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On Nearly Frobenius Structures

William Davis, Olivia Dumitrescu

TL;DR

This work extends the relationship between 2D TQFTs and Frobenius algebras to the infinite-dimensional, counital/counital-lacking realm of nearly Frobenius algebras, establishing an equivalence with Almost TQFTs. It reframes the 2D theory in terms of colored ribbon graphs and Edge Contraction axioms, enabling a combinatorial/classification approach that parallels, and generalizes, the Atiyah–Segal and Teleman frameworks. The authors derive explicit formulas for the TQFT maps $oldsymbol{oldomega}_{g,n,m}$ using Euler-like insertions $oldsymbol{E}^g$ and iterated coproducts, and connect these to recursion phenomena (Catalan-type recursions) relevant to enumerative geometry and topological recursion. The work further clarifies how ribbon-graph formulations and combinatorial data encode the same invariants as the cobordism-based TQFTs, and outlines directions for extending Teleman’s CohFT classification to nearly Frobenius structures. Overall, it provides a coherent algebraic/combinatorial foundation for Almost TQFTs and their connections to enumerative geometry via ribbon graphs and edge-contraction calculus.

Abstract

Nearly Frobenius structures and 2-dimensional Almost TQFTs were introduced and shown to be in categorical equivalence in arXiv:1907.05470 in the attempt to extend the Atiyah-Segal's definition to the category of infinite dimensional vector spaces. In this paper, we investigate nearly Frobenius structures and we give a classification result for Almost TQFTs in dimension 2. In particular, the TQFTs functorial axioms become equivalent to the Edge Contraction/Construction axioms of colored ribbon graphs recently introduced by the authors.

On Nearly Frobenius Structures

TL;DR

This work extends the relationship between 2D TQFTs and Frobenius algebras to the infinite-dimensional, counital/counital-lacking realm of nearly Frobenius algebras, establishing an equivalence with Almost TQFTs. It reframes the 2D theory in terms of colored ribbon graphs and Edge Contraction axioms, enabling a combinatorial/classification approach that parallels, and generalizes, the Atiyah–Segal and Teleman frameworks. The authors derive explicit formulas for the TQFT maps using Euler-like insertions and iterated coproducts, and connect these to recursion phenomena (Catalan-type recursions) relevant to enumerative geometry and topological recursion. The work further clarifies how ribbon-graph formulations and combinatorial data encode the same invariants as the cobordism-based TQFTs, and outlines directions for extending Teleman’s CohFT classification to nearly Frobenius structures. Overall, it provides a coherent algebraic/combinatorial foundation for Almost TQFTs and their connections to enumerative geometry via ribbon graphs and edge-contraction calculus.

Abstract

Nearly Frobenius structures and 2-dimensional Almost TQFTs were introduced and shown to be in categorical equivalence in arXiv:1907.05470 in the attempt to extend the Atiyah-Segal's definition to the category of infinite dimensional vector spaces. In this paper, we investigate nearly Frobenius structures and we give a classification result for Almost TQFTs in dimension 2. In particular, the TQFTs functorial axioms become equivalent to the Edge Contraction/Construction axioms of colored ribbon graphs recently introduced by the authors.

Paper Structure

This paper contains 10 sections, 13 theorems, 66 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Frobenius algebras and TQFTs
  4. Topological Quantum Field Theory
  5. Cohomological Field Theories.
  6. A classification result
  7. Nearly Frobenius algebras
  8. On Ribbon TQFTs
  9. Frobenius structures
  10. Atiyah Sewing for Counital Nearly Frobenius Algebras

Key Result

Theorem 1.1

Suppose $A$ is a Frobenius algebra with counit $\varepsilon$, coproduct $\delta$, and Euler element $\mathbf{e}$. Then, the value of a 2D TQFT is given by

Figures (4)

  • Figure 1: Frobenius relation of cobordisms.
  • Figure 2: Sewing two cobordisms with agreeing numbers of boundary components.
  • Figure 3: Partial sewing of $\Sigma_{0,2,0}$ onto the output side of a cobordism.
  • Figure 4: Normal form of a cobordism with $(g,n,m)=(3,4,3)$.

Theorems & Definitions (37)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Definition 2.1
  • Remark 2.2
  • Example 2.3
  • Example 2.4
  • Remark 2.5
  • Example 2.6
  • Definition 2.7
  • ...and 27 more