M Theory Lesson 23
Some people spend a lot of time worrying about 4-manifold invariants. For example, will we ever find a combinatorial formulation of the Seiberg-Witten invariants? The usual idea here is to consider a spin foam formulation along the lines of the Crane-Yetter classical invariant for 4-manifolds. A more careful use of higher categorical structures should improve the performance of the spin foam geometry. What if we used our 2-operad combinatorics?
The 3-spheres alone form the usual 4-discs 1-operad, which is well understood. Think of the internal 3-spheres as the leaves of a 1-level Batanin tree. The second level leaves must effectively attach labels to each internal 3-sphere. This must be associated with the embedded link.