Reviews in Mathematical Physics

We introduce a general class of long-range magnetic potentials and derive high velocity limits for the corresponding scattering operators in quantum mechanics, in the case of two dimensions. We analyze the high velocity limits that we obtain in the presence of an obstacle and we uniquely reconstruct from them the electric potential and the magnetic field outside the obstacle, that are accessible to the particles. We additionally reconstruct the inaccessible fluxes (magnetic fluxes produced by fields inside the obstacle) modulo 2π, which give a proof of the Aharonov–Bohm effect. For every magnetic potential A in our class, we prove that its behavior at infinity can be characterized in a natural way; we call it the long-range part of the magnetic potential. Under very general assumptions, we prove that can be uniquely reconstructed for every . We characterize properties of the support of the magnetic field outside the obstacle that permit us to uniquely reconstruct either for all or for in a subset of 𝕊¹. We also give a wide class of magnetic fields outside the obstacle allowing us to uniquely reconstruct the total magnetic flux (and for all ). This is relevant because, as it is well-known, in general the scattering operator (even if it is known for all velocities or energies) does not define uniquely the total magnetic flux (and ). We analyze additionally injectivity (i.e. uniqueness without giving a method for reconstruction) of the high velocity limits of the scattering operator with respect to . Assuming that the magnetic field outside the obstacle is not identically zero, we provide a class of magnetic potentials for which injectivity is valid.

We give a simplified definition of topological T-duality that applies to arbitrary torus bundles. The new definition does not involve Chern classes or spectral sequences, only gerbes and morphisms between them. All the familiar topological conditions for T-duals are shown to follow. We determine necessary and sufficient conditions for existence of a T-dual in the case of affine torus bundles. This is general enough to include all principal torus bundles as well as torus bundles with arbitrary monodromy representations. We show that isomorphisms in twisted cohomology, twisted K-theory and of Courant algebroids persist in this general setting. We also give an example where twisted K-theory groups can be computed by iterating T-duality.

Through chaos decomposition, we improve the Varadhan estimate for the rate of convergence of the centered approximate self-intersection local time of planar Brownian motion.