Graphs on Surfaces and the Partition Function of String Theory

TL;DR

This paper explores the connection between graph theory on surfaces and the partition function in discrete string theory, providing exact combinatorial formulas for various closed surfaces.

Contribution

It offers a formal mathematical framework linking graph embeddings on surfaces to the computation of the string theory partition function, including explicit calculations for specific surfaces.

Findings

Exact formulas for the partition function on the sphere and torus

Methodology for counting non-isomorphic triangulations

Extension to non-orientable surfaces like the projective plane

Abstract

Graphs on surfaces is an active topic of pure mathematics belonging to graph theory. It has also been applied to physics and relates discrete and continuous mathematics. In this paper we present a formal mathematical description of the relation between graph theory and the mathematical physics of discrete string theory. In this description we present problems of the combinatorial world of real importance for graph theorists. The mathematical details of the paper are as follows: There is a combinatorial description of the partition function of bosonic string theory. In this combinatorial description the string world sheet is thought as simplicial and it is considered as a combinatorial graph. It can also be said that we have embeddings of graphs in closed surfaces. The discrete partition function which results from this procedure gives a sum over triangulations of closed surfaces. This is known as the vacuum partition function. The precise calculation of the partition function depends on combinatorial calculations involving counting all non-isomorphic triangulations and all spanning trees of a graph. The exact computation of the partition function turns out to be very complicated, however we show the exact expressions for its computation for the case of any closed orientable surface. We present a clear computation for the sphere and the way it is done for the torus, and for the non-orientable case of the projective plane.