Asymptotic expansions of Witten-Reshetikhin-Turaev invariants for some simple 3-manifolds

R. 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 simple family of rational homology 3-spheres, obtained by integer surgery around (2,n) type torus knots. In particular, we find a closed formula for a formal power series Z_\infty(M)\inZ[[h]] in h=q-1 from which Z^*_{r-2}(M,\emptyset) may be derived for all sufficiently large primes r. We show that this formal power series may be viewed as the asymptotic expansion, around q=1, of a multi-valued holomorphic function of q with 1 contained on the boundary of its domain of definition. For these particular manifolds, most of which are not Z-homology spheres, this extends work of T. Ohtsuki and H. Murakami in which the existence of power series with rational coefficients related to Z^*_k(M,\emptyset) was demonstrated for integral homology spheres. The coefficients in the formal power series Z_\infty(M) 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: knot theory, manifold invariants, Feynman integrals, p-adic convergence, topological field theory

AMS subject classification: 57N10 (11B68 57M25 58G18)

Length: 24 pages

Reference: Invited contribution for the Special Issue on `Quantum geometry and diffeomorphism invariant quantum field theory' J. Math. Phys. 36 (1995) 6106-6129. MR1355900 (97a:57015) (review by Louis H. Kauffman .)

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Last updated on September 4th, 1996.