Edge contraction on dual ribbon graphs and 2D TQFT

TL;DR

This paper introduces a new axiomatic framework for 2D Topological Quantum Field Theories (TQFTs) using cell graphs and edge-contraction operations, linking graph transformations to Frobenius algebra structures.

Contribution

It formulates a novel set of axioms for 2D TQFTs based on cell graphs and edge contractions, establishing a functorial relationship with Frobenius algebras.

Findings

Edge-contraction axioms depend only on topological type, not specific graph.

Constructs a functor from cell graphs to Frobenius algebra endofunctors.

Associates cell graphs with elements in symmetric tensor algebra over A*.

Abstract

We present a new set of axioms for 2D TQFT formulated on the category of cell graphs with edge-contraction operations as morphisms. We construct a functor from this category to the endofunctor category consisting of Frobenius algebras. Edge-contraction operations correspond to natural transformations of endofunctors, which are compatible with the Frobenius algebra structure. Given a Frobenius algebra A, every cell graph determines an element of the symmetric tensor algebra defined over the dual space A*. We show that the edge-contraction axioms make this assignment depending only on the topological type of the cell graph, but not on the graph itself. Thus the functor generates the TQFT corresponding to A.