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Geometric Topology and Field Theory on 3-Manifolds

  • Kishore MaratheEmail author
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Part of the Contributions in Mathematical and Computational Sciences book series (CMCS, volume 1)

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.

Keywords

Gauge Theory Modulus Space Hopf Algebra Topological String Mapping Class Group 
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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© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.Max Planck Inst. for Mathematics in the SciencesLeipzigGermany
  2. 2.Department of Mathematics, Brooklyn CollegeCUNYBrooklynUSA

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