Stratification of the half-density quantization of the Jeffrey-Weitsman-Witten invariants

Abstract: In this paper we review the quantization procedure employed by Jeffrey and Weitsman in their formalization of Witten's three manifold invariants. In so doing, we formalize the stratification of the Lagrangian leaf which corresponds to the flat connections of a bounding surface which extend into the handlebody. The smooth structure of this space motivates a procedure for a stratified quantization: use volumes and corresponding half-densities on the stratified tangent bundle. The pairing is then defined such that the proposed BKS pairing still yields the torsion in the integrand of the Jeffrey-Weitsman-Witten invariant. The stratification yields a natural covariant constant half-density which results in a reduction of the structure group for reducible connections. The conditions of their integral are refined in the context of the stratification and this change reproduces the invariant exactly for a plethora of 3-manifolds and extends the invariant for some.