**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.*

ruthel@ma.huji.ac.il