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Almost TQFTs via colored ribbon graphs

William Davis, Olivia Dumitrescu

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.

Almost TQFTs via colored ribbon graphs

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.
Paper Structure (24 sections, 27 theorems, 90 equations, 28 figures)

This paper contains 24 sections, 27 theorems, 90 equations, 28 figures.

Table of Contents

  1. Introduction
  2. Enumerative Geometry of Catalan numbers
  3. Frobenius structures
  4. Frobenius Algebras
  5. Nearly Frobenius Algebras
  6. Topological Quantum Field Theory
  7. Background
  8. A classification result
  9. Almost TQFT
  10. Colored Cell Graphs and Edge Axioms
  11. Colored Cell Graphs
  12. Edges between Input Vertices
  13. Edges between Output Vertices
  14. Topologically-inspired Axioms
  15. Frobenius Structures on Graphs
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $A$ be a nearly Frobenius algebra with coproduct $\delta$ and the Euler map $\mathbf{E}$. Then the ribbon TQFT defined in def:ribbonTQFT associated to a colored cell graph $\gamma$ of type $(g,n,m)$ with $n,m\geq 1$, is independent of the graph and is given by Where $\delta^0$ is the identity map (for the case where $m=1$).

Figures (28)

  • Figure 1: Pairs of pants decomposition of Riemann surfaces leading to topological recursion
  • Figure 2: Generating set of morphisms in the category $\mathbf{2Cob}$
  • Figure 3: Commutativity and cocommutativity relations of the twist.
  • Figure 4: Relations involving sewing discs
  • Figure 5: Associativity and coassociativity relations of cobordisms.
  • ...and 23 more figures

Theorems & Definitions (81)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 2.4
  • Definition 3.1
  • Remark 3.2
  • ...and 71 more