dorsal/arxiv
View SchemaOn the Connectivity of Cobordisms and Half-Projective TQFT's
| Authors | Thomas Kerler |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9603017 |
| URL | https://arxiv.org/abs/q-alg/9603017 |
| DOI | 10.1007/s002200050487 |
| Journal | Commun.Math.Phys. 198 (1998) 535-590 |
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.
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"abstract": "We consider a generalization of the axioms of a TQFT, so called\nhalf-projective TQFT\u0027s, with an anomaly, $x^{\\mu}$, in the composition law.\n$\\mu$ is a coboundary on the cobordism categories with non-negative, integer\nvalues. The element $x$ of the ring over which the TQFT is defined does not\nhave to be invertible. In particular, it may be 0. This modification makes it\npossible to extend quantum-invariants, which vanish on $S^1\\times S^2$, to\nnon-trivial TQFT\u0027s. (A TQFT in the sense of Atiyah with this property has to be\ntrivial all together). Under a few natural assumptions the notion of a\nhalf-projective TQFT is shown to be the only possible generalization. Based on\nseparate work with Lyubashenko on connected TQFT\u0027s, we construct a large class\nof half-projective TQFT\u0027s with $x=0$. Their invariants vanish on $S^1\\times\nS^2$, and they coincide with the Hennings invariant for non-semisimple Hopf\nalgebras. Several toplogical tools that are relevant for vanishing properties\nof such TQFT\u0027s are developed. They are concerned with connectivity properties\nof cobordisms, as for example maximal non-separating surfaces. We introduce in\nparticular the notions of ``interior\u0027\u0027 homotopy and homology groups, and of\ncoordinate graphs, which are functions on cobordisms with values in the\nmorphisms of a graph category. For applications we will prove that\nhalf-projective TQFT\u0027s with $x=0$ vanish on cobordisms with infinite interior\nhomology, and we argue that the order of divergence of the TQFT on a cobordism\nin the ``classical limit\u0027\u0027 can be estimated by the rank of its maximal free\ninterior group.",
"arxiv_id": "q-alg/9603017",
"authors": [
"Thomas Kerler"
],
"categories": [
"q-alg",
"hep-th",
"math.QA"
],
"doi": "10.1007/s002200050487",
"journal_ref": "Commun.Math.Phys. 198 (1998) 535-590",
"title": "On the Connectivity of Cobordisms and Half-Projective TQFT\u0027s",
"url": "https://arxiv.org/abs/q-alg/9603017"
},
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