Quantum invariants of 3-manifolds via link surgery presentations and non-semi-simple categories

Abstract

In this paper we construct invariants of 3-manifolds "à la Reshetikhin-Turaev" in the setting of non-semi-simple ribbon tensor categories. We give concrete examples of such categories which lead to a family of 3-manifold invariants indexed by the integers. We prove this family of invariants has several notable features, including: they can be computed via a set of axioms, they distinguish homotopically equivalent manifolds that the standard Reshetikhin-Turaev-Witten invariants do not, and they allow the statement of a version of the Volume Conjecture and a proof of this conjecture for an infinite class of links.

46 pages, 45 figures, new introduction, some misprints corrected and an example detailed. An appendix added to correct a proof

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Cited by in corpus (45)

• Some remarks on the unrolled quantum group of sl(2)
• Rozansky-Witten geometry of Coulomb branches and logarithmic knot invariants
• 3-Dimensional TQFTs From Non-Semisimple Modular Categories
• On ribbon categories for singlet vertex algebras
• The trace on projective representations of quantum groups
• Non-Semisimple Quantum Invariants and TQFTs from Small and Unrolled Quantum Goups
• Logarithmic Hennings invariants for restricted quantum sl(2)
• Invertible braided tensor categories
• Relations between Witten-reshetikhin-turaev and non semi-simple sl(2) 3-manifold invariants
• Non-Semisimple Extended Topological Quantum Field Theories
• Tensor structure on the Kazhdan-Lusztig category for affine $\mathfrak{gl}(1|1)$
• Mapping Class Group Representations From Non-Semisimple TQFTs
• Ribbon tensor structure on the full representation categories of the singlet vertex algebras
• Renormalized Hennings Invariants and 2+1-TQFTs
• Rigid tensor structure on big module categories for some $W$-(super)algebras in type $A$
• Modular Categories and TQFTs Beyond Semisimplicity
• Colored HOMFLY-PT polynomials of quasi-alternating $3$-braid knots
• Modified graded Hennings invariants from unrolled quantum groups and modified integral
• On the volume conjecture for polyhedra
• Pseudo-Hermitian Levin-Wen models from non-semisimple TQFTs
• Extended TQFTs From Non-Semisimple Modular Categories
• Kuperberg and Turaev-Viro Invariants in Unimodular Categories
• The ADO Invariants are a q-Holonomic Family
• Kauffman bracket skein modules of small 3-manifolds
• A Hermitian TQFT from a non-semisimple category of quantum sl(2)-modules
• Universal quantum computation using Ising anyons from a non-semisimple Topological Quantum Field Theory
• Quantum holonomic link invariants derived from stated skein algebras
• Non-semisimple Levin-Wen Models and Hermitian TQFTs from quantum (super)groups
• Non semi-simple sl(2) quantum invariants, spin case
• Categories of Weight Modules for Unrolled Restricted Quantum Groups at Roots of Unity
• A Hennings type invariant of $3$-manifolds from a topological Hopf superalgebra
• Modified trace from pivotal Hopf $G$-coalgebra
• Quantum Invariants of 3-Manifolds Arising from Non-Semisimple Categories
• Modified Turaev-Viro Invariants from quantum sl(2|1)
• G-links invariants, Markov traces and the semi-cyclic Uqsl2-modules
• Holonomy braidings, biquandles and quantum invariants of links with $S{L}_2(\mathbb{C})$ flat connections
• Non-Semisimple 3-Manifold Invariants Derived From the Kauffman Bracket
• Density and unitarity of the Burau representation from a non-semisimple TQFT
• Yokota type invariants derived from non-integral highest weight representations of $U_q(s{l}_2)$
• Hennings TQFTs for Cobordisms Decorated With Cohomology Classes
• A supergroup series for knot complements
• Non-Semisimple TQFT's and BPS $q$-Series
• Vassiliev invariants for knots from the ADO polynomials
• Towards a q-series for osp(2|2n)
• Anomaly-free TQFTs from the super Lie algebra $\mathfrak{sl}(2|1)$