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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeApr 14th 2015
   • (edited Apr 14th 2015)
   • PermaLink
   Author: Urs
   Format: MarkdownItexMaybe I am slowly beginning to see a connection between the parabolic geometry as in the references at _[[BGG resolution]]_ and the story of the boquet of super-spacetimes and super $p$-branes that I am after. So one neat observation ([here](http://ncatlab.org/nlab/show/super-Cartan+geometry#StabilizerOfSupersymmetryJointWith3Cocycle)) that got me thinking is that given one of the 3-cocycles that are superstring WZW terms on a super-Minkowski spacetime $\mathbb{R}^{d-1,1\vert N}$, then the _joint_ stabilizer group in $GL(\mathbb{R}^{d-1,1\vert N})$ of the 3-cocycle _and_ the Lie bracket is the spin-cover $Spin(d-1,1)$ of the Lorentz group. This is neat because it gives a general abstract reason to equip $d$-dimensional supermanifolds with orthogonal structure, hence with super-pseudo-Riemannian structure (as opposed to other structure). But now this means the following: it means that the tangent bundle of a Lorentzian supermanifold is not just a bundle of super-vector spaces, but a bundle of supersymmetry Lie algebras: the transition functions of the tangent bundle are not just any old super-linear maps, but they preserve the super Lie algebra structure on the typical fiber $\mathbb{R}^{d-1,1\vert N}$ -- and in addition they preserve the superstring 3-cocycle on that super Lie algebra. Now this fact that the tangent bundle is actually a bundle of Lie algebras is a phenomenon shared with parabolic geometry, see e.g. [Čap-Souček 07, p. 12](http://ncatlab.org/nlab/show/BGG%20resolution#CapSoucek07). Also, these BGG sequences of bundles of Lie algebra cohomology classes begin to look like something familiar from that brane-bouquet perspective, where it is these exceptional super Lie algebra cohomology classes that one wants to globalize over the Cartan geometry. I don't know yet what the upshot is, though.

   Maybe I am slowly beginning to see a connection between the parabolic geometry as in the references at BGG resolution and the story of the boquet of super-spacetimes and super $p$-branes that I am after.

   So one neat observation (here) that got me thinking is that given one of the 3-cocycles that are superstring WZW terms on a super-Minkowski spacetime $\mathbb{R}^{d-1,1\vert N}$, then the joint stabilizer group in $GL(\mathbb{R}^{d-1,1\vert N})$ of the 3-cocycle and the Lie bracket is the spin-cover $Spin(d-1,1)$ of the Lorentz group.

   This is neat because it gives a general abstract reason to equip $d$-dimensional supermanifolds with orthogonal structure, hence with super-pseudo-Riemannian structure (as opposed to other structure).

   But now this means the following: it means that the tangent bundle of a Lorentzian supermanifold is not just a bundle of super-vector spaces, but a bundle of supersymmetry Lie algebras: the transition functions of the tangent bundle are not just any old super-linear maps, but they preserve the super Lie algebra structure on the typical fiber $\mathbb{R}^{d-1,1\vert N}$ – and in addition they preserve the superstring 3-cocycle on that super Lie algebra.

   Now this fact that the tangent bundle is actually a bundle of Lie algebras is a phenomenon shared with parabolic geometry, see e.g. Čap-Souček 07, p. 12.

   Also, these BGG sequences of bundles of Lie algebra cohomology classes begin to look like something familiar from that brane-bouquet perspective, where it is these exceptional super Lie algebra cohomology classes that one wants to globalize over the Cartan geometry.

   I don’t know yet what the upshot is, though.