This Week's Finds in Mathematical Physics (1-50)

We provide an explicit formula for the invariant of 4-manifolds introduced by Crane and Yetter (in hep-th/9301062). A consequence of our result is the existence of a combinatorial formula for the signature of a 4-manifold in terms of local data from a triangulation. Potential physical applications of our result exist in light of the fact that the Crane–Yetter invariant is a rigorous version of ideas of Ooguri on B ∧ F theory. They also have shown that Broda’s original construction, and also a souped-up construction of his, give a partition function that depends only on the signature: 2) Louis Crane, Louis H. Kauffman and David N. Yetter, “On the Classicality of Broda’s SU(2) Invariants of 4-Manifolds”, available as hep-th/9309102. Abstract: Recent work of Roberts has shown that the first surgical 4-manifold invariant of Broda and (up to an unspecified normalization factor) the state-sum invariant arising from the TQFT of Crane–Yetter are equivalent to the signature of the 4-manifold. Subsequently Broda defined another surgical invariant in which the 1and 2handles are treated differently. We use a refinement of Roberts’ techniques developed by the authors in hep-th/9309063 to show that the “improved” surgical invariant of Broda also depends only on the signature and Euler character. Recent work of Roberts has shown that the first surgical 4-manifold invariant of Broda and (up to an unspecified normalization factor) the state-sum invariant arising from the TQFT of Crane–Yetter are equivalent to the signature of the 4-manifold. Subsequently Broda defined another surgical invariant in which the 1and 2handles are treated differently. We use a refinement of Roberts’ techniques developed by the authors in hep-th/9309063 to show that the “improved” surgical invariant of Broda also depends only on the signature and Euler character. Now let me say just a little bit about what this episode might mean for physics as well as mathematics. The key is the “B ∧ F ” theory alluded to above. This is a quantum field theory that makes sense in 4 dimensions. I have found that the nicest place to read about it is: 3) Gary Horowitz, “Exactly soluble diffeomorphism-invariant theories”, Commun. Math. Phys. 125 (1989), 417–437. This theory is a kind of simplified version of 4d quantum gravity that is a lot closer in character to Chern–Simons theory. Like Chern–Simons theory, there are no “local degrees of freedom” — every solution looks pretty much like every other one as long as we don’t take a big tour of space and notice that funny things happen when we go around a noncontractible loop, which is the sort of thing that can only exist if space has a nontrivial topology. 4d quantum gravity, on the other hand, should have loads of local degrees of freedom — the local curving of spacetime!