Compute the microscopic Gram matrix and prove projective flatness of the Berry curvature

Calculate the physical L² Gram matrix of the higher-genus Laughlin many-body states and prove that the resulting microscopic Berry curvature is projectively flat, so that the pointwise Berry curvature—and not only the averaged Hall conductance—is determined by the conformal-block construction.

Background

The paper constructs a projectively flat automorphy bundle over the Jacobian and uses its characteristic classes to obtain the averaged Hall conductance 1/k. This calculation is cohomological and does not use the physical L² inner product of the many-body wave functions.

The microscopic Berry connection depends on the L² Gram matrix of the Laughlin states. Establishing a higher-genus screening estimate relating this Gram matrix to the metric used in the automorphy-bundle construction would identify the conformal-block connection with the physical Berry connection and would allow one to determine the pointwise Berry curvature.