Fourier-Mukai Transform and Adiabatic Curvature of Spectral Bundles for Landau Hamiltonians on Riemann Surfaces

Abstract

We study the family of Landau Hamiltonians on a Riemann surface S by means of a Nahm transform and an integral functor related to the Fourier-Mukai transform associated to its jacobian variety J(S). This approach allows us to explicitly determine the spectral bundles associated to the holomorphic Landau levels. As a first main result we prove that these spectral bundles are holomorphic stable bundles with respect to the canonical polarization of J(S) determined by the theta divisor . The spectral bundles are endowed with a natural connection and the adiabatic charge transport properties of the corresponding Landau levels are determined by the adiabatic curvature of , which coincides with the curvature of the determinant bundle det . By means of the theory of analytic torsion and determinant bundles developed by Bismut, Gillet and Soule we compute the adiabatic curvature of the spectral bundles on an arbitrary Riemann surface. We prove that all the holomorphic Landau levels have the same charge transport coefficients but their adiabatic curvatures differ by a term which involves the relative analytic torsion of different powers of the canonical bundle of S twisted by a fixed line bundle.