Topological quantum field theories on manifolds with a boundary

We consider a class of exactly soluble topological quantum field theories on manifolds with a boundary that are invariant on-shell under diffeomorphisms which preserve the boundary. After showing that the functional integral of the two-point function with boundary conditions yields precisely the linking number, we use it to derive topological properties of the linking number. Considering gauge fixing, we obtain exact results of the partition function (Ray-Singer torsion of manifolds with a boundary) and theN-point functions in closed expressions.