Integrality and Gauge Dependence of Hennings TQFTs

[1305.7498] Integrality and Gauge Dependence of Hennings TQFTs Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Geometric Topology arXiv:1305.7498 (math) [Submitted on 31 May 2013 ( v1 ), last revised 26 May 2016 (this version, v3)] Title: Integrality and Gauge Dependence of Hennings TQFTs Authors: Qi Chen , Thomas Kerler View a PDF of the paper titled Integrality and Gauge Dependence of Hennings TQFTs, by Qi Chen and 1 other authors View PDF HTML (experimental) Abstract: We provide a general construction of integral TQFTs over a general commutative ring, $\mathbf{k}$, starting from a finite Hopf algebra over $\mathbf{k}$ which is Frobenius and double balanced. These TQFTs specialize to the Hennings invariants of the respective doubles on closed 3-manifolds. We show the construction applies to index 2 extensions of the Borel parts of Lusztig's small quantum groups for all simple Lie types, yielding integral TQFTs over the cyclotoic integers for surfaces with boundary. We further establish and compute isomorphisms of TQFT functors constructed from Hopf algebras that are related by a strict gauge transformation in the sense of Drinfeld. Formulas for the natural isomorphisms are given in terms of the gauge twist element. These results are combined and applied to show that the Hennings invariant associated to quantum-$sl_2$ takes values in the cyclotomic integers. Using prior results of Chen et al we infer integrality also of the Witten-Reshetikhin-Turaev $SO(3)$ invariant for rational homology spheres. As opposed to most other approaches the methods described in this article do not invoke calculations of skeins, knots polynomials, or representation theory, but follow a combinatorial construction that uses only the elements and operations of the underlying Hopf algebras. Comments: Algebraic criteria for underlying Hopf algebra and commutative ring have been corrected, including several lemmas, propositions, and theorems depending on these. (The prior Dedekind condition did not suffice due to possible non-trivial Picard groups. It has been replaced by a Frobenius condition) Subjects: Geometric Topology (math.GT) ; Quantum Algebra (math.QA) MSC classes: 57R56, 57M27, 81R50 Cite as: arXiv:1305.7498 [math.GT] (or arXiv:1305.7498v3 [math.GT] for this version) https://doi.org/10.48550/arXiv.1305.7498 Focus to learn more arXiv-issued DOI via DataCite Journal reference: J. Pure Appl. Algebra 21 (2017) no. 11, 2689-2751 Submission history From: Thomas Kerler [ view email ] [v1] Fri, 31 May 2013 18:17:38 UTC (477 KB) [v2] Tue, 11 Feb 2014 03:22:31 UTC (486 KB) [v3] Thu, 26 May 2016 22:35:14 UTC (489 KB) Full-text links: Access Paper: View a PDF of the paper titled Integrality and Gauge Dependence of Hennings TQFTs, by Qi Chen and 1 other authors View PDF HTML (experimental) TeX Source view license Current browse context: math.GT < prev | next > new | recent | 2013-05 Change to browse by: math math.QA References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... 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