- #31
kneemo
- 117
- 3
Greetings Alejandro and MTd2 :)
Indeed, this recent work out of Perimeter involving the octonions is quite interesting.
Acquiring a full generation from \mathbb{C}\otimes\mathbb{O} (i.e. the bioctonion algebra) goes back to the work of Gursey (reference [7] in Furey's paper), with further work being done by Catto, who showed the bioctonions give rise to a non-associative grassmann algebra and used their 3x3 Jordan algebra in an E6 unification model (http://arxiv.org/abs/hep-th/0302079" ). As a finite-dimensional composition algebra over \mathbb{C}, the bioctonions are maximal, as Springer and Veldkamp have shown the following:
Theorem
A finite dimensional vector space V over a field \mathbb{F}=\mathbb{R},\mathbb{C} can be endowed with a composition algebra structure if and only if \mathrm{dim}_{\mathbb{F}}(V)=1,2,4,8.Note: A composition algebra is an algebra \mathbb{A}=(V,\bullet) admitting an identiy element, with a non-degenerate quadratic form \eta (norm) satisfying
\forall x,y\in\mathbb{A}\quad\eta(x\bullet y)=\eta(x)\eta(y).
Over the reals, it turns out there are two non-isomorphic dimension eight composition algebras: the octonions and the split-octonions. Over the complex numbers, for a given dimension all composition algebras are isomorphic. The split-octonions underlie supergravity theories arising from toroidal compactifications of M-theory (most famously in N=8 supergravity in D=4), while the ordinary octonions so far have no direct interpretation in M-theory.
It is, however, quite easy to show the octonions and split-octonions are real subalgebras of the bioctonion algebra. In this sense, they are unified, and this unification can be uplifted to their corresponding 2x2 and 3x3 Jordan algebras, which are subalgebras of the 2x2 and 3x3 Jordan algebras over the bioctonions. The 3x3 Jordan algebra over the bioctonions was studied by Kaplansky and Wright http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.mmj/1029001946" and is called the exceptional Jordan C*-algebra. This algebra played a pivotal role in proving that each Jordan-Banach algebra (JB-algebra) is the self-adjoint part of a unique Jordan C*-algebra.
Earlier this year, I used the exceptional Jordan C*-algebra and its Freudenthal triple system (FTS) to study extremal black holes in homogeneous supergravities based on the octonions and split-octonions http://arxiv.org/abs/1005.3514" . It was shown the bioctonions are essential in the study of M-theory on an 8-torus, which gives a D=3 supergravity with E8(8) U-duality group. Their utility arises in constructing the 57-dimensional space for which E8(8) is non-linearly realized, as the norm form on this space contains complex light-like solutions. This ultimately forces one to use the FTS over the bioctonions to complexify the 57-dimensional space, giving a realization of complex E8 on this space in which all light-like solutions are contained.
So what does this mean physically? Well, for one it hints at a new theory in which the split-octonion supergravity theories (e.g. N=8, D=4 SUGRA) arising from M-theory compactified on k-dimensional Tori are unified with the octonionic "magic" supergravity theories studied by Ferrara and Gunaydin http://arxiv.org/abs/hep-th/0606211" (which as of yet have no M-theory interpretation). If we are lucky, it might also shed some light on the proposed finiteness of N=8 supergravity.
Indeed, this recent work out of Perimeter involving the octonions is quite interesting.
Acquiring a full generation from \mathbb{C}\otimes\mathbb{O} (i.e. the bioctonion algebra) goes back to the work of Gursey (reference [7] in Furey's paper), with further work being done by Catto, who showed the bioctonions give rise to a non-associative grassmann algebra and used their 3x3 Jordan algebra in an E6 unification model (http://arxiv.org/abs/hep-th/0302079" ). As a finite-dimensional composition algebra over \mathbb{C}, the bioctonions are maximal, as Springer and Veldkamp have shown the following:
Theorem
A finite dimensional vector space V over a field \mathbb{F}=\mathbb{R},\mathbb{C} can be endowed with a composition algebra structure if and only if \mathrm{dim}_{\mathbb{F}}(V)=1,2,4,8.Note: A composition algebra is an algebra \mathbb{A}=(V,\bullet) admitting an identiy element, with a non-degenerate quadratic form \eta (norm) satisfying
\forall x,y\in\mathbb{A}\quad\eta(x\bullet y)=\eta(x)\eta(y).
Over the reals, it turns out there are two non-isomorphic dimension eight composition algebras: the octonions and the split-octonions. Over the complex numbers, for a given dimension all composition algebras are isomorphic. The split-octonions underlie supergravity theories arising from toroidal compactifications of M-theory (most famously in N=8 supergravity in D=4), while the ordinary octonions so far have no direct interpretation in M-theory.
It is, however, quite easy to show the octonions and split-octonions are real subalgebras of the bioctonion algebra. In this sense, they are unified, and this unification can be uplifted to their corresponding 2x2 and 3x3 Jordan algebras, which are subalgebras of the 2x2 and 3x3 Jordan algebras over the bioctonions. The 3x3 Jordan algebra over the bioctonions was studied by Kaplansky and Wright http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.mmj/1029001946" and is called the exceptional Jordan C*-algebra. This algebra played a pivotal role in proving that each Jordan-Banach algebra (JB-algebra) is the self-adjoint part of a unique Jordan C*-algebra.
Earlier this year, I used the exceptional Jordan C*-algebra and its Freudenthal triple system (FTS) to study extremal black holes in homogeneous supergravities based on the octonions and split-octonions http://arxiv.org/abs/1005.3514" . It was shown the bioctonions are essential in the study of M-theory on an 8-torus, which gives a D=3 supergravity with E8(8) U-duality group. Their utility arises in constructing the 57-dimensional space for which E8(8) is non-linearly realized, as the norm form on this space contains complex light-like solutions. This ultimately forces one to use the FTS over the bioctonions to complexify the 57-dimensional space, giving a realization of complex E8 on this space in which all light-like solutions are contained.
So what does this mean physically? Well, for one it hints at a new theory in which the split-octonion supergravity theories (e.g. N=8, D=4 SUGRA) arising from M-theory compactified on k-dimensional Tori are unified with the octonionic "magic" supergravity theories studied by Ferrara and Gunaydin http://arxiv.org/abs/hep-th/0606211" (which as of yet have no M-theory interpretation). If we are lucky, it might also shed some light on the proposed finiteness of N=8 supergravity.
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Wait, wait, there is one, and very important: that this separation is consistent with Freund–Rubin compactification, 
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of the sBootstrap, but it is fringe physics.