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Non-semisimple open-closed 3d TFT

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Pith's one-line read The paper constructs an open-closed three-dimensional TFT from any spherical finite tensor category with a two-sided modified trace, with finite-dimensional state spaces described by admissible skein modules.

desk verdict A genuine non-semisimple open-closed 3d TFT with finite-dimensional state spaces; the spanning-tree red-loop device is the real new idea, and the main risk is the long proof of Prop. 5.8. read the letter →

arxiv 2608.08057 v1 pith:D3MH3AQT submitted 2026-08-08 math.QA math-phmath.GTmath.MP

Ingo Runkel , Yilong Wang This is my paper · ORCID
classification math.QAmath-phmath.GTmath.MP
keywords sphericalfinitetensorcategorymodifiedtraceopen-closedtopologicalfieldtheoryadmissibleskeinmodulebulk-admissiblegraphchromaticmapsuniversalconstructionnon-semisimple3dTFT
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

State-sum constructions of three-dimensional topological field theories normally require semisimple input categories. This paper removes that restriction: starting from a spherical finite tensor category, possibly non-semisimple, equipped with a two-sided modified trace, it constructs an open-closed three-dimensional topological field theory with values in vector spaces. The construction first defines a new invariant of compact oriented 3-manifolds whose boundary carries an admissible $\mathcal{C}$-coloured graph, then applies the universal construction. The payoff is that every such category now yields a concrete 3d TFT whose state spaces are finite-dimensional and are quotients of admissible skein modules, so skein theory gives an explicit description of the quantum state spaces.

What carries the argument

The load-bearing machinery is the $(2,3)$-graph of a PLCW decomposition of a 3-manifold: one vertex for each 3-cell plus a boundary vertex, and one edge for each 2-cell. Given a spanning tree of this graph, the construction adds red loops only along the belts of 2-cells outside the tree; those red loops are then converted into blue projective strands by chromatic maps, the red-to-blue procedure, turning the boundary into an admissible bichrome graph on a handlebody that is evaluated by the multi-handlebody invariant $F$. The spanning-tree choice is what avoids red loops that bound disks, which would evaluate to zero in many non-semisimple categories. Invariance under changing the decomposition, tree, or interior points is transported by sliding moves, cutting moves, and red capping and digging moves, and the universal construction [BHMV95] converts the resulting invariant into the TFT.

What would settle it

Carry out the universal-construction pairing on a once-punctured torus for a non-semisimple spherical category in which red loops bounding disks evaluate to zero, for instance representations of a finite-dimensional unimodular unibalanced Hopf algebra over a field where the dimension of the algebra is zero. The surjection from the admissible skein module predicts a specific finite dimension for $V$; a smaller dimension, or any nonzero class of the skein module lying in the radical of the pairing, would contradict Proposition 5.8 and with it Theorem 5.10.

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Extended reading notes

Core claim

The paper's central claim is Theorem 5.10: for any spherical finite tensor category $\mathcal{C}$ with a chosen two-sided modified trace, the invariant $\tau_{\mathcal{C}}$ extends through the universal construction to a symmetric monoidal functor $V:\mathrm{Bord}^{\mathrm{nc}}_{\mathrm{oc}}(\mathcal{C})\to\mathrm{Vect}_k$, an open-closed 3-dimensional TFT. The invariant $\tau_{\mathcal{C}}(M,\Gamma)$ is defined for a compact oriented 3-manifold $M$ with boundary and a bulk-admissible $\mathcal{C}$-coloured graph $\Gamma$ in $\partial M$, meaning each connected component of each boundary component carries a projectively coloured edge. Theorem 4.8 states that $\tau_{\mathcal{C}}$ depends only on the diffeomorphism class of $(M,\Gamma)$, not on the auxiliary PLCW decomposition, spanning tree, or interior points. The main structural step is Proposition 5.8, which says every vector in every state space is a linear combination of collar elements; from this, Proposition 5.9 gives a surjection $E_{(\Sigma,L)}:S_{\mathrm{adm}}(\Sigma,L)\to V(\Sigma,L)$ from the admissible skein module to the state space, and finite-dimensionality follows.

Load-bearing premise

Everything rests on Proposition 5.8, that every vector in every state space is a linear combination of cylinder-shaped collar elements; if that spanning statement fails, the state spaces need not be finite-dimensional and monoidality of $V$ is not established.

Editorial extensions

If this is right

  • Every spherical finite tensor category with a two-sided modified trace, semisimple or not, determines an open-closed 3d TFT with finite-dimensional state spaces.
  • For any marked surface $(\Sigma,L)$, the state space $V(\Sigma,L)$ is a quotient of the admissible skein module $S_{\mathrm{adm}}(\Sigma,L)$, so skein-theoretic computations give concrete upper bounds on state-space dimensions.
  • The new invariant $\tau_{\mathcal{C}}$ reproduces the closed 3-manifold invariant of [CGPT18] when a ball is removed from a closed manifold, so the known non-semisimple closed invariants fit inside the open-closed theory.
  • When $\mathcal{C}$ is semisimple, $\tau_{\mathcal{C}}$ is a non-zero scalar multiple of the alterfold invariant, and the open-closed state spaces agree with alterfold state spaces up to isomorphism (Remark 5.13).
  • Even though the bordism category is not rigid, the universal construction here runs through finite-dimensional vector spaces, so the TFT is genuinely finite-dimensional on every object.

Reading between the lines

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

  • The spanning-tree trick is transferable: any state-sum or surgery construction that decorates 2-skeleton belts with red loops can prune loops using a spanning tree, which may remove spurious zero evaluations in categories whose global dimension or cointegral evaluation vanishes.
  • If the expected agreement with the admissible-skein TFT on closed surfaces holds, then the surjection $E_{(\Sigma,L)}$ should actually be an isomorphism on closed surfaces, giving a skein-theoretic presentation of the TFT rather than a radical quotient; this is testable surface-by-surface.
  • Following the usual state-sum/surgery correspondence, $V$ should coincide with the surgery-based TFT associated to the Drinfeld center of $\mathcal{C}$ from [DGG+19]; a proof would extend that correspondence to open-closed non-semisimple bordisms.
  • Explicit low-genus computations for categories like representations of small quantum groups at roots of unity would give concrete dimensions of the new state spaces and test the collar-element spanning statement against known admissible skein module dimensions.
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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 defines an invariant τC of compact oriented 3-manifolds with boundary decorated by a bulk-admissible C-coloured graph, where C is a spherical finite tensor category with a two-sided modified trace. The construction combines the multi-handlebody invariant F of CGPT18 with the chromatic maps of CGPV23, and introduces a new spanning-tree rule for adding red loops when passing from a PLCW decomposition to a handlebody. The authors prove that τC is independent of the PLCW decomposition, the spanning tree, and the auxiliary interior points (Theorem 4.8), that it agrees with the closed-manifold invariant KC of CGPT18 after removing a ball (Proposition 4.11), and that it is invariant under cutting moves (Proposition 4.17). Via the universal construction, τC is then extended to a symmetric monoidal functor V: Bord^nc_oc(C) -> Vect_k (Theorem 5.10). The key technical step is Proposition 5.8, which asserts that every state of a ∂-marked surface is a linear combination of collar elements; this yields a surjection from the finite-dimensional admissible skein module to the state space (Proposition 5.9).

Significance. If correct, the paper provides a genuinely non-semisimple open-closed 3d TFT with finite-dimensional state spaces and an explicit skein-theoretic description, going beyond the semisimple alterfold and Turaev-Viro frameworks. The main strengths are the detailed classical proof of the new invariant, the novel use of spanning trees to avoid red loops that would evaluate to zero, and the clear delineation of the algebraic input from earlier work. The paper also gives a transparent comparison with the closed-manifold invariant of CGPT18 and indicates expected but not fully proved relations with the alterfold theory and with the noncompact TFT of CGPV23. The proofs are lengthy, especially the proof of Proposition 5.8, but I did not find a concrete error; the main risk is the complexity of the PLCW and handlebody arguments, which are not independently verified here.

minor comments (5)
  1. [§5.3, beginning of the proof of Proposition 5.8] The sentence 'By Lemma 5.6, we may assume that M is connected' is not immediate from the statement of Lemma 5.6, which only proves injectivity of the map Ψ. The reduction can likely be justified by applying the connected statement to each connected component of M and using the containment R1⊗~V2 + ~V1⊗R2 ⊂ R1⊔2 shown in Lemma 5.6, but this should be spelled out for the reader.
  2. [§4.2, Definition 4.5] The bullet point 'Γ intersects δ^1(Σ) only on the edges of Γ' should read 'only at points on the edges of Γ'; as written it suggests that the intersection is contained in the edges rather than consisting of transverse intersection points on those edges.
  3. [§5.2, Example 5.11] In the chain of isomorphisms (5.12), the equality dim C(P,1) = dim Sadm(D,P) is asserted without proof. A one-line justification, for instance by exhibiting a basis of Sadm(D,P) represented by the single P-labelled strand with a morphism P -> 1, would make the dimension comparison transparent.
  4. [§5.3, after Eq. (5.17)] The passage from cutting along the meridian disks D(M1 \ Σ) to a collar element is very compressed: it is stated that evaluating the resulting 3-balls with F yields a linear combination of admissible graphs on Σ, but the role of the newly introduced dual-basis coupons in making the ball graphs admissible is not explicitly justified. A short explanatory sentence would help.
  5. [Remarks 4.12 and 5.13] The comparisons with the alterfold invariant are explicitly labelled as remarks and several verifications are left to the reader. Since these comparisons are not needed for the main theorem, this is acceptable, but the sentence 'We skip the details' in Remark 5.13 should perhaps be accompanied by a precise statement of which of the listed assertions are proven and which are expected equivalences.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main invariant is built from externally cited handlebody and bichrome-graph inputs, and the universal-construction TFT is proved via an independent spanning/collar argument.

full rationale

The derivation chain is not circular. The new invariant τC is defined in Definition 4.7 as Fbi evaluated on a handlebody-neighbourhood of the 1-skeleton with a bichrome graph determined by a spanning tree. Fbi itself is imported from [CGPT18] and [CGPV23], and the paper's Theorem 4.8 proves independence of the PLCW decomposition, spanning tree, and point choices by reducing to sliding moves and red capping/cutting moves that are established there (Lemmas 3.14, 3.18, Corollaries 3.15 and 3.16). Proposition 4.10 shows that on handlebodies τC agrees with F; this is a comparison theorem, not a definitional identity, since the red loops added for non-tree 2-cells are removed via the red capping move. Proposition 4.11 similarly compares two Fbi-based closed-manifold invariants and is a consistency statement, not a prediction forced by construction. The universal construction in Section 5.2 is the standard BHMV radical-quotient construction; the main burden, Proposition 5.8, is proved internally by producing a collar element u and verifying, via τC-invariance under cutting and red capping, that [M,Γ]-u lies in the radical. That verification does not presuppose the spanning statement. Finite-dimensionality of the state spaces (Proposition 5.9) uses the surjection onto V(Σ,L) from the admissible skein module, whose finite-dimensionality is cited from [RST24, Prop. 5.6]; that cited result is parameter-free, does not mention the TFT V, and is therefore independent support even though one author is shared. Self-citations to the alterfold papers [LMW+23a,b,24a] occur only in Remarks 4.12 and 5.13 stating expected comparisons, and are not inputs to the main theorem. No equation was found in which a predicted quantity is equal by construction to a fitted or cited input, and no load-bearing claim reduces to an unverified self-citation.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No fitted numerical parameters appear. The input (C,t,π1) is an algebraic structure, not a tuned quantity. The main external dependencies are the modified-trace theorem, chromatic maps, and the multi-handlebody invariant F; these are taken as black boxes. No new postulated entities such as particles or dimensions are introduced.

assumptions (5)
  • domain assumption Every pivotal unimodular finite tensor category admits a nonzero right modified trace, unique up to scalar (Theorem 2.1, from [GKP18]).
    Invoked in Section 2.1 to define the spherical input (C, t, π1) and the normalized embedding ι1 in (2.5).
  • domain assumption Chromatic spaces C_{P,G} are non-empty and their elements satisfy the chromatic-map identity (Theorem 2.6, from [CGPV23]).
    Used in Section 2.3 and Section 3.3 to define the red-blue modification map RB and its invariance properties.
  • domain assumption There exists a unique multiplicative handlebody invariant F satisfying the cutting move (Theorem 3.8, from [CGP23]), and it extends to bichrome graphs as Fbi (Corollary 3.12).
    The new invariant τ_C is defined as Fbi of a decorated handlebody; topological invariance of τ_C inherits from Fbi.
  • standard math Any two PLCW decompositions of a compact manifold are related by a finite sequence of elementary subdivisions (Kirillov, [Kir10], Thm 8).
    Used in Lemma 4.16 to prove τ_C is independent of the PLCW decomposition.
  • standard math Any triangulation of the boundary of a compact PL-manifold extends to a triangulation of the whole manifold (Armstrong, [Arm67], Cor 1).
    Used in Corollary 5.15 and the proof of Proposition 5.8 to construct collar elements and extend triangulations across glued bordisms.

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Pith. "Pith review of Non-semisimple open-closed 3d TFT." pith.science (2026). https://pith.science/paper/D3MH3AQT

@misc{pith2026260808057,
  author       = {Pith},
  title        = {Pith review of: Non-semisimple open-closed 3d TFT},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/D3MH3AQT}},
  note         = {Machine review of arXiv:2608.08057}
}
read the original abstract

Given a spherical finite tensor category C, not necessarily semisimple, and a two-sided modified trace on its projective ideal, we define an open-closed three-dimensional topological field theory with values in vector spaces. The bordism category has as morphisms three-dimensional bordisms with corners, whose boundary is partitioned into the gluing boundary, parametrised by source and target surface, and the unparametrised free boundary. The free boundary is equipped with an embedded C-coloured graph satisfying an admissibility condition. Our construction starts from a new three-manifold invariant based on the multi-handlebody invariant of [arXiv:1809.07991] and the chromatic maps of [arXiv:2302.04509]. The open-closed topological field theory is then obtained via the universal construction.

Figures

Figures reproduced from arXiv: 2608.08057 by the authors.

Figure 1
Figure 1. The cutting move along the disk D. ∂D the edges of Γ which intersect ∂D, say, labelled by X1, ..., Xn. Then, attach the cut points to two new coupons in ∂cutD(H), one on each side of the cut. By assumption, P := X1 ⊗ · · · ⊗ Xn ∈ ProjC , which means we can colour the pair of coupons joining the two sets of the cut points by a dual basis pair as in the definition of Λt P in (2.10), see [PITH_FULL_IMAGE:figures/full_… view at source ↗
Figure 2
Figure 2. a) PLCW decomposition of a genus-g handlebody represented as g handles attached to a 3-ball. The black lines represent the 1-cells, and the gray lines are only used to indicate the shape of the handlebody. b) For j = 1, ..., g, the j-th handle has two 0-cells, four 1-cells X (1) j,± , Y (1) j,± , three 2-cells W (2) j,± and B (2) j and one 3-cell A (3) j . All the handles are attached to a ball, which has additional… view at source ↗

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Reference graph

Works this paper leans on

44 extracted references · 22 canonical work pages

  1. [1]

    Armstrong, Extending triangulations , doi:10.2307/2035442 Proc.\ Amer.\ Math.\ Soc.\ 18 (1967) 701--704

    M.\,A. Armstrong, Extending triangulations , doi:10.2307/2035442 Proc.\ Amer.\ Math.\ Soc.\ 18 (1967) 701--704

    work page doi:10.2307/2035442 1967

  2. [2]

    Barrett, B.\,W

    J.\,W. Barrett, B.\,W. Westbury, Spherical categories , doi:10.1006/aima.1998.1800 Adv.\ Math.\ 143 (1999) 357--375 , arXiv:hep-th/9310164 hep-th/9310164 [hep-th]

    arXiv 1999

  3. [3]

    The equality of 3-manifold invariants

    J.\,W. Barrett, B.\,W. Westbury, The equality of 3-manifold invariants , doi:10.1017/S0305004100073825 Math.\ Proc.\ Cambridge Philos.\ Soc.\ 118 (1995) 503--510 , arXiv:hep-th/9406019 hep-th/9406019 [hep-th]

    work page Pith review arXiv 1995

  4. [4]

    Berger, A

    J. Berger, A. M. Gainutdinov, I. Runkel, Monadic cointegrals and applications to quasi-Hopf algebras , doi:10.1016/j.jpaa.2021.106678 J.\ Pure Appl.\ Algebra 225 (2021), 106678 , arXiv:2003.13307 2003.13307 [math.QA]

    arXiv 2021

  5. [5]

    Blanchet, F

    C. Blanchet, F. Costantino, N. Geer, B. Patureau-Mirand, Non-semisimple TQFTs, Reidemeister torsion and Kashaev's invariants , doi:10.1016/j.aim.2016.06.003 Adv.\ Math.\ 301 (2016), 1--78 , arXiv:1404.7289 1404.7289 [math.GT]

    arXiv 2016

  6. [6]

    Blanchet, N

    C. Blanchet, N. Habegger, G. Masbaum, P. Vogel, Topological quantum field theories derived from the Kauffman bracket , doi:10.1016/0040-9383(94)00051-4 Topology 34 (1995), no. 4, 883--927

    doi:10.1016/0040-9383(94)00051-4 1995

  7. [7]

    Brugui\`eres, A

    A. Brugui\`eres, A. Virelizier, Hopf monads , arXiv:math/0604180 math/0604180 [math.QA]

    arXiv

  8. [8]

    The double of a Hopf monad

    A. Brugui\`eres, A. Virelizier, Quantum double of Hopf monads and categorical centers , doi:10.1090/S0002-9947-2011-05342-0 Trans.\ Amer.\ Math.\ Soc.\ 364 (2012) 1225--1279 , arXiv:0812.2443 0812.2443 [math.QA]

    work page Pith review arXiv 2012

Show all 44 references
  1. [9]

    Costantino, N

    F. Costantino, N. Geer, B. Ha\"ioun, B. Patureau-Mirand, Graded spherical skein 2+1-G-HQFT and modified Turaev-Viro invariants , arXiv:2601.02617 2601.02617 [math.GT]

  2. [10]

    Costantino, N

    F. Costantino, N. Geer, B. Patureau-Mirand, Quantum invariants of 3-manifolds via link surgery presentations and non-semi-simple categories , doi:10.1112/jtopol/jtu006 J.\ Topol.\ 7 (2014), no. 4, 1005--1053 , arXiv:1202.3553 1202.3553 [math.GT]

    2014 arXiv

  3. [11]

    Costantino, N

    F. Costantino, N. Geer, B. Patureau-Mirand, Admissible skein modules , arXiv:2302.04493 2302.04493 [math.GT]

    arXiv

  4. [12]

    Costantino, N

    F. Costantino, N. Geer, B. Patureau-Mirand, V. Turaev, Kuperberg and Turaev-Viro invariants in unimodular categories , doi:10.2140/pjm.2020.306.421 Pacific J.\ Math.\ 306 (2020), no. 2, 421--450 , arXiv:1809.07991 1809.07991 [math.QA]

    2020 arXiv

  5. [13]

    Costantino, N

    F. Costantino, N. Geer, B. Patureau-Mirand, A. Virelizier, Noncompact (2+1)-TQFTs from non-semisimple spherical categories , arXiv:2302.04509 2302.04509 [math.GT]

    arXiv

  6. [14]

    B. Day, R. Street, Centres of monoidal categories of functors , in Categories in algebra, geometry and mathematical physics , doi:10.1090/conm/431/08273 Contemp.\ Math.\ 431, Amer.\ Math.\ Soc., Providence, RI, 2007, 187--202

    2007 doi

  7. [15]

    De Renzi, A

    M. De Renzi, A. M. Gainutdinov, N. Geer, B. Patureau-Mirand, I. Runkel, 3-Dimensional TQFTs from non-semisimple modular categories , doi:10.1007/s00029-021-00737-z Sel.\ Math.\ New Ser.\ 28 (2022) 42 , arXiv:1912.02063 1912.02063 [math.GT]

    2022 arXiv

  8. [16]

    C. L. Douglas, C. Schommer-Pries, N. Snyder, Dualizable tensor categories , arXiv:1312.7188 1312.7188 [math.QA]

    arXiv

  9. [17]

    Etingof, S

    P. Etingof, S. Gelaki, D. Nikshych, V. Ostrik, Tensor categories , doi:10.1090/surv/205 Mathematical Surveys and Monographs 205, Amer.\ Math.\ Soc., Providence, RI, 2015

    2015 doi

  10. [18]

    Etingof, D

    P. Etingof, D. Nikshych, V. Ostrik, An analogue of Radford's S^4 formula for finite tensor categories , arXiv:math/0404504 math/0404504 [math.QA]

    arXiv

  11. [19]

    Grinberg, An introduction to graph theory , arXiv:2308.04512 2308.04512 [math.HO]

    D. Grinberg, An introduction to graph theory , arXiv:2308.04512 2308.04512 [math.HO]

    arXiv

  12. [20]

    N. Geer, J. Kujawa, B. Patureau-Mirand, Generalized trace and modified dimension functions on ribbon categories , arXiv:1001.0985 1001.0985 [math.RT]

    arXiv

  13. [21]

    N. Geer, J. Kujawa, B. Patureau-Mirand, M-traces in (non-unimodular) pivotal categories , doi:10.1007/s10468-021-10044-y Algebr.\ Represent.\ Theory 25 (2022), no. 3, 759--776 . arXiv:1809.00499 1809.00499 [math.RT]

    2022 arXiv

  14. [22]

    N. Geer, B. Patureau-Mirand, Quantum topology via modified traces: From semisimple to non-semisimple , to appear

  15. [23]

    N. Geer, B. Patureau-Mirand, A. Virelizier, Traces on ideals in pivotal categories , doi:10.4171/QT/36 Quantum Topol.\ 4 (2013), no. 1, 91--124 , arXiv:1103.1660 1103.1660 [math.QA]

    2013 arXiv

  16. [24]

    Hofer, I

    A. Hofer, I. Runkel, Modular functors from non-semisimple 3d TFTs , arXiv:2405.18038 2405.18038 [math.QA]

    arXiv

  17. [25]

    Juh\'asz, Defining and classifying TQFTs via surgery , doi:10.4171/QT/108 Quantum Topol.\ 9 (2018), no

    A. Juh\'asz, Defining and classifying TQFTs via surgery , doi:10.4171/QT/108 Quantum Topol.\ 9 (2018), no. 2, 229--321 , arXiv:1408.0668 1408.0668 [math.GT]

    2018 arXiv

  18. [26]

    Kerler, V

    T. Kerler, V. V. Lyubashenko, Non-semisimple topological quantum field theories for 3-manifolds with corners , Lecture Notes in Mathematics 1765, Springer-Verlag, Berlin, 2001

    2001

  19. [27]

    Kirillov, Jr., On piecewise linear cell decompositions , doi:10.2140/agt.2012.12.95 Algebr.\ Geom.\ Topol.\ 12 (2012), no

    A. Kirillov, Jr., On piecewise linear cell decompositions , doi:10.2140/agt.2012.12.95 Algebr.\ Geom.\ Topol.\ 12 (2012), no. 1, 95--108 , arXiv:1009.4227 1009.4227 [math.GT]

    2012 arXiv

  20. [28]

    A. D. Lauda, H. Pfeiffer, Open-closed strings: two-dimensional extended TQFTs and Frobenius algebras , doi:10.1016/j.topol.2007.11.005 Topology Appl.\ 155 (2008), no. 7, 623--666 , arXiv:math/0510664 math/0510664 [math.QA]

    2008 arXiv

  21. [29]

    Z. Liu, S. Ming, Y. Wang, J. Wu, 3-alterfolds and quantum invariants , arXiv:2307.12284 2307.12284 [math.GT]

    arXiv

  22. [30]

    Z. Liu, S. Ming, Y. Wang, J. Wu, Alterfold topological quantum field theory , doi:10.1007/s11425-026-05590-9 SCIENCE CHINA Mathematics (2026) , arXiv:2312.06477 2312.06477 [math-ph]

    2026 arXiv

  23. [31]

    Z. Liu, S. Ming, Y. Wang, J. Wu, Alterfold Theory and Topological Modular Invariance , doi:10.1007/s00220-026-05590-9 Comm.\ Math.\ Phys.\ 407 (2026), no. 5, Paper No. 102 . arXiv:2412.12702 2412.12702 [math.QA]

    2026 arXiv

  24. [32]

    Lyubashenko, Modular transformations for tensor categories , doi:10.1016/0022-4049(94)00045-K J.\ Pure Appl.\ Algebra 98 (1995), no

    V. Lyubashenko, Modular transformations for tensor categories , doi:10.1016/0022-4049(94)00045-K J.\ Pure Appl.\ Algebra 98 (1995), no. 3, 279--327

    1995 doi

  25. [33]

    M\"uller, L

    L. M\"uller, L. Woike, The distinguished invertible object as ribbon dualizing object in the Drinfeld center , arXiv:2401.02792 2401.02792 [math.QA]

    arXiv

  26. [34]

    Runkel, C

    I. Runkel, C. Schweigert, Y. H. Tham, Excision for spaces of admissible skeins , arXiv:2407.09302 2407.09302 [math.QA]

    arXiv

  27. [35]

    Reutter, K

    D. Reutter, K. Walker, From non-unital skein theory to modified traces and non-semisimple topological field theories https://www.davidreutter.com/#h.p_e6SqJNffk9Sy, talk by D. Reutter at ``Lecture and seminar series on modified traces in algebra and topology https://him-applic...

    2020

  28. [36]

    Shibata, K

    T. Shibata, K. Shimizu, Modified traces and the Nakayama functor , doi:10.1007/s10468-021-10102-5 Algebr.\ Represent.\ Theory 26 (2023), no. 2, 513--551 , arXiv:2109.10044 2109.10044 [math.QA]

    2023 arXiv

  29. [37]

    Shimizu, The monoidal center and the character algebra , doi:10.1016/j.jpaa.2016.12.037 J.\ Pure Appl.\ Algebra 221 (2017), no

    K. Shimizu, The monoidal center and the character algebra , doi:10.1016/j.jpaa.2016.12.037 J.\ Pure Appl.\ Algebra 221 (2017), no. 9, 2338--2371 , arXiv:1504.01178 1504.01178 [math.QA]

    2017 arXiv

  30. [38]

    Shimizu, Non-degeneracy conditions for braided finite tensor categories , doi:10.1016/j.aim.2019.106778 Adv.\ Math.\ 355 (2019), 106778 , arXiv:1602.06534 1602.06534 [math.QA]

    K. Shimizu, Non-degeneracy conditions for braided finite tensor categories , doi:10.1016/j.aim.2019.106778 Adv.\ Math.\ 355 (2019), 106778 , arXiv:1602.06534 1602.06534 [math.QA]

    2019

  31. [39]

    Shimizu, Ribbon structures of the Drinfeld center of a finite tensor category , doi:10.2996/kmj46106 Kodai Math.\ J.\ 46 (2023), no

    K. Shimizu, Ribbon structures of the Drinfeld center of a finite tensor category , doi:10.2996/kmj46106 Kodai Math.\ J.\ 46 (2023), no. 1, 75--114 , arXiv:1707.09691 1707.09691 [math.QA]

    2023 arXiv

  32. [40]

    Shimizu, Integrals for finite tensor categories , doi:10.1007/s10468-018-9777-5 Algebr.\ Represent.\ Theory 22 (2019), no

    K. Shimizu, Integrals for finite tensor categories , doi:10.1007/s10468-018-9777-5 Algebr.\ Represent.\ Theory 22 (2019), no. 2, 459--493 , arXiv:1702.02425 1702.02425 [math.CT]

    2019 arXiv

  33. [41]

    Steffen, C

    M.-N. Steffen, C. Schweigert, A boundary characterization of Turaev-Viro TQFTs , arXiv:2508.13759 2508.13759 [math.QA]

    arXiv

  34. [42]

    Turaev, A

    V. Turaev, A. Virelizier, Monoidal Categories and Topological Field Theory , doi:10.1007/978-3-319-49834-8 Progress in Mathematics 322, Birkh\"auser, Cham, 2017

    2017 doi

  35. [43]

    V. G. Turaev, O. Ya. Viro, State sum invariants of 3-manifolds and quantum 6j-symbols , doi:10.1016/0040-9383(92)90015-A Topology 31 (1992), no. 4, 865--902

    1992 doi

  36. [44]

    West, Introduction to Graph Theory , Prentice Hall, 2001

    D.\,B. West, Introduction to Graph Theory , Prentice Hall, 2001

    2001

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.