On an integral representation for the genus series for $2$-cell embeddings

Abstract

An integral representation for the genus series for maps on oriented surfaces is derived from the combinatorial axiomatisation of 2-cell embeddings in orientable surfaces. It is used to derive an explicit expression for the genus series for dipoles. The approach can be extended to vertex-regular maps in general and, in this way, may shed light on the genus series for quadrangulations. The integral representation is used in conjunction with an approach through the group algebra, CO,, of the symmetric group [11] to obtain a factorisation of certain Gaussian integrals. 1. A POWER SERIES REPRESENTATION FOR THE GENUS SERIES A map is a 2-cell embedding of a connected unlabelled graph ', with loops and multiple edges allowed, in a closed surface X, without boundary, which is assumed throughout to be oriented. The deletion of ' separates X into regions homeomorphic to open discs, called the faces of the map, and the number of edges bordering a face is called its degree. A map is rooted by distinguishing a mutually incident vertex, edge and face. The genus series for a class of maps is the formal generating series for the number of inequivalent maps with respect to genus, and the numbers of vertices, edges and faces. It is assumed hereinafter that maps are rooted. The general approach adopted here combines ideas of Jackson and Visentin [11] with those of 't Hooft [8] and Bessis, Itzykson and Zuber [3] who, in the above terminology, derived the genus series for diagrams akin to a class of maps by techniques from conformal field theory. Although [81 and [3] are important papers, they have remained largely inaccessible to combinatorialists because of their uncertainty about the automorphisms of these diagrams as combinatorial structures. In this paper, an explicit construction is given for an integral representation for the genus series for general maps directly from the combinatorial axiomatisation for embeddings on oriented surfaces. Moreover, we also develop methods which are extensible to vertex-regular maps (vertices have the same degree) and thence, by restriction, to quadrangulations (maps whose faces are bounded by four edges). This is done by examining dipoles (maps with two vertices) in detail. Although the argument is an algebraic one, based on the ring of formal power series, to assert that particular series belong to the ring, it is necessary to Received by the editors July 1, 1992 and, in revised form, August 15, 1993. 1991 Mathematics Subject Classification. Primary 05A1 5, 20C1 5; Secondary 57N37. (?) 1994 American Mathematical Society 0002-9947/94 $1.00 + S.25 per page