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Lecture Note

31/2006

Topics in Physical Mathematics: Geometric Topology and Field Theory

Kishore Marathe

Abstract

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 $S U (n)$ 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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Received:: 03.08.06

Published:: 03.08.06

MSC Codes:: 57R56, 81T13, 81T30, 83C05

PACS:: 11.15.Tk, 11.25.Tq, 11.30.Pb

Keywords:: geometric topology, field theory, knot invariants