Title:

Nonassociative deformations of nongeometric flux backgrounds and field theory

In this thesis we describe the nonassociative geometry probed by closed strings in flat nongeometric Rflux backgrounds, and develop suitable quantization techniques. For this, we propose a Courant sigmamodel on an open membrane with target space M, which we regard as a topological sector of closed string dynamics on Rspace. We then reduce it to a twisted Poisson sigmamodel on the boundary of the membrane with target space the cotangent bundle T M. The pertinent twisted Poisson structure is provided by a U(1) gerbe in momentum space, which geometrizes Rspace. From the membrane perspective, the path integral over multivalued closed string fields in Qspace (i.e. the Tfold endowed with a nongeometric Q flux which is Tdual to the Rflux), is equivalent to integrating over open strings in Rspace. The corresponding boundary correlation functions reproduce Kontsevich's global deformation quantization formula for the twisted Poisson manifolds, which we take as our proposal for quantization. We calculate the corresponding nonassociative star product and its associator, and derive closed formulas for the case of a constant Rflux. We then develop various versions of the Seiberg{Witten map, which relate our nonassociative star products to associative ones and add fluctuations to the Rflux background. We also propose a second quantization method based on quantizing the dual of a Lie 2algebra via convolution in an integrating Lie 2group. This formalism provides a categori cation of Weyl's quantization map, and leads to a consistent quantization of Nambu{Poisson 3brackets. We show that the convolution product coincides with the star product obtained by Kontsevich's formula, and clarify its relation with the twisted convolution products for topological nonassociative torus bundles. As a first step towards formulating quantum gravity on nongeometric spaces, we develop a third quantization method to study nonassociative deformations of geometry in Rspace, which is analogous to noncommutative deformations of geometry (i.e. noncommutative gravity). We find that the symmetries underlying these nonassociative deformations generate the nonabelian Lie algebra of translations and Bopp shifts in phase space. Using a suitable cochain twist, we construct the quasiHopf algebra of symmetries that deforms the algebra of functions, and the exterior differential calculus in Rspace. 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 2cyclicity and 3cyclicity. In this setting, we consider extensions to nonconstant Rflux backgrounds as well as more generic twisted Poisson structures emerging from nonparabolic monodromies of closed strings. As a first application of our nonassociative star product quantization, we develop nonassociative quantum mechanics based on phase space state functions, wherein 3cyclicity is instrumental for proving consistency of the formalism. We calculate the expectation values of area and volume operators, and find coarsegraining of the string background due to the Rflux. For a second application, we construct nonassociative deformations of fields, and study perturbative nonassociative scalar field theories on Rspace. We nd that nonassociativity induces modi cations to the usual classi cation of Feynman diagrams into planar and nonplanar graphs, which are controlled by 3cyclicity. The example of '4 theory is studied in detail and the oneloop contributions to the twopoint function are calculated.
