| dc.description.abstract | In this thesis we describe the nonassociative geometry probed by closed strings in
flat non-geometric R-flux 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
fields in Q-space (i.e. the T-fold endowed with a non-geometric Q-
flux which is
T-dual to the R-flux), 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-flux. We then 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 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 first 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 find 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 differential calculus in R-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. In
this setting, we consider extensions to non-constant R-flux backgrounds as well as
more generic twisted Poisson structures emerging from non-parabolic 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
3-cyclicity is instrumental for proving consistency of the formalism. We calculate
the expectation values of area and volume operators, and find coarse-graining of
the string background due to the R-flux. For a second application, we construct
nonassociative deformations of fields, and study perturbative nonassociative scalar
field 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. | en_US |
