The Pontryagin class for pre-Courant algebroids

Liu, Zhangju; Sheng, Yunhe; Xu, Xiaomeng

doi:10.1016/j.geomphys.2016.02.007

Cited by 10 publications

(15 citation statements)

References 29 publications

(58 reference statements)

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“…Our results imply that a DFT algebroid is a special case of a pre-DFT algebroid in which imposing that the image of the derivation is in the kernel of the anchor reduces it directly to a Courant algebroid, without passing through the intermediate structures of ante-Courant and pre-Courant algebroids. All cases may be characterized in terms of an underlying L ∞ -algebra structure [41,66]. In Appendix A.…”

Section: Summary Of Results and Outlinementioning

confidence: 99%

“…We further provide the local coordinate expressions of these axioms, and discuss them in the spirit of the main text of this paper. Then we present the notions of a pre-Courant algebroid [63] and of a 4-form twisted Courant algebroid [64], whose equivalence is discussed in [66]. Finally, we introduce the notions of an ante-Courant algebroid and a pre-DFT algebroid as natural generalizations of the pre-Courant algebroid structure, and further discuss their relation to the metric algebroid of [39] and their description in terms of graded geometry.…”

Section: A From Courant Algebroids To Dft Algebroidsmentioning

confidence: 99%

“…This definition shows that the violation of the Jacobi identity is controlled by a 4-form 25 T . As discussed in [66], the two definitions are essentially equivalent.…”

Section: A From Courant Algebroids To Dft Algebroidsmentioning

confidence: 99%

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Double field theory and membrane sigma-models

Chatzistavrakidis

Jonke

Khoo

et al. 2018

J. High Energ. Phys.

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We investigate geometric aspects of double field theory (DFT) and its formulation as a doubled membrane sigma-model. Starting from the standard Courant algebroid over the phase space of an open membrane, we determine a splitting and a projection to a subbundle that sends the Courant algebroid operations to the corresponding operations in DFT. This describes precisely how the geometric structure of DFT lies in between two Courant algebroids and is reconciled with generalized geometry. We construct the membrane sigma-model that corresponds to DFT, and demonstrate how the standard T-duality orbit of geometric and non-geometric flux backgrounds is captured by its action functional in a unified way. This also clarifies the appearence of noncommutative and nonassociative deformations of geometry in non-geometric closed string theory. Gauge invariance of the DFT membrane sigma-model is compatible with the flux formulation of DFT and its strong constraint, whose geometric origin is explained. Our approach leads to a new generalization of a Courant algebroid, that we call a DFT algebroid and relate to other known generalizations, such as pre-Courant algebroids and symplectic nearly Lie 2-algebroids. We also describe the construction of a gauge-invariant doubled membrane sigma-model that does not require imposing the strong constraint. 1 a.chatzistavrakidis@gmail.com 2 larisa@irb.hr 3 Fech.Scen.Khoo@irb.hr 4 R.J.Szabo@hw.ac.uk 29)19 In this subsection we simplify the notation for the components of the map ρ + by denoting (ρ + ) I J as ρ I J .

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Section: Summary Of Results and Outlinementioning

confidence: 99%

Section: A From Courant Algebroids To Dft Algebroidsmentioning

confidence: 99%

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Double field theory and membrane sigma-models

Chatzistavrakidis

Jonke

Khoo

et al. 2018

J. High Energ. Phys.

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“…which is the natural extension on the operator (2.1). Following Stiénon & Xu [50] and Liu, Sheng & Xu [39] one can construct a quasi-cochain complex associated with a pre-Courant algebroid. It is important to note that in general D is not 'homological', i.e., D 2 = 0, due to the fact that the pre-bracket does not satisfy the Jacobi identity.…”

Section: Definition 21 (Vaisman [51]) a Courant Vector Bundle Is A mentioning

confidence: 99%

Pre-Courant algebroids

Bruce

Grabowski

2019

Journal of Geometry and Physics

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Pre-Courant algebroids are 'Courant algebroids' without the Jacobi identity for the Courant-Dorfman bracket. We examine the corresponding supermanifold description of pre-Courant algebroids and some direct consequences thereof. In particular, we define symplectic almost Lie 2-algebroids and show how they correspond to pre-Courant algebroids. We give the definition of (sub-)Dirac structures and study the naïve quasi-cochain complex within the setting of supergeometry. Moreover, the framework of supermanifolds allows us to economically define and work with pre-Courant algebroids equipped with a compatible non-negative grading. VB-Courant algebroids are natural examples of what we call weighted pre-Courant algebroids and our approach drastically simplifies working with them.Dedicated to the memory of James Alfred Bruce MSC (2010): 17A32; 17B99; 53D17; 58A50.The Jacobiator associated with the Courant-Dorfman pre-bracket on a symplectic almost Lie 2-algebroid has some interesting properties which we state in the following two propositions.Proposition 3.7. Let J Θ be the Jacobiator associated with a symplectic almost Lie 2-algebroid. Then J Θ is totally (graded) skew-symmetric, i.e.,for all σ, ψ and λ ∈ A 1 (F 2 ).Proof. Directly from the definition of the Jacobiator and application of the Jacobi identity for the Poisson bracket, we have J Θ (σ, ψ, λ) + (−1) ( σ+1)( ψ+1) J Θ (ψ, σ, λ) = (−1) ψ {{Σ, {σ, ψ}}, λ}, where we use the shorthand Σ = 1 2 {Θ, Θ}. As {ψ, σ} = ψ, σ is a function on M , the right-hand side of the above vanishes (see Definition 3.6). Directly from the Jacobi identity we haveUsing the skew-symmetry of the Poisson bracket and the Jacobi identity again, we arrive at J Θ (σ, ψ, λ) + (−1) ( ψ+1)( λ+1) J Θ (σ, λ, ψ) = (−1) ψ ({Σ, {σ, {ψ, λ}}} − {σ, {Σ, {ψ, λ}}}) = 0,

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“…According to [21] (see also [1,13]), an anchored vector bundle E with anchor ρ is a Courant vector bundle if there is a pseudo-euclidian metric g in the fibers of E such that ρ•#•ρ * : T * M → T M vanishes, where # : E * → E is the musical isomorphism induced by g.…”

Section: Introductionmentioning

confidence: 99%

Almost Lie Algebroids and Characteristic Classes

Popescu¹,

Popescu²

2019

SIGMA

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Almost Lie algebroids are generalizations of Lie algebroids, when the Jacobiator is not necessary null. A simple example is given, for which a Lie algebroid bracket or a Courant bundle is not possible for the given anchor, but a natural extension of the bundle and the new anchor allows a Lie algebroid bracket. A cohomology and related characteristic classes of an almost Lie algebroid are also constructed. We prove that these characteristic classes are all pull-backs of the characteristic classes of the base space, as in the case of a Lie algebroid.

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The Pontryagin class for pre-Courant algebroids

Cited by 10 publications

References 29 publications

Double field theory and membrane sigma-models

Double field theory and membrane sigma-models

Pre-Courant algebroids

Almost Lie Algebroids and Characteristic Classes

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