Witten-Reshetikhin-Turaev invariants of Seifert manifolds

Ruth Lawrence and Lev Rozansky

Abstract: For Seifert homology spheres, we derive a holomorphic function of K whose value at integer K is the sl_2 Witten-Reshetikhin-Turaev invariant, Z_K, at q=\exp{2\pi{}i/K}. This function is expressed as a sum of terms, which can be naturally corresponded to the contributions of flat connections in the stationary phase expansion of the Witten-Chern-Simons path integral. The trivial connection contribution is found to have an asymptotic expansion in powers of K\inv which, for K an odd prime power, converges K--adically to the exact total value of the invariant Z_K at that root of unity. Evaluations at rational K are also discussed. Using similar techniques, an expression for the coloured Jones polynomial of a torus knot is obtained, providing a trivial connection contribution which is an analytic function of the colour. This demonstrates that the stationary phase expansion of the Chern-Simons-Witten theory is exact for Seifert manifolds and for torus knots in S^3. The possibility of generalising such results is also discussed.

Keywords: knot theory, manifold invariants, Feynman integrals, p-adic convergence, topological field theory

AMS subject classification: 57Nxx 57Mxx 58Gxx

Length: 33 pages

Reference: Commun. Math. Phys. 205 (1999) 287-314. MR1712599 (2001e:58029)

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Last updated on April 9th, 1997.