The bundle of pure spinors
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 a 32-ple 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 .

This is how the sigma-model for is constructed. The target space is , but with the gauge symmetry .