<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mi>H</mml:mi></mml:math>-twisted Lie algebroids

Grützmann, Melchior

doi:10.1016/j.geomphys.2010.10.016

Cited by 24 publications

(25 citation statements)

References 10 publications

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“…Recall that in the two-dimensional case these conditions are equivalent to the vanishing of the Schouten-Nijenhuis bracket for a bivector field, which is the condition for a Poisson structure on M or the Lie algebroid axioms for the cotangent bundle T * M equipped with the Koszul-Schouten bracket, while in the three-dimensional case they are equivalent to the axioms of a Courant algebroid. In the four-dimensional case the conditions found in [39,41] define a higher algebroid structure, called a Lie algebroid up to homotopy in [39] or more generally an H-twisted Lie algebroid in [41] (see also Refs. [42,43]).…”

Section: Topological Threebrane Sigma-modelsmentioning

confidence: 99%

“…In a second step, we employ the construction of topological field theories of AKSZ-type in four worldvolume dimensions (an open threebrane), studied in [39,40]. The classical master equation for the corresponding threebrane sigma-model yields a set of conditions, which are then used to define the higher structure of a Lie algebroid up to homotopy [39]see also [41][42][43] for a somewhat more general construction. Such threebrane sigma-models were already used in [44] to relate AKSZ theories of M2-branes in topological M-theory to exceptional generalized geometry and fluxes.…”

Section: Introductionmentioning

confidence: 99%

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Fluxes in exceptional field theory and threebrane sigma-models

Chatzistavrakidis

Jonke

Lüst

et al. 2019

J. High Energ. Phys.

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Starting from a higher Courant bracket associated to exceptional generalized geometry, we provide a systematic derivation of all types of fluxes and their Bianchi identities for four-dimensional compactifications of M-theory. We show that these fluxes may be understood as generalized Wess-Zumino terms in certain topological threebrane sigma-models of AKSZ-type, which relates them to the higher structure of a Lie algebroid up to homotopy. This includes geometric compactifications of M-theory with G-flux and on twisted tori, and also its compactifications with non-geometric Q-and R-fluxes in specific representations of the U-duality group SL(5) in exceptional field theory.

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Section: Topological Threebrane Sigma-modelsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Fluxes in exceptional field theory and threebrane sigma-models

Chatzistavrakidis

Jonke

Lüst

et al. 2019

J. High Energ. Phys.

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“…The cotangent bundle

M = T^{*} M

can likewise be replaced by any Lie algebroid

scriptL

over M , with

ρ \in Γ (M, ⋀^{2} {scriptL}^{*})

a Lie algebroid two‐cochain which defines a central extension of

scriptL

. Then

H = d ρ

defines a Lie algebroid three‐cocycle with values in the kernel of the anchor map of the extension and hence can be used to endow

scriptL

with the structure of an H ‐twisted Lie algebroid, which is in particular a 2‐term

L_{\infty}

‐algebroid. However, here we stick to this concrete and simple example as it will capture the essential features that we wish to describe in this contribution.…”

Section: Magnetic Poisson Structuresmentioning

confidence: 99%

“…These brackets do not generally define a Poisson algebra but rather an H-twisted Poisson structure on M, with twisting given by the three-form H = dρ on M that we shall call a 'magnetic charge'. This means that the Schouten bracket of the bivector θ ρ with itself, which governs the associativity of the brackets defined by (3), is given by the trivector [θ ρ , θ ρ ] S = 3 θ ρ (dσ ρ ), (5) where θ ρ denotes the natural contraction of forms to vectors by the non-degenerate bivector θ ρ . It vanishes if and only if H = 0, while it generically gives a nonassociative algebra with Jacobiators…”

Section: Introductionmentioning

confidence: 99%

Quantization of Magnetic Poisson Structures

Szabo

2019

Fortschritte der Physik

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We describe three perspectives on higher quantization, using the example of magnetic Poisson structures which embody recent discussions of nonassociativity in quantum mechanics with magnetic monopoles and string theory with non‐geometric fluxes. We survey approaches based on deformation quantization of twisted Poisson structures, symplectic realization of almost symplectic structures, and geometric quantization using 2‐Hilbert spaces of sections of suitable bundle gerbes. We compare and contrast these perspectives, describing their advantages and shortcomings in each case, and mention many open avenues for investigation.

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“…This defines a Lie algebroid up to homotopy on T*E. 27 and 19 We take the following canonical function of degree 3,…”

Section: Example 42 (H 4 -Twisted Courant Algebroid) We Consider Anmentioning

confidence: 99%

Canonical functions, differential graded symplectic pairs in supergeometry, and Alexandrov-Kontsevich-Schwartz-Zaboronsky sigma models with boundaries

Ikeda

2014

Journal of Mathematical Physics

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Articles you may be interested inThe classical exchange algebra of a Green-Schwarz sigma model on supercoset target space with Z 4 m grading Consistent boundary conditions for Alexandrov-Kontsevich-Schwartz-Zaboronsky (AKSZ) sigma models and the corresponding boundary theories are analyzed. As their mathematical structures, we introduce a generalization of differential graded symplectic manifolds, called twisted QP manifolds, in terms of graded symplectic geometry, canonical functions, and QP pairs. We generalize the AKSZ construction of topological sigma models to sigma models with Wess-Zumino terms and show that all the twisted Poisson-like structures known in the literature can actually be naturally realized as boundary conditions for AKSZ sigma models. C 2014 AIP Publishing LLC. [http://dx.idea is inspired by analysis of the consistency of boundary conditions of AKSZ sigma models, 2,8,25,41 which are topological sigma models constructed by supergeometric methods and a topological open membrane. 22,38 A similar structure appears in the Batalin-Vilkovisky (BV) formalism of string field theory. 21 Another motivation comes from the canonical transformation in symplectic supergeometry, which is also called twisting. 40 It can be viewed as a higher analogue of a Poisson function, 33,45 and it defines a generalization of the Dirac structure 12 . Moreover, a canonical function describes the boundary condition structures of the AKSZ sigma models, which have played key roles in the derivation of the deformation quantization from the Poisson sigma model. 30 and 7 A canonical function leads to the concept of a QP pair, which is a certain tower of two (twisted) differential graded symplectic manifolds. This unifies the various concepts that were separately analyzed above, and it includes many geometric structures such as the Lie (2-)algebras, the (twisted or quasi) Poisson structures, the (homotopy) Lie algebroids, the (twisted) Courant algebroids, the Nambu-Poisson structures, and others. Some of these will be used below as examples.Analysis of a canonical transformation on a QP manifold naturally leads to what we call a twisted QP manifold. It is a mathematical framework for a unified understanding of the so-called Wess-Zumino terms, together with twisted Poisson-like structures. A general method to get a twisted QP manifold is given by the deformation theory. Some examples will be presented to illuminate this twisting process. Moreover a new geometric structure, the strong Courant algebroid, is proposed.The defining structure of a QP pair guarantees consistency of the bulk structure and the boundary conditions. In general, a quantum theory on a manifold X in n + 1 dimensions with given boundaries may have the same structure as the corresponding quantum theory on the boundary ∂X in n dimensions. 3 When applying this so-called bulk-boundary correspondence of quantum field theories to the AKSZ sigma models, we find that it is necessary to extend the AKSZ sigma models to the "twisted" AKSZ sigma models. This is a general...

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H-twisted Lie algebroids

Cited by 24 publications

References 10 publications

Fluxes in exceptional field theory and threebrane sigma-models

Fluxes in exceptional field theory and threebrane sigma-models

Quantization of Magnetic Poisson Structures

Canonical functions, differential graded symplectic pairs in supergeometry, and Alexandrov-Kontsevich-Schwartz-Zaboronsky sigma models with boundaries

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