Algebraic and arithmetic geometry in string theory

Abstract

In the first part we review how elements of algebraic geometry can be used to give an algebraic formula for the string partition function. In the second part we generalize these ideas to include arithmetic surfaces, i.e. surfaces defined over an algebraic number field K. We will calculate explicitly the volume of the lattice formed by K-rational tangent vectors at a K-rational point in moduli space, with respect to the Polyakov measure.