### Thursday at the Streetfest III

#### Posted by Guest

Gorbunov decided to talk about Gerbes of Chiral Differential Operators instead of what was in his abstract, motivation being CFT and the study of A and B differentials and also Witten’s half-twisted case leading to an infinite dimensional $H(V)=\oplus {V}_{i}$ with the property of being a vertex algebra (which Gorbunov defined nicely for us but I won’t repeat) such that the weighted sum of dimensions gives the elliptic genus of $M$.

I guess if you’re at Strings 2005 you’ll hear all about recent work by Witten and Kapustin on this stuff. Anyway, then we went onto vertex algebroids! For $A$ a commutative associative ring, $T$ an $A$-module and $\Omega $ a $T$ and $A$ module, a $d:A\to \Omega $ and a Courant bracket (remember Bouwknegt’s talk) there is a complicated looking list of axioms involving also maps $c:T\otimes T\to \Omega $ and $\gamma :A\otimes T\to \Omega $.

The theorem of Gorbunov et al. is that, given any complex analytic $M$ and bundle $E$ there exists a certain gerbe so that we get a vertex algebra ${H}^{*}(M,{\Omega}^{\mathrm{ch}}(M,E))$.

Yetter raced us through an unbelievable amount of stuff after struggling with the laptop setup, starting with a 5 minute summary of knot polynomials, tangles, Joyal and Street and other things leading up to the paper of Mei-Chi Shum on tortile tensor categories. As far as I know this important paper isn’t available online - sorry. Yetter stressed the importance of the fact that these structures allow us to understand why quantum groups have something to do with low dimensional topology.

He then went on to talk about Kirby calculus, 3 and 4 manifold invariants, deformation theory, bottom tangles … and by the time he got to some recent results of his own unfortunately my right hand rebelled against the torture and my brain sympathised and refused to take any more in.

Cisinski also changed his talk, and spoke about Batanin weak higher groupoids and homotopy types. This was a racy but precise journey through some sophisticated $\omega $-operad theory and theorems on Quillen equivalences (any reader who knows what these are would probably do a better job than me in discussing these ideas).

Maybe we’ll come back later and discuss Breen’s talk on monoidal braided n-categories. John Baez gave us a very entertaining talk on numbers as cardinalities - this talk is on the Streetfest website. John is first up this morning (Friday) … must go and stretch my fingers.

Marni

Posted at July 14, 2005 11:16 PM UTC
## Re: Thursday at the Streetfest III

The sad thing about Yetter’s talk is that he was getting to the really interesting part just at the end, when the audience was unable to take any more in. He explained it to me afterwards and it goes roughly like this.

There’s an known way to use Vassiliev invariants to compute the linking matrix of a framed link. This is the matrix that gives the linking numbers of each pair of circles in the link, including the “self-linking” numbers that say how the framing of each link twists around.

This is nice because you can get any compact oriented 4-manifold from doing surgery on a link, and the linking matrix is then an invariant of the 4-manifold. In fact, it’s a very famous invariant: the quadratic form on the second cohomology group of the 4-manifold! Over the real numbers you get this quadratic form just by taking the wedge product of two closed 2-forms and integrating the result to get a number, but over the integers it holds a lot more information: in fact, some theorem of Freedman says that simply-connected 4-manifolds are completely classified by their quadratic forms, maybe together with a wee bit more information….

Anyway, Yetter thinks he’s managed to get a more refined invariant of this sort that can even distinguish between smooth manifolds that are homeomorphic! This is a long-held dream in quantum topology, to find invariants of 4-manifolds that detect different smooth structures. People can already detect different smooth structures using differential equations, thanks to Donaldson and Seiberg-Witten. But the dream is to find a purely combinatorial way to do it, and Yetter seems to have done it!