Moore--Read construction and explicit monodromies of Laughlin states on Riemann surfaces

We revisit the Moore–Read construction of the $\nu =1\mathrm{/}k$ Laughlin states on compact Riemann surfaces, deriving their conformal blocks from $U(1{)}_k$ Chern–Simons theory through the CS/WZW correspondence. We compute their explicit monodromies under quasi-hole transport and Aharonov–Bohm flux insertion, and conjecture that these coincide with the corresponding adiabatic holonomies. Under this identification, we recover classical results on Laughlin states including the flux-averaged Hall conductance $1\mathrm{/}k$ via a vector bundle of Laughlin states over $Jac(\mathrm{\Sigma })$. We also relate our construction to the recently developed algebro-geometric approach to higher genus Laughlin states.