On the existence of bibundles

Murray, Michael K.; Roberts, David; Stevenson, Danny

doi:10.1112/plms/pds028

Cited by 4 publications

(15 citation statements)

References 13 publications

(68 reference statements)

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“…We show a sufficient condition for reduction is that M be a bibundle (a right bundle equipped with a right-equivariant structure map) [15][16][17] that further more obeys a topological factorisation condition. This factorisation condition requires that M admits a reduction of its structure group to Topological T-duality via QP-manifolds Our approach here will be to situate the discussion in the context of QP-manifolds.…”

Section: Generalisations Of T-dualitymentioning

confidence: 99%

“…A third equivalent definition can be given in terms of a bundle with left action ⊲ plus (left) structure map ˆ [Ñ], related to the previous by ˆ [Ñ] = ( [Ñ]) −1 where inverse denotes the inverse element in Aut(D). In our discussion we will restrict attention to Type 1 bibundles in the terminology of [17]. We give an economic characterisation thereof, alternative to the one of [17] 9 : Definition 3 ("Type 1 bibundle").…”

Section: A Class Of Bibundles With Drinfel'd Double Fibrementioning

confidence: 99%

“…In our discussion we will restrict attention to Type 1 bibundles in the terminology of [17]. We give an economic characterisation thereof, alternative to the one of [17] 9 : Definition 3 ("Type 1 bibundle"). A principal right D-bundle D ֒→ M ։ B is said to be a type 1 bibundle if it admits a reduction of its structure group onto the centre Z(D).…”

Section: A Class Of Bibundles With Drinfel'd Double Fibrementioning

confidence: 99%

“…provide the central elements in G (resp. G) which are also central in D. 9 To be absolutely clear, by Type 1 bibundles we mean the (Aut(D), D) bibundles of [17] whose type map M → Out(D) equals 1 everywhere. This implies the existence of a principal Z(D)-subbundle explicitly given by equation (4.39); we refer to that paper for more details.…”

Section: A Class Of Bibundles With Drinfel'd Double Fibrementioning

confidence: 99%

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A QP perspective on topology change in Poisson-Lie T-duality

Arvanitakis¹,

Blair²,

Thompson³

2021

Preprint

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We describe topological T-duality and Poisson-Lie T-duality in terms of QP (differential graded symplectic) manifolds and their canonical transformations. Duality is mediated by a QPmanifold on doubled non-abelian "correspondence" space, from which we can perform mutually dual symplectic reductions, where certain canonical transformations play a vital role. In the presence of spectator coordinates, we show how the introduction of bibundle structure on correspondence space realises changes in the global fibration structure under Poisson-Lie duality. Our approach can be directly translated to the worldsheet to derive dual string current algebras. Finally, the canonical transformations appearing in our reduction procedure naturally suggest a Fourier-Mukai integral transformation for Poisson-Lie T-duality.

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Section: Generalisations Of T-dualitymentioning

confidence: 99%

Section: A Class Of Bibundles With Drinfel'd Double Fibrementioning

confidence: 99%

Section: A Class Of Bibundles With Drinfel'd Double Fibrementioning

confidence: 99%

Section: A Class Of Bibundles With Drinfel'd Double Fibrementioning

confidence: 99%

See 2 more Smart Citations

A QP perspective on topology change in Poisson-Lie T-duality

Arvanitakis¹,

Blair²,

Thompson³

2021

Preprint

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“…We caution the reader that there is an a priori difference between the concordance relation and the relation on simplicial maps given by simplicial homotopy. For example, it is not true in general that the set of isomorphism classes of principal groupoid bundles is in a bijective correspondence with the set of concordance classes of principal groupoid bundles (see for instance [38]). It is of course well known that there is a bijection between the set of isomorphism classes and concordance classes of ordinary principal bundles with structure group a topological group.…”

Section: Introductionmentioning

confidence: 99%

Classifying theory for simplicial parametrized groups

Stevenson

2012

Preprint

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In this paper we describe a classifying theory for families of simplicial topological groups. If B is a topological space and G is a simplicial topological group, then we can consider the non-abelian cohomology H(B, G) of B with coefficients in G. If G is a topological group, thought of as a constant simplicial group, then the set H(B, G) is the usual set of isomorphism classes of principal G-bundles, or Gtorsors, on B. For more general simplicial groups G, the set H(B, G) parametrizes the set of equivalence classes of higher G-torsors. In this paper we consider a more general setting where G is replaced by a simplicial group in the category of spaces over B. The main result of the paper is that under suitable conditions on B and G there is an isomorphism between H(B, G) and the set of isomorphism classes of parametrized principal bundles on B, with structure group given by the fiberwise geometric realization of G.

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A QP perspective on topology change in Poisson–Lie T-duality

Arvanitakis

Blair

Thompson

2023

J. Phys. A: Math. Theor.

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We describe topological T-duality and Poisson-Lie T-duality in terms of QP (differential graded symplectic) manifolds and their canonical transformations. Duality is mediated by a QP-manifold on doubled non-abelian ``correspondence'' space, from which we can perform mutually dual symplectic reductions, where certain canonical transformations play a vital role. In the presence of spectator coordinates, we show how the introduction of a bibundle structure on correspondence space realises changes in the global fibration structure under Poisson-Lie duality. Our approach can be directly translated to the worldsheet to derive dual string current algebras. Finally, the canonical transformations appearing in our reduction procedure naturally suggest a Fourier-Mukai integral transformation for Poisson-Lie T-duality.

show abstract

On the existence of bibundles

Cited by 4 publications

References 13 publications

A QP perspective on topology change in Poisson-Lie T-duality

A QP perspective on topology change in Poisson-Lie T-duality

Classifying theory for simplicial parametrized groups

A QP perspective on topology change in Poisson–Lie T-duality

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