Counting ribbon graphs using quantum field theory

Enumeration of rooted ribbon graphs started with the work by Tutte in the 1960’s. We call such graphs one-rooted graphs and introduce a notion of more general N-rooted ribbon graphs. This definition is motivated by the bijective correspondence we establish between the N-rooted ribbon graphs with e edges and the (e-N+1)-loop Feynman diagrams of a certain quantum field theory. This result is used to obtain explicit expressions and relations for the generating functions of N-rooted maps and for the numbers of N-rooted maps with a given number of edges using the path integral approach applied to the corresponding quantum field theory. This is a joint work with K. Gopala Krishna and Patrick Labelle