Clifford bundle

Abstract: 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 T2-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.

Abstract: 2. Generalities [2; l]. If xGA, vi, v2(E.Xx, let (vu v2) denote the inner product that defines the Riemannian metric. If a: [0, l]—»A is a curve, let a':t-^a'(t) denote the tangent vector field to a. If v: H»(i)GI,(i) is a vector-field along a, let Av denote the covariant derivative of v along a [l]. Let us recall how it is defined by Elie Cartan's method of orthonormal moving frames: Suppose U is an open set of A and w, (léi,j, k • ■ • 5¡w = dim X, summation convention) 1-differential forms in U with