## 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.