A polynomial invariant of integral homology 3-spheres

In 1988 Witten [W] proposed invariants Zk(M) ∈ ℂ (what we call, quantum G invariants) for a 3-manifold M and any integer k associated with a compact simple Lie group G. The invariant Zk(M) is formally expressed by an integral (Feynman path integral) over the (infinite dimensional) quotient space of the all connections in G-bundles on M modulo gauge transformations. If one believes in Feynman path integrals, one can expect the asymptotic formula of Zk(M) for large k predicted by perturbation theory. As in [W], the asymptotic formula (which is a power series in k−1) is given by a sum of contributions from flat connections, since the integral contains an integrand which is wildly oscillatory apart from flat connections for large k. More precise forms of the asymptotic formula are studied in [AS1], [AS2] and [Ko].