A course on noncommutative geometry in string theory

Blumenhagen, Ralph

doi:10.1002/prop.201400014

Cited by 15 publications

(18 citation statements)

References 44 publications

(95 reference statements)

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“…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 [11][12][13] 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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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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“…A general treatment of nonassociative -products in this context can be found in [20] (see also the contribution of V. Kupriyanov to these proceedings). Reviews of noncommutativity and nonassociativity in non-geometric closed string theory can be found in [22,27,26,10] (see also the contributions of P. Schupp and I. Bakas to these proceedings).…”

Section: Introductionmentioning

confidence: 99%

Perspectives on Nonassociative Geometry

Szabo¹,

Barnes²,

Schenkel³

2016

Proceedings of Proceedings of the Corfu Summer Institute 2015 — PoS(CORFU2015)

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We review aspects of our formalism for differential geometry on noncommutative and nonassociative spaces which arise from cochain twist deformation quantization of manifolds. We work in the simplest setting of trivial vector bundles and flush out the details of our approach providing explicit expressions for all bimodule operations, and for connections and curvature. As applications, we describe the constructions of physically viable action functionals for Yang-Mills theory and Einstein-Cartan gravity on noncommutative and nonassociative spaces, as first steps towards more elaborate models relevant to non-geometric flux deformations of geometry in closed string theory.

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“…E.g., the emergence of the noncommutative Moyal product in quantized string theory is discussed in a particularly clear exposition in Ref. [11], in the derivation leading to Eq. (32) We thus observe that space-time quantization is not necessary, a priori, in order to combine general relativity and quantum mechanics.…”

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From Dirac theories in curved space‐times to a variation of Dirac's large–number hypothesis

Jentschura

2014

Annalen der Physik

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Some physicists, even today, may think that it is impossible to connect relativistic quantum mechanics and general relativity [1][2][3]. This impression, however, is false, as demonstrated, among many other recent works, in an article published not long ago in this journal [4]. In order to understand the subtlety, let us remember the meaning of first [5,6], second [7,8], and third quantization (see Refs. [9][10][11]). In first quantization, what was previously a well-defined particle trajectory now becomes smeared and defines a probability density of the quantum mechanical particle. In second quantization, what was previously a well-defined functional value of a physical field at a given space-time point now becomes a field operator, which is subject to quantum fluctuations. E.g., quantum fluctuations about the flat space-time metric can be expressed in terms of the graviton field operator, for which an explicit expression can be found in Eq. (5) of Ref. [8].In third quantization, what was previously a well-defined space-time point now becomes an operator, defining a conceivably noncommutative entity, which describes space-time quantization. E.g., the emergence of the noncommutative Moyal product in quantized string theory is discussed in a particularly clear exposition in Ref. [11], in the derivation leading to Eq. (32) We thus observe that space-time quantization is not necessary, a priori, in order to combine general relativity and quantum mechanics. Indeed, before we could ever conceive to observe effects due to space-time quantization, we should first consider the leading-order coupling of the Dirac particle to a curved spacetime. Space-time curvature is visible on the classical level, and can be treated on the classical level [3]. Deviations from perfect Lorentz symmetry caused by effects other than space-time curvature may result in conceivable anisotropies of spacetime; they have been investigated by Kostelecky et al. in a series of papers (see Refs. [13][14][15] and references therein) and would be visible in tiny deviations of the dispersion relations of the relativistic particles from the predictions of Dirac theory. Measurements on neutrinos [15] can be used in order to constrain the Lorentz-violating parameters. Other recent studies concern the modification of gravitational effects under slight global violations of Lorentz symmetry [16]. Brill and Wheeler [17] were among the first to study the gravitational interactions of Dirac particles, and they put a special emphasis on neutrinos. The motivation can easily be guessed: Neutrinos, at the time, were thought to be massless and transform according to the fundamental ( ) representations of the Lorentz group. Yet, their interactions have a profound impact on cosmology [18]. Being (almost) massless, their gravitational interactions can, in principle, only be formulated on a fully relativistic footing. This situation illustrates a pertinent dilemma: namely, the gravitational potential, unlike the the Coulomb potential, cannot simply be inserted into th...

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A course on noncommutative geometry in string theory

Cited by 15 publications

References 44 publications

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

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

Perspectives on Nonassociative Geometry

From Dirac theories in curved space‐times to a variation of Dirac's large–number hypothesis

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