Nonassociative deformations of non-geometric flux backgrounds and field theory

In this thesis we describe the nonassociative geometry probed by closed strings in at non-geometric R- ux backgrounds, and develop suitable quantization techniques. For this, we propose a Courant sigma-model 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 sigma-model 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 R-space. From the membrane perspective, the path integral over multivalued closed string elds in Q-space (i.e. the T-fold endowed with a non-geometric Q- ux which is T-dual to the R- ux), is equivalent to integrating over open strings in R-space. 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 R- ux. We then develop various versions of the Seiberg{Witten map, which relate our nonassociative star products to associative ones and add uctuations to the R- ux background. We also propose a second quantization method based on quantizing the dual of a Lie 2-algebra via convolution in an integrating Lie 2-group. This formalism provides a categori cation of Weyl's quantization map, and leads to a consistent quantization of Nambu{Poisson 3-brackets. 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 rst step towards formulating quantum gravity on non-geometric spaces, we develop a third quantization method to study nonassociative deformations of geometry in R-space, which is analogous to noncommutative deformations of geometry (i.e. noncommutative gravity). We nd that the symmetries underlying these nonassociative deformations generate the non-abelian Lie algebra of translations and Bopp shifts in phase space. Using a suitable cochain twist, we construct the quasi-Hopf algebra of symmetries that deforms the algebra of functions, and the exterior di erential calculus in R-space. We de ne a suitable integration on these nonassociative spaces, and nd that the usual cyclicity of associative noncommutative deformations is replaced by weaker notions of 2-cyclicity and 3-cyclicity. In this setting, we consider extensions to non-constant R- ux backgrounds as well as more generic twisted Poisson structures emerging from non-parabolic monodromies of closed strings. As a rst application of our nonassociative star product quantization, we develop nonassociative quantum mechanics based on phase space state functions, wherein 3-cyclicity is instrumental for proving consistency of the formalism. We calculate the expectation values of area and volume operators, and nd coarse-graining of the string background due to the R- ux. For a second application, we construct nonassociative deformations of elds, and study perturbative nonassociative scalar eld theories on R-space. We nd that nonassociativity induces modi cations to the usual classi cation of Feynman diagrams into planar and non-planar graphs, which are controlled by 3-cyclicity. The example of '4 theory is studied in detail and the one-loop contributions to the two-point function are calculated

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

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.Comment: 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 Physic

Membrane Sigma-Models and Quantization of Non-Geometric Flux Backgrounds

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.Comment: 52 pages; v2: references adde