Non-associative deformations of geometry in double field theory

Abstract

Non-geometric string backgrounds were proposed to be related to a nonassociative deformation of the space-time geometry. In the flux formulation of double field theory (DFT), the structure of mathematically possible non-associative deformations is analyzed in detail. It is argued that on-shell there should not be any violation of associativity in the effective DFT action. For imposing either the strong or the weaker closure constraint we discuss two possible non-associative deformations of DFT featuring two different ways how on-shell associativity can still be kept.

Nonassociative geometry in quasi-Hopf representation categories I: Bimodules and their internal homomorphisms

“…The physical origins underlying this nonassociative deformation have been elucidated in various ways: by regarding closed strings as boundary excitations of more fundamental membrane degrees of freedom in the non-geometric frame [20], in terms of matrix theory compactifications [21], and in double field theory [22]; they may be connected to the Abelian gerbes underlying the generalized manifolds in double geometry [23,24]. Explicit star product realizations of the nonassociative geometry were obtained via Kontsevich's deformation quantization of twisted Poisson manifolds [20] and by integrating higher Lie algebra structures [20,25].…”

Perspectives on Nonassociative Geometry

“…We focus on the special case of most physical relevance: the cochain twist quantization of a classical manifold; this construction is reviewed in Section 2. The formalism is powerful enough to capture the cases of constant non-geometric fluxes as well as non-constant ones such as those which arise in the flux formulation of double field theory [13]; in fact, our constructions in the remainder of this paper are completely general and can be applied to a much broader framework without specific reference to string theory. We further restrict to trivial vector bundles over these noncommutative and nonassociative spaces with diagonal action of the pertinent Hopf algebra of symmetries of the non-geometric background.…”

“…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).…”

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.…”