Example: Maxwell theoryFor 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 .Therefore in this case:Then Koszul duality (25) tells us that the space of vertex operators is:The relation between and is the following:where is some ideal.The ideal can be described rather explicitly, in the following manner. Because of the quadratic relation (*), is proportional to :(this is the definition of ). It turns out, as a consequence of (*), that:
Simply put, this ideal consists of the elements of the algebra which vanish in the vacuum.