Quantized Nambu–Poisson manifolds and <i>n</i>-Lie algebras

DeBellis, Joshua; Sämann, Christian; Szabo, Richard J.

doi:10.1063/1.3503773

Cited by 49 publications

(93 citation statements)

References 62 publications

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“…In a quantum theory, it is necessary to understand the relation (1.2) as an operator relation. However, the representation of the Lie 3-algebra relation as transformations on vector spaces or maybe some kind of generalization is still an open question (see [23,24] for some different approaches). Since the difficulties are mainly due to the insistence of the fundamental identity, it motivates us to look for a 3-bracket geometry of the form (1.2) but with a 3-bracket where the fundamental identity is not required 3 .…”

Section: Introductionmentioning

confidence: 99%

D1-strings in large RR 3-form flux, quantum Nambu geometry and M5-branes in theC-field

Chu

Sehmbi

2012

J. Phys. A: Math. Theor.

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We consider D1-branes in a RR flux background and show that there is a low energy -large flux double scaling limit where the D1-branes action is dominated by a Chern-Simons-Myers coupling term. As a classical solution to the matrix model, we find a novel quantized geometry characterized by a quantum Nambu 3-bracket. Infinite dimensional representations of the quantum Nambu geometry are constructed which demonstrate that the quantum Nambu geometry is intrinsically different from the ordinary Lie algebra type noncommutative geometry. Matrix models for the IIB string, IIA string and M-theory in the corresponding backgrounds are constructed. A classical solution of a quantum Nambu geometry in the IIA Matrix string theory gives rise to an expansion of the fundamental strings into a system of multiple D4-branes and the fluctuation is found to describe an action for a non-abelian 3-form field strength which is a natural non-abelian generalization of the PST action for a single D4-brane. In view of the recent proposals [1, 2] of the M5-branes theory in terms of the D4-branes, we suggest a natural way to include all the KK modes and propose an action for the the multiple M5-branes in a constant C-field. The worldvolume of the M5-branes in a C-field is found to be described by a quantum Nambu geometry with self-dual parameters. It is intriguing that our action is naturally formulated in terms of a 1-form gauge field living on a six dimensional quantum Nambu geometry.

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Section: Introductionmentioning

confidence: 99%

D1-strings in large RR 3-form flux, quantum Nambu geometry and M5-branes in theC-field

Chu

Sehmbi

2012

J. Phys. A: Math. Theor.

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“…Our result suggests that the correct language to express the quantum geometry of the M5-brane in the presence of a constant C-field is in terms of a 3-bracket, rather than a commutator. Quite recently representations of the relation [19] are studied.…”

Section: C-field Modification To Basu-harvey Equation and The Quantummentioning

confidence: 99%

Open multiple M2-branes I: quantum geometry of the M5-brane in a C-field

Chu

Smith

2010

Gen Relativ Gravit

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We show that the Nahm equation and the Basu-Harvey equation which provide the fuzzy funnel description can be understood as the boundary condition of the F1-strings or M2-branes probing the intersecting branes system. In particular the noncommutative geometry of a D3-brane in the presence of a constant B-field can be identified with a constant shift in the B-field modified Nahm equation. Since the C-field modified Basu-Harvey equation can be derived from the known behaviour of the system in the Bion description, in order for this modification to be consistent with the boundary condition of the M2-branes, we derive a new type of quantum geometry on the M5-brane worldvolume. Unlike the D-brane case, this is naturally expressed in terms of a relation between a 3-bracket of the M5-brane worldvolume coordinates and the C-field.

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“…[24] and references therein). Explicit formulas can be obtained from the fact that all Kontsevich diagrams factorize and their weights can be expressed in terms of three diagrams (up to permutations), two involving the bivector field Θ and one involving the trivector field Π [52].…”

Section: Pos(icmp 2013)007mentioning

confidence: 99%

“…For further details about the quantization of generic Nambu-Poisson structures, see e.g. [24] and references therein. We will set up a framework involving a suitable generalization of geometric quantization for these higher bracket structures; this involves the notion of multisymplectic manifolds.…”

Section: Geometry Of N-algebrasmentioning

confidence: 99%

Nonassociative geometry and twist deformations in non-geometric string theory

Mylonas¹,

Schupp²,

Szabo

2014

Proceedings of Third International Satellite Conference on Mathematical Methods in Physics — PoS(ICMP 2013)

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We describe nonassociative deformations of geometry probed by closed strings in non-geometric flux compactifications of string theory. We show that these non-geometric backgrounds can be geometrised through the dynamics of open membranes whose boundaries propagate in the phase space of the target space compactification, equiped with a twisted Poisson structure. The effective membrane target space is determined by the standard Courant algebroid over the target space twisted by an abelian gerbe in momentum space. Quantization of the membrane sigma-model leads to a proper quantization of the non-geometric background, which we relate to Kontsevich's formalism of global deformation quantization that constructs a noncommutative nonassociative star product on phase space. We construct Seiberg-Witten type maps between associative and nonassociative backgrounds, and show how they may realise a nonassociative deformation of gravity. We also explain how this approach is related to the quantization of certain Lie 2-algebras canonically associated to the twisted Courant algebroid, and cochain twist quantization using suitable quasi-Hopf algebras of symmetries in the phase space description of R-space which constructs a Drinfel'd twist with non-trivial 3-cocycle. We illustrate and apply our formalism to present a consistent phase space formulation of nonassociative quantum mechanics.

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Quantized Nambu–Poisson manifolds and n-Lie algebras

Cited by 49 publications

References 62 publications

D1-strings in large RR 3-form flux, quantum Nambu geometry and M5-branes in theC-field

D1-strings in large RR 3-form flux, quantum Nambu geometry and M5-branes in theC-field

Open multiple M2-branes I: quantum geometry of the M5-brane in a C-field

Nonassociative geometry and twist deformations in non-geometric string theory

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