Adiabatic Quantum Transport: Quantization and Fluctuations
  Quillen's local index theorem is used to study the charge transport coefficients (adiabatic curvature) associated to the ground state of a Schr\"odinger operator for a charged (spinless) particle on a closed, multiply connected surface. The formula splits the adiabatic curvature into an explicit integral part and a fluctuating part which has a natural interpretation in terms of quantum chaos. 
J.E.Avron, R. Seiler and P. Zograf