Communications in Mathematical Physics

, Volume 180, Issue 3, pp 653–670 | Cite as

Graph invariants of Vassiliev type and application to 4D quantum gravity

  • Nobuharu Hayashi


We consider graph invariants of Vassiliev type extended by the quantum group link invariants. When they are expanded byx whereq=e x , the expansion coefficients are known as the Vassiliev invariants of finite type. In the present paper, we define tangle operators of graphs 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 extended byU q (sl(2),C)), which are subject to a quantum group analog of the spinor identity. In terms of them, we obtain the graph invariants of Vassiliev type expressed to be identified with Chern Simons vacuum expectation values of Wilson loops including intersection points. We also consider the 4d canonical quantum gravity of Ashtekar. It is verified that the graph invariants of Vassiliev type satisfy constraints of the quantum gravity in the loop space representation of Rovelli and Smolin.


Quantum Gravity Wilson Loop Hopf Algebra Quantum Group Space Representation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • Nobuharu Hayashi
    • 1
  1. 1.Institute of PhysicsUniversity of TokyoKomaba, TokyoJapan

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