
Mathematics 2012
Membrane SigmaModels and Quantization of NonGeometric Flux BackgroundsAbstract: We develop quantization techniques for describing the nonassociative geometry probed by closed strings in flat nongeometric Rflux backgrounds M. Starting from a suitable Courant sigmamodel on an open membrane with target space M, regarded as a topological sector of closed string dynamics in Rspace, we derive a twisted Poisson sigmamodel on the boundary of the membrane whose target space is the cotangent bundle T^*M and whose quasiPoisson structure coincides with those previously proposed. We argue that from the membrane perspective the path integral over multivalued closed string fields in Qspace is equivalent to integrating over open strings in Rspace. The corresponding boundary correlation functions reproduce Kontsevich's deformation quantization formula for the twisted Poisson manifolds. For constant Rflux, we derive closed formulas for the corresponding nonassociative star product and its associator, and compare them with previous proposals for a 3product of fields on Rspace. We develop various versions of the SeibergWitten map which relate our nonassociative star products to associative ones and add fluctuations to the Rflux background. We show that the Kontsevich formula coincides with the star product obtained by quantizing the dual of a Lie 2algebra via convolution in an integrating Lie 2group associated to the Tdual 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 NambuPoisson 3brackets.
