Almost TQFTs via colored ribbon graphs

TL;DR

This paper extends 2D TQFT axioms to colored ribbon graphs, classifies ribbon TQFTs, and links these structures to generalized Catalan number recursions via Almost TQFTs.

Contribution

It introduces a new framework for ribbon TQFTs using Edge Contraction Axioms and classifies these theories, extending previous Frobenius algebra results.

Findings

Ribbon TQFTs are classified via extended axioms.

Edge Contraction Axioms are equivalent to functorial TQFT axioms.

Generalized Catalan numbers recursion is twisted by Almost TQFT.

Abstract

In this paper, we introduce ribbon TQFTs via Edge Contraction/Construction Axioms of colored ribbon graphs as an extension of the 2D TQFT axioms for ribbon graphs formulated in arXiv:1508.05922. We investigate nearly Frobenius structures and Almost TQFTs defined in arXiv:1907.05470 together with ribbon TQFTs. We give a classification result for ribbon TQFTs that extends the one obtained for Frobenius algebras in arXiv:1508.05922. In particular, the Edge Contraction/Construction Axioms of colored ribbon graphs in this work become equivalent to the functorial Axioms of TQFTs governed by the sewing principle of Atiyah and Segal discussed in arXiv:2510.03128 and arXiv:1907.05470. As an application, we obtain that the recursion of generalized Catalan numbers can be twisted by Almost TQFT for co-unital nearly Frobenius algebra.