Donaldson on Curvature and Physics

I trotted out of Drury Lane and off to Imperial College's Clore Lecture Theatre this afternoon to see Field's medallist Simon Donaldson talk on "Curvature and Geometry in Physics". The talk is the first of a series this week to mark the launch of IC's very own Institute for Mathematical Sciences, of which Donaldson will be the first president. Tomorrow Michael Duff, soon to be Principal of the Faculty of Physical Sciences at IC, will give a talk entitled "A Layman's Guide to M-Theory" - hopefully we will hear his favourite meaning for the M. But the rest of the week's talks are less relevant to theoretical physics, it's almost as if there are other mathematical sciences beyond high energy physics :)

Michael Duff introduced Lord May of Oxford, head of the Royal Society, who said hello and then told us that "Simon Donaldson needs no introduction", but, of course, we had already had two. With a slight buzz of interference the microphone system was switched to SD who began by telling us that he was there to fight the pure mathematician's corner, but had in deference to the name of the new institute included a science, namely physics, in his title. This was to be expected. SD used a series of very clear slides, making three or less points per slide, and began the talk by asking the audience to consider the manner in which a line field gives rise to integral curves and whether or not there was an analog for plane fields giving rise to integral surfaces. The answer in general was no. Then to captivate the mind of the physicists in the audience SD exhibited the example of a trolley moving on a flat surface, and then transported it to a plane field. In the flat case the trolley moved around a quadrilateral returns to its starting point, but in the plane field example the trolley is generally displaced along the normal to the plane field at the start/end point. The curvature was defined at this point as a skew-symmetric map from the plane field, P, to the normal to the plane field P': PxP->P'. And from this starting point four relations to curvature were summarised as

1. Deviation from the flat model
2. Intrinsic/Extrinsic
3. Parallel transport
4. Integrability conditions for an overdetermined set of variables

No. 2 was elaborated on and the difference between the two types of curvature was described as whether or not a fly (presumably a mathematician constrained to the manifold [the mathematican having by supposition the dimension of the manifold, without any harm coming to him or her] and not really a fly) living on the surface would be able to deduce if it was living on a plane or not. If the 'fly' could tell, e.g. if it lived on a sphere, then the curvature is intrinsic. If the 'fly' could not tell, e.g. if it lived on the curved side of a cylinder, then the curvature is extrinsic. This was good to know.

SD then made some comments about general relativity because it is the most common place curvature turns up in physics, arising from the pseudo-Riemannian metric, g, on the space-time manifold. But SD decided to focus on curvature in electromagnetism instead. The geometric interpretation of EM comes through complex line bundles, L, over flat space-time, R(3,1); so that L~CxR(3,1), but not as he emphasised canonically. There is an association between the scalar and vector potentials of EM and the field of subspaces inside L (connection) and furthermore that the Maxwell field strength, F, is asssociated to the curvature of L. SD continued that the complex line bundles were only apparent in quantum mechanics, where the wavefunction was really a section of L, and the Hilbert space of wavefunctions is really a set of sections of L. Gamma(L) was defined, here for use later, to be the set of sections of L. This part of the talk was rounded off by a comment that in string theory there is a more complicated bundle.

The focus was then shifted to curvature and geometry, commencing with topology and the Poincare conjecture in three dimensions and SD's quiet endorsement of Grigori Perelman's work. The Ricci flow of the metric with respect to a parameter, theta, was given in a short equation as partial{g}/partial{theta}=-Ric(g_theta) (here Ric means the Ricci curvature defined from the metric) and the question of whether one could deform an arbitrary three-dimensional metric to Ric=constant of the three sphere was described.

Next, complex algebraic geometry (CAG) was looked at. SD commenced by saying that CAG involved making an association between a set and a ring. For example the set of points {x,y} making a unit circle is associated to the ring {Set of all polynomials on x, y with complex coefficients}/{}. Projective geometry was briefly described as the association between a vector z~Kz where K is a non-zero real(?) coefficient (as an example of the results of a projection was the equivalency of the parabola, ellipse and the hyperbola, and this motivated the definition). An example set was given using the projection as X={z: f(z)=0}/{z~Kz}, where f is a homogeneous polynomial. SD then declared that in projective geometry one has sections of line bundles as oppose to functions, such as L -> X and that a ring can be constructed as R= Direct Sum from k=0 to infinity of Gamma(kth order tensor product of L). At this point the example of toric varieties was discussed and entirely lost on me but it looked interesting nonetheless, and can be added to my list of interesting things to look-up one day, which is frankly far too long. When I came to, the basic principle of the talk was being given inside a red box, so it must be important. It is that there is an association between positive curvature (high g) and Gamma(L) being large (many sections).

Some more was said. Then a recap was given, which consisted of a picture with a line bundle, L, over a set then from the left hand side of L emerged an arrow to algebraic geometry and the ring defined as the direct product sum of Gamma(L)'s, and from the right came an arrow to differential geometry, and a line of arrows from curvature of L = F, to g (suppose positive definite), to curvature of g (Ricci curvature). Finally the question why was asked and the answer was given in the example of Calabi-Yau metrics, where the set was given as X={a^5+b^5+c^5+d^5+e^5=0}/{Some equivalency relation, not given}.

Afterwards a reception was held on the eight floor and some very nice food was enjoyed by Vid and myself and I had a glass of wine. We didn't talk about curvature.