An algebraic construction of a solution to the mean field equations on hyperelliptic Curves and its diabatic limit

In this paper, we give an algebraic construction of the solution to the following mean field equation $$ Delta psi+e^{psi}=4pisum_{i=1}^{2g+2}delta_{P_{i}}, $$ on a genus $ggeq 2$ hyperelliptic curve $(X,ds^{2})$ where $ds^{2}$ is a canonical metric on $X$ and ${P_{1},cdots,P_{2g+2}}$ is the set of Weierstrass points on $X.$ Furthermore, we study the rescaled equation $$ Delta psi+gamma e^{psi}=4pigamma sum_{i=1}^{2g+2}delta_{P_{i}} $$ and its adiabatic limit at $gamma=0$.