Szngularitzes 79T is the negative of the Chevalley involution w, so that the elements of the Lie
algebra of K(E1o) are "T-antisymmetric", kT = -k (which is equivalent to them
being fixed under w: w(k) = +k).
Current tests of the SUGRA/Elo correspondence
An El0 group element g(t) is parametrized by an infinite number of coordinates.
When decomposing (the Lie algebra of) El0 with respect to (the Lie algebra of) the
GL(10) subgroup defined by the horizontal line in the Dynkin diagram of Elo, the
various components of g(t) can be graded by their GL(10) level l. At the l = 0 level
g(t) is parametrized by a GL(10) matrix kj, to which is associated (in the coset
space GL(lO)/SO(lO)) a symmetric matrix gaJ = (ek)5(ek)$. [The indices i,j,...
take ten values.] At the level l = 1, one finds a 3-form Aijk. At the level C = 2,
a 6-form Aijklmn, and at the level l = 3 a 9-index object A9 with Young-tableau
symmetry (8, l}.
The coset action (1) then defines a coupled set of equations of motion for
gij(t), Aijk(t), Aili z...is(t), A~l~z...~g(t),~~.. By explicit calculations, it was shown
that these coupled equations of motion could be identified, modulo terms which
correspond to potential walls of height at least 30, to the SUGRAll equations of
motion. This identification between the coset dynamics and the SUGRAll one is
obtained by means of a dictionary which maps: (1) gij(t) to the spatial components
of the 11-dimensional metric Gij(t,xo) in a certain coframe (Ndt,Qi), (2) Aijk(t)
to the mixed temporal-spatial ('electric') components of the 11-dimensional field
strength F = dA in the same coframe, (3) the conjugate momentum of Aili z...i6(t)
to the dual (using ~ilzz"'ilo) of the spatial ('magnetic') frame components of .F = dA,
and (4) the conjugate momentum of Ai li2...ig (t) to the do dual (on jlc) of the struc-
ture constants Cjk of the coframe 8, i.e. doi = $CjkQj A Qk. Here all the SUGRA
field variables are considered at some fixed (but arbitrary) spatial point XO.
The fact that at levels l = 2 (AG), and .t = 3 (Ag) the dictionary between
SUGRA and coset variables is such that the first spatial gradients of the SUGRA
variables G, A are mapped onto (time derivatives of) coset variables suggested the
conjecture that the infinite tower of coset variables could fully encode all the spa-
tial derivatives of the SUGRA variables, thereby explaining how a one-dimensional
coset dynamics could correspond to an 11-dimensional one. Some evidence for this
conjecture comes from the fact that among the infinite number of generators of
El0 there do exist towers of generators that have the appropriate GL(10) index
structure for representing the infinite sequence of spatial gradients of the various
SUGRA variables.
It is not known how to extend this dictionary beyond the level C = 3 (corre-
sponding to Ag). The difficulty in extending the dictionary might be due (similarly
to what happens in the ADS/CFT case) to the non-existence of a common domain
of validity for the two descriptions.
However, Ref. [9] found evidence for a nice compatibility between some high-level
contributions in the coset action, corresponding to imaginary roots ( (a,a) < 0,
..