Topological T-duality and T-folds

Bouwknegt, Peter; Pande, Ashwin S.

doi:10.4310/atmp.2009.v13.n5.a6

Cited by 12 publications

(20 citation statements)

References 14 publications

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“…This suggests that T-duality and doubled geometry is the natural framework to investigate closed string noncommutative geometry. This point was noticed already some time ago for tori with constant B-fields in [60,61], while noncommutative correspondence spaces associated to T-folds are described in [16,13] in the context of open strings. In [63] this was demonstrated by applying T-duality along one direction of a twisted three-torus with non-vanishing geometric flux, which reveals that there is a non-trivial commutation relation between the coordinates in the doubled geometry that is determined by the winding number in the base direction.…”

Section: Introduction and Summary Background And Contextmentioning

confidence: 81%

“…However, T-dualising along two cycles gives rise to a space with non-geometric Q-flux whereby one of the cycles is only periodic up to T-duality, which mixes momentum and winding modes; the resulting geometry is thus only well-defined locally and is called a T-fold. An approach to describing these backgrounds mathematically in the context of open string theory was put forward in [66,28,32,13] using the language of noncommutative geometry:…”

Section: Introduction and Summary Background And Contextmentioning

confidence: 99%

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Membrane sigma-models and quantization of non-geometric flux backgrounds

Mylonas

Schupp

Szabo

2012

J. High Energ. Phys.

125

282

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We develop quantization techniques for describing the nonassociative geometry probed by closed strings in flat non-geometric R-flux backgrounds M . Starting from a suitable Courant sigma-model on an open membrane with target space M , regarded as a topological sector of closed string dynamics in R-space, we derive a twisted Poisson sigma-model on the boundary of the membrane whose target space is the cotangent bundle T * M and whose quasi-Poisson structure coincides with those previously proposed. We argue that from the membrane perspective the path integral over multivalued closed string fields in Q-space is equivalent to integrating over open strings in R-space. The corresponding boundary correlation functions reproduce Kontsevich's deformation quantization formula for the twisted Poisson manifolds. For constant R-flux, we derive closed formulas for the corresponding nonassociative star product and its associator, and compare them with previous proposals for a 3-product of fields on R-space. We develop various versions of the Seiberg-Witten map which relate our nonassociative star products to associative ones and add fluctuations to the R-flux background. We show that the Kontsevich formula coincides with the star product obtained by quantizing the dual of a Lie 2-algebra via convolution in an integrating Lie 2-group associated to the T-dual doubled geometry, and hence clarify the relation to the twisted convolution products for topological nonassociative torus bundles. We further demonstrate how our approach leads to a consistent quantization of Nambu-Poisson 3-brackets. *

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Section: Introduction and Summary Background And Contextmentioning

confidence: 81%

Section: Introduction and Summary Background And Contextmentioning

confidence: 99%

Membrane sigma-models and quantization of non-geometric flux backgrounds

Mylonas

Schupp

Szabo

2012

J. High Energ. Phys.

125

282

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“…In contrast to these analyses, here we work in a controlled setting with (doubled) twisted tori and quantised fluxes, without any linear approximations and with an exact effective field theory description of the string geometry. Noncommutative and nonassociative geometries were suggested as global (algebraic) descriptions of T-fold and R-flux non-geometries respectively in the mathematical framework of topological T-duality in [46][47][48][49], which strictly speaking only applies to the worldvolumes of D-branes, but it was further suggested that such a description should also apply to the closed string background itself. Such a suggestion requires further clarification, insofar that in closed string theory itself there is no immediate evidence for such nonassociative structures.…”

Section: Introductionmentioning

confidence: 99%

Noncommutative gauge theories on D-branes in non-geometric backgrounds

Hull

Szabo

2019

J. High Energ. Phys.

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We investigate the noncommutative gauge theories arising on the worldvolumes of D-branes in non-geometric backgrounds obtained by T-duality from twisted tori. We revisit the low-energy effective description of D-branes on three-dimensional T-folds, examining both cases of parabolic and elliptic twists in detail. We give a detailed description of the decoupling limits and explore various physical consequences of the open string non-geometry. The T-duality monodromies of the non-geometric backgrounds lead to Morita duality monodromies of the noncommutative Yang-Mills theories induced on the D-branes. While the parabolic twists recover the wellknown examples of noncommutative principal torus bundles from topological T-duality, the elliptic twists give new examples of noncommutative fibrations with non-geometric torus fibres. We extend these considerations to D-branes in backgrounds with R-flux, using the doubled geometry formulation, finding that both the non-geometric background and the D-brane gauge theory necessarily have explicit dependence on the dual coordinates, and so have no conventional formulation in spacetime.

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“…Such a description is provided by generalized geometry [17,18,19,20,21]. In some situations, as encountered for example in T-fold backgrounds [22,23,24,25,26], it is even more elegant to use an anti-symmetric bi-vector together with a metric to describe T-dual backgrounds. An immediate question is about the geometric analogues e.g.…”

Section: Mathematical Backgroundmentioning

confidence: 99%

Lie algebroids, non‐associative structures and non‐geometric fluxes

Deser¹

2013

Fortschritte der Physik

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In the first part of this article, the geometry of Lie algebroids as well as the Moyal-Weyl star product and some of its generalizations in open string theory are reviewed. A brief introduction to T-duality and non-geometric fluxes is given. Based on these foundations, more recent results are discussed in the second part of the article. On the world-sheet level, we will analyse closed string theory with flat background and constant H-flux. After an odd number of T-dualities, correlation functions allow to extract a three-product having a pattern similar to the Moyal-Weyl product. We then focus on the target space and the local appearance of the various fluxes. An algebra based on vector fields is proposed, whose structure functions are given by the fluxes. Jacobi-identities for vector fields allow for the computation of Bianchi-identities. Based on the latter, we give a proof for a special Courant algebroid structure on the generalized tangent bundle, where the fluxes are realized by the commutation relations of a basis of sections. As reviewed in the first part of this work, in the description of non-geometric Q- and R-fluxes, the B-field gets replaced by a bi-vector \beta, which is supposed to serve as the dual object to B under T-duality. A natural question is about the existence of a differential geometric framework allowing the construction of actions manifestly invariant under coordinate- and gauge transformations, which couple the \beta-field to gravity. It turns out that Lie algebroids are the right language to answer this question positively. We conclude by giving an outlook on future directions.Comment: 105 pages, Ph.D. thesis of the autho

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Topological T-duality and T-folds

Cited by 12 publications

References 14 publications

Membrane sigma-models and quantization of non-geometric flux backgrounds

Membrane sigma-models and quantization of non-geometric flux backgrounds

Noncommutative gauge theories on D-branes in non-geometric backgrounds

Lie algebroids, non‐associative structures and non‐geometric fluxes

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