Cohomology and Extensions
A super-translation group is a supergroup generalization of the translation group, hence of the additive Lie group . Its underlying supermanifold is a super-Euclidean space or super-Minkowski spacetime.
Given a super Poincaré Lie algebra extension of a orthogonal Lie algebra for some spin representation , then the corresponding super-translation Lie algebra is the quotient
The underlying super vector space of this is
where is the vector space underlying the given spin representation.
The super Lie algebra structure is mildly non-abelian with the only non-trivial bracket being that between two spinors and given by the bilinear pairing (the charge conjugation matrix) between two spinors:
where is a basis for the translation generators in .
As a central extension of the superpoint
The super-translation Lie algebra is a super-Lie algebra extension of the abelian super Lie algebra which is just the superpoint by the super Lie algebra cocycles
for , where are the basis elements of .
This simple but maybe noteworthy fact has been highlighted in the context of the brane scan in (CAIP 99, section 2.1).
This mechanism plays a role in string theory when realizing as a central extension of , for this formalizes aspects of the idea that type IIA string theory with a D0-brane condensate is 11-dimensional supergravity/M-theory (FSS 13).
In dimension 1
The additive group structure on is given on generalized elements in (i.e. in the logic internal to) the topos of sheaves on the category SCartSp? of cartesian superspaces by
Recall how the notation works here: by the Yoneda embedding we have a full and faithful functor
and we also have the theorem, discussed at supermanifolds, that maps from some into is given by a tuple of even section and odd sections . The above notation specifies the map of supermanifolds by displaying what map of sets of maps from some test object it corresponds to under the Yoneda embedding.
Now, or each SDiff there is a group structure on the hom-set given by precisely the above formula for this given
where etc and where the addition and product on the right takes place in the function super algebra .
Since the formula looks the same for all , one often just writes it without mentioning as above.
Discussion in the context of the brane scan is in section 2.1 of
and more generally in the context of The brane bouquet in