Witten-Reshetikhin-Turaev invariants of 3-manifolds as holomorphic functions

R.J. Lawrence

Abstract: For any Lie algebra, g, and integral level, k, there is defined an invariant, Z^*_k(M,L), of embeddings of links L in 3-manifolds M, known as the Witten-Reshetikhin-Turaev invariant. It is known that for links in S^3, Z^*_k(S^3,L) is a polynomial in q=\exp{2\pi{}i\over(k+2)}, namely the generalised Jones polynomial of the link L. This paper investigates the invariant Z^*_{r-2}(M,\emptyset) when g=sl(2) for a rational homology 3-spheres, showing that the correct way to view these invariants is as eminating from a single (multi-valued) holomorphic function, Z_\infty(M), of log q whose domain contains q=1 on the boundary, rather than as a family of complex numbers indexed by roots of unity. This extends previous work of the author where the existence of such holomorphic functions was demonstrated for a particular class of 3-manifolds obtained from S^3 by surgery around simple knots. The coefficients in the formal power series in h=q-1 obtained from the expansion of Z_\infty(M) around q=1 are expected to be identical to those obtained from a perturbative expansion of the Witten-Chern-Simons path integral formula for Z^*(M,\emptyset).

Keywords: topological field theory, manifold invariants, p-adic convergence, knot theory, finite type invariants

AMS subject classification: 57M25 40G99 05A30 11B68 81Q30

Length: 15 pages

Reference: `Proceedings on Geometry and Physics, Aarhus, Denmark', Eds. J.E. Andersen, J. Dupont, H. Pedersen, A. Swann Lecture Notes in Pure and Applied Mathematics 184 (1997) 363-377. MR1423181 (98b:57031) (review by Hitoshi Murakami .)

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Last updated on March 18th, 2006.