On this page:
1 General definition and generators
2 Structure of commutation relations

Superconformal algebra

1 General definition and generators

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 .

2 Structure of commutation relations

The commutation relations are parametrized by the structure constants.

For our present purpose, we have to know the following facts:
  1. First of all, acts by the adjoint representation:

  2. We need to know that the commutator of and falls into :

  3. 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”.