Lecture note 31/2006
Topics in Physical Mathematics: Geometric Topology and Field Theory
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Submission date: 03. Aug. 2006
Marathe, K. B.: Topics in physical mathematics
London : Springer, 2010. - XXII, 442 p.
ISBN 978-1-84882-938-1 - ISBN 978-1-84882-939-8
MSC-Numbers: 57R56, 81T13, 81T30, 83C05
PACS-Numbers: 11.15.Tk, 11.25.Tq, 11.30.Pb
Keywords and phrases: geometric topology, field theory, knot invariants
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In recent years the interaction between geometic topology and classical and quantum field theories has attracted a great deal of attention from both the mathematicians and physicists. We discuss some topics from low dimensional topology where this has led to new viewpoints as well as new results. They include categorification of knot polynomials 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 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.