Topological quantum field theories derived from the Kauffman bracket (Q1910457)

The authors give a purely topological construction of the TQFTs associated to invariants satisfying the Kauffman bracket relations, that is, essentially, of the TQFTs corresponding to Jones' original \(V\)-polynomial. They renormalize the invariants \(\theta_p\) obtained in their previous paper [ibid. 31, No. 4, 685-699 (1992; Zbl 0771.57004)] to construct a series of invariants \(\langle\;\rangle_p\) of banded links in closed 3-manifolds, and then use these invariants to define, in a ``universal'' way, modules \(V_p(\Sigma)\) \((p\geq 1)\), associated to surfaces \(\Sigma\) (which may have banded links, too). The authors prove the finiteness and multiplicativity properties of the \(V_p(\Sigma)\) \((p\geq 1)\), using the language of bimodules over algebroids. It turns out that the ranks of their modules are given by Verlinde's formula. Finally, they describe an action of a Heisenberg type group \(\Gamma (\Sigma)\) on the modules \(V_{2p}(\Sigma)\). This action is used to obtain a natural decomposition of \(V_{2p}(\Sigma)\) into a non-trivial tensor product, or into a direct sum of subspaces (whose ranks are computed explicitly), associated to spin structures or cohomology classes on the surface.