3-cocycles, non-associative star-products and the magnetic paradigm of R-flux string vacua

Abstract

Abstract:We consider the geometric and non-geometric faces of closed string vacua arising by T-duality from principal torus bundles with constant H-flux and pay attention to their double phase space description encompassing all toroidal coordinates, momenta and their dual on equal footing. We construct a star-product algebra on functions in phase space that is manifestly duality invariant and substitutes for canonical quantization. The 3-cocycles of the Abelian group of translations in double phase space are seen to account for non-associativity of the star-product. We also provide alternative cohomological descriptions of non-associativity and draw analogies with the quantization of point-particles in the field of a Dirac monopole or other distributions of magnetic charge. The magnetic field analogue of the R-flux string model is provided by a constant uniform distribution of magnetic charge in space and non-associativity manifests as breaking of angular symmetry. The Poincaré vector comes to rescue angular symmetry as well as associativity and also allow for quantization in terms of operators and Hilbert space only in the case of charged particles moving in the field of a single magnetic monopole.

Nonassociative geometry and twist deformations in non-geometric string theory

“…On the other hand, a consistent nonassociative version of quantum mechanics based on the phase space quantization of the R-flux string model can be developed following [53] and shown rigorously to lead to minimal volume uncertainty relations ∆x i ∆x j ∆x k ≥ 3 2h 2 R i jk characterizing a coarse-graining of the nongeometric R-flux background; this is consistent with the fact that these backgrounds do not allow the introduction of point-like objects [26]. The parallels between nonassociative parabolic R-flux string vacua and the dynamics of charged particles in uniform magnetic charge distributions is elucidated in [9].…”

“…with G h the Heisenberg group integrating h via the exponential map, whose associator integrates j and defines a 3-cocycle in the Chevalley-Eilenberg cohomology of G h with values in R. The roles of 3-cocycles of Lie algebra and Lie group cohomology in the quantization of nonassociative R-space is elucidated in [9]. By computing convolution type products in this Lie 2-group, one can mimick the standard approach based on Weyl quantization in the associative setting of operator algebras (see e.g.…”

“…By computing convolution type products in this Lie 2-group, one can mimick the standard approach based on Weyl quantization in the associative setting of operator algebras (see e.g. [64] for a review) to rederive the nonassociative star products (4.2) for the phase space description of R-space; this algebraic approach is taken in [52,9].…”

Higher form gauge fields and their nonassociative symmetry algebras

“…In fact, our interest in studying the nonassociativity of FDA dual algebras has been prompted by a recent paper [10], where flux backgrounds in closed string theory are described by nonassociative structures in double phase space, controlled again by Chevalley-Eilenberg cohomology (for a very partial list of references see for example [11][12][13][14]). Since flux backgrounds involve p-forms, it seems that algebraic structures describing p-forms tend to exhibit nonassociativity, depending on nontrivial cohomology classes, both for flux backgrounds and in FDA's dual algebras.…”

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).…”