Nonassociative geometry and twist deformations in non-geometric string theory

Mylonas, Dionysios; Schupp, Peter J.; Szabo, Richard J.

doi:10.48550/arxiv.1402.7306

Cited by 21 publications

(27 citation statements)

References 54 publications

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“…in harmony with the expectations that on-shell conformal field theory correlation functions should see no traces of nonassociativity [2,10]. Analogous graded cyclicity conditions also hold for differential forms on M of arbitrary degree [9].…”

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confidence: 62%

Nonassociative field theory on non‐geometric spaces

Mylonas¹,

Szabo²

2014

Fortschritte der Physik

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We describe quasi-Hopf twist deformations of flat closed string compactifications with non-geometric Rflux using a suitable cochain twist, and construct nonassociative deformations of fields and differential calculus. We report on our new findings in using this formalism to construct perturbative nonassociative field theories on these backgrounds. We describe the modifications to the usual classification of Feynman diagrams into planar and non-planar graphs. The example of ϕ 4 theory is studied in detail and the one-loop contributions to the two-point function are calculated. Theory and Gravity, Corfu, Greece, September 8-15, 2013. Non-geometric backgrounds arise as consistent string vacua in p-form flux compactifications via Tduality transformations. Consider the standard example of closed strings propagating in a three-torus endowed with non-vanishing constant 3-form flux H = dB. Employing T-duality along all three directions takes H to its T-dual 3-vector flux R and results in a purely non-geometric background where transition functions between patches cannot even be defined locally [6]. This "R-space" exhibits an intriguing nonassociative deformation of geometry which is consistent with the original nonassociative deformations of spacetime discovered in [3], where standard conformal field theory approaches were used to study closed strings propagating in a constant H-flux background; we refer to the lecture notes [2] of these proceedings for further details of these approaches and for a more exhaustive list of references. Based on talk given by D.M. at the Workshop on Noncommutative FieldA geometrization of R-space is provided by the phase space of its T-dual background. This geometry is induced by regarding the fundamental degrees of freedom in the non-geometric background as membranes in a Courant σ-model whose boundary dynamics are described by a closed string quasi-Poisson σ-model with target space the cotangent bundle of the original membrane spacetime, which can be quantized using Kontsevich's deformation quantization [8]. This yields a nonassociative star product of fields on R-space that reproduces the nonassociative geometry discovered in [3,6], and leads to Seiberg-Witten maps which untwist the nonassociative product to a family of associative noncommutative star products; we refer to the lectures of P. Schupp from these proceedings for further details of this approach. It is also equivalent to a strict deformation quantization approach that is based on integrating a pertinent Lie 2-algebra to a Lie 2-group, which leads to the formulation of the nonassociative star products discussed by D. Lüst in these proceedings.These deformations can also be acquired by twisting the Hopf algebra of symmetries of R-space to a quasi-Hopf algebra using a suitable cochain twist [9]; see also [10] for a review and for further references. The advantage of this technique is that it is algorithmic in the sense that once a twist has been found, it can be used to deform all geometric structures on R-space. In this co...

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confidence: 62%

Nonassociative field theory on non‐geometric spaces

Mylonas¹,

Szabo²

2014

Fortschritte der Physik

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“…In this series of papers we will generalize the above constructions, and also extend them into a more general framework of nonassociative geometry. Let us give some background motivation for this extension from string theory, in order to clarify the physical origins of the problems we study in this largely purely mathematical paper; see [Lus11,MSS13a,Blu14] for brief reviews of the aspects of non-geometric string theory discussed below.…”

Section: Nonassociative Geometry In Non-geometric String Theorymentioning

confidence: 99%

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

Barnes

Schenkel

Szabo

2015

Journal of Geometry and Physics

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MSC:16T05 17B37 46L87 53D55Keywords: Noncommutative/nonassociative differential geometry Quasi-Hopf algebras Braided monoidal categories Internal homomorphisms Cochain twist quantization a b s t r a c t We systematically study noncommutative and nonassociative algebras A and their bimodules as algebras and bimodules internal to the representation category of a quasitriangular quasi-Hopf algebra. We enlarge the morphisms of the monoidal category of A-bimodules by internal homomorphisms, and describe explicitly their evaluation and composition morphisms. For braided commutative algebras A the full subcategory of symmetric A-bimodule objects is a braided closed monoidal category, from which we obtain an internal tensor product operation on internal homomorphisms. We describe how these structures deform under cochain twisting of the quasi-Hopf algebra, and apply the formalism to the example of deformation quantization of equivariant vector bundles over a smooth manifold. Our constructions set up the basic ingredients for the systematic development of differential geometry internal to the quasi-Hopf representation category, which will be tackled in the sequels to this paper, together with applications to models of noncommutative and nonassociative gravity such as those anticipated from non-geometric string theory.

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“…This paper is the second part in a series of articles whose goal is to systematically develop a formalism for differential geometry on noncommutative and nonassociative spaces. The main physical inspiration behind this work is sparked by the recent observations from closed string theory that certain non-geometric flux compactifications experience a nonassociative deformation of the spacetime geometry [BHM06, BP11, Lus10, BDLPR11, MSS12, BFHLS14] (see [Lus11,MSS13,Blu14] for reviews and further references), together with the constructions of [MSS14,ASz15] which show that the corresponding nonassociative algebras and their basic geometric structures can be obtained by cochain twist quantization, and hence are commutative and associative quantities when regarded as objects in a suitable braided monoidal category. See the first paper in this series [BSS15], hereafter referred to as Part I, for further motivation and a more complete list of relevant references.…”

Section: Introductionmentioning

confidence: 99%

Nonassociative geometry in quasi-Hopf representation categories II: Connections and curvature

Barnes

Schenkel

Szabo

2016

Journal of Geometry and Physics

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We continue our systematic development of noncommutative and nonassociative differential geometry internal to the representation category of a quasitriangular quasi-Hopf algebra. We describe derivations, differential operators, differential calculi and connections using universal categorical constructions to capture algebraic properties such as Leibniz rules. Our main result is the construction of morphisms which provide prescriptions for lifting connections to tensor products and to internal homomorphisms. We describe the curvatures of connections within our formalism, and also the formulation of Einstein-Cartan geometry as a putative framework for a nonassociative theory of gravity.

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Nonassociative geometry and twist deformations in non-geometric string theory

Cited by 21 publications

References 54 publications

Nonassociative field theory on non‐geometric spaces

Nonassociative field theory on non‐geometric spaces

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

Nonassociative geometry in quasi-Hopf representation categories II: Connections and curvature

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