On the Connectivity of Cobordisms and Half-Projective TQFT's

Abstract

We consider a generalization of the axioms of a TQFT, so called half-projective TQFT's, with an anomaly, $x^{\mu }$, in the composition law. $\mu$ is a coboundary on the cobordism categories with non-negative, integer values. The element $x$ of the ring over which the TQFT is defined does not have to be invertible. In particular, it may be 0. This modification makes it possible to extend quantum-invariants, which vanish on $S^1\times S^2$, to non-trivial TQFT's. (A TQFT in the sense of Atiyah with this property has to be trivial all together). Under a few natural assumptions the notion of a half-projective TQFT is shown to be the only possible generalization. Based on separate work with Lyubashenko on connected TQFT's, we construct a large class of half-projective TQFT's with $x=0$. Their invariants vanish on $S^1\times S^2$, and they coincide with the Hennings invariant for non-semisimple Hopf algebras. Several toplogical tools that are relevant for vanishing properties of such TQFT's are developed. They are concerned with connectivity properties of cobordisms, as for example maximal non-separating surfaces. We introduce in particular the notions of ``interior'' homotopy and homology groups, and of coordinate graphs, which are functions on cobordisms with values in the morphisms of a graph category. For applications we will prove that half-projective TQFT's with $x=0$ vanish on cobordisms with infinite interior homology, and we argue that the order of divergence of the TQFT on a cobordism in the ``classical limit'' can be estimated by the rank of its maximal free interior group.

55 pages, Latex

References in corpus (3)

• Genealogy of Nonperturbative Quantum-Invariants of 3-Manifolds: The Surgical Family
• Decomposition of the adjoint representation of the small quantum $s{l}_2$
• Bridged Links and Tangle Presentations of Cobordism Categories

Cited by in corpus (12)

• Genealogy of Nonperturbative Quantum-Invariants of 3-Manifolds: The Surgical Family
• 3-Dimensional TQFTs From Non-Semisimple Modular Categories
• On the relation between the WRT invariant and the Hennings invariant
• Renormalized Hennings Invariants and 2+1-TQFTs
• Modular Categories and TQFTs Beyond Semisimplicity
• On Two Invariants of Three Manifolds from Hopf Algebras
• Three-Dimensional 2-Framed TQFTs and Surgery
• Resolutions of p-Modular TQFT's and Representations of Symmetric Groups
• Integrality and Gauge Dependence of Hennings TQFTs
• Homology TQFT's and the Alexander-Reidemeister Invariant of 3-Manifolds via Hopf Algebras and Skein Theory
• Non-semisimple link and manifold invariants for symplectic fermions
• Quantum invariants of 3-manifolds associated to restricted quantum groups