Berry connection in quantum Hall effect

One of the main features of the quantum Hall states is that the corresponding quasiparticles carry fractional or even non-abelian statistics. Equivalently, the ground state must be degenerate on higher genus surfaces. To derive the statistics and qualify non-abelian nature of a state requires to compute the Berry matrix. This is a challenging problem to solve in general. For example, the standard way to obtain the fractional statistics of the Laughlin state uses the plasma analogy. We will consider the problem of computing the Berry connection for quantum Hall states without quasiholes but on surfaces of genus greater than zero.