Notas sobre superspacio (Salam-Strathdee)
Last weekend in the library xeroxing the bibliography.
I have found the original definition of superspace very clear,
directly in Salam Strathdee NPB 76, p. 477. One gets
space-time coordinates, then one adds Grassman variables
at will. Initially we are not restricted to have the same
number of Grassman and Space-Time degrees of freedom.
But in most interesting cases the number coincides, and then
we can do the substitution between grassman variables
\theta and differentials dx. In other cases one proceeds
ad hoc via dimensional reduction technique, it seems.
The main source of confusion, to me, is that grassmann variables
relate to the supercharge, but have nothing with the fermions
of the theory, i.e. with the representation. Fermions and
bosons are pieces of the superfield. The superfield can
be seen as a differential form, ie an element of \Omega^*.
As explained by Urs, the supercharge operator acts in
\Omega^* very like (d+ *d*); the differential forms are
bosons or fermions depending of their degree.
I believe this answers my question about SUSY breaking: it
divides the differential complex \Omega^* in its
components \Omega^0, \Omega^1,...,\Omega^n; or at least
in the even and odd graded pieces.
The superfield can be scalar or vector-valued. This paves
the way to Witten's analysis of the non-linear sigma model,
NPB 202, p 253-316, section 11. There the superfield takes
values over the coordinates of a target manifold, so that
one of the bosonic fields ranges again over this set of
coordinates. It is to be noticed that there are not
evident restrictions in the dimension of the target
manifold at this level. The practical man should
like to have as many superchargues as the dimension
of the target.
BTW, the whole mechanism of differential forms remembers me
of the Dirac-Kahler equation, where the fermionic degrees
of freedom increase by doubling and then they are scattered
in different pieces of the whole differential form. Has
somebody taken a look to this?
yours,
Alejandro Rivero
Marginal note
While reading Witten's 1982, the feeling is very like attending
Lucas' new trilogy. One sees him as the new Feymann, the
heritage of american theoretists, the promise...
If he is mirroring the career path of Anakin Skywalker, will
we come to see the return of the Jedi?