While looking back at some gradings for exceptional Lie algebras, I re-discovered an old jewel of a paper (hep-th/9704054) by Itzhak Bars, known for his work on supergravity and the 13-dimensional S-theory. Essentially, Bars argues that a 14-dimensional theory could lie behind non-perturbative string theory and its dualities. He shows how to embed type A,B,C, heterotic and type-I superalgebras covariantly in the framework of 14 dimensions with signature (11,3) and SO(11,3) symmetry and that in lower dimensions one can embed three sets of 32 supercharges as different projections of 64 supercharges, which form three distinct superalgebras.(adsbygoogle = window.adsbygoogle || []).push({});

The SO(11,3) symmetry has also been used by Nesti and Percacci in arXiv:0909.4537 in a GraviGUT model, that has ties with Lisi's E8 theory, as can be seen from the grading of E8(-24)'s algebra: 14*+64*+so(11,3)+R+64+14. The gradings clearly shows the 64 supercharges, (11,3) signature symmetry and some extra G2 symmetry. A supersymmetric form of Lisi's theory might very well be equivalent to Bars' 14-dimensional theory that would contain M-theory and F-theory.

Lisi has also expressed interest in the other non-compact form of E8, namely E8(8) which admits the algebraic grading: 14*+64*+so(7,7)+R+64+14. Itzhak Bars hasn't explored this (7,7) signature theory yet, but in light of E8(C) with grading: 14*+64*+so(14,C)+C+64+14, it should be dual to the SO(11,3) theory in the sense that E8(8) and E8(-24) are just real non-compact forms of E8(C). The complex E8(C) theory, in this sense, would be more fundamental.

Other gradings of E8 have been used extensively in the construction of 57-dimensional extremal black hole charge spaces in D=3 (e.g. E8(C)=1*+56*+E7(C)+C+56+1) (hep-th/0008063). There, the E8(8) and E8(-24) gradings amount to a choice of split-octonion or octonion variables inside the algebra of complexified octonions. The construction of a Jordan C*-algebra and its extended Freudenthal triple system leads to the full E8(C) symmetry, that is ultimately necessary to accommodate solutions where the E7 quartic invariant takes negative values and leads to non-real values for the 57D conformal invariant.

Does anybody have any thoughts on such a 14-dimensional theory?

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# A Case for 14-Dimensions

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