The String Coffee Table

Kevin Costello wrote:

Hi Urs

I’ve been looking at your blog, and came across some comments you made a few months back about open topological strings. I just thought I’d point out a paper I wrote about this stuff, math.QA/0412149, in case you’re interested.

There’s a Segal style definition of closed topological string theory, as follows. There’s a category, whose objects are the integers, and whose morphisms are the singular chains on the moduli spaces of Riemann surfaces with $n$ incoming and $m$ outgoing boundaries. A functor from this to chain complexes, which is compatible with differentials, should be a closed topological string theory.

This is the type of structure that arises in for example Gromov-Witten theory.

Segal calls such a gadget a ‘string background’. Some people (including me) use the phrase ‘topological conformal field theory’ (TCFT) but possibly this is a misnomer.

This is a much richer structure than simply a Frobenius algebra, because of the complexity of the topology of the moduli spaces of curves. We find a Frobenius algebra when we take $H_0$.

Anyway, there’s a straightforward way to modify this definition for open strings and open-closed strings.

What I show in my paper is that if we do the open analog of this, then an open TCFT is the same as an $A_\infty$ category satisfying a Calabi-Yau condition. This is a kind of categorification of the ribbon graph picture of moduli space.

I also show how to construct the closed theory from the open, and that when we do this to the Fukaya category we should recover Gromov-Witten theory. Applying this to an $A_\infty$ version of derived category of sheaves on a Calabi-Yau gives the B model.

Cheers – and sorry for the long mail Kevin Costello

I’ll have a look at that as soon as possible. Right now I am preparing for a vacation.