It is important to remember that the classical SUGRA background has a large algebra of symmetries, the superconformal algebra a.k.a. . We will denote it :
It has the subalgebra which consists of those generators which preserve a fixed point:
Indeed, is a coset space:
Let us use a calligraphic index to parametrize the generators of :
The rest of the generators of are denoted:
The superindex corresponds to some -grading which we do not need now, we just have to remember that the generators generate , and generate the rest of .
The commutation relations are parametrized by the structure constants.
For our present purpose, we have to know the following facts:
- First of all, acts by the adjoint representation:
- We need to know that the commutator of and falls into :
- The commutator is proportional to , and moreover there is a basis where the structure constants are proportional to the gamma-matrices:
We see that the superconformal algebra is somewhat similar to the algebra (11), or maybe we should say “to a sum of two copies of it”.