Chiral bosonization, determinants and the string partition function

We study the bosonization of chiral fermion theories on arbitrary compact Riemann surfaces. We express the fermionic and bosonic correlation functions in terms of theta functions and prove their equality. This is used to obtain explicit expressions for a class of chiral determinants relevant to string theory. The anomaly structure of these determinants and their behaviour on degenerate Riemann surfaces is analysed. We apply these results to multi-loop calculations of the bosonic string.