Published Native American Drama, 1970–2011

E11, brane dynamics and duality symmetries

“…We will compute the dynamics in the absence of the background IIB supergravity fields and so we will consider the non-linear realisation of the semi-direct product of the Cartan involution subalgebra I c (E 11 ) with the vector representation, that is, I c (E 11 ) ⊗ s l 1 . This algebra was worked out in collaboration with Michaella Pettit [33]. At level zero I c (E 11 ) is SO(1, 9) ⊗ SO(2).…”

“…with the Minkowski metric corresponding to the fact that we are working with the algebra SO(1, 9) ⊗ SO(2) at lowest level. The group element used to construct the non-linear realisation is of the form g = g l g h where g l = e The Cartan forms can be written as which results, using reference [33] in the variations…”

“…The seven dimensional theory emerges when we decompose E 11 into GL(7) ⊗ Sl(5) which is the algebra that emerges when we delete node seven in the E 11 Dynkin diagram. The generators in this decomposition at low level are easily computed using the Nutma programme SimpLie [30] and are given by The E 11 ⊗ s l 1 algebra in this decomposition has been worked out at low levels by Michaella Pettit and the author and will be given elsewhere [33]. The non-linear realisation I c (E 11 ) ⊗ s l 1 is constructed from g = g L g h where…”

“…The Cartan forms which belong to the vector representation of the E 11 algebra can be written in the form Using equation (2.9), the group element of equation (6.1.5) and the results of reference [33], we find that the Cartan forms are given at low levels by…”

“…Finally, in section seven discuss some general features of brane dynamics that emerge from the non-linear realisation. The calculations in the IIB theory and in seven dimensions use the Cartan involution invariant subalgebra of E 11 ⊗ s l 1 in the decomposition appropriate to these theories; these results will be published elsewhere [33].…”