On the Genus Distribution of $(p,q,n)$-Dipoles

Abstract

There are many applications of the enumeration of maps in surfaces to other areas of mathematics and the physical sciences. In particular, in quantum field theory and string theory, there are many examples of occasions where it is necessary to sum over all the Feynman graphs of a certain type. In a recent paper of Constable et al. on pp-wave string interactions, they must sum over a class of Feynman graphs which are equivalent to what we call $(p,q,n)$-dipoles. In this paper we perform a combinatorial analysis that gives an exact formula for the number of $(p,q,n)$-dipoles in the torus (genus 1) and double torus (genus 2).