Abstract

Quantum Hall effect occurs is real-world materials and is characterised by a precise quantization of Hall conductance, which can take integer of fractional values. This phenomenon has a geometric origin and best understood when considering QH states on Riemann surfaces. Main goal of the theory is to compute adiabatic phases corresponding to various geometric deformations (associated with the line bundle, metric and complex structure moduli), in the limit of a large number of particles. I will define the objects involved, and then give a complete solution of the problem for the integer QH states and for the Laughlin states in the fractional QHE , by computing the generating functional for these states. In the integer QH our method relies on methods borrowed form Kahler geometry, such as Bergman kernel expansion for high powers of holomorphic line bundle, and the answer is expressed in terms of energy functionals in Kahler geometry. We explain the relation of geometric phases to Quillen theory of determinant line bundles, using Bismut-Gillet-Soule anomaly formulas. For the much harder Laughlin states no rigorous methods are available, yet we still can compute the generating functional by physics methods such as path integral.