  It has long been known that in principle, the genus g vacuum amplitude for
bosonic strings or superstrings in 26 or 10 dimensions can be entirely
determined from conditions of holomorphy. Moreover, this has been done in
practice for bosonic strings of low genus. Here we describe in a unified way
how to determine the bosonic string and superstring vacuum amplitude in genus 1
and 2 via holomorphy. The main novelty is the superstring analysis in genus 2,
where we use holomorphy to get a new understanding of some of the results that
previously have been obtained by more explicit calculations.
