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

TL;DR

This paper develops a quantization framework for nonassociative geometries arising from non-geometric flux backgrounds in string theory, connecting membrane sigma-models, deformation quantization, and nonassociative star products.

Contribution

It introduces a membrane-based approach to quantize non-geometric flux backgrounds, deriving explicit nonassociative star products and relating them to Lie 2-algebras and T-duality.

Findings

Derived explicit formulas for nonassociative star products in R-flux backgrounds.

Connected the quantization procedure to Kontsevich's deformation quantization.

Established a link between nonassociative structures and Lie 2-algebras.

Abstract

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.