Nonassociative geometry and twist deformations in non-geometric string theory

Abstract

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.

28 pages; Based on lectures given at ICMP 2013, 21-26 October, 2013, Londrina, Brazil; to appear in Proceedings of Science

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• Quantized Nambu-Poisson Manifolds and n-Lie Algebras
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• Topological Open Membranes
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Cited by in corpus (11)

• Quantum Gravity, Dynamical Phase Space and String Theory
• Double Field Theory and Membrane Sigma-Models
• On the Non-commutativity of Closed String Zero Modes
• Non-associativity in non-geometric string and M-theory backgrounds, the algebra of octonions, and missing momentum modes
• $G_2$-structures and quantization of non-geometric M-theory backgrounds
• Symplectic realisation of electric charge in fields of monopole distributions
• Gauge Theory on Twisted $\kappa$-Minkowski: Old Problems and Possible Solutions
• T-duality, Quotients and Currents for Non-Geometric Closed Strings
• Towards a world-sheet description of doubled geometry in string theory
• Working with Nonassociative Geometry and Field Theory
• The ground state of non-associative hydrogen and upper bounds on the magnetic charge of elementary particles