Monday, June 20, 2005

Generalized Complex Geometry I

This morning, a balmy, humid morning (they say we're getting plenty of hot air from France at the moment) I braved the excruciating temperatures of the tube and ventured over to Imperial College's Blackett Laboratory to hear Marco Gualtieri, a former student of Hitchin, give the first of three introductory talks on Generalized Geometry and Physics. Gualtieri's PhD thesis Generalized Complex Geometry is available online here. Generalised geometry is a popular seminar topic at the moment, and you can find Lubos Motl's description of the Harvard Postdoc Journal Club's discussion of it here.

The talk had been billed as an introduction to generalized geometry, and this appealed to such a large audience that room 1004 was packed to the gills. The first two-thirds of this first of three talks was devoted to setting up the linear algebra. As such it isn't really possible to write it up on the blog, as I still haven't investigated Techexplorer and Mathplayer enough yet :( Nevertheless we did learn some interesting things that it might be possible to express in words, without trying to repeat Lubos' post.

We commenced with a real 2n-dimensional vector space, E, with a non-degenerate inner-product < , >, an (n,n) bilinear form, which acts naturally on T + T*, the direct product of tangent and the cotangent bundles. From here one can form a Clifford algebra CL(E)~CL(n,n,R)=Mat(2^n,R), presumably as {CL(E),CL(F)}=-k< E,F >. If we consider an element of E, X+x, where X is in T and x is in T*, the natural product with a form r being defined as (X+x).r=i_X.r+x^r, where ^ is meant to be a wedge product. Then [(X+x)^2].r=< X+x,X+x >r, and the spin group is given by the even powers of (X+x), so that the parity of r is preserved under its action. The Lie algebra of SO(n,n) can be decomposed into three distinct parts, the endomorphims of T, and components in T* and T. These were labelled A, B and \beta by Gualtieri, with B:T->T* and \beta:V*->V and the specific action of the "B-field" was checked by exponentiating the 2-by-2 matrix whose bottom left entry was B and the remaining entries are zero, and acting with this matrix on the natural column vector formed by X+x. The vector is mapped to X+(x+i_X.B), so the action of the B-field is to translate points parallel to the T* direction, an amount determined by X.

Okay, I've looked back over this entry so far and it's more illegible than most of my other posts, so I'm going to advise any readers interested in the mathematical details, which is probably both of you :), to look at Gualtieri's thesis, for which there is a link above. From now on in, I will just describe the route taken in the talk.

We looked next at the exterior derivative, and wondered about its action on T+T*, and were motivated to use the "derived bracket construction", which is a nested set of two commutators, containing the exterior derivative, and two elements of T+T*. If this derived bracket construction was antisymmetrised with respect to the two T components of the elements of T+T* that it contained then we were told that we obtained the "Courant bracket". We then learned a few things about this bracket, including the fact that it doesn't satisfy the Jacobi identity, and so is not a Lie bracket, as we may have naively been thinking. So the question was raised by Gualtieri of just what the bracket might be and we were led down a path of constructing a "twisted Courant bracket" that would better suit our purposes.

This was motivated by first applying the exponentiated B-field (we recall that this is a group element of SO(n,n,R)) to two elements prior to putting them in the Courant bracket. It turns out that this gives us back the bracket, multiplied by exp(B), plus a 3-form term i_Xi_YdB. The next step is to form the twisted Courant bracket by adding a closed 3-form to the Courant bracket in such a way as to cancel out the dB contribution given by the new bracket's action on the elements of exp{B}.(T+T*).

We finished our first hour on generalized geometry with some geometry! A generalised Riemann structure was built on T+T*. This was started by picking a positive definite sub-bundle C+. This was pictured as the vertical axis on a usual Cartesian grid, where T* and T formed the null boundaries of the light cone. An equivalent negative definite bundle, C-, was picked to be orthgonal to C+. From there C+ was expressed as the usual Riemannian metric, g, plus a 2-form, b, and a general metric G was formed, which when restricted to T became the usual Riemannian metric, and in the general case a different symmetric volume form was found which Gualtieri has called the Born-Infeld volume to "coincide with physics terminology." From here we quickly made use of the Hodge dual and defined "Mukai pairing" and briefly thought about a twisted exterior derivative and its associated twisted Laplacian, before coming to the end.

Unfortunately I have to move house on Wednesday when part II is scheduled to take place, so I will miss the next installment. Fortunately for you poor readers it means you won't have to put up with another near-illegible post. Well at least not until after Wednesday :)

I hope it's less humid on Wednesday.