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Topological Quantum Field Theories and Operator Algebras

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Quantum Field Theory and Noncommutative Geometry

Part of the book series: Lecture Notes in Physics ((LNP,volume 662))

Abstract

We have seen much fruitful interactions between 3-dimensional topology and operator algebras since the stunning discovery of the Jones polynomial for links [19] arising from his theory of subfactors [18] in theory of operator algebras. In this paper, we review the current status of theory of “quantum” topological invariants of 3-manifolds arising from operator algebras. The original discovery of topological invariants arising from operator algebras was for knots and links, as above, rather than 3-manifolds, but here we concentrate on invariants for 3-manifolds. On the way of studying such topological invariants, we naturally go through topological invariants of knots and links. From operator algebraic data, we construct not only topological invariants of 3-manifolds, but also topological quantum field theories of dimension 3, in the sense of Atiyah [2], as the title of this paper shows, but for simplicity of expositions, we consider mainly complex number-valued topological invariants of oriented compact manifolds of dimension 3 without boundary.

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Tokyo, Komaba, 153-8914, Tokyo, JAPAN

    Y. Kawahigashi

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  1. Y. Kawahigashi
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Ursula Carow-Watamura Yoshiaki Maeda Satoshi Watamura

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Kawahigashi, Y. Topological Quantum Field Theories and Operator Algebras. In: Carow-Watamura, U., Maeda, Y., Watamura, S. (eds) Quantum Field Theory and Noncommutative Geometry. Lecture Notes in Physics, vol 662. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11342786_13

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