Example: Maxwell theory

For example, consider the supersymmetric Maxwell theory in 10D in flat space. In the coset space approach, the flat space can be described as a group manifold of the Lie group , which is generated by supersymmetries plus Poincare translations.

Solutions of the equations of motion correspond to the ghost number 1 vertex operators. The BRST complex is based on the functions of and :

The space of functions (or, rather, the Taylor series) is dual to the universal enveloping algebra .

This is true for any group with the Lie algebra . The universal enveloping algebra can be identified with the left-invariant differential operators on of finite order, i.e. expressions of the form:

where . Then given a function on , we can compute the derivative of at the unit :

This requires the knowledge of the coefficients of the Taylor series of at .
Therefore we observe the duality relation between the functions on and the elements of the universal enveloping algebra .

Therefore in this case:
 (26)
Then Koszul duality (25) tells us that the space of vertex operators is:
 (27)
The relation between and is the following:
 (28)
where is some ideal.

This is just to say that the algebra “is a solution” to the Yang-Mills constraint:
(*)
in the following sense. If we put the generator of the supersymmetry transformation, then the constraint is satisfied. The definition (20), (22) of is such that is the most general (“universal”) Lie algebra satisfying this constraint; therefore any algebra satisfying the Yang-Mills constraint should be a factoralgebra of by some ideal.

The ideal can be described rather explicitly, in the following manner. Because of the quadratic relation (*), is proportional to :
 (29)
(this is the definition of ). It turns out, as a consequence of (*), that:
 (30)

So defined generates the ideal

Simply put, this ideal consists of the elements of the algebra which vanish in the vacuum.