The String Coffee Table

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 $d: A \rightarrow \Omega$ and a Courant bracket (remember Bouwknegt’s talk) there is a complicated looking list of axioms involving also maps $c: T \otimes T \rightarrow \Omega$ and $\gamma: A \otimes T \rightarrow \Omega$.

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