Non-Geometric Fluxes, Quasi-Hopf Twist Deformations and Nonassociative Quantum Mechanics

Abstract

We analyse the symmetries underlying nonassociative deformations of geometry in non-geometric R-flux compactifications which arise via T-duality from closed strings with constant geometric fluxes. Starting from the non-abelian Lie algebra of translations and Bopp shifts in phase space, together with a suitable cochain twist, we construct the quasi-Hopf algebra of symmetries that deforms the algebra of functions and the exterior differential calculus in the phase space description of nonassociative R-space. In this setting nonassociativity is characterised by the associator 3-cocycle which controls non-coassociativity of the quasi-Hopf algebra. We use abelian 2-cocycle twists to construct maps between the dynamical nonassociative star product and a family of associative star products parametrized by constant momentum surfaces in phase space. We define a suitable integration on these nonassociative spaces and find that the usual cyclicity of associative noncommutative deformations is replaced by weaker notions of 2-cyclicity and 3-cyclicity. Using this star product quantization on phase space together with 3-cyclicity, we formulate a consistent version of nonassociative quantum mechanics, in which we calculate the expectation values of area and volume operators, and find coarse-graining of the string background due to the R-flux.

38 pages; v2: typos corrected, reference added; v3: typos corrected, comments about cyclicity added in section 4.2, references updated; Final version to be published in Journal of Mathematical Physics

References in corpus (10)

• Four-dimensional String Compactifications with D-Branes, Orientifolds and Fluxes
• Double Field Theory
• Flux Compactification
• A Gravity Theory on Noncommutative Spaces
• T-duality and closed string non-commutative (doubled) geometry
• Twisted Gauge Theories
• Noncommutative D=4 gravity coupled to fermions
• Non-geometric strings, symplectic gravity and differential geometry of Lie algebroids
• Non-associative Deformations of Geometry in Double Field Theory
• Quantized Nambu-Poisson Manifolds and n-Lie Algebras

Cited by in corpus (57)

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• Consistent Compactification of Double Field Theory on Non-geometric Flux Backgrounds
• Symplectic embeddings, homotopy algebras and almost Poisson gauge symmetry
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• Nonassociative geometry in quasi-Hopf representation categories II: Connections and curvature
• Nonassociative differential geometry and gravity with non-geometric fluxes
• $G_2$-structures and quantization of non-geometric M-theory backgrounds
• Triproducts, nonassociative star products and geometry of R-flux string compactifications
• Towards a Theory of Nonassociative Gravity
• Symplectic realisation of electric charge in fields of monopole distributions
• Poisson gauge theory
• On T-duality transformations for the three-sphere
• Nonassociative Weyl star products
• Smooth 2-Group Extensions and Symmetries of Bundle Gerbes
• On current algebras, generalised fluxes and non-geometry
• Geometry and 2-Hilbert Space for Nonassociative Magnetic Translations
• Testing non-associative quantum mechanics
• States in non-associative quantum mechanics: Uncertainty relations and semiclassical evolution
• Photon self-interaction on deformed spacetime
• T-duality, Quotients and Currents for Non-Geometric Closed Strings
• Locally non-geometric fluxes and missing momenta in M-theory
• Nonassociative geometry and twist deformations in non-geometric string theory
• The quasi-nonassociative exceptional $F(4)$ deformed quantum oscillator
• $L_{\infty }$-Algebras of Einstein-Cartan-Palatini Gravity
• Star products on graded manifolds and ${\alpha }^{\prime }$-corrections to Courant algebroids from string theory
• Nonassociative geometric and quantum information flows and R-flux deformations of wormhole solutions in string gravity
• Non-geometric Kaluza-Klein monopoles and magnetic duals of M-theory R-flux backgrounds
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• Global Double Field Theory is Higher Kaluza-Klein Theory
• Magnetic monopoles and nonassociative deformations of quantum theory
• Towards a world-sheet description of doubled geometry in string theory
• Recurrence relations for symplectic realization of (quasi)-Poisson structures
• Weak associativity and deformation quantization
• Quantization of Magnetic Poisson Structures
• Small magnetic charges and monopoles in non-associative quantum mechanics
• Nonassociative Ricci flows, star product and R-flux deformed black holes, and swampland conjectures
• Nonassociative black ellipsoids distorted by R-fluxes and four dimensional thin locally anisotropic accretion disks
• Monopole star products are non-alternative
• Nearly associative deformation quantization
• Open-string T-duality and applications to non-geometric backgrounds
• Nonassociative geometry of nonholonomic phase spaces with star R-flux string deformations and (non) symmetric metrics
• Working with Nonassociative Geometry and Field Theory
• The Anholonomic Frame and Connection Deformation Method for constructing off-diagonal solutions in (modified) Einstein gravity and nonassociative geometric flows and Finsler-Lagrange-Hamilton theories
• Marginal deformations and quasi-Hopf algebras
• Integrable Hopf twists, marginal deformations and generalised geometry
• Homotopy Lie Algebras of Gravity and their Braided Deformations
• Matrix Quantization of Classical Nambu Brackets and Super $p$-Branes
• The ground state of non-associative hydrogen and upper bounds on the magnetic charge of elementary particles
• Open-String Non-Associativity in an R-flux Background
• Moments and saturation properties of eigenstates
• Lagrangian formulation for electric charge in a magnetic monopole distribution
• Alternative multiplications and non-associativity in physics
• Star Products that can not be induced by Drinfel'd Twists
• $L_{\infty }$-algebras and membrane sigma models