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Geometric Topology and Field Theory on 3-Manifolds

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The Mathematics of Knots

Part of the book series: Contributions in Mathematical and Computational Sciences ((CMCS,volume 1))

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Abstract

In recent years the interaction between geometric topology and classical and quantum field theories has attracted a great deal of attention from both the mathematicians and physicists. This interaction has been especially fruitful in low dimensional topology. In this article We discuss some topics from the geometric topology of 3-manifolds with or without links where this has led to new viewpoints as well as new results. They include in addition to the early work of Witten, Casson, Bott, Taubes and others, the categorification of knot polynomials by Khovanov. Rozansky, Bar-Natan and Garofouladis and a special case of the gauge theory to string theory correspondence in the Euclidean version of the theories, where exact results are available. We show how the Witten-Reshetikhin-Turaev invariant in SU(n) Chern-Simons theory on S 3 is related via conifold transition to the all-genus generating function of the topological string amplitudes on a Calabi-Yau manifold. This result can be thought of as an interpretation of TQFT as TQG (Topological Quantum Gravity). A brief discussion of Perelman’s work on the geometrization conjecture and its relation to gravity is also included.

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Authors and Affiliations

  1. Max Planck Inst. for Mathematics in the Sciences, Leipzig, Germany

    Kishore Marathe

  2. Department of Mathematics, Brooklyn College, CUNY, 2900 Bedford Avenue, Brooklyn, NY, 11210, USA

    Kishore Marathe

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  1. Kishore Marathe
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  1. Department of Mathematics, University of Heidelberg, Im Neuenheimer Feld 288, Heidelberg, 69120, Germany

    Markus Banagl

  2. Department of Mathematics, University of Heidelberg, Im Neuenheimer Feld 288, Heidelberg, 69120, Germany

    Denis Vogel

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Marathe, K. (2011). Geometric Topology and Field Theory on 3-Manifolds. In: Banagl, M., Vogel, D. (eds) The Mathematics of Knots. Contributions in Mathematical and Computational Sciences, vol 1. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15637-3_8

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