Now we define and —
collections of vector fields on
by this formula:
(there is an imbiguity called “shift transformations” —
see next slide).
We can now continue describing the SUGRA data. We got:
satisfying the following properties:
commutes with the action of
- is “fixed modulo ” in the following sense: for any point let be the natural projection , then(in other words, only the vertical component of is non-obvious; the projection to is tautological)
- SUGRA constraints:where:
and are some sections of and
, and some sections of (i.e. vertical vector fields); they are essentially “curvatures”
Notice that satisfying the SUGRA constraints does depend on the vertical component of .