dorsal/arxiv
View SchemaGraph Invariants of Vassiliev Type and Application to 4D Quantum Gravity
| Authors | Nobuharu Hayashi |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9503010 |
| URL | https://arxiv.org/abs/q-alg/9503010 |
| DOI | 10.1007/BF02099627 |
Abstract
We consider a special class of Kauffman's graph invariants of rigid vertex isotopy (graph invariants of Vassiliev type). They are given by a functor from a category of colored and oriented graphs embedded into a 3-space to a category of representations of the quasi-triangular ribbon Hopf algebra $U_q(sl(2,\bf C))$. Coefficients in expansions of them with respect to $x$ ($q=e^x$) are known as the Vassiliev invariants of finite type. In the present paper, we construct two types of tangle operators of vertices. One of them corresponds to a Casimir operator insertion at a transverse double point of Wilson loops. This paper proposes a non-perturbative generalization of Kauffman's recent result based on a perturbative analysis of the Chern-Simons quantum field theory. As a result, a quantum group analog of Penrose's spin network is established taking into account of the orientation. We also deal with the 4-dimensional canonical quantum gravity of Ashtekar. It is verified that the graph invariants of Vassiliev type are compatible with constraints of the quantum gravity in the loop space representation of Rovelli and Smolin.
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"abstract": "We consider a special class of Kauffman\u0027s graph invariants of rigid vertex\nisotopy (graph invariants of Vassiliev type). They are given by a functor from\na category of colored and oriented graphs embedded into a 3-space to a category\nof representations of the quasi-triangular ribbon Hopf algebra $U_q(sl(2,\\bf\nC))$. Coefficients in expansions of them with respect to $x$ ($q=e^x$) are\nknown as the Vassiliev invariants of finite type. In the present paper, we\nconstruct two types of tangle operators of vertices. One of them corresponds to\na Casimir operator insertion at a transverse double point of Wilson loops. This\npaper proposes a non-perturbative generalization of Kauffman\u0027s recent result\nbased on a perturbative analysis of the Chern-Simons quantum field theory. As a\nresult, a quantum group analog of Penrose\u0027s spin network is established taking\ninto account of the orientation. We also deal with the 4-dimensional canonical\nquantum gravity of Ashtekar. It is verified that the graph invariants of\nVassiliev type are compatible with constraints of the quantum gravity in the\nloop space representation of Rovelli and Smolin.",
"arxiv_id": "q-alg/9503010",
"authors": [
"Nobuharu Hayashi"
],
"categories": [
"q-alg",
"gr-qc",
"hep-th",
"math.QA"
],
"doi": "10.1007/BF02099627",
"title": "Graph Invariants of Vassiliev Type and Application to 4D Quantum Gravity",
"url": "https://arxiv.org/abs/q-alg/9503010"
},
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