Non-geometric fluxes, quasi-Hopf twist deformations, and nonassociative quantum mechanics

Abstract

We analyse the symmetries underlying nonassociative deformations of geometry in nongeometric 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. *

Perspectives on Nonassociative Geometry

“…This property holds for Abelian twists, as in Example 2.1, and also for the nonassociative deformation of Example 2.2, see [25].…”

“…Physically consistent models with novel properties in the context of quantum mechanics were constructed in [25] using this formalism, and of Euclidean scalar quantum field theory in [23]. To extend these considerations to more complicated field theories, a general systematic formalism was developed in [6,7] for differential geometry on noncommutative and nonassociative spaces internal to the representation category of any quasi-Hopf algebra, generalizing and extending earlier work [15,8,2].…”

“…is a completely antisymmetric real-valued tensor of rank 3 which arises from a constant non-geometric R-flux of closed string theory [25]. In this example we have…”

“…The cochain twist deformation quantization techniques originally developed by [25] were motivated by the search for a systematic way to generalize notions of differential geometry to such non-geometric backgrounds, and in particular to construct nonassociative deformations of field theory and ultimately gravity (see also [3]); this approach is different in spirit to the nonassociative twist deformation of the geometric f-flux frame considered in [18], which does not seem to be of relevance for non-geometric string theory, nor does it agree with the string theory inspired nonassociative torus bundles of [14,19] which reproduce the classical limit only up to Morita equivalence. Physically consistent models with novel properties in the context of quantum mechanics were constructed in [25] using this formalism, and of Euclidean scalar quantum field theory in [23].…”

“…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).…”