A TQFT for Wormhole cobordisms over the field of rational functions

Abstract

We consider a cobordism category whose morphisms are punctured connect sums of $S^1 \times S^2$'s (wormhole spaces) with embedded admissibly colored banded trivalent graphs. We define a TQFT on this cobordism category over the field of rational functions in an indeterminant $A.$ For $r$ large, we recover, by specializing $A$ to a primitive 4rth root of unity, the Witten-Reshetikhin-Turaev TQFT restricted to links in wormhole spaces. Thus, for $r$ large, the $r$th Witten-Reshetikhin-Turaev invariant of a link in some wormhole space, properly normalized, is the value of a certain rational function at $e^{\frac{\pi i}{2r}}.$ We relate our work to Hoste and Przytycki's calculation of the Kauffman bracket skein module of $S^1 \times S^2.$