Topological quantum field theories for surfaces with spin structure

Refined quantum invariants for closed three-manifolds with links and spin structures are extended to a Topological Quantum Field Theory. By a `universal construction ', one associates, to surfaces with structure, modules which are shown to be free of finite rank. These modules satisfy the multiplicativity axiom of TQFT in an extended Z=2-graded sense, and their ranks are given by a spin refined version of the `Verlinde formula'. The relationship with the `unspun' theory is given by a natural `transfer map'. Introduction A Topological Quantum Field Theory (TQFT) in dimension 3 is a functor from a 2 + 1dimensional cobordism category to a category of modules, satisfying certain axioms. This terminology was introduced by Atiyah [1] following Witten's [32] interpretation, in terms of quantum field theory, of the Jones polynomial invariant of links in the 3-sphere. The TQFT-axioms imply that the functor is determined by its values on closed bordisms. These lie in the ground ring, and are 3-...