I'm looking for a status report on analogues of quantum invariants of knots, for the 2-knots (homotopy classes of spheres / other Riemann surfaces embedded into 4-manifolds).
I'm mostly interested in this subject from the point of view of theoretical physics. One of the beautiful inspiring results from the end of 20th century was the discovery by E.Witten that the Jones polynomial (and its generalizations) admit a manifestly topologically-invariant, 3-dimensional definition in terms of expectation values of quantum physical observables corresponding to the knots, in a quantized version of the Chern-Simons field theory.
Naively, by analogy, invariants of 2-knots can be obtained by considering quantum observables corresponding to 2-surfaces in some kind of generally covariant 4-dimensional theory. Examples also include quantum General Relativity (though I'm fairly certain nobody was able to define it up to date).
I'm looking for a quick overview of known 2-knot invariants, including those which admit manifestly topologically invariant 4-dimensional definitions in terms of quantum physical observables in some 4-dimensional generally covariant physical theory. Or, for a unequivocal statement that those haven't been discovered yet.