Sergei Gukov caught me by surprise with the first talk of the morning (only because I was too lazy to read the schedule). Since everybody else was talking about categorification, I assumed he would as well. Man, was I wrong.
The actual topic was “arithmetic topology.” I’ll be honest here, this is not a field of topology that I even knew existed before this morning (and which I am still slightly suspicious of).
As far as I can tell, this means that you do topology (or number theory) pretending that there is a dictionary between 3- manifolds and number fields similar to the one between ideals and varieties (even those this is, in fact, obviously false).
In a nutshell, what Gukov was proposing is to take a TQFT and whack your favorite 3-manifold with it to get a the partition function or, more accurately, a perturbative expansion of such.
This perturpative expansion is a power series of (where may actually be denoting a coupling constant. This is about where my knowledge of mathematical physics starts to give out) and so you can look at the subfield of generated by these.
In some cases ( Chern-Simons theory) this will just be , we will simply curse our luck and move on, since no interesting arithmetic happens.
(This section of the talk contained the following classic exchange:
Gukov: …and this is the thesis of Dror Bar-Natan.
Bar-Natan (from the audience): Sergei, I must protest. I admit to writing a Ph.D. thesis, but none of this stuff is in it. )
But in other cases ( Chern-Simons theory, for example), you will actually get other complex numbers, and one could hope the number theory will be interesting.
It seems that this hasn’t been investigated all that fully, but the punch-line of the talk was a hint of more interesting things to come, though.
Conjecture: Let M be hyperbolic. Then the coefficients of the expansion of the partition function generate the same number field as the traces of all elements of , where is a subgroup of the isometry group of hyperbolic 3-space such that .
It’s a little unclear how much this really connects in with the main body of arithmetic topology (different audience members had different opinions on the matter), but I, at least, am of the opinion it’s a interesting question. What say ye, O mathematical physicist readers?