Establish the Moore–Read monodromy–Berry holonomy correspondence on compact Riemann surfaces

Prove that the explicit monodromies of the higher-genus Moore–Read conformal blocks for Laughlin states coincide with the corresponding adiabatic Berry holonomies under quasi-hole transport and Aharonov–Bohm flux insertion, thereby establishing the Moore–Read identification beyond the presently computed conformal-block monodromies.

Background

The paper derives explicit conformal blocks for ν=1/k Laughlin states on compact Riemann surfaces and computes their monodromies under quasi-hole transport and variation of the flat part of the background magnetic connection. These calculations reproduce the expected fractional statistics, Aharonov–Bohm phases, and finite Heisenberg algebra, and yield the cohomological averaged Hall conductance.

However, the conformal-block connection is not automatically the microscopic Berry connection obtained from the physical L² Gram matrix of normalized many-body states. Identifying the two requires a higher-genus screening argument. Consequently, the claimed agreement between conformal-block monodromy and adiabatic holonomy remains conjectural in the general compact-surface setting.