The most general gauge-invariant marginal deformation of four-dimensional
abelian BF-type topological field theory is studied. It is shown that the
deformed quantum field theory is topological and that its observables compute,
in addition to the usual linking numbers, smooth intersection indices of
immersed surfaces which are related to the Euler and Chern characteristic
classes of their normal bundles in the underlying spacetime manifold. Canonical
quantization of the theory coupled to non-dynamical particle and string sources
is carried out in the Hamiltonian formalism and explicit solutions of the
Schroedinger equation are obtained. The wavefunctions carry a one-dimensional
unitary representation of the particle-string exchange holonomies and of
non-topological string-string intersection holonomies given by adiabatic limits
of the worldsheet Euler numbers. They also carry a multi-dimensional projective
representation of the deRham complex of the underlying spatial manifold and
define a generalization of the presentation of its motion group from Euclidean
space to an arbitrary 3-manifold. Some potential physical applications of the
topological field theory as a dual model for effective vortex strings are
discussed.