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.


Suppose you have a 4d TQFT with surface operators (2-knots). Then it can be decategorified to a 3d TQFT with line operators (1-knots); that is, let's say $M$ is a 3-manifold and $L\subset M$ is a 1-knot. Then for any 4d TQFT $Z_4$ with surface operators, $Z_3(M,L):=Z_4(M\times S^1,L\times S^1)$ should be a 3d TQFT. Therefore, in this sense, your question is about categorifying 3d TQFTs to 4dTQFTs.

There are several knot polynomials which have been categorified (i.e. lifted to a 4d TQFT) in the past few decades. The first was Khovanov homology, which is a categorification of Jones polynomial, giving invariants for knot cobordisms (2-knots). Similarly, there are categorifications of colored Jones polynomials and the HOMFLY-PT polynomial. These mathematical constructions depend on the projection of links, so they are not manifestly 3-dimensional.

There is a proposal by Witten describing Khovanov homology as a space of BPS states. This physics based description of Khovanov homology is manifestly 3-dimensional, in the similar spirit as Witten's paper on the Jones polynomial 30 years ago.

While many of these knot invariants have been categorified, so far they are all done in $\mathbb{R}^3$ (or $S^3$). Considering the fact that Jones polynomial can be generalized to an invariant of links inside any 3-manifolds (Witten-Reshetikhin-Turaev (WRT) invariant), we would like to categorify those invariants as well. However, categorifying quantum 3-manifold invariants is still incomplete and largely open. There are some recent proposals, but I guess still a concrete categorifiation is far from being done.


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