is sometimes referred to as the [8] how the Hamiltonian can be obtained acting with B- Roytenberg bracket [10]. In that approach, phase space has transformations on diagonal generalized metric. The same ˘ been changed, so that the momentum algebra gives rise to the method could be replicated with the twisting matrix eB, that H-flux, after which the generalized currents were defined in would give rise to a different Hamiltonian, whose further terms of the open string fields. The bracket obtained this way analysis can provide interesting insights in the role that the corresponds to the Courant bracket that was firstly twisted Courant bracket twisted by both B and θ plays in understand- by B field, and then by a bi-vector θ. The matrix of that twist ing T-duality. is given by Data Availability Statement This manuscript has no associated data δμ + αμ κθμν or the data will not be deposited. [Authors’ comment: No data was used R θˆ Bˆ ν ν for this paper.] e = e e = ν . (9.2) 2Bμν δμ Open Access This article is licensed under a Creative Commons Attri- In our approach, we obtained the transformations that twists bution 4.0 International License, which permits use, sharing, adaptation, θ distribution and reproduction in any medium or format, as long as you the Courant bracket at the same time by B and , resulting in a give appropriate credit to the original author(s) and the source, pro- C˚ -twisted Courant bracket. As a consequence, the C˚ -twisted vide a link to the Creative Commons licence, and indicate if changes Courant bracket is defined in terms of auxiliary fields B˚ (5.26) were made. The images or other third party material in this article and θ˚ (5.30), that are themselves function of α. This is not the are included in the article’s Creative Commons licence, unless indi- cated otherwise in a credit line to the material. If material is not case in [11,12]. The Roytenberg bracket calculated therein included in the article’s Creative Commons licence and your intended can be also obtained following our approach by twisting with use is not permitted by statutory regulation or exceeds the permit- the matrix ted use, you will need to obtain permission directly from the copy- right holder. To view a copy of this licence, visit http://creativecomm ˘ C2 κ(CSθ) ons.org/licenses/by/4.0/. eC = AeB = , (9.3) Funded by SCOAP3. 2BCS 1 demanding that the background fields are infinitesimal B ∼ , θ ∼ and keeping the terms up to 2. With these conditions, eC (9.3) becomes exactly eR (9.2), and the bracket References becomes the Roytenberg bracket. Analyzing the C˚ -twisted Courant bracket, we recognized 1. T. Courant, Dirac manifolds. Trans. Am. Math. Soc. 319, 631–661 that certain terms can be seen as new brackets on the space of (1990) 2. Z.-J. Liu, A. Weinstein, P. Xu, Manin triples for Lie bialgebroids. generalized vectors, that we named star brackets. We demon- J. Differ. Geom. 45, 547–574 (1997) strated that they are closely related to projections on isotropic 3. A. Alekseev, T. Strobl, Current algebras and differential geometry. spaces. It is well established that the Courant bracket does not JHEP 03, 035 (2005) satisfy the Jacobi identity in general case. The sub-bundles 4. C. Hull, B. Zwiebach, The gauge algebra of double field theory and Courant brackets. JHEP 09, 090 (2009) on which the Jacobi identity is satisfied are known as Dirac 5. C. Hull, B. Zwiebach, Double field theory. JHEP 0909, 099 (2009) structures, which as a necessary condition need to be sub- 6. W. Siegel, Two-vierbein formalism for string-inspired axionic sets of isotropic spaces. Therefore, the star brackets might gravity. Phys. Rev. D47, 5453–5459 (1993) provide future insights into integrability conditions for the 7. W. Siegel, Superspace duality in low-energy superstrings. Phys. ˚ Rev. D48, 2826–2837 (1993) C-twisted Courant bracket [28]. 8. Lj Davidović, I. Ivanišević, B. Sazdović, Courant bracket as T- In the end, we obtained the Courant bracket twisted at the dual invariant extension of Lie bracket. JHEP 03, 109 (2021). same time by B and θ by considering the generator in the arXiv:2010.10662 basis spanned by ι˘ and k˘, equivalent to undoing A transfor- 9. P. Ševera, A. Weinstein, Poisson geometry with a 3-form background. Prog. Theor. Phys. Suppl. 144, 145–154 (2001) mation, used to simplify calculations. With the introduction 10. D. Roytenberg, Quasi-Lie bialgebroids and twisted Poisson mani- ˘ ˘μν of new fields Bμν and θ , this bracket has a similar form folds. Lett. Math. Phys. 61, 123 (2002) as C˚ -twisted Courant bracket, whereby the Lie, Schouten– 11. N. Halmagyi, Non-geometric string backgrounds and worldsheet Nijenhuis and Koszul brackets became their twisted counter- algebras. JHEP 07, 137 (2008) 12. N. Halmagyi, Non-geometric backgrounds and the first order string parts. sigma model. arXiv:0906.2891 It has already been established that B-twisted and θ- 13. I. Ivanišević, Lj Davidović, B. Sazdović, Courant bracket found twisted Courant brackets appear in the generator algebra out to be T-dual to Roytenberg one. Eur. Phys. J. C80, 571 (2020) defined in bases related by self T-duality [13]. When the 14. M. Grana, Flux compactifications in string theory: a comprehensive 423 θ review. Phys. Rep. , 91–158 (2006) Courant bracket is twisted by both B and , it is self T-dual, 15. Lj. Davidović, I. Ivanišević, B. Sazodivć, Twisted C bracket (in and as such, represent the self T-dual extension of the Lie preparation) 123 Eur. Phys. J. C (2021) 81:685 Page 15 of 15 685

16. E. Plauschinn, Non-geometric backgrounds in string theory. Phys. 24. T. Buscher, A symmetry of the string background field equations. Rep. 798, 1–122 (2019) Phys. Lett. B 194, 51 (1987) 17. M. Grana, R. Minasian, M. Petrini, D. Waldram, T-duality, gen- 25. Lj Davidović, B. Sazdović, The T-dual symmetries of a bosonic eralized geometry and non-geometric backgrounds. JHEP 04, 075 string. Eur. Phys. J. C78, 600 (2018) (2009) 26. Y.Kosmann-Schwarzbach, From Poisson algebras to Gerstenhaber 18. R. Blumenhagen, A. Deser, E. Plauschinn, F. Rennecke, Bianchi algebras. Annales de l’institut Fourier 46, 1243–1274 (1996) identities for non-geometric fluxes from quasi-Poisson structures 27. J.A. de Azcarraga, A.M. Perelomov, J.C. Perez Bueno, The to Courant algebroids. Fortschr. Phys. 60, 1217–1228 (2012) Schouten–Nijenhuis bracket, cohomology and generalized Pois- 19. K. Becker, M. Becker, J. Schwarz, String Theory and M-Theory: son structures. J. Phys. A29, 7993–8110 (1996) A Modern Introduction (Cambridge University Press, Cambridge, 28. Lj. Davidović, I. Ivanišević, B. Sazdović, The integrability proper- 2007) ties of the Courant bracket twisted by elements of O(D, D) group 20. B. Zwiebach, A First Course in String Theory (Cambridge Univer- (in preparation) sity Press, Cambridge, 2004) 21. M. Gualtieri, Generalized complex geometry (2003). arXiv:0401221 [math] 22. E. Alvarez, L. Alvarez-Gaume, Y. Lozano, An introduction to T- duality in string theory. Nucl. Phys. Proc. Suppl. 41, 1–20 (1995) 23. A. Giveon, M. Parrati, E. Rabinovici, Target space duality in string theory. Phys. Rep. 244, 77–202 (1994)

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