**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.*

ruthel@ma.huji.ac.il