Topological quantum field theories for surfaces with spin structure (Q1919624)

The authors propose an answer to the question of how a formula which asserts that the sum of the spin invariants of a closed 3-manifold is equal to the ``unspun'' invariant can be generalized on the level of TQFTs. A partial answer to this question was given by the same authors in their previous paper where they started from the 3-manifold invariant, extended it by a universal construction to a functor on a cobordism category and then proved the TQFT-axioms. In the present paper they use the same strategy but start from the spin-refined invariants. The aim is to find out what a ``Spin TQFT'' should be, and to understand its relationship with ``unspun'' theory. There are three main results. The first is that the \(V^s_{8k}\)-modules associated to surfaces are free of finite rank. The second main result concerns the \(V^s_{8k}\)-modules associated to disjoint unions. It turns out that the functors \(V^s_{8k}\) do not satisfy the usual multiplicativity axiom of TQFT but an ``extended'' tensor product formula. The third main result is a complete answer to the above mentioned question which is given by a transfer map from the unspun theory to the spin theory. The transfer map identifies the spin submodules with the ``zero graded parts'' of the \(V^s_{8k}\)-modules.