August 16: Naturality in 2+1 dimensions

Josh Batson

A 2+1 TQFT assigns vector spaces to surfaces, and linear maps to cobordisms between them. One of Atiyah's axioms is that these maps be natural: given a diffeomorphism between cobordisms f:W-->W', we get a commuting diagram: maps between the vector spaces on the ends intertwining the maps induced by W and W'. This means that, in a suitable sense, the TQFT commutes with a representation of the mapping class group. A neat application of representation theory (due to Donaldson) shows that the behavior of such TQFTs are dramatically restricted (essentially by Shur's lemma): they all contain the same information. In particular, the invariants they produce for closed 3-manifolds with H_1(W)=Z are entirely determined by the Alexander polynomial of W! The specific example of a 2+1 TQFT out of which pops the Alexander polynomial is given by applying the push-pull construction Jon Bloom talked about a week ago to the moduli spaces of flat U(1) connections. We'll go through this in detail. The corollary of this lovely example and the naturality argument above is a pair of formulae, one computing the Casson invariant of W in terms of its alexander polynomial, and one computing the (dimensionally reduced) Seiberg-Witten invariants in terms of the same. If one can prove naturality for these theories (which Tim Nguyen tells me is harder than Donaldson makes it out to be), then these otherwise difficult-to-prove results fall out for free! It's such free lunches, brought to you by the power of naturality (and Clark Barwick's grant), which will be provided on Tuesday.