SUGRA data
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:


    • 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 .