Generalised BPS conditions

We write down two E11 invariant conditions which at low levels reproduce the known half BPS conditions for type II theories. These new conditions contain, in addition to the familiar central charges, an infinite number of further charges which are required in an underlying theory of strings and branes. We comment on the application of this work to higher derivative string corrections

Representations of G+++ and the role of space-time

We consider the decomposition of the adjoint and fundamental representations of very extended Kac-Moody algebras G+++ with respect to their regular A type subalgebra which, in the corresponding non-linear realisation, is associated with gravity. We find that for many very extended algebras almost all the A type representations that occur in the decomposition of the fundamental representations also occur in the adjoint representation of G+++. In particular, for E8+++, this applies to all its fundamental representations. However, there are some important examples, such as An+++, where this is not true and indeed the adjoint representation contains no generator that can be identified with a space-time translation. We comment on the significance of these results for how space-time can occur in the non-linear realisation based on G+++. Finally we show that there is a correspondence between the A representations that occur in the fundamental representation associated with the very extended node and the adjoint representation of G+++ which is consistent with the interpretation of the former as charges associated with brane solutions.Comment: 45 pages, 9 figures, 9 tables, te

Generalised Space-time and Gauge Transformations

We consider the generalised space-time introduced by the author in 2003 in the context of the non-linear realisation of the semi-direct product of E11 and its first fundamental representation. For all the fields we propose gauge transformations which are compatible with the underlying E11 structure. A crucial role is played by the generalised vielbein that the generalised space-time possess. We work out the explicit form of the gauge transformations, at low levels, in four, five and eleven dimensions.Comment: 33 page

E11 and Spheric Vacuum Solutions of Eleven- and Ten dimensional Supergravity Theories

In view of the newly conjectured Kac-Moody symmetries of supergravity theories placed in eleven and ten dimensions, the relation between these symmetry groups and possible compactifications are examined. In particular, we identify the relevant group cosets that parametrise the vacuum solutions of AdS x S type.Comment: discussion improve

Duality Symmetries and G^{+++} Theories

We show that the non-linear realisations of all the very extended algebras G^{+++}, except the B and C series which we do not consider, contain fields corresponding to all possible duality symmetries of the on-shell degrees of freedom of these theories. This result also holds for G_2^{+++} and we argue that the non-linear realisation of this algebra accounts precisely for the form fields present in the corresponding supersymmetric theory. We also find a simple necessary condition for the roots to belong to a G^{+++} algebra.Comment: 35 pages. v2: 2 appendices added, other minor corrections. v3: tables corrected, other minor changes, one appendix added, refs. added. Version published in Class. Quant. Gra

Kac-Moody Symmetries of Ten-dimensional Non-maximal Supergravity Theories

A description of the bosonic sector of ten-dimensional N=1 supergravity as a non-linear realisation is given. We show that if a suitable extension of this theory were invariant under a Kac-Moody algebra, then this algebra would have to contain a rank eleven Kac-Moody algebra, that can be identified to be a particular real form of very-extended D_8. We also describe the extension of N=1 supergravity coupled to an abelian vector gauge field as a non-linear realisation, and find the Kac-Moody algebra governing the symmetries of this theory to be very-extended B_8. Finally, we discuss the related points for the N=1 supergravity coupled to an arbitrary number of abelian vector gauge fields

Generalised geometry, eleven dimensions and E11

We construct the non-linear realisation of E11 and its first fundamental representation in eleven dimensions at low levels. The fields depend on the usual coordinates of space-time as well as two form and five form coordinates. We derive the terms in the dynamics that contain the three form and six form fields and show that when we restricted their field dependence to be only on the usual space-time we recover the correct self-duality relation. Should this result generalise to the gravity fields then the non-linear realisation is an extension of the maximal supergravity theory, as previously conjectured. We also comment on the connections between the different approaches to generalised geometry.Comment: 17 pages, Trivial typos corrected in version one and a substantial note added which gives the equation of motion relating the gravity field to its dua

E11, generalised space-time and equations of motion in four dimensions

We construct the non-linear realisation of the semi-direct product of E11 and its first fundamental representation at low levels in four dimensions. We include the fields for gravity, the scalars and the gauge fields as well as the duals of these fields. The generalised space-time, upon which the fields depend, consists of the usual coordinates of four dimensional space-time and Lorentz scalar coordinates which belong to the 56-dimensional representation of E7. We demand that the equations of motion are first order in derivatives of the generalised space-time and then show that they are essentially uniquely determined by the properties of the E11 Kac-Moody algebra and its first fundamental representation. The two lowest equations correctly describe the equations of motion of the scalars and the gauge fields once one takes the fields to depend only on the usual four dimensional space-time

Tracking in a space variant active vision system

Without the ability to foveate on and maintain foveation, active vision for applications such as surveillance, object recognition and object tracking are difficult to build. Although foveation in cartesian coordinates is being actively pursued by many, multi-resolution high accuracy foveation in log polar space has not been given much attention. This paper addresses the use of foveation to track a single object as well as multiple objects for a simulated space variant active vision system. Complex logarithmic mapping is chosen firstly because it provides high resolution and wide angle viewing. Secondly, the spatially variant structure of log polar space leads to an object increasing in size as it moves towards the fovea. This is important as we know which object is closer to the fovea at any instant in time.<br /

E_{11} origin of Brane charges and U-duality multiplets

We derive general equations which determine the decomposition of the G^{+++} multiplet of brane charges into the sub-algebras that arise when the non-linearly realised G^{+++} theory is dimensionally reduced on a torus. We apply this to calculate the low level E_8 multiplets of brane charges that arise when the E_{8}^{+++}, or E_{11}, non-linearly realised theory is dimensionally reduced to three dimensions on an eight dimensional torus. We find precise agreement with the U-duality multiplet of brane charges previously calculated, thus providing a natural eleven dimensional origin for the "mysterious" brane charges found that do not occur as central charges in the supersymmetry algebra. We also discuss the brane charges in nine dimensions and how they arise from the IIA and IIB theories.Comment: 30 pages, plain te