Nonperturbative structures in hyperelliptic universal moduli space

We investigate the nonperturbative structures of string theory by studying the properties of Hyperelliptic Universal Moduli Space, which consists of a family of hyperelliptic Riemann surfaces. A coordinate of such space can be inherited by the branch points and all the correlation functions and chiral determinants of Cauchy-Riemann operators can be computed in terms of branch points, using the branch point operator method. We observe certain phase transitions based on the renormalization group equation for the grand partition function and such phase structures provide the nonperturbative information of theories defined on such a space