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

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Pith's one-line read Rank-two Frobenius algebras over Dedekind domains can be projective but not free, and this paper constructs explicit examples including one over $\mathbb{Z}[\sqrt{-5}]$.

desk verdict Correct, focused construction of rank-two Frobenius algebras over Dedekind rings; the link homology part is programmatic and the paper needs minor but real fixes before acceptance. read the letter →

arxiv 2502.04502 v2 pith:QEP4O5EN submitted 2025-02-06 math-ph math.CTmath.MPmath.QA

Fabian Espinoza , Mee Seong Im , Mikhail Khovanov This is my paper · ORCID
classification math-phmath.CTmath.MPmath.QA MSC 11R0457K1618M0516L60
keywords two-dimensionalTQFTcommutativeFrobeniusalgebraDedekinddomainprojectivenonfreemoduleidealclassgrouplinkhomologyranktwoZ[√-5]
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper establishes that two-dimensional topological quantum field theories can have state spaces that are projective but not free modules over the ground ring. Concretely, it constructs rank-two commutative Frobenius algebras over Dedekind domains $O$, of the form $A = O \oplus \mu X$ where $\mu$ is a nonprincipal ideal of order two in the class group, and it reduces the existence question to explicit integrality equations. It solves those equations in several families and gives a fully explicit example over $O = \mathbb{Z}[\sqrt{-5}]$ with $\mu = (2, 1+\sqrt{-5})$. If correct, the result shows that nonfree modules are not an obstruction to 2D TQFT structure and opens a route toward link homology theories whose state spaces carry nontrivial ideal class group data.

What carries the argument

The load-bearing object is the pair $(A, \varepsilon)$ with $A = O \oplus \mu X$ and trace $\varepsilon$: the trace is encoded by $\varepsilon(X) \in z^{-1}\mu$ because $\operatorname{Hom}_O(\mu,O) \cong z^{-1}\mu$, and the Frobenius condition is the requirement that $\widetilde{\varepsilon}: A \to A^*$ be an isomorphism. The argument reduces this to the determinant $\widetilde{\Delta} = \varepsilon(1)\varepsilon(X^2) - \varepsilon(X)^2$ and the integrality inclusions in (2.16)-(2.17), then solves the resulting linear system for the auxiliary parameters $c, d, c', d'$. The nonprincipal ideal $\mu$ of order two in $\operatorname{Cl}(O)$ is what makes $A$ projective but not free.

What would settle it

Compute the matrix entries for the claimed $\mathbb{Z}[\sqrt{-5}]$ example ($\mu=(2,1+\sqrt{-5})$, $z=2$, $a=1-\sqrt{-5}$, $\varepsilon(1)=1$, $\varepsilon(X)=(1+\sqrt{-5})/2$, $b=\sqrt{-5}-6$) and check the inclusions $\varepsilon(X^2) \in \widetilde{\Delta} O$, $\varepsilon(X) \in \widetilde{\Delta} \mu$; the algebra is Frobenius exactly when these inclusions and $\widetilde{\Delta} \neq 0$ hold.

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Extended reading notes

Core claim

The central discovery is a normal form and existence proof for commutative Frobenius algebras of rank two over a Dedekind domain that are projective but not free. The paper proves that any such algebra with $A \cong O \oplus \mu X$, $\mu^2=(z)$, has multiplication $X^2 = z^{-1}aX + z^{-1}b$ and is determined by parameters $a \in z^{-1}\mu$, $b \in z^{-1}O$, $\varepsilon(1) \in O$, $\varepsilon(X) \in z^{-1}\mu$, subject to the integrality conditions $\varepsilon(X^2) \in \widetilde{\Delta} O$, $\varepsilon(X) \in \widetilde{\Delta} \mu$, $\varepsilon(1) \in \widetilde{\Delta} z O$. It then produces solutions: a full family with $\varepsilon(X)=0$ classified by $a \in O$ and $b, \varepsilon(1) \in O^\times$ for any nonprincipal $\mu$ with square principal, and an explicit example over $O=\mathbb{Z}[\sqrt{-5}]$ in which $\varepsilon(X)$ is neither zero nor a unit. These data make $A$ into a genuine Frobenius $O$-algebra, hence define a 2D TQFT with nonfree circle state space.

Load-bearing premise

The construction relies on the standard fact that the dual of the rank-one ideal $\mu$ is the ideal $z^{-1}\mu$ when $\mu^2=(z)$; if $\operatorname{Hom}_O(\mu,O)$ were not this inverse ideal, the trace parametrization and integrality equations would not hold.

Editorial extensions

If this is right

  • Every rank-two Frobenius algebra over a Dedekind domain fits the normal form $A = O \oplus \mu X$ with $\mu$ of order two in the ideal class group, so existence forces $\operatorname{Cl}(O)$ to have 2-torsion.
  • Over any Dedekind domain with a nonprincipal ideal whose square is principal, the $\varepsilon(X)=0$ family yields infinitely many such Frobenius algebras, with trace parameters $b, \varepsilon(1) \in O^\times$.
  • Over $\mathbb{Z}[\sqrt{-5}]$, the explicit example with $\varepsilon(X) = (1+\sqrt{-5})/2$ shows solutions exist where $\varepsilon(X)$ is neither zero nor a unit, so the phenomenon is not confined to trivial trace values.
  • If an additional condition (such as $b$ invertible) makes $\ker(m) \cong A$ as an $A$-module, the Frobenius algebra can be fed into the standard rank-two Frobenius-extension link homology construction to produce complexes over $O$; tensoring with the fraction field recovers a Lee-type homology.
  • The paper's rank-$N$ graded examples $A_N = O \oplus \mu X \oplus \cdots \oplus \mu^{N-1} X^{N-1}$ extend the nonfree phenomenon to higher rank commutative graded Frobenius algebras.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Going beyond the paper's own claims, the order-two condition on $\mu$ can be read as a cohomological obstruction: the state-space module must be self-dual, so only self-inverse ideal classes can appear in a nonfree rank-two 2D TQFT.
  • As an extension, the explicit $\mathbb{Z}[\sqrt{-5}]$ example could serve as a test case for lifting link homology to a theory over the ring of integers of a number field, with the class group appearing as grading or decoration data rather than as an obstruction.
  • A testable extension would be to search algorithmically for further solutions by fixing $O$ and $\mu$ and solving the single equation (3.24); the paper's loose parameter choices suggest such solutions are abundant.
  • Implicit in the link-homology discussion, if any of these Frobenius algebras produces Reidemeister-invariant homology groups over $O$, the resulting Euler characteristic would be valued in a class-group-twisted module, giving a new invariant sensitive to ideal class group data.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 5 minor

Summary. The paper constructs commutative Frobenius algebras of rank two over a Dedekind domain O whose underlying O-module is projective but not free. Writing A = O1 ⊕ µX for a non-principal ideal µ with µ² = (z), the authors define multiplication by X² = z^{-1}aX + z^{-1}b and parametrize O-linear traces by the pair (ε(1), ε(X)). They derive integrality conditions (2.16)-(2.17) and linear equations (2.25)-(2.28) that are equivalent to nondegeneracy of the trace, solve these equations in the cases ε(X)=0 and ε(X)≠0, and give a concrete example over Z[√-5] with µ = (2,1+√-5), z=2, a=1-√-5, ε(X)=(1+√-5)/2, ε(1)=1, and b=√-5-6. A final section discusses possible applications to link homology, noting explicitly that Reidemeister II invariance requires an isomorphism A ⊗ µ ≅ A that is not established.

Significance. If correct, the paper supplies explicit examples of rank-two Frobenius algebras over Dedekind domains that are projective but not free, hence 2D TQFTs with nonfree circle state spaces. The central derivation is detailed and checkable: the integrality conditions follow from explicit matrix equations, and the concrete example can be verified directly from (2.41)-(2.45). The corollary that the nonfree summand must have order two in the ideal class group is a clean structural restriction. The paper is honest about the programmatic nature of the link homology part, which is presented as a direction rather than a proved invariant.

minor comments (5)
  1. [§3.1, Proposition 3.1] The conclusion "a ∈ O" should read "a ∈ µ". The closure condition (2.47) requires a ∈ µ for the multiplication u1X · u2X to land in A = O1 ⊕ µX; allowing an arbitrary a ∈ O would not preserve A. This does not affect the existence example in Section 3.3, but the classification statement should be corrected.
  2. [§2.1, Proposition 2.1(II)] The sentence "We already know from (2.34) that t = ε(X)^2 − ε(1)ε(X^2) ≠ 0" is a misstatement: (2.34) asserts the nonvanishing of the determinant expression ε(X)^2 − ε(1)ε(X^2), not of t = ε(X^2). The subsequent argument does not rely on this sentence, but it should be rephrased.
  3. [§2.1, Remark 2.2, Eq. (2.49)] The transformation law for b under X ↦ X + λ1 appears incorrect. Substituting X = X' − λ1 into X² = aX + b gives X'^2 = (a + 2λ1)X' + (b − aλ1 − λ1²), so the correct change is b ↦ b − aλ1 − λ1², not b ↦ b + λ1². Please correct or clarify the intended convention.
  4. [§3.3] The verification "One can check that all integrality conditions are satisfied by this solution" is asserted rather than shown. Since this is the paper's central concrete example, it would be helpful to display the resulting parameters c, d, c′, d′, t and confirm relations (2.41)–(2.45) explicitly.
  5. [§3.1, proof of Proposition 3.1] In the proof, the statement "The last two equations (2.43) and (2.44) give c′ε(1) = 0, d′t = 1" does not follow from (2.43) and (2.44) as written; (2.43) is d′εX = −dε(1). The intended reference is likely to (2.27) and (2.28), or the conclusion can be obtained from c′ = z^{-1}d = 0. Please clarify.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the examples are constructed from explicit parameters and independently verified against the derived integrality conditions.

full rationale

The paper's central claim is an existence statement: rank-two commutative Frobenius algebras over Dedekind domains that are projective but not free. The derivation starts from a general ansatz A = O1 ⊕ µX with µ²=(z), then imposes closure of multiplication and integrality of the Frobenius structure. The conditions (2.16)–(2.17) are derived from the requirement that the induced map ε̃: A → A* be an isomorphism, i.e. that its inverse sends A* into A; this is a definitional requirement of a Frobenius algebra, not a fit to the target conclusion. The paper then explicitly exhibits parameters in §3.3 over Z[√−5] (z=2, µ=(2,1+√−5), a=1−√−5, ε(1)=±1, ε(X)=(1+√−5)/2, and b computed from (3.24)) and states the integrality conditions are satisfied; this is an honest verification of a concrete example. The standard facts used — Hom_O(µ,O)≅µ^{-1}, µµ=(z), and the class-group parametrization of rank-one projectives — are external mathematical theorems, not consequences of the paper's conclusion. Self-citations such as [GIK+23] and [Kho00,BN05,KR22] are used for background facts about TQFT state spaces and the standard rank-two Frobenius link-homology construction, neither of which is the paper's central claim. The prose is appropriately cautious about open points (e.g., 'We do not know a classification...' in §4), which further indicates no result is being assumed through a self-citation chain. One minor typo in Proposition 3.1 ('a ∈ O' should be 'a ∈ µ') is a correctness issue in a subcase, not a circularity. Overall, no parameter is fitted to an outcome that the paper claims to derive, and no load-bearing premise reduces to a self-citation.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The construction depends on standard algebraic number theory and module theory; no new physical or algebraic entities are postulated. The parameters a, b, ε(1), ε(X) are structure constants chosen to satisfy derived integrality conditions, not fitted to data. The main load-bearing axioms are the structure theorem for projective modules over Dedekind domains, the dual-ideal identification, and the equivalence between 2D TQFTs and Frobenius algebras. Chebotarev's theorem is used only to justify taking µ prime in one classification case.

free parameters (4)
  • a (rescaled coefficient in X² = z^{-1}aX + z^{-1}b) = a = 1-√-5 in the Z[√-5] example; generally any element of µ
    Chosen by hand; controls the X term in multiplication. Its integrality a ∈ µ is derived, but the specific value is arbitrary.
  • b (rescaled coefficient) = b = √-5-6 for the ε(1) = 1, s = 1 example; in the εX = 0 case any unit of O
    Chosen by hand; controls the constant term of X² and must satisfy b ∈ O, with b ∈ O× in several families.
  • ε(1) (trace of unit) = 1 in the example; arbitrary element of O, a unit in the εX = 0 case
    Chosen by hand; the trace can be rescaled by units, so this is a gauge choice in the construction.
  • ε(X) (trace of X) = (1+√-5)/2 in the example; 0 or 1 in other families
    Chosen in z^{-1}µ; different values give different Frobenius structures, and the paper classifies several cases.
assumptions (6)
  • standard math Structure theorem for finitely generated projective modules over a Dedekind domain: M ≅ O^{n-1} ⊕ P for a rank-one projective P.
    Used in Section 1 and Section 2.1 to write A ≅ O ⊕ µX and to deduce that P must have order at most two in Cl(O).
  • standard math Dual of an ideal in a Dedekind domain: Hom_O(µ, O) ≅ µ^{-1}, and if µ² = (z) then µ^{-1} = z^{-1}µ.
    Used in (2.3)-(2.4) to parametrize traces and compute A*. This is the load-bearing premise for the integrality conditions.
  • standard math Classification of ideals in Dedekind domains: any ideal can be generated by two elements, and for µ² = (z), multiplication µ ⊗ µ → (z) gives an isomorphism µ ⊗ µ ≅ O.
    Used in Section 2.1 around (2.47) and in Section 4 for dot cancellation and kernel computations.
  • standard math Every element of the ideal class group of a number field can be represented by a prime ideal (Chebotarev density theorem).
    Used in Section 3.2.1 to simplify the ε(X) = 1 classification by assuming µ is prime; not needed for the explicit examples.
  • domain assumption 2D oriented TQFTs over a ring R are equivalent to commutative Frobenius R-algebras.
    Used to translate the constructed Frobenius algebras into TQFT statements in the introduction and Section 4; standard but a theorem rather than a definition.
  • domain assumption The standard Khovanov link homology cube construction applies to any rank-two Frobenius extension, as in [Kho06, BN05, KR22].
    Used in the application section; the paper does not re-derive this and explicitly leaves full invariance open.

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Pith. "Pith review of Two-dimensional topological quantum field theories of rank two over Dedekind domains." pith.science (2026). https://pith.science/paper/QEP4O5EN

@misc{pith2026250204502,
  author       = {Pith},
  title        = {Pith review of: Two-dimensional topological quantum field theories of rank two over Dedekind domains},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QEP4O5EN}},
  note         = {Machine review of arXiv:2502.04502}
}
read the original 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.

Figures

Figures reproduced from arXiv: 2502.04502 by the authors.

Figure 4.0
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 D1 on the right hand side of the RI move. Right: Depicting the tensor product A ⊗ µ by a dot on a strand; two adjacent dots can be removed due to the isomorphism µ⊗O µ ≅ O. A Frobenius algebra A ≅ R[X]/(X2−aX−b) over the ground ring R, of rank two with a basis {1, X} and a fixed Frobenius structur… view at source ↗
Figure 4.0
Figure 4.0. 2. Left: RI’ move, a variation on the Reidemeister I move. The tensor product A ⊗ µ is depicted by a dot on a strand. Middle: two adjacent dots on a strand can be canceled due to an isomorphism µ ⊗O µ ≅ O. Right: dots can be moved out of the strands to the plane, with the canceling cobordism on a pair of dots shown. C(D) as above, using the tensor powers of A over O. The complex is singly-graded (no q-grading) and h… view at source ↗

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Works this paper leans on

32 extracted references · 29 canonical work pages

  1. [1]

    Aphelli, Is every element in the class group is represented by a prime ideal?, Mathematics Stack Exchange, 2021, https://math.stackexchange.com/q/4122212 (version: 2021-04-30)

    work page arXiv 2021

  2. [2]

    Dror Bar-Natan, Khovanov's homology for tangles and cobordisms, Geom. Topol. 9 (2005), 1443--1499

    work page 2005

  3. [3]

    Sabin Cautis and Joel Kamnitzer, Knot homology via derived categories of coherent sheaves. II . sl _m case , Invent. Math. 174 (2008), no. 1, 165--232

    work page 2008

  4. [4]

    Qi Chen and Thang Le, Almost integral TQFT s from simple L ie algebras , Algebr. Geom. Topol. 5 (2005), 1291--1314

    work page 2005

  5. [5]

    Keith Conrad, Ideal classes and the K ronecker bound , Online lecture notes for Algebraic Number Theory (preprint), https://kconrad.math.uconn.edu/blurbs/gradnumthy/classgroupKronecker.pdf (2019), 1--16

    work page 2019

  6. [6]

    Freed, Short-range entanglement and invertible field theories, arXiv preprint arXiv:1406.7278 https://arxiv.org/abs/1406.7278 (2014), 1--60

    Daniel S. Freed, Short-range entanglement and invertible field theories, arXiv preprint arXiv:1406.7278 https://arxiv.org/abs/1406.7278 (2014), 1--60

    arXiv 2014

  7. [7]

    Paul Gustafson, Mee Seong Im, Remy Kaldawy, Mikhail Khovanov, and Zachary Lihn, Automata and one-dimensional TQFT s with defects , Lett. Math. Phys. 113 (2023), no. 5, Paper No. 93, 38

    work page 2023

  8. [8]

    High Energy Phys

    Sergei Gukov, Pavel Putrov, and Cumrun Vafa, Fivebranes and 3-manifold homology, J. High Energy Phys. (2017), no. 7, 071, front matter+80

    work page 2017

Show all 32 references
  1. [9]

    Stavros Garoufalidis, Peter Scholze, Campbell Wheeler, and Don Zagier, The H abiro ring of a number field , arXiv preprint arXiv:2412.04241 https://www.arxiv.org/abs/2412.04241 (2024), 1--71

    2024 arXiv

  2. [10]

    Luc Guyot, Finding prime ideals for ideal classes in arbitrary dedekind domains, MathOverflow, 2024, https://mathoverflow.net/q/482280 (version: 2024-11-17)

    2024

  3. [11]

    Kazuo Habiro, On the quantum sl_2 invariants of knots and integral homology spheres , Invariants of knots and 3-manifolds ( K yoto, 2001), Geom. Topol. Monogr., vol. 4, Geom. Topol. Publ., Coventry, 2002, pp. 55--68

    2001

  4. [12]

    , Cyclotomic completions of polynomial rings, Publ. Res. Inst. Math. Sci. 40 (2004), no. 4, 1127--1146

    2004

  5. [13]

    Masaki Kashiwara, Crystalizing the q -analogue of universal enveloping algebras , Comm. Math. Phys. 133 (1990), no. 2, 249--260

    1990

  6. [14]

    , On crystal bases of the Q -analogue of universal enveloping algebras , Duke Math. J. 63 (1991), no. 2, 465--516

    1991

  7. [15]

    Mikhail Khovanov, A categorification of the J ones polynomial , Duke Math. J. 101 (2000), no. 3, 359--426

    2000

  8. [16]

    , sl (3) link homology , Algebr. Geom. Topol. 4 (2004), 1045--1081

    2004

  9. [17]

    , Link homology and F robenius extensions , Fund. Math. 190 (2006), 179--190

    2006

  10. [18]

    Mikhail Khovanov and Lev Rozansky, Matrix factorizations and link homology, Fund. Math. 199 (2008), no. 1, 1--91

    2008

  11. [19]

    Mikhail Khovanov and Louis-Hadrien Robert, Link homology and F robenius extensions II , Fund. Math. 256 (2022), no. 1, 1--46

    2022

  12. [20]

    Eun Soo Lee, An endomorphism of the K hovanov invariant , Adv. Math. 197 (2005), no. 2, 554--586

    2005

  13. [21]

    George Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), no. 2, 447--498

    1990

  14. [22]

    auser Classics, Birkh\

    , Introduction to quantum groups, Modern Birkh\"auser Classics, Birkh\"auser/Springer, New York, 2010, Reprint of the 1994 edition

    2010

  15. [23]

    Ciprian Manolescu, Link homology theories from symplectic geometry, Adv. Math. 211 (2007), no. 1, 363--416

    2007

  16. [24]

    190, Springer-Verlag, New York, 2005

    Maruti Ram Murty and Jody Esmonde, Problems in algebraic number theory, second ed., Graduate Texts in Mathematics, vol. 190, Springer-Verlag, New York, 2005

    2005

  17. [25]

    Turaev, Invariants of 3 -manifolds via link polynomials and quantum groups , Invent

    Nicolai Reshetikhin and Vladimir G. Turaev, Invariants of 3 -manifolds via link polynomials and quantum groups , Invent. Math. 103 (1991), no. 3, 547--597

    1991

  18. [26]

    David E. V. Rose and Paul Wedrich, Deformations of colored sl _N link homologies via foams , Geom. Topol. 20 (2016), no. 6, 3431--3517

    2016

  19. [27]

    11 (2020), no

    Louis-Hadrien Robert and Emmanuel Wagner, A closed formula for the evaluation of foams, Quantum Topol. 11 (2020), no. 3, 411--487

    2020

  20. [28]

    Hans Sachs, Is every ideal class represented by a prime?, Mathematics Stack Exchange, 2016, https://math.stackexchange.com/q/1883790 (version: 2016-08-05)

    2016

  21. [29]

    Christopher Schommer-Pries, Invertible topological field theories, J. Topol. 17 (2024), no. 2, Paper No. e12335, 64

    2024

  22. [30]

    Vladimir Turaev and Paul Turner, Unoriented topological quantum field theory and link homology, Algebr. Geom. Topol. 6 (2006), 1069--1093

    2006

  23. [31]

    Ben Webster, Knot invariants and higher representation theory, Mem. Amer. Math. Soc. 250 (2017), no. 1191, v+141

    2017

  24. [32]

    Edward Witten, Quantum field theory and the J ones polynomial , Comm. Math. Phys. 121 (1989), no. 3, 351--399

    1989

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