The worldsheet theory has spinor ghosts and .
Because, as we explained, they are in some sense infinitesimal objects,
we should tell them to live in some nontrivial bundle over space-time. The transition functions of that bundle
should respect the pure spinor constraints
. This means that the
transition functions should belong to:
It is useful to consider the corresponding principal bundle, which we will call .
It is constructed in the following way.
The description of the SUGRA data starts with specifying a distribution
. Invariantly, this is defined
as at the singular locus .
Moreover, we must have specified, for every point ,
an -orbit of
in . This orbit
is the fiber of over
We will formulate the sigma-model so that the target space is , but there
is a gauge symmetry projecting .