Combination topological field theory

Abstract

This work contains the construction of a topological quantum field theory (TQFT, or TFT) based on combinatorial information which consists of directed metric graphs with vertices labeled by metric ribbon graphs. A canonical map between such objects and smooth Riemann surfaces is established using the theory of quadratic differentials investigated by Strebel and others. The surfaces derived have natural decomposition into finite and infinite length cylinders enumerated by the edges of the directed metric graph. Moreover, the surfaces have a gluing operation which agrees with a natural connecting operation on the level of graphs. Finally, the cylindrical decomposition gives the surfaces the structure of a model surface originally investigated by M. Schwartz. He offers a functor on the category of such surfaces which satisfies the properties of a TFT. Combining this functor with the combinatorial information gives the construction presented herein.