Vafa describes Topological String Theory

So earlier on this week, we, in London, were treated to a series of three talks by Cumrun Vafa describing Topological String Theory (TST). The interested throngs gathered in Imperial College on Monday for the first talk and included students from as far away as Cambridge and even Chicago. Now I have been doing my homework and making sure I read around and learn as much as I can about the language of string theory, and so I thought it would be a short pleasant step to TST. Alas not: there's always more to learn.

Vafa's plan for the three talks went:

I. What is TST?
II. Dualities & topological strings.
III. Black holes, topological M-theory & topological strings.

It all began well enough. Vafa told us that he knew the audience consisted of a mix between mathematicians and physicists and so his talks would of necessity be quite general. This sounded excellent. Vafa began by describing the concept of localisation in mathematics, where problems which are too hard to compute in general are reduced to an understandable subset. Of course in string theory the correlation functions are too hard to compute in general but a subset are exact. In string theory we map, X, from a Riemann surface into a 10-dimensional manifold, M^10 which is the product of a Minkowski spacetime R^4 which we concede exists and a missing 6-dimensional manifold M^6, which many are less sure of. The localisation that gives TST restricts the image of X to a point in R^4 crossed with M^6. So TST is the study of maps from the Riemann surface to M^6.

Vafa told us there were two types of TST:

- The A-model (the IIA superstring) where the localisation restriction is \bar{\partial X}=0, holomorphic maps.
- The B-model (the IIB superstring) where the localisation restriction is dX=0, constant maps.

Typically M^6 must be Ricci flat and Kahler, these two conditions give us a Calabi-Yau manifold (or complex 3-fold, as Vafa preferred). He went on to tell us that the moduli space of a CY n-fold naturally splits into the product of a complex manifold (with hodge number h^{1,n-1}) and a Kahler manifold (h^{1,1}). Now the A-model computations depend only on the Kahler manifold, while the B-model depends only on the complex manifold.

Then we had an aside about mirror symmetry. Vafa asked us to consider a map to an S^1 (the circle) target manifold of radius R. At a fixed time, the closed string is also a circle so in this case we are imagining a very simple map from one circle to a second which has radius R. In the spacetime image the energy of the unexcited string is determined by how many times it wraps the circular dimension, E~w.R, where w is an integer called the winding number. There is also the possibility of momentum states on the string which are subject to p~n/R, where n is the integer momentum number. Vafa now had us primed for T-duality. Since the string may have both winding states and momentum states, E+p is unaltered by T-duality, which exchanges R<->1/R, and w<->n.

The ante was then upped a little as we moved from an S^1 target space to a CY 1-fold (=T^2, the 2-dimensional torus). Defining the torus as a rectangle, of sides R_1 and R_2, with opposite edges being identified, Vafa defined two new quantities: \tau=iR_2/R_1 (complex structure parameter) and, the complexified area iA=iR_1R_2 (Kahler parameter). Now under a T-duality about the circle of length R_1, these quantities are mapped to each other i.e. \tau->iA, and iA->\tau. That is, T^2 is mirror to itself, and the complex and Kahler structures are exchanged. So we can see that A-model and B-model are interchanged, in this example and Vafa encouraged us to wonder whether or not this is true for a more general target manifold, or if T^2 is special.

It turned out that T^2 is special. Vafa told us that for a CY n-fold target space, T-duality changes the hodge numbers as h^{p,q}<->h^{n-p,q}, hence for T^2 (n=2, p=q=1) h^{1,1)<->h{1,1}. This was the turning point in the talk for me, because from here on in deductions were made from "facts" that I had not seen before. So I was very uncomfortable. For example, Vafa next considered the A-model and its Kahler manifold and gave an argument that the study of TST naturally leads one to be interested in CY 3-folds, without detailed knowledge of string theory. It went like this:

Fact: "dim(\bar{\partial X})"=(n-3)(1-g) + C_1(M)|(image)

where C_1(M)|(image) isthe first Chern class restricted to the image.

Deduction: If we are interested in the most straightforward maps, those with dim=0,then we could consider n=3, C_1(M)|(image)=0, which are CY 3-folds.

This is nice to know, but I don't feel like I've understood anything because I don't really know where the starting fact came from. Vafa looked at the B-model in a similar fashion and then wound up his first talk, before being lampooned with questions by the much-better-informed-than-I audience. So my notes will run dry here, but I'll give my wish list of terminology/facts so that a friendly reader can help me out, and in future I can remember what I don't know :)

- The Gromov-Witten invariant
- That the integral of the top Chern class defined over a vector bundle equals the GW invariant
- classical geometry is related to the periods of a holomorphic n-form
- Ray-Singer torsion
- Kodaira-Spencer theory of gravity

So that was the first talk, so despite the star billing I didn't attend the others, besides I'm just entering the final quarter of my second-year so it is time to start "working hard", as I was told by someone who clearly has my levels of work figured out.

If anyone reads this and went to any of the two other talks, comments on how they went are very welcome :)