Genealogy of Non-perturbative Quantum-Invariants of 3-Manifolds: The Surgical Family

@article{Kerler1995GenealogyON,
  title={Genealogy of Non-perturbative Quantum-Invariants of 3-Manifolds: The Surgical Family},
  author={Thomas Kerler},
  journal={arXiv: Quantum Algebra},
  year={1995}
}
  • T. Kerler
  • Published 21 January 1996
  • Mathematics, Physics
  • arXiv: Quantum Algebra
We study the relations between the invariants $\tau_{RT}$, $\tau_{HKR}$, and $\tau_L$ of Reshetikhin-Turaev, Hennings-Kauffman-Radford, and Lyubashenko, respectively. In particular, we discuss explicitly how $\tau_L$ specializes to $\tau_{RT}$ for semisimple categories and to $\tau_{HKR}$ for Tannakian categories. We give arguments for that $\tau_L$ is the most general invariant that stems from an extended TQFT. We introduce a canonical, central element, {\sf Q}, for a quasi-triangular Hopf… 
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  • T. Kerler
  • Mathematics
    Canadian Journal of Mathematics
  • 2003
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References

SHOWING 1-10 OF 43 REFERENCES
On the Connectivity of Cobordisms and Half-Projective TQFT's
Abstract:We consider a generalization of the axioms of a TQFT, the so-called half-projective TQFT's, where we inserted an anomaly, , in the composition law. Here μ0 is a coboundary (in a group
INVARIANTS OF 3-MANIFOLDS DERIVED FROM FINITE DIMENSIONAL HOPF ALGEBRAS
This paper studies invariants of 3-manifolds derived from certain finite dimensional Hopf algebras via regular isotopy invariants of unoriented links in the blackboard framing. The invariants are
The Grothendieck Festschrift
The many diverse articles presented in these three volumes, collected on the occasion of Alexander Grothendieck’s sixtieth birthday and originally published in 1990, were offered as a tribute to one
ON ALGEBRAIC STRUCTURES IMPLICIT IN TOPOLOGICAL QUANTUM FIELD THEORIES
We show that any 3D topological quantum field theory satisfying physically reasonable factorizability conditions has associated to it in a natural way a Hopf algebra object in a suitable tensor
Chern-Simons theory with finite gauge group
We construct in detail a 2+1 dimensional gauge field theory with finite gauge group. In this case the path integral reduces to a finite sum, so there are no analytic problems with the quantization.
The Institute for Advanced Study
THE Institute for Advanced Study in the United States was founded in 1930 by a grant of 5,000,000 dollars from Louis Bamberger and Mrs. Felix Fuld. Dr. Abraham Flexner, director of the Institute, has
Topological quantum field theories
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Commun. Math. Phys
  • Commun. Math. Phys
  • 1995
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