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A polynomial invariant of integral homology 3-spheres

Published online by Cambridge University Press:  24 October 2008

Tomotada Ohtsuki
Tomotada Ohtsuki
Affiliation:
Department of Mathematical Sciences, University of Tokyo, Hongo, Tokyo (113), Japan
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Extract

In 1988 Witten [W] proposed invariants Zk(M) ∈ ℂ (what we call, quantum G invariants) for a 3-manifold M and any integer k associated with a compact simple Lie group G. The invariant Zk(M) is formally expressed by an integral (Feynman path integral) over the (infinite dimensional) quotient space of the all connections in G-bundles on M modulo gauge transformations. If one believes in Feynman path integrals, one can expect the asymptotic formula of Zk(M) for large k predicted by perturbation theory. As in [W], the asymptotic formula (which is a power series in k−1) is given by a sum of contributions from flat connections, since the integral contains an integrand which is wildly oscillatory apart from flat connections for large k. More precise forms of the asymptotic formula are studied in [AS1], [AS2] and [Ko].

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 117 , Issue 1 , January 1995 , pp. 83 - 112
Copyright
Copyright © Cambridge Philosophical Society 1995

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References

REFERENCES

[AM] Akbulut, S. and McCarthy, J. D.. Casson's Invariant for Oriented Homology 3-Spheres. An Exposition. Mathematical Notes 36 (Princeton University Press, 1990.)Google Scholar
[AS1] Axelrod, S. and Singer, I. M.. Chern-Simons Perturbation Theory. Proceedings of the XXth international conference on differential geometric methods in theoretical physics, World Scientific, 1991.Google Scholar
[AS2] Axelrod, S. and Singer, I. M.. Chern-Simons Perturbation Theory II (preprint).Google Scholar
[D] Drinfeld, V. G.. On the almost cocommutative Hopf algebras. Algebra and Analysis 1 (1989), 30–47.Google Scholar
[IR] Ireland, K. and Rosen, M.. A Classical Introduction to Modern Number Theory, 2nd ed., (Springer-Verlag, 1990).Google Scholar
[J] Jones, V. F. R.. A polynomial invariant for knots via von Neumann algebras. Bull. Amer. Math. Soc. 12 (1985), 103–111.CrossRefGoogle Scholar
[K] Kirby, R.. The calculus of framed links in S 3, Invent. Math. 45 (1978), 35–56.CrossRefGoogle Scholar
[KM] Kirby, R. and Melvin, P.. The 3-manifold invariants of Witten and Reshetikhin-Turaev for sl(2, ℂ), Invent. Math. 105 (1991), 473–545.Google Scholar
[Ko] Kontsevich, M.. Feynman diagrams and low-dimensional topology (preprint).Google Scholar
[L] Lawrence, R. J., A Universal Link Invariant Using Quantum Groups, Differential geometric methods in theoretical physics (Chester 1988), 55–63.Google Scholar
[Mu] Murakami, H.. Quantum SU(2)-invariants dominate Casson's SU(2)-invariant, Math. Proc. Camb. Phil. Soc. 115 (1994), 253–281.Google Scholar
[O] Ohtsuki, T.. Colored ribbon Hopf algebras and universal invariants of framed links, J. Knot Theory and its Ramification, 2 (1993), 211–232.Google Scholar
[RT] Reshetikhin, N. Yu and Turaev, V. G.. Invariants of 3-manifolds via link polynomials and quantum groups. Invent Math. 103 (1991), 547–597.Google Scholar
[S] Serre, J. P.. Cours d'arithmétique (Presses Universitaires de France, Paris, 1970).Google Scholar
[W] Witten, E.. Quantum field theory and the Jones polynomial, Commun. Math. Phys. 121 (1989), 351–399.CrossRefGoogle Scholar