Membrane sigma-models and quantization of non-geometric flux backgrounds

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. *

Sigma models for genuinely non-geometric backgrounds

“…Our approach is different than the above ones and complements previous work in the string theory literature [3][4][5][6][7][8]. The starting point is Courant algebroids (CAs), which are structures introduced in Ref.…”

“…These fluxes come in several types, in particular standard ones such as NSNS flux, torsion or geometric flux and RR fluxes, but also non-standard types such as non-geometric fluxes in the NSNS [1] and RR [2] sectors. The latter can be described with techniques from the differential geometry of Lie and Courant algebroids [3][4][5][6][7][8] which are also used in Hitchin's generalized complex geometry [9]. They often appear in (generalized) T-or S-duals of standard geometries [10][11][12][13].…”

UV-IR mixing in nonassociative Snyder ϕ4 theory

“…First, from the underlying mathematics, like L ∞ algebras (see [45] and references within [46]), new structures arise, for example the star-product algebra of functions, which were studied through nongeometric strings, probing noncommutative and nonassociative deformations of closed string background geometries [47][48][49], see also the celebrated paper by Kontsevich [50]. Second, the quantization of these backgrounds through explicit constructions of phase space star products were provided in [51,52], and subsequently applied to construct nonassociative theories [53]. In this article, we show the active role originating from the nonassociativity of the star product.…”

Perspectives on Nonassociative Geometry

“…In the standard T-duality orbit H → f → Q → R relating geometric and non-geometric fluxes, Q-flux backgrounds experience a noncommutative but strictly associative deformation while the purely non-geometric R-flux backgrounds witness a noncommutative and nonassociative geometry. Nonassociativity in this setting can be encoded by certain triproducts of fields on configuration space predicted by off-shell amplitudes in conformal field theory [12] and in double field theory [13], or by nonassociative -products from deformation quantization of twisted Poisson structures in the phase space formulation of nonassociative R-space [24,4,25]; the equivalence between these two approaches was demonstrated and extended in [3]. A general treatment of nonassociative -products in this context can be found in [20] (see also the contribution of V. Kupriyanov to these proceedings).…”