A polynomial invariant of integral homology 3-spheres

Abstract

In 1988 Witten [W] proposed invariants Z k (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 Z k (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 Z k (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].