Today we have had a series of talks half of which were concerned with generalised geometry. Nigel Hitchin was due to speak this morning but was unable to due to illness, but he sent along his slides in his absence and Chris Hull and Richard Thomas filled the gap and made a very good presentation of his talk. The talk was entitled "B-fields, gerbes and generalized geometry" and the first half was a survey of the recent progress in the generalised geometry programme (see this related post about an introductory talk I saw given by Marco Gualtieri, a former student of Hitchin's). The second talk was concerned with looking at the different types of geometry classified by the programme, and was presented in by Richard Thomas. The slides from both parts are available online (part 1, part 2). Since much of the talk can be understood best by reading Gualtieri's thesis I will not say much apart from commenting on how well the two speakers did presenting someone else's slides. It really was impressive, bearing in mind they only had two days to prepare for it. Of course Richard Thomas did take advantage of the situation in order to add comedic value to his presentation, for example proclaiming that one of the slides was so straightforward that he was sure all of us in the audience understood it and so he wasn't going to spend any time talking about it (with a big grin), and more of the same throughout. In all it was a sterling effort from the two stand-in speakers.
After lunch, Claus Hertling, talked to us about "Oscillating integrals and nilpotent orbits of twistor structures", followed by Chris Hull's second talk of the day entitled "Generalised Geometry and Duality" based on his paper "A Geometry for Non-Geometric String Backgrounds". Hull's talk was probably the most accessible talk of the meeting so far for those of us with only an introductory background in physics.
Hull commenced by motivating us that a special approach to the geometry of a spacetime containing strings was needed. He did this by reminding us that the string excitations contain gravity and so strings have the possibility of altering the background spacetime. From there Hull discussed T-dualities which interchange the winding(string)/wrapping(brane) modes with momentum modes, commenting that compactifications on mirror symmetric Calabi-Yau manifolds leads to the same physics but usually on different geometric backgrounds. The picture he wasnted to analyse was whether it would be possible to combine the two geometric backgrounds, so that the whole geometry under mirror symmetry is unaltered along with the physics. His proposal began by doubling the coordinate manifold and using one copy of the coordinates to express the physics in terms of the degrees of freedom coming from the momentum modes, and the other copy of the coordinates to express the same physics in terms of the winding modes. The extra degrees of freedom were to be halved using a set of duality conditions relating the fields in the theory, e.g. dA=*dA'+... and the actual spacetime coordinates used to specify physics were to be singled out by a choice that Hull called polarisation. Since this proposal doubles the tangent space and is a further doubling of Hitchin's generalised geometry, Hull suggested the procedure might be called generalised generalised geometry, but then rejected this for the pithier T-fold geometry. The two sets of coordinates were to be glued together and the transition functions would be T-duality functions. Hull asked the question whether one would be able to patch together the sets of coordinates in this way to give a spacetime manifold and then told us that in general it would not be possible. The end picture was that while local spacetime would be covered by a coordinate patch, globally the manifold structure would be lost, and Hull seemed to be suggesting that doing away with the global spacetime manifold was not such a bad thing, if it meant that there was a better equivalence between geometry and physics. It should go without saying that this was my understanding of the broad picture of Hull's talk, that I may have got the wrong end of the stick, that Hull's arguments can be found in his paper, and that if anyone (especially those who have heard this talk) have any comments/amendments then I would be grateful if they posted them as a comment :)
After the coffee break, where I drank my fifth coffee of the day and got hold of some of the precious chocolate chip cookies that the organisers have been supplying us with, José Figueroa-O'Farrill gave a talk on "Gauged WZ terms in sigma models with boundary" which was based on his paper with Noureddine Mohammedi, "Gauging the Wess-Zumino term of a sigma model with boundary". Before commencing his talk José told us about the new research partnership ERP, Edinburgh Research Partnership and asked us to visit their website, in particular for the vacancies currently available. So consider this information dutifully passed on.
As if five talks weren't enough for one day, we were treated to an extra evening talk after dinner at 8pm entitled "How to Knit a Scarf" presented by Alastair King. Little did I suspect, as I should have done, that this wouldn't be a workshop and there was no wool involved :) Indeed it was a gentle (by the symposium standards) category theory talk, giving the physicists in the audience a gentle appreciation of the "natural" approach of the category theorists. A scarf turns out to be a quiver diagram composed out of many copies of three nodes (A_3) arranged in a triangle and with arrows having an anticlockwise orientation on the basis diagram. It was very colourful, and looked like it would be fun to work with, but that is the limit of my appreciation at the moment :( Alastair King also modified Terry Gannon's comments earlier in the week drawing a comic parallel between categorists and beavers to a quote of his own that ran "Category theorists are not like beavers". The surreal debate trundles on, and I live in fear of what will happen when category theorists discover the jumper.