$T$-duality for torus bundles with $H$-fluxes via noncommutative topology. {II}. The high-dimensional case and the $T$-duality group

Mathai, Varghese; Rosenberg, Jonathan

doi:10.4310/atmp.2006.v10.n1.a5

Cited by 65 publications

(112 citation statements)

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“…The T-dual complex structure was determined in [22] to be 13) so that the Poisson bivector is read off to be δP = BI + I t B, which is not necessarily vanishing, as expected. Moreover, if we insist that the β-transformed D0-brane be a generalized D0-brane with respect to I ′ after monodromy, we obtain the constraint β = β (0,2) , in terms of the decomposition with respect to I.…”

Section: Torus With H-flux and Mirror Symmetrymentioning

confidence: 84%

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T-duality with H-flux: Non-commutativity, T-folds and structure

Grange

Schäfer‐Nameki

2007

Nuclear Physics B

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Various approaches to T-duality with NSNS three-form flux are reconciled. Non-commutative torus fibrations are shown to be the open-string version of T-folds. The non-geometric T-dual of a three-torus with uniform flux is embedded into a generalized complex six-torus, and the non-geometry is probed by D0-branes regarded as generalized complex submanifolds. The noncommutativity scale, which is present in these compactifications, is given by a holomorphic Poisson bivector that also encodes the variation of the dimension of the world-volume of D-branes under monodromy. This bivector is shown to exist in SU(3) × SU(3) structure compactifications, which have been proposed as mirrors to NSNS-flux backgrounds. The two SU(3)-invariant spinors are generically not parallel, thereby giving rise to a non-trivial Poisson bivector. Furthermore we show that for non-geometric T-duals, the Poisson bivector may not be decomposable into the tensor product of vectors. IntroductionCompactifications with H-flux are known to give rise to topology changes and even to nongeometric situations when T-duality is performed along directions which have non-trivial support of the NSNS H-flux [1,2,3,4,5,6,7,8]. Non-geometry occurs for example in the very simple situation of a three-torus endowed with an H-flux proportional to its volume form. Consider namely the three-torus as a trivial T 2 -fibration over a circle. Upon T-duality along the fibre, the metric picks up a factor that makes it shrink under monodromy around the base circle. The monodromy around the base is a non-trivial element of the O(2, 2; Z) group acting on the two-torus. This prevents a three-dimensional global Riemannian description from existing. Further T-dualizing along the base leads to more pathological situations, where points do not exist even in a local coordinate patch, and the fibres are conjectured to become non-associative [9,10]. We will restrict ourselves to the case of two T-dualities, and assume that local coordinate patches do exist. Progress in the description of non-associative T-duals was achieved in the recent paper [11], which also contains observations on the open-string metric and non-commutativity for two T-dualities that have some overlap with ours.Essentially three conjectures have been put forward for the description of the T-dual of a torus with H-flux: 1 (I) Field of non-commutative tori: Mathai and Rosenberg proposed that T-dualizing along a two-torus with non-zero H-flux yields a fibration by (or more precisely: field of) noncommutative tori. In particular, this fibration is encoded in a closed one-form, which is obtained by integrating the NSNS flux along the fibre directions [7,12,13].(II) T-folds: these are spaces where T-dualities can act as transition functions between local patches [8]. The T-dualized directions are doubled, and T-duality transformations may patch the doubled fibres together. A sigma model with a T-fold as its target space was proposed, and its boundary conditions were studied in [14,15,16,17,18].(III) G × G struc...

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Section: Torus With H-flux and Mirror Symmetrymentioning

confidence: 84%

“…In particular, this fibration is encoded in a closed one-form, which is obtained by integrating the NSNS flux along the fibre directions [7,12,13].…”

Section: Introductionmentioning

confidence: 99%

T-duality with H-flux: Non-commutativity, T-folds and structure

Grange

Schäfer‐Nameki

2007

Nuclear Physics B

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“…Presumably there should be some analog of twisted K-theory for the general flat fiber theories that we are studying, including also the non-geometric fluxes. Matching this onto the work of Mathai and collaborators [46,47,48,49,50,35,51,52] would be very interesting. Similarly, exploiting the connections between the base-fiber approach described here and spaces with generalized complex structure (see e.g.…”

Section: Advantages and Puzzlesmentioning

confidence: 99%

Toroidal orientifolds in IIA with general NS-NS fluxes

Ihl

Robbins

Wrase

2007

J. High Energy Phys.

184

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Type IIA toroidal orientifolds offer a promising toolkit for model builders, especially when one includes not only the usual fluxes from NS-NS and R-R field strengths, but also fluxes that are T-dual to the NS-NS three-form flux. These new ingredients are known as metric fluxes and non-geometric fluxes, and can help stabilize moduli or can lead to other new features. In this paper we study two approaches to these constructions, by effective field theory or by toroidal fibers twisted over a toroidal base. Each approach leads us to important observations, in particular the presence of D-terms in the four-dimensional effective potential in some cases, and a more subtle treatment of the quantization of the general NS-NS fluxes. Though our methods are general, we illustrate each approach on the example of an orientifold of T 6 /Z 4 .

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“…His theory has been remarkably successful, giving rise to many examples of noncommutative manifolds, which have become extremely useful both in mathematics and mathematical physics. In a recent paper [3] Echterhoff, Nest, and Oyono-Oyono defined noncommutative principal torus bundles, inspired by fundamental results in [17], as well as the T-duals of certain continuous trace algebras [13,14]. They also classified all noncommutative principal torus bundles in terms of (noncommutative) fibre products of principal torus bundles and group C * -algebras of lattices in simply-connected 2-step nilpotent Lie groups, cf.…”

Section: Introductionmentioning

confidence: 99%