A brief review of E theory

Abstract

I begin with some memories of Abdus Salam who was my PhD supervisor. After reviewing the theory of non-linear realisations and Kac-Moody algebras, I explain how to construct the non-linear realisation based on the Kac-Moody algebra E 11 and its vector representation. I explain how this field theory leads to dynamical equations which contain an infinite number of fields defined on a spacetime with an infinite number of coordinates. I then show that these unique dynamical equations, when truncated to low level fields and the usual coordinates of spacetime, lead to precisely the equations of motion of eleven dimensional supergravity theory. By taking different group decompositions of E 11 we find all the maximal supergravity theories, including the gauged maximal supergravities, and as a result the non-linear realisation should be thought of as a unified theory that is the low energy effective action for type II strings and branes. These results essentially confirm the E 11 conjecture given many years ago.

Tensor Hierarchy Algebra Extensions of Over-Extended Kac–Moody Algebras

“…for all a, b, c ∈ S(g (n) ). 1579 (11) 1285 (14) 125 (17) 1955 (22) 510( 25) 9( 28) 1020( 30) 982 (33) 4787 (12) 1980 (15) 91 (18) 2652 (20) 4194( 23) 578( 26) 3( 29) 3954 (31) 1574 (34) 46 (37) 2188 (42) 198 (45) 1043( 50) 383(53) 348(61) 2954 (10) 7035 (13) 1581 (16) 27 (19) 7221 (21) 4594 (24) 322 (27) 6384 (32) 1285 (35) 13 (38) 3199 (40) 2458 (43) 112( 46 5856 (11) 5376 (14) 633( 17)…”

“…The hope of understanding extended geometry for infinite-dimensional, in particular hyperbolic, structure groups is the main motivation of the present work, and we hope that it will lead to a reformulation of gravity or supergravity where the hyperbolic Belinskii-Khalatnikov-Lifshitz group [33][34][35] not only emerges in extreme situations, but is an integral part of the formulation of the theory. This may eventually put the E 10 [35] and E 11 [11,36,37] proposals on a firm ground, and provide a mechanism for the emergence of space(-time).…”

Electric-Magnetic Duality in Gravity and Higher-Spin Fields

“…The second-level 6-form field, A a 1 •••a 6 , has equations of motion, which can be dual to those of the 3-form field A a 1 a 2 a 3 , and provides another way to describe the degrees of freedom of A a 1 a 2 a 3 . These fields lead to at least one spacetime coordinate for each in the E 11 ⊗ s l 1 non-linear realization [125,129],…”

Massive dual gravity in N spacetime dimensions

“…If massless the dual of h µν is a tensor T The indices for these T fields are symmetrized, in an obvious way, according to the corresponding Young tableaux. 1 Various individual fields of this type appear in string theories [9][10][11][12], and in "M-theory" and "E-theory" [13][14][15][16][17]. For a recent review of duality for gravity and higher-spin fields, with an emphasis on massless models in higher dimensions, see [18].…”