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Two-dimensional topological quantum field theories of rank two over Dedekind domains

Fabian Espinoza, Mee Seong Im, Mikhail Khovanov

TL;DR

We address the problem of constructing 2D TQFTs over a Dedekind domain $\mathcal{O}$ whose circle state space $A$ is projective but not free, focusing on rank-two Frobenius $\mathcal{O}$-algebras. The main approach derives explicit integrality conditions for a basis $\{1,X\}$ with $X^2= aX+b$, under $A\cong\mathcal{O}\oplus\mu$ with $\mu^2\cong(z)$, and presents two principal solution families: $\varepsilon(X)=0$ and $\varepsilon(X)\neq 0$, including normal forms obtained via rescalings and twists. The paper provides parametric constructions of Frobenius $\mathcal{O}$-algebras that are not free, gives concrete examples (e.g., over $\mathbb{Z}[\sqrt{-5}]$) with nontrivial $\varepsilon(X)$, and discusses how such algebras can yield a (singly graded) link homology theory, while highlighting RI and II move obstructions and proposing decorated or higher-rank variants. Overall, these results broaden integral TQFT frameworks by leveraging ideal-class group data and suggest new connections to classical link homologies through decorated and higher-rank generalizations.

Abstract

We give examples of Frobenius algebras of rank two over ground Dedekind rings which are projective but not free and discuss possible applications of these algebras to link homology.

Two-dimensional topological quantum field theories of rank two over Dedekind domains

TL;DR

We address the problem of constructing 2D TQFTs over a Dedekind domain whose circle state space is projective but not free, focusing on rank-two Frobenius -algebras. The main approach derives explicit integrality conditions for a basis with , under with , and presents two principal solution families: and , including normal forms obtained via rescalings and twists. The paper provides parametric constructions of Frobenius -algebras that are not free, gives concrete examples (e.g., over ) with nontrivial , and discusses how such algebras can yield a (singly graded) link homology theory, while highlighting RI and II move obstructions and proposing decorated or higher-rank variants. Overall, these results broaden integral TQFT frameworks by leveraging ideal-class group data and suggest new connections to classical link homologies through decorated and higher-rank generalizations.

Abstract

We give examples of Frobenius algebras of rank two over ground Dedekind rings which are projective but not free and discuss possible applications of these algebras to link homology.

Paper Structure

This paper contains 10 sections, 5 theorems, 114 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Rank two TQFTs
  3. Integrality conditions and equations
  4. Some solutions
  5. Case epsilonXEQUALSzero
  6. Case varepsilonXnotZero not equal to zero
  7. Case varepsilonone
  8. Case varepsilon(X) is invertible in Ostar
  9. Other values of epsilon(X)
  10. Towards link homology

Key Result

Corollary 1.1

For a commutative Frobenius $\mathcal{O}$-algebra $A$ as above, rank one projective $\mathcal{O}$-module $P$ has order at most two in the ideal class group $\mathsf{Cl}(\mathcal{O})$.

Figures (2)

  • Figure 4.0.1: Left: One of the two flavors of the Reidemeister I move of link diagrams. Middle: two resolutions of the curl diagram $D_1$ on the right hand side of the RI move. Right: Depicting the tensor product $A\otimes \mu$ by a dot on a strand; two adjacent dots can be removed due to the isomorphism $\mu\otimes_{\mathcal{O}}\mu\cong \mathcal{O}$.
  • Figure 4.0.2: Left: RI' move, a variation on the Reidemeister I move. The tensor product $A\otimes \mu$ is depicted by a dot on a strand. Middle: two adjacent dots on a strand can be canceled due to an isomorphism $\mu\otimes_{\mathcal{O}}\mu\cong \mathcal{O}$. Right: dots can be moved out of the strands to the plane, with the canceling cobordism on a pair of dots shown.

Theorems & Definitions (15)

  • Corollary 1.1
  • Proposition 2.1
  • proof
  • Remark 2.2: Twistings
  • Proposition 3.1
  • proof
  • Remark 3.2
  • Remark 3.3
  • Proposition 3.4
  • proof
  • ...and 5 more