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## Thursday, June 25, 2009

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Quasiconformal Realizations of E_{6(6)}, E_{7(7)}, E_{8(8)}

As mentioned at RF, Gunaydin and Pavlyk have posted a new paper arXiv:0904.0784v1 [hep-th] on non-compact exceptional groups as quasiconformal groups over the split cubic Jordan algebras. The exceptional groups E_{6(6)}, E_{7(7)}, E_{8(8)} are known to arise from toroidal compactifications of D=11 supergravity down to d dimensions, where the global non-compact symmetry group of the maximally extended supergravity is given by E_{11-d(11-d)} (arXiv:hep-th/0409263v1). It is believed the discrete subgroups E_{6(6)}(Z), E_{7(7)}(Z), E_{8(8)}(Z) yield the symmetries of the non-perturbative spectra of toroidally compactified M-theory (arXiv:hep-th/9410167v2).

Algebraically, E_{6(6)}, E_{7(7)}, E_{8(8)} arise as transformation groups of the Freudenthal triple system over the split octonions. E_{6(6)} is the subgroup of the automorphism group of the Freudenthal triple system, E_{7(7)}, which preserves the cubic form of the split exceptional Jordan algebra. Geometrically, E_{6(6)} is the collineation group of the split Moufang plane OP^2_s, the space of projectors of the split exceptional Jordan algebra. E_{7(7)} and E_{8(8)} act as conformal and quasiconformal groups, respectively, over this space.

Gunaydin and Pavlyk show in their new paper that E_{6(6)}, E_{7(7)}, E_{8(8)} individually act as quasiconformal groups over the split cubic Jordan algebras J(3,C_s), J(3,H_s) and J(3,O_s). This implies they also act as quasiconformal groups over the split projective planes CP^2_s, HP^2_s and OP^2_s, respectively. Physically, this yields a new type of duality, a duality of U-dualities, so to speak. For example, M-theory compactified on an 8-torus with D=3 U-duality group E_{6(6)} is dual to M-theory compactified on a 6-torus with D=5 U-duality group E_{6(6)} as the symmetries of the non-perturbative spectra are equivalent.

As mentioned at RF, Gunaydin and Pavlyk have posted a new paper arXiv:0904.0784v1 [hep-th] on non-compact exceptional groups as quasiconformal groups over the split cubic Jordan algebras. The exceptional groups E_{6(6)}, E_{7(7)}, E_{8(8)} are known to arise from toroidal compactifications of D=11 supergravity down to d dimensions, where the global non-compact symmetry group of the maximally extended supergravity is given by E_{11-d(11-d)} (arXiv:hep-th/0409263v1). It is believed the discrete subgroups E_{6(6)}(Z), E_{7(7)}(Z), E_{8(8)}(Z) yield the symmetries of the non-perturbative spectra of toroidally compactified M-theory (arXiv:hep-th/9410167v2).

Algebraically, E_{6(6)}, E_{7(7)}, E_{8(8)} arise as transformation groups of the Freudenthal triple system over the split octonions. E_{6(6)} is the subgroup of the automorphism group of the Freudenthal triple system, E_{7(7)}, which preserves the cubic form of the split exceptional Jordan algebra. Geometrically, E_{6(6)} is the collineation group of the split Moufang plane OP^2_s, the space of projectors of the split exceptional Jordan algebra. E_{7(7)} and E_{8(8)} act as conformal and quasiconformal groups, respectively, over this space.

Gunaydin and Pavlyk show in their new paper that E_{6(6)}, E_{7(7)}, E_{8(8)} individually act as quasiconformal groups over the split cubic Jordan algebras J(3,C_s), J(3,H_s) and J(3,O_s). This implies they also act as quasiconformal groups over the split projective planes CP^2_s, HP^2_s and OP^2_s, respectively. Physically, this yields a new type of duality, a duality of U-dualities, so to speak. For example, M-theory compactified on an 8-torus with D=3 U-duality group E_{6(6)} is dual to M-theory compactified on a 6-torus with D=5 U-duality group E_{6(6)} as the symmetries of the non-perturbative spectra are equivalent.

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## 3 comments:

Er, where the post about water in Mas has gone?.

About this posts I wonder how this new dualities affect (that is, if they affect them at all) the F-theory GUT local models.

Wow. Reloading the page the post about Mars (and indeed, the one about the superheavy element)have reapeared (and my previous comment dissapeared).

A new reload and the two posts are again missing and my previous comment is here again. I am viewing your blog with google chrome, I am not sure if similar behaviour happens with other browsers.

Hi Javier

No worries, your browser works just fine. I actually deleted the two posts because they were only temporary.

As for the connection to F-theory GUT local models, this is an open question. One difference is the appearance of compact exceptional groups, rather than the non-compact ones discussed here. Fortunately, one can always complexify the cubic Jordan algebras and recover compact E_6, E_7 and E_8 from their Freudenthal triple systems. I am just not aware of any supergravity theories corresponding to these complexified structures.

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