11 Dec 2002, 7 Feb 2004
Bernard to Predrag emails summarized text:
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Your Magic Triangle is distinctly different from mine. The first occurence
of my triangle is Nuffield workshop summer 1980, preprint fall 1980:
B. L. Julia, ``Group disintegrations'', in
S. Hawking and M. Rocek, eds.
{\em Superspace and Supergravity}
(Cambridge Univ. Press, Cambridge 1981)
It was stimulated by a number of ideas, the remark by Gibbons and Hawking
in 1979 on Gravitational instantons and Ehlers symmetry, the vague but
provocating remarks of Morel and Thierry Mieg quoted in my paper. My
version was was definitely not motivated by yours, the N=8 column is much
older and the desire to understand the other N's came in order to fit in
the N=1 or 0 case ie GR then to understand the E_9 or affine case which I
had in mind since 1978.
My first reference to your work is in the first paper of 1981 quoted
above, about negative dimensions. As you confirm, your E_7 result was
written up after my first paper with Cremmer.
Your work is not related to mine as far as I can see even now; you have no
real forms and the real forms that I have do not even match with the
entire Tits-Freudenthal subsquare. The intersection of the two triangles
is the [3x3] Magic Square which I knew long before those times and
probably from similar sources, (I heard F. Gursey in a Princeton seminar
on exceptional groups, about 1975). F_4 is a possibility for the Magic
Square of Kugo Sierra and Townsend, but not G_2.
Bernard L. Julia
ENS-CNRS/LPT
24 rue Lhomond
75005 Paris FRANCE
Tel. (01)47077146. FAX. (01)43367666. caius(at)lpt.ens.fr
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references with Julia's triangle:
``Disintegration (i.e. Oxidation) for En Cosets''
E. Cremmer, B. Julia, H. Lu, C.N. Pope
Higher-dimensional Origin of D=3 Coset Symmetries
hep-th/9909099
Applied Math vol 21 (1985) 355.
(I have not seen this one myself)
Pierre Henry-Labordere, Bernard Julia, Louis Paulot
Borcherds symmetries in M-theory
JHEP 0204 (2002) 049
hep-th/0203070
B.L. Julia
Magics of M-gravity
Fortsch. Phys. 49 (2001) 551-555
hep-th/0105031