On the geometry of some unitary Riemann surface braid group representations and Laughlin-type wave functions

In this note we construct the simplest unitary Riemann surface braid \r\ngroup representations geometrically by means of stable holomorphic vector bundles over complex tori and the prime form on Riemann surfaces. \r\nGeneralised Laughlin wave functions are then introduced. The genus one \r\ncase is discussed in some detail also with the help of noncommutative geometric tools, and an application of Fourier-Mukai-Nahm techniques is also given, explaining the emergence of an intriguing Riemann surface braid group duality.