Non-semisimple link and manifold invariants for symplectic fermions

[2307.06069] Non-semisimple link and manifold invariants for symplectic fermions Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Quantum Algebra arXiv:2307.06069 (math) [Submitted on 12 Jul 2023 ( v1 ), last revised 14 Apr 2026 (this version, v2)] Title: Non-semisimple link and manifold invariants for symplectic fermions Authors: Johannes Berger , Azat M. Gainutdinov , Ingo Runkel View a PDF of the paper titled Non-semisimple link and manifold invariants for symplectic fermions, by Johannes Berger and 2 other authors View PDF HTML (experimental) Abstract: We consider the link and three-manifold invariants from arXiv:1912.02063 , which are defined in terms of certain non-semisimple finite ribbon categories $\mathcal{C}$ together with a choice of tensor ideal and modified trace. If the ideal is all of $\mathcal{C}$, these invariants agree with those defined by Lyubashenko in the 90's. We show that in that case the invariants depend on the objects labelling the link only through their simple composition factors, so that in order to detect non-trivial extensions one needs to pass to proper ideals. We compute examples of link and three-manifold invariants for $\mathcal{C}$ being the category of $N$ pairs of symplectic fermions. Using a quasi-Hopf algebra realisation of $\mathcal{C}$, we find that the Lyubashenko-invariant of a lens space is equal to the order of its first homology group to the power $N$, a relation we conjecture to hold for all rational homology spheres. For $N \ge 2$, $\mathcal{C}$ allows for tensor ideals $\mathcal{I}$ with a modified trace which are different from all of $\mathcal{C}$ and from the projective ideal. Using the theory of pull-back traces and symmetrised cointegrals, we show that the link invariant obtained from $\mathcal{I}$ can distinguish a continuum of indecomposable but reducible objects which all have the same composition series. Comments: 77 pages; v2: Final version published in Quantum Topology, with more general framework of locally finite tensor categories for Sec 2, new Example 2.2, references corrected in Sec 3.1, Prop 3.13 was corrected, a typo in Prop 4.2 fixed, new Remarks 2.12 & 4.8 & 5.4 Subjects: Quantum Algebra (math.QA) ; Geometric Topology (math.GT) MSC classes: 16T99 Cite as: arXiv:2307.06069 [math.QA] (or arXiv:2307.06069v2 [math.QA] for this version) https://doi.org/10.48550/arXiv.2307.06069 Focus to learn more arXiv-issued DOI via DataCite Journal reference: Quantum Topology (2026) Related DOI : https://doi.org/10.4171/QT/256 Focus to learn more DOI(s) linking to related resources Submission history From: Azat Gainutdinov [ view email ] [v1] Wed, 12 Jul 2023 10:38:58 UTC (96 KB) [v2] Tue, 14 Apr 2026 11:35:45 UTC (100 KB) Full-text links: Access Paper: View a PDF of the paper titled Non-semisimple link and manifold invariants for symplectic fermions, by Johannes Berger and 2 other authors View PDF HTML (experimental) TeX Source view license Current browse context: math.QA < prev | next > new | recent | 2023-07 Change to browse by: math math.GT References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

Related documents

• On the Hofer-Zehnder conjecture for semipositive symplectic manifolds#128267
• Exact group invariant scar towers in two dimensional gauge theories#1110515
• The Dijkgraaf-Witten invariants from second Chern classes#365116
• A Chern-Class Obstruction to Measurable Eigensections of Analytic Quasi-Periodic Bundle Cocycles#1077505
• Exact solutions of hidden free fermion models#1083604