Skeins, \(SU(N)\) three-manifold invariants and TQFT (Q1567105)

In [Math. Ann. 307, No. 1, 109-138 (1997; Zbl 0953.57009)], \textit{Y. Yokota} defined a skein theoretic version of \(SU(N)\) 3-manifold invariants. That is, for a 3-manifold \(M\) (possibly with boundary), he defined its skein space, \({\mathcal S}_N(M)\), as the quotient of the vector space of formal linear combinations of framed oriented links in \(M\) (components of which may be circles or arcs whose boundary points are in \(\partial{M}\)), by particular linear relations (skein relations) amongst links differing only locally (along with some stabilization relations). Using the surgery representation of 3-manifolds and special elements of \({\mathcal S}_N(S^1\times{}D^2)\) which are `plumbed' into components of a link, Yokota obtained 3-manifold invariants, which are precisely the \(SU(N)\) quantum invariants. Starting from a skein theoretic formulation of any invariant of links in 3-manifolds, satisfying certain basic conditions, one may construct a topological quantum field theory (TQFT) [\textit{M. Atiyah}, The geometry and physics of knots (1990; Zbl 0729.57002)], by a formal procedure as outlined in [\textit{C. Blanchet, N. Habegger, G. Masbaum} and \textit{P. Vogel}, Topology 34, No. 4, 883-927 (1995; Zbl 0887.57009)]. The main conditions are that the vector spaces \(V(\Sigma)\) associated to closed surfaces \(\Sigma\) are finite dimensional, change to the dual under reversal of orientation, and to the tensor product under disjoint union. The current paper verifies these conditions directly from the skein theoretic formulation, by a careful analysis of explicit bases for the skein spaces. Since the representation theory and structure constants (including quantum \(6j\) symbols) of the quantum group \(SU_q(N)\) play a central role in the resulting TQFT, it is not surprising that the combinatorics of the theory is highly non-trivial.