Cosmological Billiards talk online

Well, moving house was much more tiring than I thought. Not only did I miss part II of the generalized complex geometry talk but also part III :( But I did get a lot of rest!

Just a short post to say that Thibault Damour's talk at Geometry and Physics after 100 years of Einstein's Relativity is available to listen to online here. His title was Symmetry and Chaos in Gravity and Supergravity and it's an excellent talk if you wish to start learning about cosmological billiards. All the talks from the meeting are available online as I found out thanks to Peter Woit's web research.

If only all talks were available online perhaps then I wouldn't miss so many of them...

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.

Solutions and Signatures

Permit me to take a brief break from doing the new paper dance as first seen performed by The Quantum Pontiff, to make some comments. (By the way, make sure you check out Mr. Bacon's link to footage from the 1927 Solvay conference: Dirac, Bohr, Heisenberg, Pauli, Schrodinger, Lorentz, Einstein, Curie and many more all in two sparse minutes of fascinating footage.) For indeed I have a new paper out today, cowritten with Peter West:

Title: M-Theory Solutions in Multiple Signatures from E_11
Authors: P. P. Cook, P. C. West
Comments: 34 pages, LaTeX2e

I think "LaTeX2e" is a fairly uninteresting comment, but it isn't the done thing to make interesting comments in this section. I won't make many comments here either, as those who are interested can read the paper in their own time. In the paper we apply Keurentjes' observation's that Weyl reflections alter the signature of the local sub-algebra generators to the solution generating group element of E_11/H_11, mentioned here previously. Principally, we find that any solution of a (1,10) theory coexists with solutions in (2,9), (5,6) and their inverses. These are Hull's M* and M'-theories. It is a puzzle, beyond the scope of the paper, to understand how so many M-theories might all fit together.

Now I am free to think about other problems, and for a trainee theoretical physicist there are many, and to try and work out just how many ways there of constructing an N element string using only the two elements {A,B} such that no substring has a difference of more than three between the numbers of A's and B's in it.

Playing Catch-Up (Or some E11 related papers on hep-th)

Summer is high, well it's not raining every day at least, and seminars are rare. I have just got back from Riga, in Latvia, where I went for a stag-do, and I hope this explains the recent lack of posting, not even a postcard I'm afraid. Riga is very pretty by the way.

In the past two weeks there have been no less than three papers with some relevance to E11 and my line of research, and I really ought to try and understand them all. In the meantime let me simply list them and make some comments:

1. Dualities and signatures of G++ invariant theories by Sophie de Buyl, Laurent Houart and Nassiba Tabti

The first paper makes use of Keurentjes' observation that Weyl reflections do not commute with the involution used to choose the local subalgebra. In an earlier paper one of the authors, Laurent Houart, together with Francois Englert and Marc Henneaux had applied the observation that a reduction from a very-extended theory, E11, to an over-extended theory, E10, by the deletion of an "end node" on the Dynkin diagram gives rise to two distinct theories. The two theories are arrived at by, in one case, applying a Weyl reflection in the deleted node's associated root before deleting the node, and in the other case by direct deletion of the node. The two resulting E10 theories have different signatures. This paper extends considerations of this idea to all the other G++ theories.

2. Hidden Symmetries and Dirac Fermions by Sophie de Buyl, Marc Henneaux and Louis Paulot

The second paper introduces spin one-half fermions into the G++ theories, but as I haven't read this thoroughly yet I will not say much. According to the introduction this results in the chaotic motion reported in the cosmological billiards picture being lost. Furthermore the null geodesic in E10/K(10), which encodes dynamics, becomes timelike once the spin one half fermions are present. I will add any further comments here later, if they come to me :)

3. IIB Supergravity Revisited by Eric A. Bergshoeff, Mees de Roo, Sven F. Kerstan and Fabio Riccioni

The third paper is also very interesting, and the idea is straightforward to describe. Back in 1983 when the IIB supersymmetry algebra was first written down, branes were not an important concept, so the only gauge fields that were considered were the two scalars, the two-form and the four-form. These couple to a string and a three-brane. Since then more emphasis has been placed on the five-brane, the seven-brane and the space-filling nine brane, which couple to a six-form, an eight-form and a ten-form respectively. The authors of this paper introduce these extra gauge fields to the IIB multiplet of fields without introducing any degrees of freedom, by asserting that a duality relation between the extra fields to the 1983 multiplet of fields. The supersymmetric variations of the new fields are given, and it is noted that the duality condition that was asserted is really a necessary condition for the algebra to close. They find that the ten-form can transform as a doublet and a quadruplet of SU(1,1) and the authors argue that no other independent ten-forms can be added. The findings for the ten-form multiplets match the previous deductions from E11 given in Very-extended Kac-Moody algebras and their interpretation at low levels by Axel Kleinschmidt, Igor Schnakenburg and Peter West. (Thanks to Sven Kerstan for kindly talking through the ideas of the paper with me, however, as ever, I proudly take responsibility for all mistakes in these notes :) )