Noncommutative principal torus bundles via parametrised strict deformation quantization

T-duality simplifies bulk–boundary correspondence: the noncommutative case

“…• simplifies the complicated (momentum space) boundary map defined at the level of the physical C * -algebras of observables; • continues to hold even when the momentum space is noncommutative (as in the IQHE), in the presence of disorder [32], in the real case (relevant for time-reversal symmetry), and in the parametrised and twisted setting of [18,19] (relevant to string theory in the presence of H-flux [4]). At the mathematical level, the meta-principle expressed by the diagram naturally generalizes a simple phenomenon already present at the level of ordinary Fourier transforms: integration in Fourier space, which can be understood topologically as a push-forward map, picks out the constant Fourier mode, so it is equivalent to the Fourier transform of a "restriction-to-zero" map.…”

T-Duality Simplifies Bulk–Boundary Correspondence: Some Higher Dimensional Cases

“…The smooth noncommutative torus can be realized as a deformation quantization of the smooth functions on a torus T = R d /Z d of dimension equal to d, by a construction due to Rieffel [58] (we use the notation T rather than T d to emphasize the group structure, thus T refers to the Pontryagin dual of T ). The parametrized case was considered in [19,20].…”

Topological T-Duality for Twisted Tori

“…In string theory it is T-dual to the three-torus T 3 with H-flux by the standard Buscher rules (see e.g. [26]), and in topological T-duality it gives the basic example of a noncommutative principal torus bundle [31,15,22]. Here we shall give a new algebraic perspective on both these T-duals by applying our formalism of topological T-duality directly to the Heisenberg nilmanifold.…”

“…of C * -algebra bundles over X. For further details and a classification of noncommutative principal torus bundles, see [15,22].…”

“…The star-products (5.7) with θ(x) = θ m (x) = m x are equivalent to the starproducts of [30,22,27]. In [15,22], it is shown that the description of Proposition 5.14 defines a noncommutative principal T 2 -bundle (cf. Example 4.15).…”