2004/01/23 by Varghese Mathai, Jonathan Rosenberg · 10 citations
Mathematics · Physics and Astronomy · #Bundle #Cohomology #Combinatorics #Crystallography #Cyclic homology #Duality (order theory) #Fundamental representation #Geometric and Algebraic Topology #Geometry #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology #Isomorphism (crystallography) #Lie algebra #Mathematical physics #Mathematics #Maximal torus #Noncommutative geometry #Pure mathematics #T-duality #Topology (electrical circuits) #Torus #hep-th #math.OA
paper · pdf · doi:10.1007/s00220-004-1159-7
published as Commun.Math.Phys.253:705-721,2004 · 16 pages, Latex2e, 1 figure
arxiv created 2004/01/23 · openalex publication_date 2004/08/26 · arxiv updated 2014/11/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
It is known that the T-dual of a circle bundle with H-flux (given by a Neveu-Schwarz 3-form) is the T-dual circle bundle with dual H-flux. However, it is also known that torus bundles with H-flux do not necessarily have a T-dual which is a torus bundle. A big puzzle has been to explain these mysterious "missing T-duals.'' Here we show that this problem is resolved using noncommutative topology. It turns out that every principal 2-torus-bundle with H-flux does indeed have a T-dual, but in the missing cases (which we characterize), the T-dual is non-classical and is a bundle of noncommutative tori. The duality comes with an isomorphism of twisted K-theories, just as in the classical case. The isomorphism of twisted cohomology which one gets in the classical case is replaced by an isomorphism of twisted cyclic homology.