Simplest example of the pure spinor formalism

Here I will give the simplest nontrivial example of the pure spinor formalism.

Consider the following
pure spinor BRST operator acting on functions of
as follows:

We assume that are polynomials in .
The simplest example of the pure spinor formalism is given by the following

Theorem 7: The cohomology is nonzero only at the ghost number 1, and counts the solutions of SUSY Maxwell equations.

We will now present some algebraic point of view on this BRST complex.