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 :) )