We develop quantization techniques for describing the nonassociative geometry probed by closed strings in flat non-geometric R-flux backgrounds M . Starting from a suitable Courant sigma-model on an open membrane with target space M , regarded as a topological sector of closed string dynamics in R-space, we derive a twisted Poisson sigma-model on the boundary of the membrane whose target space is the cotangent bundle T * M and whose quasi-Poisson structure coincides with those previously proposed. We argue that from the membrane perspective the path integral over multivalued closed string fields in Q-space is equivalent to integrating over open strings in R-space. The corresponding boundary correlation functions reproduce Kontsevich's deformation quantization formula for the twisted Poisson manifolds. For constant R-flux, we derive closed formulas for the corresponding nonassociative star product and its associator, and compare them with previous proposals for a 3-product of fields on R-space. We develop various versions of the Seiberg-Witten map which relate our nonassociative star products to associative ones and add fluctuations to the R-flux background. We show that the Kontsevich formula coincides with the star product obtained by quantizing the dual of a Lie 2-algebra via convolution in an integrating Lie 2-group associated to the T-dual doubled geometry, and hence clarify the relation to the twisted convolution products for topological nonassociative torus bundles. We further demonstrate how our approach leads to a consistent quantization of Nambu-Poisson 3-brackets. *

We analyse the symmetries underlying nonassociative deformations of geometry in nongeometric 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. *

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.

We describe quasi-Hopf twist deformations of flat closed string compactifications with non-geometric Rflux using a suitable cochain twist, and construct nonassociative deformations of fields and differential calculus. We report on our new findings in using this formalism to construct perturbative nonassociative field theories on these backgrounds. We describe the modifications to the usual classification of Feynman diagrams into planar and non-planar graphs. The example of ϕ 4 theory is studied in detail and the one-loop contributions to the two-point function are calculated. Theory and Gravity, Corfu, Greece, September 8-15, 2013. Non-geometric backgrounds arise as consistent string vacua in p-form flux compactifications via Tduality transformations. Consider the standard example of closed strings propagating in a three-torus endowed with non-vanishing constant 3-form flux H = dB. Employing T-duality along all three directions takes H to its T-dual 3-vector flux R and results in a purely non-geometric background where transition functions between patches cannot even be defined locally [6]. This "R-space" exhibits an intriguing nonassociative deformation of geometry which is consistent with the original nonassociative deformations of spacetime discovered in [3], where standard conformal field theory approaches were used to study closed strings propagating in a constant H-flux background; we refer to the lecture notes [2] of these proceedings for further details of these approaches and for a more exhaustive list of references. Based on talk given by D.M. at the Workshop on Noncommutative FieldA geometrization of R-space is provided by the phase space of its T-dual background. This geometry is induced by regarding the fundamental degrees of freedom in the non-geometric background as membranes in a Courant σ-model whose boundary dynamics are described by a closed string quasi-Poisson σ-model with target space the cotangent bundle of the original membrane spacetime, which can be quantized using Kontsevich's deformation quantization [8]. This yields a nonassociative star product of fields on R-space that reproduces the nonassociative geometry discovered in [3,6], and leads to Seiberg-Witten maps which untwist the nonassociative product to a family of associative noncommutative star products; we refer to the lectures of P. Schupp from these proceedings for further details of this approach. It is also equivalent to a strict deformation quantization approach that is based on integrating a pertinent Lie 2-algebra to a Lie 2-group, which leads to the formulation of the nonassociative star products discussed by D. Lüst in these proceedings.These deformations can also be acquired by twisting the Hopf algebra of symmetries of R-space to a quasi-Hopf algebra using a suitable cochain twist [9]; see also [10] for a review and for further references. The advantage of this technique is that it is algorithmic in the sense that once a twist has been found, it can be used to deform all geometric structures on R-space. In this co...

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.